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APPLIEDDIFFERENTIALGEOMETRYA Modern Introduction


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APPLIEDDIFFERENTIALGEOMETRYA Modern IntroductionVladimir G <strong>Ivancevic</strong>Defence Science and Technology Organisation, AustraliaTijana T <strong>Ivancevic</strong>The University of Adelaide, AustraliaWorld ScientificN E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I


Published byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HEBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.APPLIED DIFFERENTIAL GEOMETRYA Modern IntroductionCopyright © 2007 by World Scientific Publishing Co. Pte. Ltd.All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.ISBN-13 978-981-270-614-0ISBN-10 981-270-614-3Printed in Singapore.


Dedicated to:Nitya, Atma and Kali


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Preface<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction is a graduate–levelmonographic textbook. It is designed as a comprehensive introduction intomethods and techniques of modern differential geometry with its variousphysical and non–physical applications. In some sense, it is a continuationof our previous book, Natural Biodynamics (World Scientific, 2006), whichcontains all the necessary background for comprehensive reading of the currentbook. While the previous book was focused on biodynamic applications,the core applications of the new book are in the realm of modern theoreticalphysics, mainly following its central line: Einstein–Feynman–Witten.Other applications include (among others): control theory, robotics, neurodynamics,psychodynamics and socio–economical dynamics.The book has six chapters. Each chapter contains both ‘pure mathematics’and related ‘applications’ labelled by the word ‘Application’.The first chapter provides a soft (‘plain–English’) introduction into manifoldsand related geometrical structures, for all the interested readers withoutthe necessary background. As a ‘snap–shot’ illustration, at the end ofthe first chapter, a paradigm of generic differential–geometric modelling isgiven, which is supposed to fit all above–mentioned applications.The second chapter gives technical preliminaries for development of themodern applied differential geometry. These preliminaries include: (i) classicalgeometrical objects – tensors, (ii) both classical and modern physicalobjects – actions, and modern geometrical objects – functors.The third chapter develops modern manifold geometry, together with itsmain physical and non–physical applications. This chapter is a neccessarybackground for comprehensive reading of the remaining chapters.The fourth chapter develops modern bundle geometry, together with itsmain physical and non–physical applications.vii


viii<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe fifth chapter develops modern jet bundle geometry, together withits main applications in non–autonomous mechanics and field physics. Allmaterial in this chapter is based on the previous chapter.The sixth chapter develops modern geometrical machinery of Feynman’spath integrals, together with their various physical and non–physical applications.For most of this chapter, only the third chapter is a neccessarybackground, assuming a basic understanding of quantum mechanics (asprovided in the above–mentioned World Scientific book, Natural Biodynamics).The book contains both an extensive Index (which allows easy connectionsbetween related topics) and a number of cited references related tomodern applied differential geometry.Our approach to dynamics of complex systems is somewhat similar tothe approach to mathematical physics used at the beginning of the 20thCentury by the two leading mathematicians: David Hilbert and John vonNeumann – the approach of combining mathematical rigor with conceptualclarity, or geometrical intuition that underpins the rigor.The intended audience includes (but is not restricted to) theoreticaland mathematical physicists; applied and pure mathematicians; control,robotics and mechatronics engineers; computer and neural scientists;mathematically strong chemists, biologists, psychologists, sociologists andeconomists – both in academia and industry.Compared to all differential–geometric books published so far, <strong>Applied</strong><strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction has much wider variety ofboth physical and non–physical applications. After comprehensive readingof this book, a reader should be able to both read and write journalpapers in such diverse fields as superstring & topological quantum field theory,nonlinear dynamics & control, robotics, biomechanics, neurodynamics,psychodynamics and socio–economical dynamics.V. <strong>Ivancevic</strong>Defence Science & Technology Organisation, Australiae-mail: Vladimir.<strong>Ivancevic</strong>@dsto.defence.gov.auT. <strong>Ivancevic</strong>School of Mathematics, The University of Adelaidee-mail: Tijana.<strong>Ivancevic</strong>@adelaide.edu.auAdelaideMay, 2006


ixAcknowledgmentsThe authors wish to thank Land Operations Division, Defence Science& Technology Organisation, Australia, for the support in developing theHuman Biodynamics Engine (HBE) and all HBE–related text in this monograph.Finally, we express our gratitude to the World Scientific PublishingCompany, and especially to Ms. Zhang Ji and Mr. Rhaimie Wahap.


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Glossary of Frequently Used SymbolsGeneral– ‘iff’ means ‘if and only if’;– ‘r.h.s’ means ‘right hand side’; ‘l.h.s’ means ‘left hand side’;– ODE means ordinary differential equation, while PDE means partial differentialequation;– Einstein’s summation convention over repeated indices (not necessarilyone up and one down) is assumed in the whole text, unless explicitly statedotherwise.SetsN – natural numbers;Z – integers;R – real numbers;C – complex numbers;H – quaternions;K – number field of real numbers, complex numbers, or quaternions.Mapsf : A → B – a function, (or map) between sets A ≡ Dom f and B ≡ Cod f;Ker f = f −1 (e B ) − a kernel of f;Im f = f(A) − an image of f;Coker f = Cod f/ Im f − a cokernel of f;Coim f = Dom f/ Ker f − a coimage of f;xi


xii<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionXf✲ Y❅❅❅❅❘Zhg❄− a commutative diagram, requiring h = g ◦ f .DerivativesC k (A, B) – set of k−times differentiable functions between sets A to B;C ∞ (A, B) – set of smooth functions between sets A to B;C 0 (A, B) – set of continuous functions between sets A to B;f ′ (x) = df(x)dx– derivative of f with respect to x;ẋ – total time derivative of x;∂ t ≡ ∂ ∂t– partial time derivative;∂ x i ≡ ∂ i ≡ ∂∂x– partial coordinate derivative;˙if = ∂ t f + ∂ x if ẋ i – total time derivative of the scalar field f = f(t, x i );u t ≡ ∂ t u, u x ≡ ∂ x u, u xx ≡ ∂ x 2u – only in partial differential equations;L x i ≡ ∂ x iL, Lẋi ≡ ∂ẋiL – coordinate and velocity partial derivatives of theLagrangian function;d – exterior derivative;d n – coboundary operator;∂ n – boundary operator;∇ = ∇(g) – affine Levi–Civita connection on a smooth manifold M withRiemannian metric tensor g = g ij ;Γ i jk– Christoffel symbols of the affine connection ∇;∇ X T – covariant derivative of the tensor–field T with respect to the vector–field X, defined by means of Γ i jk ;T ;x i ≡ T |x i – covariant derivative of the tensor–field T with respect to thecoordinate basis {x i };T˙≡ DTdt≡ ∇Tdt– absolute (intrinsic, or Bianchi) derivative of the tensor–field T upon the parameter t; e.g., acceleration vector is the absolute timederivative of the velocity vector, a i = ˙¯v i ≡ Dvidt; note that in general,a i ≠ ˙v i – this is crucial for proper definition of Newtonian force;L X T – Lie derivative of the tensor–field T in direction of the vector–fieldX;[X, Y ] – Lie bracket (commutator) of two vector–fields X and Y ;[F, G], or {F, G} – Poisson bracket, or Lie–Poisson bracket, of two functionsF and G.


Glossary of Frequently Used SymbolsxiiiSmooth Manifolds, Fibre Bundles and Jet SpacesUnless otherwise specified, all manifolds M, N, ... are assumed C k −smooth,real, finite–dimensional, Hausdorff, paracompact, connected and withoutboundary, 1 while all maps are assumed C k −smooth. We use the symbols⊗, ∨, ∧ and ⊕ for the tensor, symmetrized and exterior products, as well asthe Whitney sum 2 , respectively, while ⌋ denotes the interior product (contraction)of (multi)vectors and p−forms, and ↩→ denotes a manifold imbedding(i.e., both a submanifold and a topological subspace of the codomainmanifold). The symbols ∂B A denote partial derivatives with respect to coordinatespossessing multi–indices B A (e.g., ∂ α = ∂/∂x α );T M – tangent bundle of the manifold M;π M : T M → M – natural projection;T ∗ M – cotangent bundle of the manifold M;π : Y → X – fibre bundle;(E, π, M) – vector bundle with total space E, base M and projection π;(Y, π, X, V ) – fibre bundle with total space Y , base X, projection π andstandard fibre V ;J k (M, N) – space of k−jets of smooth functions between manifolds M andN;J k (X, Y ) – k−-jet space of a fibre bundle Y → X; in particular, inmechanics we have a 1–jet space J 1 (R, Q), with 1–jet coordinate mapsjt 1 s : t ↦→ (t, x i , ẋ i ), as well as a 2–jet space J 2 (R, Q), with 2–jet coordinatemaps jt 2 s : t ↦→ (t, x i , ẋ i , ẍ i );jxs k – k−jets of sections s i : X → Y of a fibre bundle Y → X;We use the following kinds of manifold maps: immersion, imbedding, submersion,and projection. A map f : M → M ′ is called the immersion ifthe tangent map T f at every point x ∈ M is an injection (i.e., ‘1–1’ map).When f is both an immersion and an injection, its image is said to be asubmanifold of M ′ . A submanifold which also is a topological subspace iscalled imbedded submanifold. A map f : M → M ′ is called submersion ifthe tangent map T f at every point x ∈ M is a surjection (i.e., ‘onto’ map).If f is both a submersion and a surjection, it is called projection or fibrebundle.1 The only 1D manifolds obeying these conditions are the real line R and the circleS 1 .2 Whitney sum ⊕ is an analog of the direct (Cartesian) product for vector bundles.Given two vector bundles Y and Y ′ over the same base X, their Cartesian product is avector bundle over X × X. The diagonal map induces a vector bundle over X called theWhitney sum of these vector bundles and denoted by Y ⊕ Y ′ .


xiv<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLie and (Co)Homology GroupsG – usually a general Lie group;GL(n) – general linear group with real coefficients in dimension n;SO(n) – group of rotations in dimension n;T n – toral (Abelian) group in dimension n;Sp(n) – symplectic group in dimension n;T (n) – group of translations in dimension n;SE(n) – Euclidean group in dimension n;H n (M) = Ker ∂ n / Im ∂ n−1 – nth homology group of the manifold M;H n (M) = Ker d n / Im d n+1 – nth cohomology group of the manifold M.Other Spaces and Operatorsi ≡ √ −1 – imaginary unit;C k (M) – space of k−differentiable functions on the manifold M;Ω k (M) – space of k−forms on the manifold M;g – Lie algebra of a Lie group G, i.e., the tangent space of G at its identityelement;Ad(g) – adjoint endomorphism; recall that adjoint representation of a Liegroup G is the linearized version of the action of G on itself by conjugation,i.e., for each g ∈ G, the inner automorphism x ↦→ gxg −1 gives a lineartransformation Ad(g) : g → g, from the Lie algebra g of G to itself;nD space (group, system) means n−dimensional space (group, system), forn ∈ N;✄ – semidirect (noncommutative) product; e.g., SE(3) = SO(3) ✄ R 3 ;∫ – interior product, or contraction, of a vector–field and a one–form;Σ – Feynman path integral symbol, denoting integration over continuousspectrum of smooth paths and summation over discrete spectrum ofMarkov chains; e.g.,∫Σ D[x] e iS[x] denotes the path integral (i.e., sum–over–histories) over all possible paths x i = x i (t) defined by the Hamilton action,∫ t1∫Σ D[Φ] e iS[Φ] denotes the path integral overS[x] = 1 2 t 0g ij ẋ i ẋ j dt, whileall possible fields Φ i = Φ i (x) defined by some field action S[Φ].CategoriesS – all sets as objects and all functions between them as morphisms;PS – all pointed sets as objects and all functions between them preserving


Glossary of Frequently Used Symbolsxvbase point as morphisms;V – all vector spaces as objects and all linear maps between them as morphisms;B – Banach spaces over R as objects and bounded linear maps betweenthem as morphisms;G – all groups as objects, all homomorphisms between them as morphisms;A – Abelian groups as objects, homomorphisms between them as morphisms;AL – all algebras (over a given number field K) as objects, all their homomorphismsbetween them as morphisms;T – all topological spaces as objects, all continuous functions between themas morphisms;PT – pointed topological spaces as objects, continuous functions betweenthem preserving base point as morphisms;T G – all topological groups as objects, their continuous homomorphisms asmorphisms;M – all smooth manifolds as objects, all smooth maps between them asmorphisms;M n – nD manifolds as objects, their local diffeomorphisms as morphisms;LG – all Lie groups as objects, all smooth homomorphisms between themas morphisms;LAL – all Lie algebras (over a given field K) as objects, all smooth homomorphismsbetween them as morphisms;T B – all tangent bundles as objects, all smooth tangent maps between themas morphisms;T ∗ B – all cotangent bundles as objects, all smooth cotangent maps betweenthem as morphisms;VB – all smooth vector bundles as objects, all smooth homomorphisms betweenthem as morphisms;FB – all smooth fibre bundles as objects, all smooth homomorphisms betweenthem as morphisms;Symplec – all symplectic manifolds (i.e., physical phase–spaces), all symplecticmaps (i.e., canonical transformations) between them as morphisms;Hilbert – all Hilbert spaces and all unitary operators as morphisms.


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ContentsPrefaceGlossary of Frequently Used Symbolsviixi1. Introduction 11.1 Manifolds and Related <strong>Geom</strong>etrical Structures . . . . . . 11.1.1 <strong>Geom</strong>etrical Atlas . . . . . . . . . . . . . . . . . . 61.1.2 Topological Manifolds . . . . . . . . . . . . . . . . 81.1.2.1 Topological manifolds withoutboundary . . . . . . . . . . . . . . . . . 101.1.2.2 Topological manifolds with boundary . 101.1.2.3 Properties of topological manifolds . . . 101.1.3 <strong>Diff</strong>erentiable Manifolds . . . . . . . . . . . . . . 121.1.4 Tangent and Cotangent Bundles of Manifolds . . 141.1.4.1 Tangent Bundle of a Smooth Manifold . 141.1.4.2 Cotangent Bundle of a SmoothManifold . . . . . . . . . . . . . . . . . 151.1.4.3 Fibre–, Tensor–, and Jet–Bundles . . . 151.1.5 Riemannian Manifolds: Configuration Spacesfor Lagrangian Mechanics . . . . . . . . . . . . . 161.1.5.1 Riemann Surfaces . . . . . . . . . . . . 171.1.5.2 Riemannian <strong>Geom</strong>etry . . . . . . . . . . 191.1.5.3 Application: Lagrangian Mechanics . 221.1.5.4 Finsler manifolds . . . . . . . . . . . . . 251.1.6 Symplectic Manifolds: Phase–Spaces forHamiltonian Mechanics . . . . . . . . . . . . . . . 25xvii


xviii<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction1.1.7 Lie Groups . . . . . . . . . . . . . . . . . . . . . . 281.1.7.1 Application: Physical Examples ofLie Groups . . . . . . . . . . . . . . . . 301.1.8 Application: A Bird View on Modern Physics . 311.1.8.1 Three Pillars of 20th Century Physics . 311.1.8.2 String Theory in ‘Plain English’ . . . . 331.2 Application: Paradigm of <strong>Diff</strong>erential–<strong>Geom</strong>etricModelling of Dynamical Systems . . . . . . . . . . . . . . 462. Technical Preliminaries: Tensors, Actions and Functors 512.1 Tensors: Local Machinery of <strong>Diff</strong>erential <strong>Geom</strong>etry . . . . 512.1.1 Transformation of Coordinates and ElementaryTensors . . . . . . . . . . . . . . . . . . . . . . . . 512.1.1.1 Transformation of Coordinates . . . . . 522.1.1.2 Scalar Invariants . . . . . . . . . . . . . 532.1.1.3 Vectors and Covectors . . . . . . . . . . 532.1.1.4 Second–Order Tensors . . . . . . . . . . 542.1.1.5 Higher–Order Tensors . . . . . . . . . . 562.1.1.6 Tensor Symmetry . . . . . . . . . . . . 572.1.2 Euclidean Tensors . . . . . . . . . . . . . . . . . . 582.1.2.1 Basis Vectors and the Metric Tensorin R n . . . . . . . . . . . . . . . . . . . 582.1.2.2 Tensor Products in R n . . . . . . . . . . 592.1.3 Covariant <strong>Diff</strong>erentiation . . . . . . . . . . . . . . 602.1.3.1 Christoffel’s Symbols . . . . . . . . . . . 602.1.3.2 Geodesics . . . . . . . . . . . . . . . . . 612.1.3.3 Covariant Derivative . . . . . . . . . . . 612.1.3.4 Covariant Form of <strong>Diff</strong>erentialOperators . . . . . . . . . . . . . . . . . 622.1.3.5 Absolute Derivative . . . . . . . . . . . 632.1.3.6 3D Curve <strong>Geom</strong>etry: Frenet–SerretFormulae . . . . . . . . . . . . . . . . . 642.1.3.7 Mechanical Acceleration and Force . . . 642.1.4 Application: Covariant Mechanics . . . . . . . . 652.1.4.1 Riemannian Curvature Tensor . . . . . 702.1.4.2 Exterior <strong>Diff</strong>erential Forms . . . . . . . 712.1.4.3 The Covariant Force Law . . . . . . . . 762.1.5 Application: Nonlinear Fluid Dynamics . . . . 782.1.5.1 Continuity Equation . . . . . . . . . . . 78


Contentsxix2.1.5.2 Forces Acting on a Fluid . . . . . . . . 802.1.5.3 Constitutive and Dynamical Equations 812.1.5.4 Navier–Stokes Equations . . . . . . . . 822.2 Actions: The Core Machinery of Modern Physics . . . . . 832.3 Functors: Global Machinery of Modern Mathematics . . . 872.3.1 Maps . . . . . . . . . . . . . . . . . . . . . . . . . 882.3.1.1 Notes from Set Theory . . . . . . . . . 882.3.1.2 Notes From Calculus . . . . . . . . . . . 892.3.1.3 Maps . . . . . . . . . . . . . . . . . . . 892.3.1.4 Algebra of Maps . . . . . . . . . . . . . 892.3.1.5 Compositions of Maps . . . . . . . . . . 902.3.1.6 The Chain Rule . . . . . . . . . . . . . 902.3.1.7 Integration and Change of Variables . . 902.3.1.8 Notes from General Topology . . . . . . 912.3.1.9 Topological Space . . . . . . . . . . . . 922.3.1.10 Homotopy . . . . . . . . . . . . . . . . . 932.3.1.11 Commutative Diagrams . . . . . . . . . 952.3.1.12 Groups and Related AlgebraicStructures . . . . . . . . . . . . . . . . . 972.3.2 Categories . . . . . . . . . . . . . . . . . . . . . . 1022.3.3 Functors . . . . . . . . . . . . . . . . . . . . . . . 1052.3.4 Natural Transformations . . . . . . . . . . . . . . 1082.3.4.1 Compositions of NaturalTransformations . . . . . . . . . . . . . 1092.3.4.2 Dinatural Transformations . . . . . . . 1092.3.5 Limits and Colimits . . . . . . . . . . . . . . . . . 1112.3.6 Adjunction . . . . . . . . . . . . . . . . . . . . . . 1112.3.7 Abelian Categorical Algebra . . . . . . . . . . . . 1132.3.8 n−Categories . . . . . . . . . . . . . . . . . . . . 1162.3.8.1 Generalization of ‘Small’ Categories . . 1172.3.8.2 Topological Structure of n−Categories . 1212.3.8.3 Homotopy Theory and Relatedn−Categories . . . . . . . . . . . . . . . 1212.3.8.4 Categorification . . . . . . . . . . . . . 1232.3.9 Application: n−Categorical Framework forHigher Gauge Fields . . . . . . . . . . . . . . . . 1242.3.10 Application: Natural <strong>Geom</strong>etrical Structures . 1282.3.11 Ultimate Conceptual Machines: Weakn−Categories . . . . . . . . . . . . . . . . . . . . 132


xx<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3. <strong>Applied</strong> Manifold <strong>Geom</strong>etry 1373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1373.1.1 Intuition behind Einstein’s <strong>Geom</strong>etrodynamics . . 1383.1.2 Einstein’s <strong>Geom</strong>etrodynamics in Brief . . . . . . . 1423.2 Intuition Behind the Manifold Concept . . . . . . . . . . 1433.3 Definition of a <strong>Diff</strong>erentiable Manifold . . . . . . . . . . . 1453.4 Smooth Maps between Smooth Manifolds . . . . . . . . . 1473.4.1 Intuition behind Topological Invariants ofManifolds . . . . . . . . . . . . . . . . . . . . . . 1483.5 (Co)Tangent Bundles of Smooth Manifolds . . . . . . . . 1503.5.1 Tangent Bundle and Lagrangian Dynamics . . . . 1503.5.1.1 Intuition behind a Tangent Bundle . . . 1503.5.1.2 Definition of a Tangent Bundle . . . . . 1503.5.2 Cotangent Bundle and Hamiltonian Dynamics . . 1533.5.2.1 Definition of a Cotangent Bundle . . . . 1533.5.3 Application: Command/Control in Human–Robot Interactions . . . . . . . . . . . . . . . . . 1543.5.4 Application: Generalized BidirectionalAssociative Memory . . . . . . . . . . . . . . . . . 1573.6 Tensor Fields on Smooth Manifolds . . . . . . . . . . . . 1633.6.1 Tensor Bundle . . . . . . . . . . . . . . . . . . . . 1633.6.1.1 Pull–Back and Push–Forward . . . . . . 1643.6.1.2 Dynamical Evolution and Flow . . . . . 1653.6.1.3 Vector–Fields and Their Flows . . . . . 1673.6.1.4 Vector–Fields on M . . . . . . . . . . . 1673.6.1.5 Integral Curves as DynamicalTrajectories . . . . . . . . . . . . . . . . 1683.6.1.6 Dynamical Flows on M . . . . . . . . . 1723.6.1.7 Categories of ODEs . . . . . . . . . . . 1733.6.2 <strong>Diff</strong>erential Forms on Smooth Manifolds . . . . . 1743.6.2.1 1−Forms on M . . . . . . . . . . . . . . 1743.6.2.2 k−Forms on M . . . . . . . . . . . . . . 1763.6.2.3 Exterior <strong>Diff</strong>erential Systems . . . . . . 1793.6.3 Exterior Derivative and (Co)Homology . . . . . . 1803.6.3.1 Intuition behind Cohomology . . . . . . 1823.6.3.2 Intuition behind Homology . . . . . . . 1833.6.3.3 De Rham Complex and HomotopyOperators . . . . . . . . . . . . . . . . . 185


Contentsxxi3.6.3.4 Stokes Theorem and de RhamCohomology . . . . . . . . . . . . . . . 1863.6.3.5 Euler–Poincaré Characteristics of M . . 1883.6.3.6 Duality of Chains and Forms on M . . 1883.6.3.7 Hodge Star Operator and HarmonicForms . . . . . . . . . . . . . . . . . . . 1903.7 Lie Derivatives on Smooth Manifolds . . . . . . . . . . . . 1923.7.1 Lie Derivative Operating on Functions . . . . . . 1923.7.2 Lie Derivative of Vector Fields . . . . . . . . . . . 1943.7.3 Time Derivative of the Evolution Operator . . . . 1973.7.4 Lie Derivative of <strong>Diff</strong>erential Forms . . . . . . . . 1973.7.5 Lie Derivative of Various Tensor Fields . . . . . . 1983.7.6 Application: Lie–Derivative Neurodynamics . . 2003.7.7 Lie Algebras . . . . . . . . . . . . . . . . . . . . . 2023.8 Lie Groups and Associated Lie Algebras . . . . . . . . . . 2023.8.1 Definition of a Lie Group . . . . . . . . . . . . . . 2033.8.2 Actions of Lie Groups on Smooth Manifolds . . . 2073.8.3 Basic Dynamical Lie Groups . . . . . . . . . . . . 2103.8.3.1 Galilei Group . . . . . . . . . . . . . . . 2103.8.3.2 General Linear Group . . . . . . . . . . 2113.8.4 Application: Lie Groups in Biodynamics . . . . 2123.8.4.1 Lie Groups of Joint Rotations . . . . . . 2123.8.4.2 Euclidean Groups of Total JointMotions . . . . . . . . . . . . . . . . . . 2163.8.4.3 Group Structure of BiodynamicalManifold . . . . . . . . . . . . . . . . . 2223.8.5 Application: Dynamical Games onSE(n)−Groups . . . . . . . . . . . . . . . . . . . 2273.8.5.1 Configuration Models for PlanarVehicles . . . . . . . . . . . . . . . . . . 2273.8.5.2 Two–Vehicles Conflict ResolutionManoeuvres . . . . . . . . . . . . . . . . 2283.8.5.3 Symplectic Reduction and DynamicalGames on SE(2) . . . . . . . . . . . . . 2303.8.5.4 Nash Solutions for Multi–VehicleManoeuvres . . . . . . . . . . . . . . . . 2333.8.6 Classical Lie Theory . . . . . . . . . . . . . . . . 2353.8.6.1 Basic Tables of Lie Groups and theirLie Algebras . . . . . . . . . . . . . . . 236


xxii<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.8.6.2 Representations of Lie groups . . . . . . 2393.8.6.3 Root Systems and Dynkin Diagrams . . 2403.8.6.4 Simple and Semisimple Lie Groupsand Algebras . . . . . . . . . . . . . . . 2453.9 Lie Symmetries and Prolongations on Manifolds . . . . . 2473.9.1 Lie Symmetry Groups . . . . . . . . . . . . . . . . 2473.9.1.1 Exponentiation of Vector Fields on M . 2473.9.1.2 Lie Symmetry Groups and General DEs 2493.9.2 Prolongations . . . . . . . . . . . . . . . . . . . . 2503.9.2.1 Prolongations of Functions . . . . . . . 2503.9.2.2 Prolongations of <strong>Diff</strong>erential Equations 2513.9.2.3 Prolongations of Group Actions . . . . 2523.9.2.4 Prolongations of Vector Fields . . . . . 2533.9.2.5 General Prolongation Formula . . . . . 2543.9.3 Generalized Lie Symmetries . . . . . . . . . . . . 2563.9.3.1 Noether Symmetries . . . . . . . . . . . 2573.9.4 Application: Biophysical PDEs . . . . . . . . . 2613.9.4.1 The Heat Equation . . . . . . . . . . . . 2613.9.4.2 The Korteveg–De Vries Equation . . . . 2623.9.5 Lie–Invariant <strong>Geom</strong>etric Objects . . . . . . . . . . 2623.9.5.1 Robot Kinematics . . . . . . . . . . . . 2623.9.5.2 Maurer–Cartan 1–Forms . . . . . . . . . 2643.9.5.3 General Structure of IntegrableOne–Forms . . . . . . . . . . . . . . . . 2653.9.5.4 Lax Integrable Dynamical Systems . . . 2673.9.5.5 Application: Burgers DynamicalSystem . . . . . . . . . . . . . . . . . . 2683.10 Riemannian Manifolds and Their Applications . . . . . . 2713.10.1 Local Riemannian <strong>Geom</strong>etry . . . . . . . . . . . . 2713.10.1.1 Riemannian Metric on M . . . . . . . . 2723.10.1.2 Geodesics on M . . . . . . . . . . . . . 2773.10.1.3 Riemannian Curvature on M . . . . . . 2783.10.2 Global Riemannian <strong>Geom</strong>etry . . . . . . . . . . . 2813.10.2.1 The Second Variation Formula . . . . . 2813.10.2.2 Gauss–Bonnet Formula . . . . . . . . . 2843.10.2.3 Ricci Flow on M . . . . . . . . . . . . . 2853.10.2.4 Structure Equations on M . . . . . . . 2873.10.3 Application: Autonomous LagrangianDynamics . . . . . . . . . . . . . . . . . . . . . . 289


Contentsxxiii3.10.3.1 Basis of Lagrangian Dynamics . . . . . 2893.10.3.2 Lagrange–Poincaré Dynamics . . . . . . 2913.10.4 Core Application: Search for Quantum Gravity . 2923.10.4.1 What is Quantum Gravity? . . . . . . . 2923.10.4.2 Main Approaches to Quantum Gravity . 2933.10.4.3 Traditional Approaches to QuantumGravity . . . . . . . . . . . . . . . . . . 3003.10.4.4 New Approaches to Quantum Gravity . 3043.10.4.5 Black Hole Entropy . . . . . . . . . . . 3103.10.5 Basics of Morse and (Co)Bordism Theories . . . . 3113.10.5.1 Morse Theory on Smooth Manifolds . . 3113.10.5.2 (Co)Bordism Theory on SmoothManifolds . . . . . . . . . . . . . . . . . 3143.11 Finsler Manifolds and Their Applications . . . . . . . . . 3163.11.1 Definition of a Finsler Manifold . . . . . . . . . . 3163.11.2 Energy Functional, Variations and Extrema . . . 3173.11.3 Application: Finsler–Lagrangian Field Theory . 3213.11.4 Riemann–Finsler Approach to Information<strong>Geom</strong>etry . . . . . . . . . . . . . . . . . . . . . . 3233.11.4.1 Model Specification and ParameterEstimation . . . . . . . . . . . . . . . . 3233.11.4.2 Model Evaluation and Testing . . . . . 3243.11.4.3 Quantitative Criteria . . . . . . . . . . 3243.11.4.4 Selection Among <strong>Diff</strong>erent Models . . . 3273.11.4.5 Riemannian <strong>Geom</strong>etry of MinimumDescription Length . . . . . . . . . . . . 3303.11.4.6 Finsler Approach to Information<strong>Geom</strong>etry . . . . . . . . . . . . . . . . . 3333.12 Symplectic Manifolds and Their Applications . . . . . . . 3353.12.1 Symplectic Algebra . . . . . . . . . . . . . . . . . 3353.12.2 Symplectic <strong>Geom</strong>etry . . . . . . . . . . . . . . . . 3363.12.3 Application: Autonomous HamiltonianMechanics . . . . . . . . . . . . . . . . . . . . . . 3383.12.3.1 Basics of Hamiltonian Mechanics . . . . 3383.12.3.2 Library of Basic Hamiltonian Systems . 3513.12.3.3 Hamilton–Poisson Mechanics . . . . . . 3613.12.3.4 Completely Integrable HamiltonianSystems . . . . . . . . . . . . . . . . . . 363


xxiv<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.12.3.5 Momentum Map and SymplecticReduction . . . . . . . . . . . . . . . . . 3723.12.4 Multisymplectic <strong>Geom</strong>etry . . . . . . . . . . . . . 3743.13 Application: Biodynamics–Robotics . . . . . . . . . . . 3753.13.1 Muscle–Driven Hamiltonian Biodynamics . . . . . 3763.13.2 Hamiltonian–Poisson Biodynamical Systems . . . 3793.13.3 Lie–Poisson Neurodynamics Classifier . . . . . . . 3833.13.4 Biodynamical Functors . . . . . . . . . . . . . . . 3843.13.4.1 The Covariant Force Functor . . . . . . 3843.13.4.2 Lie–Lagrangian Biodynamical Functor . 3853.13.4.3 Lie–Hamiltonian Biodynamical Functor 3913.13.5 Biodynamical Topology . . . . . . . . . . . . . . . 4013.13.5.1 (Co)Chain Complexes in Biodynamics . 4013.13.5.2 Morse Theory in Biodynamics . . . . . 4053.13.5.3 Hodge–De Rham Theory inBiodynamics . . . . . . . . . . . . . . . 4153.13.5.4 Lagrangian–Hamiltonian Dualityin Biodynamics . . . . . . . . . . . . . . 4193.14 Complex and Kähler Manifolds and TheirApplications . . . . . . . . . . . . . . . . . . . . . . . . . 4283.14.1 Complex Metrics: Hermitian and Kähler . . . . . 4313.14.2 Calabi–Yau Manifolds . . . . . . . . . . . . . . . . 4363.14.3 Special Lagrangian Submanifolds . . . . . . . . . 4373.14.4 Dolbeault Cohomology and Hodge Numbers . . . 4383.15 Conformal Killing–Riemannian <strong>Geom</strong>etry . . . . . . . . . 4413.15.1 Conformal Killing Vector–Fields and Forms on M 4423.15.2 Conformal Killing Tensors and LaplacianSymmetry . . . . . . . . . . . . . . . . . . . . . . 4433.15.3 Application: Killing Vector and Tensor Fieldsin Mechanics . . . . . . . . . . . . . . . . . . . . . 4453.16 Application: Lax–Pair Tensors in Gravitation . . . . . . 4483.16.1 Lax–Pair Tensors . . . . . . . . . . . . . . . . . . 4503.16.2 <strong>Geom</strong>etrization of the 3–Particle Open TodaLattice . . . . . . . . . . . . . . . . . . . . . . . . 4523.16.2.1 Tensorial Lax Representation . . . . . . 4533.16.3 4D Generalizations . . . . . . . . . . . . . . . . . 4563.16.3.1 Case I . . . . . . . . . . . . . . . . . . . 4563.16.3.2 Case II . . . . . . . . . . . . . . . . . . 4573.16.3.3 Energy–Momentum Tensors . . . . . . . 457


Contentsxxv3.17 <strong>Applied</strong> Unorthodox <strong>Geom</strong>etries . . . . . . . . . . . . . . 4583.17.1 Noncommutative <strong>Geom</strong>etry . . . . . . . . . . . . 4583.17.1.1 Moyal Product and NoncommutativeAlgebra . . . . . . . . . . . . . . . . . . 4583.17.1.2 Noncommutative Space–Time Manifolds 4593.17.1.3 Symmetries and <strong>Diff</strong>eomorphisms onDeformed Spaces . . . . . . . . . . . . . 4623.17.1.4 Deformed <strong>Diff</strong>eomorphisms . . . . . . . 4653.17.1.5 Noncommutative Space–Time <strong>Geom</strong>etry 4673.17.1.6 Star–Products and ExpandedEinstein–Hilbert Action . . . . . . . . . 4703.17.2 Synthetic <strong>Diff</strong>erential <strong>Geom</strong>etry . . . . . . . . . . 4733.17.2.1 Distributions . . . . . . . . . . . . . . . 4743.17.2.2 Synthetic Calculus in Euclidean Spaces 4763.17.2.3 Spheres and Balls as Distributions . . . 4783.17.2.4 Stokes Theorem for Unit Sphere . . . . 4803.17.2.5 Time Derivatives of Expanding Spheres 4813.17.2.6 The Wave Equation . . . . . . . . . . . 4824. <strong>Applied</strong> Bundle <strong>Geom</strong>etry 4854.1 Intuition Behind a Fibre Bundle . . . . . . . . . . . . . . 4854.2 Definition of a Fibre Bundle . . . . . . . . . . . . . . . . . 4864.3 Vector and Affine Bundles . . . . . . . . . . . . . . . . . . 4914.3.1 The Second Vector Bundle of the Manifold M . . 4954.3.2 The Natural Vector Bundle . . . . . . . . . . . . . 4964.3.3 Vertical Tangent and Cotangent Bundles . . . . . 4984.3.3.1 Tangent and Cotangent BundlesRevisited . . . . . . . . . . . . . . . . . 4984.3.4 Affine Bundles . . . . . . . . . . . . . . . . . . . . 5004.4 Application: Semi–Riemannian <strong>Geom</strong>etrical Mechanics 5014.4.1 Vector–Fields and Connections . . . . . . . . . . . 5014.4.2 Hamiltonian Structures on the Tangent Bundle . 5034.5 K−Theory and Its Applications . . . . . . . . . . . . . . 5084.5.1 Topological K−Theory . . . . . . . . . . . . . . . 5084.5.1.1 Bott Periodicity Theorem . . . . . . . . 5094.5.2 Algebraic K−Theory . . . . . . . . . . . . . . . . 5104.5.3 Chern Classes and Chern Character . . . . . . . . 5114.5.4 Atiyah’s View on K−Theory . . . . . . . . . . . . 5154.5.5 Atiyah–Singer Index Theorem . . . . . . . . . . . 518


xxvi<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction4.5.6 The Infinite–Order Case . . . . . . . . . . . . . . 5204.5.7 Twisted K−Theory and the Verlinde Algebra . . 5234.5.8 Application: K−Theory in String Theory . . . 5264.5.8.1 Classification of Ramond–RamondFluxes . . . . . . . . . . . . . . . . . . . 5264.5.8.2 Classification of D−Branes . . . . . . . 5284.6 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . 5294.7 Distributions and Foliations on Manifolds . . . . . . . . . 5334.8 Application: Nonholonomic Mechanics . . . . . . . . . 5344.9 Application: <strong>Geom</strong>etrical Nonlinear Control . . . . . . 5374.9.1 Introduction to <strong>Geom</strong>etrical Nonlinear Control . . 5374.9.2 Feedback Linearization . . . . . . . . . . . . . . . 5394.9.3 Nonlinear Controllability . . . . . . . . . . . . . . 5474.9.4 <strong>Geom</strong>etrical Control of Mechanical Systems . . . 5544.9.4.1 Abstract Control System . . . . . . . . 5544.9.4.2 Global Controllability of Linear ControlSystems . . . . . . . . . . . . . . . . . . 5554.9.4.3 Local Controllability of Affine ControlSystems . . . . . . . . . . . . . . . . . . 5554.9.4.4 Lagrangian Control Systems . . . . . . 5564.9.4.5 Lie–Adaptive Control . . . . . . . . . . 5664.9.5 Hamiltonian Optimal Control and MaximumPrinciple . . . . . . . . . . . . . . . . . . . . . . . 5674.9.5.1 Hamiltonian Control Systems . . . . . . 5674.9.5.2 Pontryagin’s Maximum Principle . . . . 5704.9.5.3 Affine Control Systems . . . . . . . . . 5714.9.6 Brain–Like Control Functor in Biodynamics . . . 5734.9.6.1 Functor Control Machine . . . . . . . . 5744.9.6.2 Spinal Control Level . . . . . . . . . . . 5764.9.6.3 Cerebellar Control Level . . . . . . . . . 5814.9.6.4 Cortical Control Level . . . . . . . . . . 5844.9.6.5 Open Liouville Neurodynamics andBiodynamical Self–Similarity . . . . . . 5874.9.7 Brain–Mind Functorial Machines . . . . . . . . . 5944.9.7.1 Neurodynamical 2−Functor . . . . . . . 5944.9.7.2 Solitary ‘Thought Nets’ and‘Emerging Mind’ . . . . . . . . . . . . . 5974.9.8 <strong>Geom</strong>etrodynamics of Human Crowd . . . . . . . 6024.9.8.1 Crowd Hypothesis . . . . . . . . . . . . 603


Contentsxxvii4.9.8.2 <strong>Geom</strong>etrodynamics of IndividualAgents . . . . . . . . . . . . . . . . . . . 6034.9.8.3 Collective Crowd <strong>Geom</strong>etrodynamics . . 6054.10 Multivector–Fields and Tangent–Valued Forms . . . . . . 6064.11 Application: <strong>Geom</strong>etrical Quantization . . . . . . . . . 6144.11.1 Quantization of Hamiltonian Mechanics . . . . . . 6144.11.2 Quantization of Relativistic HamiltonianMechanics . . . . . . . . . . . . . . . . . . . . . . 6174.12 Symplectic Structures on Fiber Bundles . . . . . . . . . . 6244.12.1 Hamiltonian Bundles . . . . . . . . . . . . . . . . 6254.12.1.1 Characterizing Hamiltonian Bundles . . 6254.12.1.2 Hamiltonian Structures . . . . . . . . . 6264.12.1.3 Marked Hamiltonian Structures . . . . . 6304.12.1.4 Stability . . . . . . . . . . . . . . . . . . 6324.12.1.5 Cohomological Splitting . . . . . . . . . 6324.12.1.6 Homological Action of Ham(M) on M 6344.12.1.7 General Symplectic Bundles . . . . . . . 6364.12.1.8 Existence of Hamiltonian Structures . . 6374.12.1.9 Classification of Hamiltonian Structures 6424.12.2 Properties of General Hamiltonian Bundles . . . . 6454.12.2.1 Stability . . . . . . . . . . . . . . . . . . 6454.12.2.2 Functorial Properties . . . . . . . . . . 6484.12.2.3 Splitting of Rational Cohomology . . . 6504.12.2.4 Hamiltonian Bundles and Gromov–Witten Invariants . . . . . . . . . . . . 6544.12.2.5 Homotopy Reasons for Splitting . . . . 6594.12.2.6 Action of the Homology of (M) onH ∗ (M) . . . . . . . . . . . . . . . . . . 6614.12.2.7 Cohomology of General SymplecticBundles . . . . . . . . . . . . . . . . . . 6644.13 Clifford Algebras, Spinors and Penrose Twistors . . . . . 6664.13.1 Clifford Algebras and Modules . . . . . . . . . . . 6664.13.1.1 The Exterior Algebra . . . . . . . . . . 6694.13.1.2 The Spin Group . . . . . . . . . . . . . 6724.13.1.3 4D Case . . . . . . . . . . . . . . . . . . 6724.13.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . 6754.13.2.1 Basic Properties . . . . . . . . . . . . . 6754.13.2.2 4D Case . . . . . . . . . . . . . . . . . . 6774.13.2.3 (Anti) Self Duality . . . . . . . . . . . . 681


xxviii<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction4.13.2.4 Hermitian Structure on the Spinors . . 6864.13.2.5 Symplectic Structure on the Spinors . . 6894.13.3 Penrose Twistor Calculus . . . . . . . . . . . . . . 6914.13.3.1 Penrose Index Formalism . . . . . . . . 6914.13.3.2 Twistor Calculus . . . . . . . . . . . . . 6984.13.4 Application: Rovelli’s Loop Quantum Gravity . 7014.13.4.1 Introduction to Loop Quantum Gravity 7014.13.4.2 Formalism of Loop Quantum Gravity . 7084.13.4.3 Loop Algebra . . . . . . . . . . . . . . . 7094.13.4.4 Loop Quantum Gravity . . . . . . . . . 7114.13.4.5 Loop States and Spin Network States . 7124.13.4.6 Diagrammatic Representation of theStates . . . . . . . . . . . . . . . . . . . 7154.13.4.7 Quantum Operators . . . . . . . . . . . 7164.13.4.8 Loop v.s. Connection Representation . 7174.14 Application: Seiberg–Witten Monopole Field Theory . 7184.14.1 SUSY Formalism . . . . . . . . . . . . . . . . . . 7214.14.1.1 N = 2 Supersymmetry . . . . . . . . . . 7214.14.1.2 N = 2 Super–Action . . . . . . . . . . . 7214.14.1.3 Spontaneous Symmetry–Breaking . . . 7234.14.1.4 Holomorphy and Duality . . . . . . . . 7244.14.1.5 The SW Prepotential . . . . . . . . . . 7244.14.2 Clifford Actions, Dirac Operators and SpinorBundles . . . . . . . . . . . . . . . . . . . . . . . 7254.14.2.1 Clifford Algebras and Dirac Operators . 7274.14.2.2 Spin and Spin c Structures . . . . . . . 7294.14.2.3 Spinor Bundles . . . . . . . . . . . . . . 7304.14.2.4 The Gauge Group and Its Equations . . 7314.14.3 Original SW Low Energy Effective Field Action . 7324.14.4 QED With Matter . . . . . . . . . . . . . . . . . 7354.14.5 QCD With Matter . . . . . . . . . . . . . . . . . 7374.14.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . 7384.14.6.1 Witten’s Formalism . . . . . . . . . . . 7404.14.7 Structure of the Moduli Space . . . . . . . . . . . 7464.14.7.1 Singularity at Infinity . . . . . . . . . . 7464.14.7.2 Singularities at Strong Coupling . . . . 7474.14.7.3 Effects of a Massless Monopole . . . . . 7484.14.7.4 The Third Singularity . . . . . . . . . . 749


Contentsxxix4.14.7.5 Monopole Condensation andConfinement . . . . . . . . . . . . . . . 7504.14.8 Masses and Periods . . . . . . . . . . . . . . . . . 7524.14.9 Residues . . . . . . . . . . . . . . . . . . . . . . . 7544.14.10 SW Monopole Equations and Donaldson Theory . 7574.14.10.1 Topological Invariance . . . . . . . . . . 7604.14.10.2 Vanishing Theorems . . . . . . . . . . . 7624.14.10.3 Computation on Kähler Manifolds . . . 7654.14.11 SW Theory and Integrable Systems . . . . . . . . 7694.14.11.1 SU(N) Elliptic CM System . . . . . . . 7714.14.11.2 CM Systems Defined by Lie Algebras . 7724.14.11.3 Twisted CM–Systems Defined byLie Algebras . . . . . . . . . . . . . . . 7734.14.11.4 Scaling Limits of CM–Systems . . . . . 7744.14.11.5 Lax Pairs for CM–Systems . . . . . . . 7764.14.11.6 CM and SW Theory for SU(N) . . . . 7794.14.11.7 CM and SW Theory for GeneralLie Algebra . . . . . . . . . . . . . . . . 7824.14.12 SW Theory and WDVV Equations . . . . . . . . 7844.14.12.1 WDVV Equations . . . . . . . . . . . . 7844.14.12.2 Perturbative SW Prepotentials . . . . . 7874.14.12.3 Associativity Conditions . . . . . . . . . 7894.14.12.4 SW Theories and Integrable Systems . . 7904.14.12.5 WDVV Equations in SW Theories . . . 7935. <strong>Applied</strong> Jet <strong>Geom</strong>etry 7975.1 Intuition Behind a Jet Space . . . . . . . . . . . . . . . . 7975.2 Definition of a 1–Jet Space . . . . . . . . . . . . . . . . . 8015.3 Connections as Jet Fields . . . . . . . . . . . . . . . . . . 8065.3.1 Principal Connections . . . . . . . . . . . . . . . . 8155.4 Definition of a 2–Jet Space . . . . . . . . . . . . . . . . . 8185.5 Higher–Order Jet Spaces . . . . . . . . . . . . . . . . . . 8225.6 Application: Jets and Non–Autonomous Dynamics . . . 8245.6.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . 8305.6.2 Quadratic Dynamical Equations . . . . . . . . . . 8315.6.3 Equation of Free–Motion . . . . . . . . . . . . . . 8325.6.4 Quadratic Lagrangian and Newtonian Systems . . 8335.6.5 Jacobi Fields . . . . . . . . . . . . . . . . . . . . . 8355.6.6 Constraints . . . . . . . . . . . . . . . . . . . . . . 836


xxx<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.6.7 Time–Dependent Lagrangian Dynamics . . . . . . 8415.6.8 Time–Dependent Hamiltonian Dynamics . . . . . 8435.6.9 Time–Dependent Constraints . . . . . . . . . . . 8485.6.10 Lagrangian Constraints . . . . . . . . . . . . . . . 8495.6.11 Quadratic Degenerate Lagrangian Systems . . . . 8525.6.12 Time–Dependent Integrable Hamiltonian Systems 8555.6.13 Time–Dependent Action–Angle Coordinates . . . 8585.6.14 Lyapunov Stability . . . . . . . . . . . . . . . . . 8605.6.15 First–Order Dynamical Equations . . . . . . . . . 8615.6.16 Lyapunov Tensor and Stability . . . . . . . . . . . 8635.6.16.1 Lyapunov Tensor . . . . . . . . . . . . . 8635.6.16.2 Lyapunov Stability . . . . . . . . . . . . 8645.7 Application: Jets and Multi–Time RheonomicDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8685.7.1 Relativistic Rheonomic Lagrangian Spaces . . . . 8705.7.2 Canonical Nonlinear Connections . . . . . . . . . 8715.7.3 Cartan’s Canonical Connections . . . . . . . . . . 8745.7.4 General Nonlinear Connections . . . . . . . . . . 8765.8 Jets and Action Principles . . . . . . . . . . . . . . . . . . 8775.9 Application: Jets and Lagrangian Field Theory . . . . . 8835.9.1 Lagrangian Conservation Laws . . . . . . . . . . . 8885.9.2 General Covariance Condition . . . . . . . . . . . 8935.10 Application: Jets and Hamiltonian Field Theory . . . . 8975.10.1 Covariant Hamiltonian Field Systems . . . . . . . 8995.10.2 Associated Lagrangian and Hamiltonian Systems 9025.10.3 Evolution Operator . . . . . . . . . . . . . . . . . 9045.10.4 Quadratic Degenerate Systems . . . . . . . . . . . 9105.11 Application: Gauge Fields on Principal Connections . . 9135.11.1 Connection Strength . . . . . . . . . . . . . . . . 9135.11.2 Associated Bundles . . . . . . . . . . . . . . . . . 9145.11.3 Classical Gauge Fields . . . . . . . . . . . . . . . 9155.11.4 Gauge Transformations . . . . . . . . . . . . . . . 9175.11.5 Lagrangian Gauge Theory . . . . . . . . . . . . . 9195.11.6 Hamiltonian Gauge Theory . . . . . . . . . . . . . 9205.11.7 Gauge Conservation Laws . . . . . . . . . . . . . 9235.11.8 Topological Gauge Theories . . . . . . . . . . . . 9245.12 Application: Modern <strong>Geom</strong>etrodynamics . . . . . . . . 9285.12.1 Stress–Energy–Momentum Tensors . . . . . . . . 9285.12.2 Gauge Systems of Gravity and Fermion Fields . . 955


Contentsxxxi5.12.3 Hawking–Penrose Quantum Gravity andBlack Holes . . . . . . . . . . . . . . . . . . . . . 9636. <strong>Geom</strong>etrical Path Integrals and Their Applications 9836.1 Intuition Behind a Path Integral . . . . . . . . . . . . . . 9846.1.1 Classical Probability Concept . . . . . . . . . . . 9846.1.2 Discrete Random Variable . . . . . . . . . . . . . 9846.1.3 Continuous Random Variable . . . . . . . . . . . 9846.1.4 General Markov Stochastic Dynamics . . . . . . . 9856.1.5 Quantum Probability Concept . . . . . . . . . . . 9896.1.6 Quantum Coherent States . . . . . . . . . . . . . 9916.1.7 Dirac’s < bra | ket > Transition Amplitude . . . . 9926.1.8 Feynman’s Sum–over–Histories . . . . . . . . . . . 9946.1.9 The Basic Form of a Path Integral . . . . . . . . . 9966.1.10 Application: Adaptive Path Integral . . . . . . 9976.2 Path Integral History . . . . . . . . . . . . . . . . . . . . 9986.2.1 Extract from Feynman’s Nobel Lecture . . . . . . 9986.2.2 Lagrangian Path Integral . . . . . . . . . . . . . . 10026.2.3 Hamiltonian Path Integral . . . . . . . . . . . . . 10036.2.4 Feynman–Kac Formula . . . . . . . . . . . . . . . 10046.2.5 Itô Formula . . . . . . . . . . . . . . . . . . . . . 10066.3 Standard Path–Integral Quantization . . . . . . . . . . . 10066.3.1 Canonical versus Path–Integral Quantization . . . 10066.3.2 Application: Particles, Sources, Fields andGauges . . . . . . . . . . . . . . . . . . . . . . . . 10116.3.2.1 Particles . . . . . . . . . . . . . . . . . . 10116.3.2.2 Sources . . . . . . . . . . . . . . . . . . 10126.3.2.3 Fields . . . . . . . . . . . . . . . . . . . 10136.3.2.4 Gauges . . . . . . . . . . . . . . . . . . 10136.3.3 Riemannian–Symplectic <strong>Geom</strong>etries . . . . . . . . 10146.3.4 Euclidean Stochastic Path Integral . . . . . . . . 10166.3.5 Application: Stochastic Optimal Control . . . . 10206.3.5.1 Path–Integral Formalism . . . . . . . . 10216.3.5.2 Monte Carlo Sampling . . . . . . . . . . 10236.3.6 Application: Nonlinear Dynamics of OptionPricing . . . . . . . . . . . . . . . . . . . . . . . . 10256.3.6.1 Theory and Simulations of OptionPricing . . . . . . . . . . . . . . . . . . 10256.3.6.2 Option Pricing via Path Integrals . . . 1029


xxxii<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction6.3.6.3 Continuum Limit and AmericanOptions . . . . . . . . . . . . . . . . . . 10356.3.7 Application: Nonlinear Dynamics of ComplexNets . . . . . . . . . . . . . . . . . . . . . . . . . 10366.3.7.1 Continuum Limit of the Kuramoto Net 10376.3.7.2 Path–Integral Approach to ComplexNets . . . . . . . . . . . . . . . . . . . . 10386.3.8 Application: Dissipative Quantum Brain Model 10396.3.9 Application: Cerebellum as a NeuralPath–Integral . . . . . . . . . . . . . . . . . . . . 10436.3.9.1 Spinal Autogenetic Reflex Control . . . 10456.3.9.2 Cerebellum – the Comparator . . . . . 10476.3.9.3 Hamiltonian Action and Neural PathIntegral . . . . . . . . . . . . . . . . . . 10496.3.10 Path Integrals via Jets: Perturbative QuantumFields . . . . . . . . . . . . . . . . . . . . . . . . . 10506.4 Sum over <strong>Geom</strong>etries and Topologies . . . . . . . . . . . . 10556.4.1 Simplicial Quantum <strong>Geom</strong>etry . . . . . . . . . . . 10576.4.2 Discrete Gravitational Path Integrals . . . . . . . 10596.4.3 Regge Calculus . . . . . . . . . . . . . . . . . . . 10616.4.4 Lorentzian Path Integral . . . . . . . . . . . . . . 10646.4.5 Application: Topological Phase Transitionsand Hamiltonian Chaos . . . . . . . . . . . . . . . 10696.4.5.1 Phase Transitions in HamiltonianSystems . . . . . . . . . . . . . . . . . . 10696.4.5.2 <strong>Geom</strong>etry of the Largest LyapunovExponent . . . . . . . . . . . . . . . . . 10726.4.5.3 Euler Characteristics of HamiltonianSystems . . . . . . . . . . . . . . . . . . 10756.4.6 Application: Force–Field Psychodynamics . . . 10796.4.6.1 Motivational Cognition in the LifeSpace Foam . . . . . . . . . . . . . . . . 10806.5 Application: Witten’s TQFT, SW–Monopoles andStrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10976.5.1 Topological Quantum Field Theory . . . . . . . . 10976.5.2 Seiberg–Witten Theory and TQFT . . . . . . . . 11036.5.2.1 SW Invariants and Monopole Equations 11036.5.2.2 Topological Lagrangian . . . . . . . . . 11056.5.2.3 Quantum Field Theory . . . . . . . . . 1107


Contentsxxxiii6.5.2.4 Dimensional Reduction and 3D FieldTheory . . . . . . . . . . . . . . . . . . 11126.5.2.5 <strong>Geom</strong>etrical Interpretation . . . . . . . 11156.5.3 TQFTs Associated with SW–Monopoles . . . . . 11186.5.3.1 Dimensional Reduction . . . . . . . . . 11226.5.3.2 TQFTs of 3D Monopoles . . . . . . . . 11246.5.3.3 Non–Abelian Case . . . . . . . . . . . . 11356.5.4 Stringy Actions and Amplitudes . . . . . . . . . . 11386.5.4.1 Strings . . . . . . . . . . . . . . . . . . 11396.5.4.2 Interactions . . . . . . . . . . . . . . . . 11406.5.4.3 Loop Expansion – Topology of ClosedSurfaces . . . . . . . . . . . . . . . . . . 11416.5.5 Transition Amplitudes for Strings . . . . . . . . . 11436.5.6 Weyl Invariance and Vertex Operator Formulation 11466.5.7 More General Actions . . . . . . . . . . . . . . . . 11466.5.8 Transition Amplitude for a Single Point Particle . 11476.5.9 Witten’s Open String Field Theory . . . . . . . . 11486.5.9.1 Operator Formulation of String FieldTheory . . . . . . . . . . . . . . . . . . 11496.5.9.2 Open Strings in Constant B−FieldBackground . . . . . . . . . . . . . . . . 11516.5.9.3 Construction of Overlap Vertices . . . . 11546.5.9.4 Transformation of String Fields . . . . . 11646.6 Application: Dynamics of Strings and Branes . . . . . . 11686.6.1 A Relativistic Particle . . . . . . . . . . . . . . . 11696.6.2 A String . . . . . . . . . . . . . . . . . . . . . . . 11716.6.3 A Brane . . . . . . . . . . . . . . . . . . . . . . . 11736.6.4 String Dynamics . . . . . . . . . . . . . . . . . . . 11756.6.5 Brane Dynamics . . . . . . . . . . . . . . . . . . . 11776.7 Application: Topological String Theory . . . . . . . . . 11806.7.1 Quantum <strong>Geom</strong>etry Framework . . . . . . . . . . 11806.7.2 Green–Schwarz Bosonic Strings and Branes . . . . 11816.7.3 Calabi–Yau Manifolds, Orbifolds and MirrorSymmetry . . . . . . . . . . . . . . . . . . . . . . 11866.7.4 More on Topological Field Theories . . . . . . . . 11896.7.5 Topological Strings . . . . . . . . . . . . . . . . . 12046.7.6 <strong>Geom</strong>etrical Transitions . . . . . . . . . . . . . . 12216.7.7 Topological Strings and Black Hole Attractors . . 1225


xxxiv<strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction6.8 Application: Advanced <strong>Geom</strong>etry and Topology ofString Theory . . . . . . . . . . . . . . . . . . . . . . . . . 12326.8.1 String Theory and Noncommutative <strong>Geom</strong>etry . . 12326.8.1.1 Noncommutative Gauge Theory . . . . 12336.8.1.2 Open Strings in the Presence ofConstant B−Field . . . . . . . . . . . . 12356.8.2 K−Theory Classification of Strings . . . . . . . . 1241Bibliography 1253Index 1295


Chapter 1IntroductionIn this introductory chapter we will firstly give a soft, ‘plain English’ introductioninto manifolds and related differential–geometric terms, withintention to make this book accessible to the wider scientific and engineeringcommunity. Secondly, we will present the paradigm of differential–geometric modelling of dynamical systems, in the form of a generic algorithmic‘recipe’ (see [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)] for background details).The readers familiar with the manifold concept can skip the first sectionand only quickly review the second one.1.1 Manifolds and Related <strong>Geom</strong>etrical StructuresThe core of both differential geometry and modern geometrical dynamicsrepresents the concept of manifold. A manifold is an abstract mathematicalspace, which locally (i.e., in a close–up view) resembles the spaces describedby Euclidean geometry, but which globally (i.e., when viewed as a whole)may have a more complicated structure. As main pure–mathematical references,we recommend popular graduate textbooks by two ex–Bourbakimembers, Serge Lang [Lang (2005); Lang (2003); Lang (2002); Lang (1999)]and Jean Dieudonne [Dieudonne (1969); Dieudonne (1988)]. Besides, thereader might wish to consult some other ‘classics’, including [Spivak (1965);Spivak (1970-75); Choquet-Bruhat and DeWitt-Morete (1982); Bott and Tu(1982); Abraham et al. (1988); De Rham (1984); Milnor (1997); Munkres(1999)], as well as free internet sources [Wikipedia (2005); Weisstein (2004);PlanetMath (2006)]. Finally, as first–order applications, we recommendthree popular textbooks in mechanics, [Abraham and Marsden (1978);Arnold (1989); Marsden and Ratiu (1999)].For example, the surface of Earth is a manifold; locally it seems to be1


2 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionflat, but viewed as a whole from the outer space (globally) it is actuallyround. A manifold can be constructed by ‘gluing’ separate Euclidean spacestogether; for example, a world map can be made by gluing many maps oflocal regions together, and accounting for the resulting distortions. 1Another example of a manifold is a circle S 1 . A small piece of a circleappears to be like a slightly–bent part of a straight line segment, but overallthe circle and the segment are different 1D manifolds (see Figure 1.1). Acircle can be formed by bending a straight line segment and gluing the endstogether. 21 On a sphere, the sum of the angles of a triangle is not equal to 180 o . A sphere is not aEuclidean space, but locally the laws of the Euclidean geometry are good approximations.In a small triangle on the face of the earth, the sum of the angles is very nearly 180 o . Asphere can be represented by a collection of two dimensional maps, therefore a sphere isa manifold.2 Locally, the circle looks like a line. It is 1D, that is, only one coordinate is neededto say where a point is on the circle locally. Consider, for instance, the top part ofthe circle (Figure 1.1), where the y−coordinate is positive. Any point in this part canbe described by the x−coordinate. So, there is a continuous bijection χ top (a mappingwhich is 1–1 both ways), which maps the top part of the circle to the open interval(−1, 1), by simply projecting onto the first coordinate: χ top (x, y) = x. Such a functionis called a chart. Similarly, there are charts for the bottom, left , and right parts of thecircle. Together, these parts cover the whole circle and the four charts form an atlas (seethe next subsection) for the circle. The top and right charts overlap: their intersectionlies in the quarter of the circle where both the x− and the y−coordinates are positive.The two charts χ top and χ right map this part bijectively to the interval (0, 1). Thus afunction T from (0, 1) to itself can be constructed, which first inverts the top chart toreach the circle and then follows the right chart back to the interval:T (a) = χ right“χ −1top (a) ”= χ right“a, p 1 − a 2 ”= p 1 − a 2 .Such a function is called a transition map. The top, bottom, left, and right charts showthat the circle is a manifold, but they do not form the only possible atlas. Charts neednot be geometric projections, and the number of charts is a matter of choice. T and theother transition functions in Figure 1.1 are differentiable on the interval (0, 1). Therefore,with this atlas the circle is a differentiable, or smooth manifold.


Introduction 3Fig. 1.1 The four charts each map part of the circle to an open interval, and togethercover the whole circle.The surfaces of a sphere 3 and a torus 4 are examples of 2D manifolds.3 The surface of the sphere S 2 can be treated in almost the same way as the circle S 1 .It can be viewed as a subset of R 3 , defined by: S = {(x, y, z) ∈ R 3 |x 2 +y 2 +z 2 = 1}. Thesphere is 2D, so each chart will map part of the sphere to an open subset of R 2 . Considerthe northern hemisphere, which is the part with positive z coordinate. The function χdefined by χ(x, y, z) = (x, y), maps the northern hemisphere to the open unit disc byprojecting it on the (x, y)−plane. A similar chart exists for the southern hemisphere.Together with two charts projecting on the (x, z)−plane and two charts projecting on the(y, z)−plane, an atlas of six charts is obtained which covers the entire sphere. This canbe easily generalized to an nD sphere S n = {(x 1 , x 2 , ..., x n) ∈ R n |x 2 1 + x2 2 + ... + x2 n = 1}.An n−sphere S n can be also constructed by gluing together two copies of R n . Thetransition map between them is defined as R n \ {0} → R n \ {0} : x ↦→ x/‖x‖ 2 . Thisfunction is its own inverse, so it can be used in both directions. As the transition mapis a (C ∞ )−smooth function, this atlas defines a smooth manifold.4 A torus (pl. tori), denoted by T 2 , is a doughnut–shaped surface of revolution generatedby revolving a circle about an axis coplanar with the circle. The sphere S 2 is aspecial case of the torus obtained when the axis of rotation is a diameter of the circle. Ifthe axis of rotation does not intersect the circle, the torus has a hole in the middle andresembles a ring doughnut, a hula hoop or an inflated tire. The other case, when theaxis of rotation is a chord of the circle, produces a sort of squashed sphere resembling around cushion.A torus can be defined parametrically by:x(u, v) = (R + r cos v) cos u, y(u, v) = (R + r cos v) sin u, z(u, v) = r sin v,where u, v ∈ [0, 2π], R is the distance from the center of the tube to the center of thetorus, and r is the radius of the tube. According to a broader definition, the generatorof a torus need not be a circle but could also be an ellipse or any other conic section.Topologically, a torus is a closed surface defined as product of two circles: T 2 = S 1 ×S 1 . The surface described above, given the relative topology from R 3 , is homeomorphicto a topological torus as long as it does not intersect its own axis.One can easily generalize the torus to arbitrary dimensions. An n−torus T n is definedas a product of n circles: T n = S 1 × S 1 × · · · × S 1 . Equivalently, the n−torus isobtained from the n−cube (the R n −generalization of the ordinary cube in R 3 ) by gluing


4 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionManifolds are important objects in mathematics, physics and control theory,because they allow more complicated structures to be expressed andunderstood in terms of the well–understood properties of simpler Euclideanspaces.The Cartesian product of manifolds is also a manifold (note that notevery manifold can be written as a product). The dimension of the productmanifold is the sum of the dimensions of its factors. Its topology is theproduct topology, and a Cartesian product of charts is a chart for the productmanifold. Thus, an atlas for the product manifold can be constructedusing atlases for its factors. If these atlases define a differential structureon the factors, the corresponding atlas defines a differential structure onthe product manifold. The same is true for any other structure definedon the factors. If one of the factors has a boundary, the product manifoldalso has a boundary. Cartesian products may be used to construct tori andcylinders, for example, as S 1 × S 1 and S 1 × [0, 1], respectively.Manifolds need not be connected (all in ‘one piece’): a pair of separatecircles is also a topological manifold(see below). Manifolds need not beclosed: a line segment without its ends is a manifold. Manifolds need notbe finite: a parabola is a topological manifold.Manifolds can be viewed using either extrinsic or intrinsic view. In theextrinsic view, usually used in geometry and topology of surfaces, an nDmanifold M is seen as embedded in an (n + 1)D Euclidean space R n+1 .Such a manifold is called a ‘codimension 1 space’. With this view it iseasy to use intuition from Euclidean spaces to define additional structure.For example, in a Euclidean space it is always clear whether a vector atsome point is tangential or normal to some surface through that point. Onthe other hand, the intrinsic view of an nD manifold M is an abstractway of considering M as a topological space by itself, without any need forsurrounding (n+1)D Euclidean space. This view is more flexible and thus itis usually used in high–dimensional mechanics and physics (where manifoldsused represent configuration and phase spaces of dynamical systems), canmake it harder to imagine what a tangent vector might be.Additional structures are often defined on manifolds. Examples of manifoldswith additional structure include:• differentiable (or, smooth manifolds, on which one can do calculus;the opposite faces together.An n−torus T n is an example of an nD compact manifold. It is also an importantexample of a Lie group (see below).


Introduction 5• Riemannian manifolds, on which distances and angles can be defined;• symplectic manifolds, which serve as the phase space in mechanicsand physics;• 4D pseudo–Riemannian manifolds which model space–time in generalrelativity.The study of manifolds combines many important areas of mathematics:it generalizes concepts such as curves and surfaces as well as ideas fromlinear algebra and topology. Certain special classes of manifolds also haveadditional algebraic structure; they may behave like groups, for instance.Historically, before the modern concept of a manifold there were severalimportant results:(1) Carl Friedrich Gauss was arguably the first to consider abstract spacesas mathematical objects in their own right. His ‘Theorema Egregium’gives a method for computing the curvature of a surface S withoutconsidering the ambient Euclidean 3D space R 3 in which the surfacelies. Such a surface would, in modern terminology, be called a manifold.(2) Non–Euclidean geometry considers spaces where Euclid’s ‘Parallel Postulate’fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai,and Riemann developed them 100 years later. Their research uncoveredtwo more types of spaces whose geometric structures differ from thatof classical Euclidean nD space R n ; these gave rise to hyperbolic geometryand elliptic geometry. In the modern theory of manifolds, thesenotions correspond to manifolds with negative and positive curvature,respectively.(3) The Euler characteristic is an example of a topological property (or topologicalinvariant) of a manifold. For a convex polyhedron in Euclidean3D space R 3 , with V vertices, E edges and F faces, Euler showed thatV − E + F = 2. Thus the number 2 is called the Euler characteristicof the space R 3 . The Euler characteristic of other 3D spaces is a usefultopological invariant, which can be extended to higher dimensions usingthe so–called Betti numbers. The study of other topological invariantsof manifolds is one of the central themes of topology.(4) Bernhard Riemann was the first to do extensive work generalizing theidea of a surface to higher dimensions. The name manifold comes fromRiemann’s original German term, ‘Mannigfaltigkeit’, which W.K. Cliffordtranslated as ‘manifoldness’. In his famous Göttingen inaugural


6 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionlecture entitled ‘On the Hypotheses which lie at the Bases of <strong>Geom</strong>etry’,Riemann described the set of all possible values of a variable withcertain constraints as a ‘manifoldness’, because the variable can havemany values. He distinguishes between continuous manifoldness anddiscontinuous manifoldness, depending on whether the value changescontinuously or not. As continuous examples, Riemann refers to notonly colors and the locations of objects in space, but also the possibleshapes of a spatial figure. Using mathematical induction, Riemannconstructs an n times extended manifoldness, or nD manifoldness, asa continuous stack of (n − 1)D manifoldnesses. Riemann’s intuitivenotion of a ‘manifoldness’ evolved into what is today formalized as amanifold.(5) Henri Poincaré studied 3D manifolds at the end of the 19th Century,and raised a question, today known as the Poincaré conjecture. HermannWeyl gave an intrinsic definition for differentiable manifolds in1912. During the 1930s, H. Whitney and others clarified the foundationalaspects of the subject, and thus intuitions dating back to thelatter half of the 19th Century became precise, and developed throughdifferential geometry (in particular, by the Lie group theory introducedby Sophus Lie in 1870, see below).1.1.1 <strong>Geom</strong>etrical AtlasNow we continue introducing manifolds. As already stated above, an atlasdescribes how a complicated space called a manifold is glued together fromsimpler pieces. Each piece is given by a chart (also known as coordinatechart or local coordinate system). 5The description of most manifolds requires more than one chart (a singlechart is adequate for only the simplest manifolds). An atlas is a specificcollection of charts which covers a manifold. An atlas is not unique asall manifolds can be covered multiple ways using different combinations of5 A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertiblemap between a subset of the manifold and a simple space such that both the map andits inverse preserve the desired structure. For a topological manifold, the simple spaceis some Euclidean space R n and interest is focused on the topological structure. Thisstructure is preserved by homeomorphisms (invertible maps that are continuous in bothdirections).In the case of a differentiable manifold, a set of charts called an atlas allows us todo calculus on manifolds. Polar coordinates, for example, form a chart for the plane R 2minus the positive x−axis and the origin. Another example of a chart is the map χ topmentioned above, a chart for the circle.


Introduction 7charts.The atlas containing all possible charts consistent with a given atlas iscalled the maximal atlas. Unlike an ordinary atlas, the maximal atlas of agiven atlas is unique.More generally, an atlas for a complicated space is constructed out ofthe following pieces of information:(i) A list of spaces that are considered simple.(ii) For each point in the complicated space, a neighborhood of thatpoint that is homeomorphic to a simple space, the homeomorphism beinga chart.(iii) We require the different charts to be compatible. At the minimum,we require that the composite of one chart with the inverse of another be ahomeomorphism (also known as a change of coordinates, or a transformationof coordinates, or a transition function, or a transition map) but weusually impose stronger requirements, such as C ∞ −smoothness. 6This definition of atlas is exactly analogous to the non–mathematicalmeaning of atlas. Each individual map in an atlas of the world gives aneighborhood of each point on the globe that is homeomorphic to the plane.While each individual map does not exactly line up with other maps that itoverlaps with (because of the Earth’s curvature), the overlap of two mapscan still be compared (by using latitude and longitude lines, for example).<strong>Diff</strong>erent choices for simple spaces and compatibility conditions give differentobjects. For example, if we choose for our simple spaces the Euclideanspaces R n , we get topological manifolds. If we also require the coordinate6 Charts in an atlas may overlap and a single point of a manifold may be representedin several charts. If two charts overlap, parts of them represent the same region ofthe manifold, just as a map of Europe and a map of Asia may both contain Moscow.Given two overlapping charts, a transition function can be defined, which goes froman open Euclidean nD ball B n = {(x 1 , x 2 , ..., x n) ∈ R n |x 2 1 + x2 2 + ... + x2 n < 1} in R nto the manifold and then back to another (or perhaps the same) open nD ball in R n .The resultant map, like the map T in the circle example above, is called a change ofcoordinates, a coordinate transformation, a transition function, or a transition map.An atlas can also be used to define additional structure on the manifold. The structureis first defined on each chart separately. If all the transition maps are compatible withthis structure, the structure transfers to the manifold.This is the standard way differentiable manifolds are defined. If the transition functionsof an atlas for a topological manifold preserve the natural differential structureof R n (that is, if they are diffeomorphisms, i.e., invertible maps that are smooth inboth directions), the differential structure transfers to the manifold and turns it into adifferentiable, or smooth manifold.In general the structure on the manifold depends on the atlas, but sometimes differentatlases give rise to the same structure. Such atlases are called compatible.


8 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionchanges to be diffeomorphisms, we get differentiable manifolds, or smoothmanifolds.We call two atlases compatible if the charts in the two atlases are allcompatible (or equivalently if the union of the two atlases is an atlas).Usually, we want to consider two compatible atlases as giving rise to thesame space. Formally, (as long as our concept of compatibility for chartshas certain simple properties), we can define an equivalence relation on theset of all atlases, calling two the same if they are compatible. In fact, theunion of all atlases compatible with a given atlas is itself an atlas, calleda complete (or maximal) atlas. Thus every atlas is contained in a uniquecomplete atlas.By definition, a smooth differentiable structure (or differential structure)on a manifold M is such a maximal atlas of charts, all related bysmooth coordinate changes on the overlaps.1.1.2 Topological ManifoldsA topological manifold is a manifold that is glued together from Euclideanspaces R n . Euclidean spaces are the simplest examples of topological manifolds.Thus, a topological manifold is a topological space that locally lookslike an Euclidean space. More precisely, a topological manifold is a topologicalspace 7 locally homeomorphic to a Euclidean space. This means that7 Topological spaces are structures that allow one to formalize concepts such as convergence,connectedness and continuity. They appear in virtually every branch of modernmathematics and are a central unifying notion. Technically, a topological space is a setX together with a collection T of subsets of X satisfying the following axioms:(1) The empty set and X are in T ;(2) The union of any collection of sets in T is also in T ; and(3) The intersection of any pair of sets in T is also in T .The collection T is a topology on X. The sets in T are the open sets, and theircomplements in X are the closed sets. The elements of X are called points. By induction,the intersection of any finite collection of open sets is open. Thus, the third Axiom can bereplaced by the equivalent one that the topology be closed under all finite intersectionsinstead of just pairwise intersections. This has the benefit that we need not explicitlyrequire that X be in T , since the empty intersection is (by convention) X. Similarly, wecan conclude that the empty set is in T by using Axiom 2. and taking a union over theempty collection. Nevertheless, it is conventional to include the first Axiom even whenit is redundant.A function between topological spaces is said to be continuous iff the inverse imageof every open set is open. This is an attempt to capture the intuition that there areno ‘breaks’ or ‘separations’ in the function. A homeomorphism is a bijection that iscontinuous and whose inverse is also continuous. Two spaces are said to be homeomor-


Introduction 9every point has a neighborhood for which there exists a homeomorphism(a bijective continuous function whose inverse is also continuous) mappingthat neighborhood to R n . These homeomorphisms are the charts of themanifold.Usually additional technical assumptions on the topological space aremade to exclude pathological cases. It is customary to require that thespace be Hausdorff and second countable.The dimension of the manifold at a certain point is the dimension of theEuclidean space charts at that point map to (number n in the definition).All points in a connected manifold have the same dimension.In topology and related branches of mathematics, a connected space isa topological space which cannot be written as the disjoint union of two ormore nonempty spaces. Connectedness is one of the principal topologicalproperties that is used to distinguish topological spaces. A stronger notionis that of a path–connected space, which is a space where any two pointscan be joined by a path. 8phic if there exists a homeomorphism between them. From the standpoint of topology,homeomorphic spaces are essentially identical.The category of topological spaces, Top, with topological spaces as objects and continuousfunctions as morphisms is one of the fundamental categories in mathematics. Theattempt to classify the objects of this category (up to homeomorphism) by invariants hasmotivated and generated entire areas of research, such as homotopy theory, homologytheory, and K–theory.8 Formally, for a topological space X the following conditions are equivalent:(1) X is connected.(2) X cannot be divided into two disjoint nonempty closed sets (this follows since thecomplement of an open set is closed).(3) The only sets which are both open and closed (open sets) are X and the empty set.(4) The only sets with empty boundary are X and the empty set.(5) X cannot be written as the union of two nonempty separated sets.The maximal nonempty connected subsets of any topological space are called theconnected components of the space. The components form a partition of the space(that is, they are disjoint and their union is the whole space). Every component is aclosed subset of the original space. The components in general need not be open: thecomponents of the rational numbers, for instance, are the one–point sets. A space inwhich all components are one–point sets is called totally disconnected.The space X is said to be path–connected iff for any two points x, y ∈ X there existsa continuous function f : [0, 1] → X, from the unit interval [0, 1] to X, with f(0) = xand f(1) = y (this function is called a path from x to y). Every path–connected spaceis connected.


10 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction1.1.2.1 Topological manifolds without boundaryThe prototypical example of a topological manifold without boundary isEuclidean space. A general manifold without boundary looks locally, as atopological space, like Euclidean space. This is formalized by requiring thata manifold without boundary is a non–empty topological space in whichevery point has an open neighborhood homeomorphic to (an open subsetof) R n (Euclidean n−space). Another way of saying this, using charts,is that a manifold without boundary is a non–empty topological space inwhich at every point there is an R n −chart.1.1.2.2 Topological manifolds with boundaryGenerally speaking, it is possible to allow a topological manifold to have aboundary. The prototypical example of a topological manifold with boundaryis the Euclidean closed half–space. Most points in Euclidean closedhalf–space, those not on the boundary, have a neighborhood homeomorphicto Euclidean space in addition to having a neighborhood homeomorphicto Euclidean closed half–space, but the points on the boundary onlyhave neighborhoods homeomorphic to Euclidean closed half–space and notto Euclidean space. Thus we need to allow for two kinds of points inour topological manifold with boundary: points in the interior and pointsin the boundary. Points in the interior will, as before, have neighborhoodshomeomorphic to Euclidean space, but may also have neighborhoods homeomorphicto Euclidean closed half–space. Points in the boundary will haveneighborhoods homeomorphic to Euclidean closed half–space. Thus a topologicalmanifold with boundary is a non-empty topological space in whichat each point there is an R n −chart or an [0, ∞) × R n−1 −chart. The set ofpoints at which there are only [0, ∞) × R n−1 −charts is called the boundaryand its complement is called the interior. The interior is always non–emptyand is a topological n−manifold without boundary. If the boundary is non–empty then it is a topological (n − 1)−manifold without boundary. If theboundary is empty, then we regain the definition of a topological manifoldwithout boundary.1.1.2.3 Properties of topological manifoldsA manifold with empty boundary is said to be closed manifold if it iscompact, and open manifold if it is not compact.All 1–manifolds are curves and all 2–manifolds are surfaces. Examples


Introduction 11of curves include circles, hyperbolas, and the trefoil knot. Sphere, cylinder,torus, projective plane, 9 Möbius strip, 10 and Klein bottle 11 are examplesof surfaces.Manifolds inherit many of the local properties of Euclidean space. Inparticular, they are locally path–connected, locally compact and locallymetrizable. Being locally compact Hausdorff spaces, they are necessarilyTychonoff spaces. Requiring a manifold to be Hausdorff may seem strange;it is tempting to think that being locally homeomorphic to a Euclideanspace implies being a Hausdorff space. A counterexample is created bydeleting zero from the real line and replacing it with two points, an openneighborhood of either of which includes all nonzero numbers in some openinterval centered at zero. This construction, called the real line with twoorigins is not Hausdorff, because the two origins cannot be separated.All compact surfaces are homeomorphic to exactly one of the 2–sphere,a connected sum of tori, or a connected sum of projective planes.A topological space is said to be homogeneous if its homeomorphismgroup acts transitively on it. Every connected manifold without boundaryis homogeneous, but manifolds with nonempty boundary are not homogeneous.It can be shown that a manifold is metrizable if and only if it is paracompact.Non–paracompact manifolds (such as the long line) are generally9 Begin with a sphere centered on the origin. Every line through the origin piercesthe sphere in two opposite points called antipodes. Although there is no way to do sophysically, it is possible to mathematically merge each antipode pair into a single point.The closed surface so produced is the real projective plane, yet another non-orientablesurface. It has a number of equivalent descriptions and constructions, but this routeexplains its name: all the points on any given line through the origin projects to thesame ‘point’ on this 1plane’.10 Begin with an infinite circular cylinder standing vertically, a manifold withoutboundary. Slice across it high and low to produce two circular boundaries, and thecylindrical strip between them. This is an orientable manifold with boundary, uponwhich ‘surgery’ will be performed. Slice the strip open, so that it could unroll to becomea rectangle, but keep a grasp on the cut ends. Twist one end 180 deg, making the innersurface face out, and glue the ends back together seamlessly. This results in a strip witha permanent half–twist: the Möbius strip. Its boundary is no longer a pair of circles,but (topologically) a single circle; and what was once its ‘inside’ has merged with its‘outside’, so that it now has only a single side.11 Take two Möbius strips; each has a single loop as a boundary. Straighten out thoseloops into circles, and let the strips distort into cross–caps. Gluing the circles togetherwill produce a new, closed manifold without boundary, the Klein bottle. Closing thesurface does nothing to improve the lack of orientability, it merely removes the boundary.Thus, the Klein bottle is a closed surface with no distinction between inside and outside.Note that in 3D space, a Klein bottle’s surface must pass through itself. Building a Kleinbottle which is not self–intersecting requires four or more dimensions of space.


12 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionregarded as pathological, so it’s common to add paracompactness to thedefinition of an n−manifold. Sometimes n−manifolds are defined to besecond–countable, which is precisely the condition required to ensure thatthe manifold embeds in some finite–dimensional Euclidean space. Note thatevery compact manifold is second–countable, and every second–countablemanifold is paracompact.Topological manifolds are usually required to be Hausdorff and second–countable. Every Hausdorff, second countable manifold of dimension nadmits an atlas consisting of at most n + 1 charts.1.1.3 <strong>Diff</strong>erentiable ManifoldsFor most applications, a special kind of topological manifold, a differentiablemanifold, is used. If the local charts on a manifold are compatible in acertain sense, one can define directions, tangent spaces, and differentiablefunctions on that manifold. In particular it is possible to use calculus ona differentiable manifold. Each point of an nD differentiable manifold hasa tangent space. This is an Euclidean space R n consisting of the tangentvectors of the curves through the point.Two important classes of differentiable manifolds are smooth and analyticmanifolds. For smooth manifolds the transition maps are smooth, thatis infinitely differentiable, denoted by C ∞ . Analytic manifolds are smoothmanifolds with the additional condition that the transition maps are analytic(a technical definition which loosely means that Taylor’s expansionTheorem 12 holds). The sphere can be given analytic structure, as can mostfamiliar curves and surfaces.In other words, a differentiable (or, smooth) manifold is a topologicalmanifold with a globally defined differentiable (or, smooth) structure. Atopological manifold can be given a differentiable structure locally by usingthe homeomorphisms in the atlas of the topological space (i.e., the homeomorphismcan be used to give a local coordinate system). The globaldifferentiable structure is induced when it can be shown that the naturalcomposition of the homeomorphisms between the corresponding openEuclidean spaces are differentiable on overlaps of charts in the atlas. Therefore,the coordinates defined by the homeomorphisms are differentiable withrespect to each other when treated as real valued functions with respect to12 The most basic example of Taylor’s Theorem is the approximation of the exponentialfunction near the origin point x = 0: e x ≈ 1 + x + x2 + x3 + · · · + xn . For technical2! 3! n!details, see any calculus textbook.


Introduction 13the variables defined by other coordinate systems whenever charts overlap.This idea is often presented formally using transition maps.This allows one to extend the meaning of differentiability to spaces withoutglobal coordinate systems. Specifically, a differentiable structure allowsone to define a global differentiable tangent space, and consequently, differentiablefunctions, and differentiable tensor–fields (including vector–fields).<strong>Diff</strong>erentiable manifolds are very important in physics. Special kinds ofdifferentiable manifolds form the arena for physical theories such as classicalmechanics (Hamiltonian mechanics and Lagrangian mechanics), generalrelativity and Yang–Mills gauge theory. It is possible to develop calculuson differentiable manifolds, leading to such mathematical machinery as theexterior calculus.Historically, the development of differentiable manifolds (as well as differentialgeometry in general) is usually credited to C.F. Gauss and hisstudent B. Riemann. The work of physicists J.C. Maxwell and A. Einsteinlead to the development of the theory transformations between coordinatesystems which preserved the essential geometric properties. Eventuallythese ideas were generalized by H. Weyl in ‘Idee der Riemannschen Fläshe’(1913) and ‘Raum, Ziet, Materie’ (‘Space Time Matter’, 1921). T. Levi–Civita applied these ideas in ‘Lezioni di calcolo differenziale assoluto’ (‘TheAbsolute <strong>Diff</strong>erential Calculus’, 1923). The approach of Weyl was essentiallyto consider the coordinate functions in terms of other coordinatesand to assume differentiability for the coordinate function. In 1963, S.Kobayashi and K. Nomizu gave the group transformation/atlas approach.Generalizations of manifoldsThe three most common generalizations of manifolds are:• orbifolds: An orbifold is a generalization of manifold allowing for certainkinds of ‘singularities’ in the topology. Roughly speaking, it is aspace which locally looks like the quotients of some simple space (e.g.,Euclidean space) by the actions of various finite groups. The singularitiescorrespond to fixed points of the group actions, and the actionsmust be compatible in a certain sense.• algebraic varieties and schemes: An algebraic variety is glued togetherfrom affine algebraic varieties, which are zero sets of polynomials overalgebraically closed fields. Schemes are likewise glued together fromaffine schemes, which are a generalization of algebraic varieties. Both


14 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionare related to manifolds, but are constructed using sheaves 13 insteadof atlases. Because of singular points one cannot assume a variety is amanifold.• CW–complexes: A CW complex is a topological space formed by gluingobjects of different dimensionality together; for this reason they generallyare not manifolds. However, they are of central interest in algebraictopology, especially in homotopy theory, where such dimensional defectsare acceptable.1.1.4 Tangent and Cotangent Bundles of Manifolds1.1.4.1 Tangent Bundle of a Smooth ManifoldThe tangent bundle of an open contractable set U ∈ R n is the smoothmanifold U × R n .The tangent bundle T M of the smooth manifold M is constructed usingthe transition maps which define the differentiable structure of M. One mayconstruct transition maps for the atlas of smooth manifolds U i × R n , where13 A sheaf F on a topological space X is an object that assigns a structure F (U) (suchas a set, group, or ring) to each open subset U of X. The structures F (U) are compatiblewith the operations of restricting the open set to smaller subsets and gluing smaller opensets to get a bigger one. A presheaf is similar to a sheaf, but it may not be possible toglue. Sheaves enable one to discuss in a refined way what is a local property, as appliedto a function.Sheaves are used in topology, algebraic geometry and differential geometry wheneverone wants to keep track of algebraic data that vary with every open set of the givengeometrical space. They are a global tool to study objects which vary locally (that is,depend on the open sets). As such, they are a natural instrument to study the globalbehavior of objects which are of local nature, such as open sets, analytic functions,manifolds, and so on.For a typical example, consider a topological space X, and for every open set U ∈ X,let F (U) be the set of all continuous functions U → R. If V is an open subset of U,then the functions on U can be restricted to V , and we get a map F (U) → F (V ). Now,‘gluing’ describes the following process: suppose the U i are given open sets with unionU, and for each i we are given an element f i ∈ F (U i ), a continuous function f i : U i → R.If these functions agree where they overlap, then we can glue them together in a uniqueway to form a continuous function f : U → R, which agrees with all the given f i . Thecollection of the sets F (U) together with the restriction maps F (U) → F (V ) then forma sheaf of sets on X. Furthermore, each F (U) is a commutative ring and the restrictionmaps are ring homomorphisms, making F a sheaf of rings on X.For a very similar example, consider a smooth manifold M, and for every open set U ofM, let F (U) be the set of smooth functions U → R. Here too, ‘gluing’ works and we geta sheaf of rings on M. Another sheaf on M assigns to every open set U of M the vectorspace of all smooth vector–fields defined on U. Restriction and gluing of vector–fieldsworks like that of functions, and we get a sheaf of vector spaces on the manifold M.


Introduction 15U i denotes one of the charts in the atlas for M. The extended atlas definesa topological manifold and the differentiablity of the transition maps definea differentiable structure on the tangent bundle manifold.The tangent bundle is where tangent vectors live, and is itself a smoothmanifold. The so–called Lagrangian is a natural energy function on thetangent bundle.Associated with every point x on a smooth manifold M is a tangentspace T x M and its dual, the cotangent space Tx ∗ M. The former consists ofthe possible directional derivatives, and the latter of the differentials, whichcan be thought of as infinitesimal elements of the manifold. These spacesalways have the same dimension n as the manifold does. The collection ofall tangent spaces can in turn be made into a manifold, the tangent bundle,whose dimension is 2n.1.1.4.2 Cotangent Bundle of a Smooth ManifoldRecall that the dual of a vector space is the set of linear functionals (i.e.,real valued linear functions) on the vector space. In particular, if the vectorspace is finite and has an inner product then the linear functionals can berealized by the functions f v (w) = 〈v, w〉.The cotangent bundle T ∗ M is the dual to the tangent bundle T M in thesense that each tangent space has a dual cotangent space as a vector space.The cotangent bundle T ∗ M is a smooth manifold itself, whose dimensionis 2n. The so–called Hamiltonian is is a natural energy function on thecotangent bundle. The total space of a cotangent bundle naturally has thestructure of a symplectic manifold (see below).1.1.4.3 Fibre–, Tensor–, and Jet–BundlesA fibre bundle is a space which locally looks like a product of two spaces butmay possess a different global structure. Tangent and cotangent bundlesare special cases of a fibre bundle. 14A tensor bundle is a direct sum of all tensor products of the tangentbundle and the cotangent bundle. 15 To do calculus on the tensor bundle14 Every fiber bundle consists of a continuous surjective map: π : E −→ B, where smallregions in the total space E look like small regions in the product space B × F. Here Bis called the base space while F is the fiber space. For example, the product space B × F ,equipped with π equal to projection onto the first coordinate, is a fiber bundle. This iscalled the trivial bundle. One goal of the theory of bundles is to quantify, via algebraicinvariants, what it means for a bundle to be non–trivial.15 Recall that a tensor is a certain kind of geometrical entity which generalizes the


16 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona connection is needed (see below). In particular, the exterior calculuson a totally antisymmetric tensor bundle allows for a generalization of theclassical gradient, divergence and curl operators.A jet bundle is a generalization of both the tangent bundle and thecotangent bundle. The Jet bundle is a certain construction which makes anew smooth fiber bundle out of a given smooth fiber bundle. It makes itpossible to write differential equations on sections of a fiber bundle in an invariantform. In contrast with Riemannian manifolds and their (co)tangentbundles, a connection is a tensor on the jet bundle.1.1.5 Riemannian Manifolds: Configuration Spaces for LagrangianMechanicsTo measure distances and angles on manifolds, the manifold must be Riemannian.A Riemannian manifold is an analytic manifold in which eachtangent space is equipped with an inner product g = 〈·, ·〉, in a mannerconcepts of scalar, vector and linear operator in a way that is independent of any chosenframe of reference. While tensors can be represented by multi-dimensional arraysof components, the point of having a tensor theory is to explain the further implicationsof saying that a quantity is a tensor, beyond that specifying it requires a numberof indexed components. In particular, tensors behave in special ways under coordinatetransformations. The tensor notation (also called the covariant formalism) was developedaround 1890 by Gregorio Ricci–Curbastro under the title ‘Absolute <strong>Diff</strong>erential<strong>Geom</strong>etry’, and made accessible to many mathematicians by the publication of TullioLevi–Civita’s classic text ‘The Absolute <strong>Diff</strong>erential Calculus’ in 1900. The tensor calculusachieved broader acceptance with the introduction of Einstein’s general relativitytheory, around 1915. General Relativity is formulated completely in the language oftensors, which Einstein had learned from Levi–Civita himself with great difficulty. Buttensors are used also within other fields such as continuum mechanics (e.g., the straintensor). Note that the word ‘tensor’ is often used as a shorthand for ‘tensor–field’, whichis a tensor value defined at every point in a manifold.The so–called ‘classical approach’ views tensors as multidimensional arrays that are nDgeneralizations of scalars, 1D vectors and 2D matrices. The ‘components’ of the tensorare the indices of the array. This idea can then be further generalized to tensor–fields,where the elements of the tensor are functions, or even differentials.On the other hand, the so–called ‘modern’ or component–free approach, views tensorsinitially as abstract geometrical objects, expressing some definite type of multi–linearconcept. Their well–known properties can be derived from their definitions, as linearmaps, or more generally; and the rules for manipulations of tensors arise as an extensionof linear algebra to multilinear algebra. This treatment has largely replacedthe component–based treatment for advanced study, in the way that the more moderncomponent–free treatment of vectors replaces the traditional component–based treatmentafter the component–based treatment has been used to provide an elementarymotivation for the concept of a vector. You could say that the slogan is ‘tensors areelements of some tensor bundle’.


Introduction 17which varies smoothly from point to point. Given two tangent vectors Xand Y, the inner product 〈X, Y 〉 gives a real number. The dot (or scalar)product is a typical example of an inner product. This allows one to definevarious notions such as length, angles, areas (or, volumes), curvature,gradients of functions and divergence of vector–fields. Most familiar curvesand surfaces, including n−spheres and Euclidean space, can be given thestructure of a Riemannian manifold.Any smooth manifold admits a Riemannian metric, which often helpsto solve problems of differential topology. It also serves as an entry levelfor the more complicated structure of pseudo–Riemannian manifolds, which(in four dimensions) are the main objects of the general relativity theory. 16Every smooth submanifold of R n (see extrinsic view above) has an inducedRiemannian metric g: the inner product on each tangent space isthe restriction of the inner product on R n . Therefore, one could definea Riemannian manifold as a metric space which is isometric to a smoothsubmanifold of R n with the induced intrinsic metric, where isometry hereis meant in the sense of preserving the length of curves.Usually a Riemannian manifold M is defined as a smooth manifoldwith a smooth section of positive–definite quadratic forms on the associatedtangent bundle T M. Then one has to work to show that it can be turnedto a metric space.Even though Riemannian manifolds are usually ‘curved’ (e.g., the space–time of general relativity), there is still a notion of ‘straight line’ on them:the geodesics. These are curves which locally join their points along shortestpaths.In Riemannian manifolds, the notions of geodesic completeness, topologicalcompleteness and metric completeness are the same: that each impliesthe other is the content of the Hopf–Rinow Theorem.1.1.5.1 Riemann SurfacesA Riemann surface, is a 1D complex manifold. Riemann surfaces can bethought of as ‘deformed versions’ of the complex plane: locally near everypoint they look like patches of the complex plane, but the global topologycan be quite different. For example, they can look like a sphere, or a torus,or a couple of sheets glued together.16 A pseudo–Riemannian manifold is a variant of Riemannian manifold where the metrictensor is allowed to have an indefinite signature (as opposed to a positive–definiteone).


18 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe main point of Riemann surfaces is that holomorphic (analytic complex)functions may be defined between them. Riemann surfaces are nowadaysconsidered the natural setting for studying the global behavior of thesefunctions, especially multi–valued functions such as the square root or thelogarithm.Every Riemann surface is a 2D real analytic manifold (i.e., a surface),but it contains more structure (specifically, a complex structure) which isneeded for the unambiguous definition of holomorphic functions. A 2D realmanifold can be turned into a Riemann surface (usually in several inequivalentways) iff it is orientable. So the sphere and torus admit complexstructures, but the Möbius strip, Klein bottle and projective plane do not.<strong>Geom</strong>etrical facts about Riemann surfaces are as ‘nice’ as possible, andthey often provide the intuition and motivation for generalizations to othercurves and manifolds. The Riemann–Roch Theorem is a prime example ofthis influence. 17Examples of Riemann surfaces include: the complex plane 18 , open subsetsof the complex plane 19 , Riemann sphere 20 , and many others.Riemann surfaces naturally arise in string theory as models of string17 Formally, let X be a Hausdorff space. A homeomorphism from an open subsetU ⊂ X to a subset of C is a chart. Two charts f and g whose domains intersect are saidto be compatible if the maps f ◦ g −1 and g ◦ f −1 are holomorphic over their domains. IfA is a collection of compatible charts and if any x ∈ X is in the domain of some f ∈ A,then we say that A is an atlas. When we endow X with an atlas A, we say that (X, A)is a Riemann surface.<strong>Diff</strong>erent atlases can give rise to essentially the same Riemann surface structure on X;to avoid this ambiguity, one sometimes demands that the given atlas on X be maximal,in the sense that it is not contained in any other atlas. Every atlas A is contained in aunique maximal one by Zorn’s lemma.18 The complex plane C is perhaps the most trivial Riemann surface. The map f(z) = z(the identity map) defines a chart for C, and f is an atlas for C. The map g(z) = z* (theconjugate map) also defines a chart on C and g is an atlas for C. The charts f and g arenot compatible, so this endows C with two distinct Riemann surface structures.19 In a fashion analogous to the complex plane, every open subset of the complex planecan be viewed as a Riemann surface in a natural way. More generally, every open subsetof a Riemann surface is a Riemann surface.20 The Riemann sphere is a useful visualization of the extended complex plane, whichis the complex plane plus a point at infinity. It is obtained by imagining that all therays emanating from the origin of the complex plane eventually meet again at a pointcalled the point at infinity, in the same way that all the meridians from the south poleof a sphere get to meet each other at the north pole.Formally, the Riemann sphere is obtained via a one–point compactification of the complexplane. This gives it the topology of a 2–sphere. The sphere admits a unique complexstructure turning it into a Riemann surface. The Riemann sphere can be characterizedas the unique simply–connected, compact Riemann surface.


Introduction 19interactions.1.1.5.2 Riemannian <strong>Geom</strong>etryRiemannian geometry is the study of smooth manifolds with Riemannianmetrics g, i.e. a choice of positive–definite quadratic form g = 〈·, ·〉 ona manifold’s tangent spaces which varies smoothly from point to point.This gives in particular local ideas of angle, length of curves, and volume.From those some other global quantities can be derived by integrating localcontributions.The manifold may also be given an affine connection, 21 which is roughlyan idea of change from one point to another. If the metric does not ‘varyfrom point to point’ under this connection, we say that the metric andconnection are compatible, and we have a Riemann–Cartan manifold. Ifthis connection is also self–commuting when acting on a scalar function, wesay that it is torsion–free, and the manifold is a Riemannian manifold.The Levi–Civita ConnectionIn Riemannian geometry, the Levi–Civita connection (named after TullioLevi–Civita) is the torsion–free Riemannian connection, i.e., a torsion–free connection of the tangent bundle, preserving a given Riemannian metric(or, pseudo–Riemannian metric). 22 The fundamental Theorem of Rieman-21 Connection (or, covariant derivative) is a way of specifying a derivative of a vector–field along another vector–field on a manifold. That is an application to tangent bundles;there are more general connections, used to formulate intrinsic differential equations.Connections give rise to parallel transport along a curve on a manifold. A connectionalso leads to invariants of curvature, and the so–called torsion.An affine connection is a connection on the tangent bundle T M of a smooth manifoldM. In general, it might have a non–vanishing torsion.The curvature of a connected manifold can be characterized intrinsically by taking avector at some point and parallel transporting it along a curve on the manifold. Althoughcomparing vectors at different points is generally not a well–defined process, an affineconnection ∇ is a rule which describes how to legitimately move a vector along a curveon the manifold without changing its direction (‘keeping the vector parallel’).22 Formally, let (M, g) be a Riemannian manifold (or pseudo–Riemannian manifold);then an affine connection is the Levi–Civita connection if it satisfies the following conditions:(1) Preserves metric g, i.e., for any three vector–fields X, Y, Z ∈ M we have Xg(Y, Z) =g(∇ X Y, Z)+g(Y, ∇ X Z), where Xg(Y, Z) denotes the derivative of a function g(Y, Z)along a vector–field X.(2) Torsion–free, i.e., for any two vector–fields X, Y, Z ∈ M we have ∇ X Y − ∇ Y X =[X, Y ], where [X, Y ] is the Lie bracket for vector–fields X and Y .


20 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionnian geometry states that there is a unique connection which satisfies theseproperties. 23In the theory of Riemannian and pseudo–Riemannian manifolds theterm covariant derivative is often used for the Levi–Civita connection. Thecoordinate expression of the connection is given by Christoffel symbols. 24Note that connection is not a tensor, except on jet bundles.The Fundamental Riemannian TensorsThe two basic objects in Riemannian geometry are the metric tensor andthe curvature tensor. The metric tensor g = 〈·, ·〉 is a symmetric second–order (i.e., (0, 2)) tensor that is used to measure distance in a space. In otherwords, given a Riemannian manifold, we make a choice of a (0, 2)−tensoron the manifold’s tangent spaces. 25 At a given point in the manifold, thistensor takes a pair of vectors in the tangent space to that point, and gives a23 The Levi–Civita connection defines also a derivative along curves, usually denotedby D. Given a smooth curve (a path) γ = γ(t) : R → M and a vector–field X = X(t)on γ, its derivative along γ is defined by: D tX = ∇ ˙γ(t) X. This equation defines theparallel transport for a vector–field X.24 The Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinateexpressions for the Levi–Civita connection derived from the metric tensor. TheChristoffel symbols are used whenever practical calculations involving geometry must beperformed, as they allow very complex calculations to be performed without confusion.In particular, if we denote the unit vectors on M as e i = ∂/∂x i , then the Christoffelsymbols of the second kind are defined by Γ k ij = 〈∇e ; e i j, e k 〉. Alternatively, using themetric tensor g ik (see below) we get the explicit expression for the Christoffel symbolsin a holonomic coordinate basis:Γ i kl = 1 2 gim „ ∂gmk∂x l+ ∂g ml∂x k− ∂g «kl∂x m .In a general, nonholonomic coordinates they include the additional commutation coefficients.The Christoffel symbols are used to define the covariant derivative of varioustensor–fields, as well as the Riemannian curvature. Also, they figure in the geodesicequation:d 2 x idt 2 + dx j dx kΓi jk = 0dt dtfor the curve x i = x i (t) on the smooth manifold M.25 The most familiar example is that of basic high–school»geometry:–the 2D Euclidean1 0metric tensor, in the usual x − y coordinates, reads: g = . The associated length0 1of a curve is given by the familiar calculus formula: L = R b pa (dx) 2 + (dy) 2 .The unit sphere in R 3 comes equipped with a natural metric induced from the ambientEuclidean»metric.–In standard spherical coordinates (θ, φ) the metric takes the form:1 0g =0 sin 2 , which is usually written as: g = dθ 2 + sinθ2 θ dφ 2 .


Introduction 21real number. This concept is just like a dot product, or inner product. Thisfunction from vectors into the real numbers is required to vary smoothlyfrom point to point. 26On any Riemannian manifold, from its second–order metric tensorg = 〈·, ·〉, one can derive the associated fourth–order Riemann curvaturetensor. This tensor is the most standard way to express curvature of Riemannianmanifolds, or more generally, any manifold with an affine connection,torsionless or with torsion. 2726 Once a local coordinate system x i is chosen, the metric tensor appears as a matrix,conventionally given by its components, g = g ij . Given the metric tensor of aRiemannian manifold and using the Einstein summation notation for implicit sums,the length of a segment of a curve parameterized by t, from a to b, is defined as:L = R qb dxa g i dx jij dt dt dt. Also, the angle θ between two tangent vectors ui , v i is definedgas: cos θ =ij u i v jq|gij u i u j ||g ij v i v j | .27 The Riemann curvature tensor is given in terms of a Levi–Civita connection ∇ (moregenerally, an affine connection, or covariant differentiation, see below) by the followingformula:R(u, v)w = ∇ u∇ vw − ∇ v∇ uw − ∇ [u,v] w,where u, v, w are tangent vector–fields and R(u, v) is a linear transformation of thetangent space of the manifold; it is linear in each argument. If u = ∂/∂x i and v = ∂/∂x jare coordinate vector–fields then [u, v] = 0 and therefore the above formula simplifies toR(u, v)w = ∇ u∇ vw − ∇ v∇ uw,i.e., the curvature tensor measures non–commutativity of the covariant derivative. Thelinear transformation w ↦→ R(u, v)w is also called the curvature transformation or endomorphism.In local coordinates x µ (e.g., in general relativity) the Riemann curvature tensor canbe written using the Christoffel symbols of the manifold’s Levi–Civita connection:R ρ σµν = ∂ µΓ ρ νσ − ∂νΓρ µσ + Γρ µλ Γλ νσ − Γρ νλ Γλ µσ .The Riemann curvature tensor has the following symmetries:R(u, v) = −R(v, u),R(u, v)w + R(v, w)u + R(w, u)v = 0.〈R(u, v)w, z〉 = −〈R(u, v)z, w〉,The last identity was discovered by Ricci, but is often called the first Bianchi identity oralgebraic Bianchi identity, because it looks similar to the Bianchi identity below. Thesethree identities form a complete list of symmetries of the curvature tensor, i.e. givenany tensor which satisfies the identities above, one can find a Riemannian manifold withsuch a curvature tensor at some point. Simple calculations show that such a tensor hasn 2 (n 2 − 1)/12 independent components.The Bianchi identity involves the covariant derivatives:∇ uR(v, w) + ∇ vR(w, u) + ∇ wR(u, v) = 0.


22 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction1.1.5.3 Application: Lagrangian MechanicsRiemannian manifolds are natural stage for the Lagrangian mechanics,which is a re–formulation of classical mechanics introduced by Joseph LouisLagrange in 1788. In Lagrangian mechanics, the trajectory of an object isderived by finding the path which minimizes the action, a quantity whichis the integral of the Lagrangian over time. The Lagrangian for classicalmechanics L is taken to be the difference between the kinetic energy T andthe potential energy V , so L = T − V . This considerably simplifies manyphysical problems.For example, consider a bead on a hoop. If one were to calculate themotion of the bead using Newtonian mechanics, one would have a complicatedset of equations which would take into account the forces thatthe hoop exerts on the bead at each moment. The same problem usingLagrangian mechanics is much simpler. One looks at all the possible motionsthat the bead could take on the hoop and mathematically finds theone which minimizes the action. There are fewer equations since one isnot directly calculating the influence of the hoop on the bead at a givenmoment.Lagrange’s EquationsThe equations of motion in Lagrangian mechanics are Lagrange’s equations,also known as Euler–Lagrange equations. Below, we sketch out thederivation of Lagrange’s equation from Newton’s laws of motion (see nextchapter for details).Consider a single mechanical particle with mass m and position vector⃗r. The applied force, ⃗ F , can be expressed as the gradient (denoted ∇) of ascalar potential energy function V (⃗r, t):⃗F = −∇V.A contracted curvature tensor is called the Ricci tensor. It is a symmetric second–ordertensor given by:R ik = ∂Γl ik∂x l− ∂Γl il∂x k+ Γl ikΓ m lm − Γ m ilΓ l km.Its further contraction gives the Ricci scalar curvature, R = g ik R ik . The Einstein tensorG ik is defined in terms of the Ricci tensor R ik and the Ricci scalar R,G ik = R ik − 1 2 g ikR.


Introduction 23Such a force is independent of third– or higher–order derivatives of ⃗r, soNewton’s Second Law forms a set of 3 second–order ODEs. Therefore,the motion of the particle can be completely described by 6 independentvariables, or degrees of freedom (DOF). An obvious set of variables is theCartesian components of ⃗r and their time derivatives, at a given instant oftime, that is position (x, y, z) and velocity (v x , v y , v z ).More generally, we can work with a set of generalized coordinates,q i , (i = 1, ..., n), and their time derivatives, the generalized velocities, ˙q i .The position vector ⃗r is related to the generalized coordinates by sometransformation equation: ⃗r = ⃗r(q i , t). The term ‘generalized coordinates’is really a leftover from the period when Cartesian coordinates were thedefault coordinate system. In the q i −coordinates the Lagrange’s equationsread:∂L∂q i = d ∂Ldt ∂ ˙q i ,where L = T − V is the system’s Lagrangian.The time integral of the Lagrangian L, denoted S is called the action: 28∫S = L dt.Let q 0 and q 1 be the coordinates at respective initial and final times t 0 andt 1 . Using the calculus of variations, it can be shown the Lagrange’s equationsare equivalent to the Hamilton’s principle: “The system undergoesthe trajectory between t 0 and t 1 whose action has a stationary value.” Thisis formally written:δS = 0,where by ‘stationary’, we mean that the action does not vary to first–orderfor infinitesimal deformations of the trajectory, with the end–points (q 0 , t 0 )28 The action principle is an assertion about the nature of motion, from which thetrajectory of a dynamical system subject to some forces can be determined. The pathof an object is the one that yields a stationary value for a quantity called the action.Thus, instead of thinking about an object accelerating in response to applied forces,one might think of them picking out the path with a stationary action. The action isa scalar (a number) with the unit of measure for Action as Energy × Time. Althoughequivalent in classical mechanics with Newton’s laws, the action principle is better suitedfor generalizations and plays an important role in modern physics. Indeed, this principleis one of the great generalizations in physical science. In particular, it is fully appreciatedand best understood within quantum mechanics. Richard Feynman’s path integralformulation of quantum mechanics is based on a stationary–action principle, using pathintegrals. Maxwell’s equations can be derived as conditions of stationary action.


24 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand (q 1 , t 1 ) fixed. 29The total energy function called Hamiltonian, denoted by H, is obtainedby performing a Legendre transformation on the Lagrangian. 30 TheHamiltonian is the basis for an alternative formulation of classical mechanicsknown as Hamiltonian mechanics (see below).In 1948, R.P. Feynman invented the path–integral formulation extendingthe principle of least action to quantum mechanics for electrons and pho-29 More generally, a Lagrangian L[ϕ i ] of a dynamical system is a function of the dynamicalvariables ϕ i (x) and concisely describes the equations of motion of the systemin coordinates x i , (i = 1, ..., n). The equations of motion are obtained by means of anaction principle, written asδSδϕ i= 0,where the action is a functionalZS[ϕ i ] = L[ϕ i (s)] d n x,(d n x = dx 1 ...dx n ).The equations of motion obtained by means of the functional derivative are identical tothe usual Euler–Lagrange equations. Dynamical system whose equations of motion areobtainable by means of an action principle on a suitably chosen Lagrangian are knownas Lagrangian dynamical systems. Examples of Lagrangian dynamical systems rangefrom the (classical version of the) Standard Model, to Newton’s equations, to purelymathematical problems such as geodesic equations and the Plateau’s problem.The Lagrangian mechanics is important not just for its broad applications, but alsofor its role in advancing deep understanding of physics. Although Lagrange sought todescribe classical mechanics, the action principle that is used to derive the Lagrange’sequation is now recognized to be deeply tied to quantum mechanics: physical action andquantum–mechanical phase (waves) are related via Planck’s constant, and the Principleof stationary action can be understood in terms of constructive interference of wavefunctions. The same principle, and the Lagrangian formalism, are tied closely to NoetherTheorem, which relates physical conserved quantities to continuous symmetries of aphysical system; and Lagrangian mechanics and Noether’s Theorem together yield anatural formalism for first quantization by including commutators between certain termsof the Lagrange’s equations of motion for a physical system.More specifically, in field theory, occasionally a distinction is made between the LagrangianL, of which the action is the time integral S = R Ldt and the Lagrangiandensity L, which one integrates over all space–time to get the 4D action:ZS[ϕ i ] = L[ϕ i (x)] d 4 x.The Lagrangian is then the spatial integral of the Lagrangian density.30 The Hamiltonian is the Legendre transform of the Lagrangian:H (q, p, t) = X i˙q i p i − L(q, ˙q, t).


Introduction 25tons. In this formulation, particles travel every possible path between theinitial and final states; the probability of a specific final state is obtained bysumming over all possible trajectories leading to it. In the classical regime,the path integral formulation cleanly reproduces the Hamilton’s principle,as well as the Fermat’s principle in optics.1.1.5.4 Finsler manifoldsFinsler manifolds represent generalization of Riemannian manifolds. AFinsler manifold allows the definition of distance, but not of angle; it isan analytic manifold in which each tangent space is equipped with a norm‖.‖ in a manner which varies smoothly from point to point. This norm canbe extended to a metric, defining the length of a curve; but it cannot ingeneral be used to define an inner product. Any Riemannian manifold (butnot a pseudo–Riemannian manifold) is a Finsler manifold. 311.1.6 Symplectic Manifolds: Phase–Spaces for HamiltonianMechanicsA symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate,2–form ω called the symplectic volume form, or Liouville measure.This condition forces symplectic manifolds to be even–dimensional.Cotangent bundles, which arise as phase–spaces in Hamiltonian mechanics,are the motivating example, but many compact manifolds also have symplecticstructure. All surfaces have a symplectic structure, since a symplecticstructure is simply a volume form. The study of symplectic manifoldsis called symplectic geometry/topology.Symplectic manifolds arise naturally in abstract formulations of classicalmechanics as the cotangent bundles of configuration manifolds: the set ofall possible configurations of a system is modelled as a manifold M, andthis manifold’s cotangent bundle T ∗ M describes the phase–space of the31 Formally, a Finsler manifold is a differentiable manifold M with a Banach normdefined over each tangent space such that the Banach norm as a function of position issmooth, usually it is assumed to satisfy the following regularity condition:For each point x of M, and for every nonzero vector X in the tangent space T × M,the second derivative of the function L : T × M → R given by L(w) = 1 2 ‖w‖2 at X ispositive definite.The length of a smooth curve γ in a Finsler manifold M is given by R ‚ ‚ dγ ‚‚ ‚ dt (t) dt.Length is invariant under reparametrization. With the above regularity condition,geodesics are locally length–minimizing curves with constant speed, or equivalentlycurves in whose energy function, R ‚ ‚ ‚dγdt (t) ‚ ‚‚ 2dt, is extremal under functional derivatives.


26 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondynamical system. 32Any real–valued differentiable function H on a symplectic manifold canserve as an energy function or Hamiltonian. 33 Associated to any Hamiltonianis a Hamiltonian vector–field. 34 The integral curves of the Hamiltonian32 Hamiltonian mechanics is a re–formulation of classical mechanics that was inventedin 1833 by William Rowan Hamilton. It arose from Lagrangian mechanics, a previousre–formulation of classical Newtonian mechanics, introduced by J.L. Lagrange in 1788. Itcan however be formulated without recourse to Lagrangian mechanics, using symplecticmanifolds. It is based on canonical Hamilton’s equations of motion:ṗ = − ∂H∂H, ˙q =∂q ∂p ,with canonical coordinate q and momentum p variables and the Hamiltonian (totalenergy) function H = H(q, p). An important special case consists of those Hamiltoniansthat are quadratic forms, that is, Hamiltonians that can be written as:H(q, p) = 1 〈p, p〉q2This Hamiltonian consists entirely of the kinetic term. If one considers a Riemannianmanifold, or a pseudo–Riemannian manifold, so that one has an invertible, non–degenerate metric, then 〈·, ·〉 q is simply the inverse of the Riemannian metric g.33 In contrast, the quantum–mechanical Hamiltonian H is the observable correspondingto the total energy of the system. It generates the time evolution of quantum states (see,e.g., [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]). If the wave–function |ψ(t)〉 represents the stateof the quantum system at time t, then its time evolution is given by the Schrödingerequation:H |ψ(t)〉 = i ∂ ∂t |ψ(t)〉 ,where is the Planck constant. If the quantum Hamiltonian H is independent of time,then the time evolution is given by:„|ψ(t)〉 = exp − iHt «|ψ(0)〉 .34 In canonical coordinates (q i , p i ) on a symplectic manifold M, the symplectic form ωcan be written as ω = P “ ” n dqi ∧dp i and thus the Hamiltonian vector–field takes the formX H = ∂H, − ∂H∂p i ∂q i . The Hamiltonian vector–field X H also induces a special operation,the Poisson bracket, which is a bilinear map turning two differentiable functions on asymplectic manifold M into a function on M. In particular, if we have two functions, fand g, then the Poisson bracket {f, g} = ω(df, dg). In canonical coordinates (q i , p i ) onthe phase–space manifold, the Poisson bracket takes the form{f, g} = ω(df, dg) =nXi=1» ∂f ∂g∂q i − ∂f –∂g∂p i ∂p i ∂q i .The time evolution of a function f on the symplectic manifold can be given as a one–parameter family of symplectomorphisms, with the time t being the parameter. Thetotal time derivative can be written as d dt f = ∂ ∂t f + {f, H} = ∂ f − {H, f}. The Poisson∂t


Introduction 27vector–field are solutions to the Hamilton–Jacobi equation. 35The Hamiltonian vector–field defines a flow on the symplectic manifold,called a Hamiltonian flow or symplectomorphism. By Liouville’s Theorem,Hamiltonian flows preserve the volume form on the phase space. 36Symplectic manifolds are special cases of a Poisson manifold; the definitionof a symplectic manifold requires that the symplectic 2–form ω benon–degenerate everywhere. If this condition is violated, the manifold maystill be a Poisson manifold. 37 Also, a symplectic manifold endowed with ametric that is compatible with the symplectic form is a Kähler manifold.bracket acts on functions on the symplectic manifold, thus giving the space of functionson the manifold the structure of a Lie algebra (see Lie groups below).35 The Hamilton–Jacobi equation (HJE) is a particular canonical transformation of theclassical Hamiltonian which results in a first order, nonlinear differential equation whosesolution describes the behavior of the system. While the canonical Hamilton’s equationsof motion represent the system of first order ODEs, two for each coordinate, the HJE isa single PDE of one variable for each coordinate. If we have a Hamiltonian of the formthen the HJE for that system is„H q 1 , . . . , q n ; ∂S∂q 1 , . . . , ∂S «∂q n ; t + ∂S∂t = 0,where S represents the classical action functional. The HJE can be used to solve severalproblems elegantly, such as the Kepler problem.36 Liouville’s Theorem, named after the French mathematician Joseph Liouville, is akey Theorem in classical statistical and Hamiltonian mechanics. It asserts that thephase–space distribution function is constant along the trajectories of the system – thatis that the density of system points in the vicinity of a given system point travellingthrough phase–space is constant with time.The Liouville equation describes the time evolution of a phase–space distributionfunction, or Liouville measure. Consider a dynamical system with coordinates q i andconjugate momenta p i (i = 1, ..., n). The time evolution of the phase–space distributionρ(p, q) is governed by the Liouville equation:dρdt = ∂ρ n ∂t + X„ ∂ρ∂qi=1i ˙qi + ∂ρ «ṗ i = 0.∂p iTime derivatives are denoted by dots, and are evaluated according to Hamilton’s equationsfor the system. The Liouville’s Theorem states that the phase–space distributionfunction ρ(p, q) is constant along any trajectory in phase space. The Theorem is oftenrestated in terms of the Poisson bracket with the Hamiltonian function H = H(q, p):∂ρ= −{ρ, H}.∂t<strong>Geom</strong>etrically, this Theorem says that Liouville measure is invariant under the Hamiltonianflow.37 Closely related to even–dimensional symplectic manifolds are the odd–dimensionalmanifolds known as contact manifolds. Any (2n + 1)−D contact manifold (M, ω) givesrise to a (2n + 2)−D symplectic manifold (M × R, d(e t ω)).


28 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThere are several natural geometric notions of submanifold of a symplecticmanifold. There are symplectic submanifolds (potentially of anyeven dimension), where the symplectic form is required to induce a symplecticform on the submanifold. The most important case of these is thatof Lagrangian submanifold, which are isotropic submanifolds of maximal dimension,namely half the dimension of the ambient manifold. Lagrangiansubmanifolds arise naturally in many physical and geometric situations. 381.1.7 Lie GroupsA Lie group is smooth manifold which also carries a group structure whoseproduct and inversion operations are smooth as maps of manifolds. Theseobjects arise naturally in describing symmetries.A Lie group is a group whose elements can be continuously parametrizedby real numbers, such as the rotation group SO(3), which can beparametrized by the Euler angles. More formally, a Lie group is an analyticreal or complex manifold that is also a group, such that the groupoperations multiplication and inversion are analytic maps. Lie groups areimportant in mathematical analysis, physics and geometry because theyserve to describe the symmetry of analytical structures. They were introducedby Sophus Lie in 1870 in order to study symmetries of differentialequations.While the Euclidean space R n is a real Lie group (with ordinary vectoraddition as the group operation), more typical examples are given by matrixLie groups, i.e., groups of invertible matrices (under matrix multiplication).For instance, the group SO(3) of all rotations in R 3 is a matrix Lie group.One classifies Lie groups regarding their algebraic properties 39 (simple,semisimple, solvable, nilpotent, Abelian), their connectedness (connected38 One major example is that the graph of a symplectomorphism in the product symplecticmanifold (M × M, ω × −ω) is Lagrangian. Their intersections display rigidityproperties not possessed by smooth manifolds; the Arnold conjecture gives the sum ofthe submanifold’s Betti numbers as a lower bound for the number of self intersections ofa smooth Lagrangian submanifold, rather than the Euler characteristic in the smoothcase.39 If G and H are Lie groups (both real or both complex), then a Lie–group–homomorphism f : G → H is a group homomorphism which is also an analytic map (onecan show that it is equivalent to require only that f be continuous). The composition oftwo such homomorphisms is again a homomorphism, and the class of all (real or complex)Lie groups, together with these morphisms, forms a category. The two Lie groupsare called isomorphic iff there exists a bijective homomorphism between them whoseinverse is also a homomorphism. Isomorphic Lie groups do not need to be distinguishedfor all practical purposes; they only differ in the notation of their elements.


Introduction 29or simply connected) and their compactness. 40To every Lie group, we can associate a Lie algebra which completelycaptures the local structure of the group (at least if the Lie group is connected).4140 An n−torus T n = S 1 × S 1 × · · · × S 1 (as defined above) is an example of a compactAbelian Lie group. This follows from the fact that the unit circle S 1 is a compact AbelianLie group (when identified with the unit complex numbers with multiplication). Groupmultiplication on T n is then defined by coordinate–wise multiplication.Toroidal groups play an important part in the theory of compact Lie groups. This isdue in part to the fact that in any compact Lie group one can always find a maximaltorus; that is, a closed subgroup which is a torus of the largest possible dimension.41 Conventionally, one can regard any field X of tangent vectors on a Lie group as apartial differential operator, denoting by Xf the Lie derivative (the directional derivative)of the scalar field f in the direction of X. Then a vector–field on a Lie group G issaid to be left–invariant if it commutes with left translation, which means the following.Define L g[f](x) = f(gx) for any analytic function f : G → R and all g, x ∈ G. Then thevector–field X is left–invariant iff XL g = L gX for all g ∈ G. Similarly, instead of R, wecan use C. The set of all vector–fields on an analytic manifold is a Lie algebra over R(or C).On a Lie group G, the left–invariant vector–fields form a subalgebra, the Lie algebrag associated with G. This Lie algebra is finite–dimensional (it has the same dimensionas the manifold G) which makes it susceptible to classification attempts. By classifyingg, one can also get a handle on the group G. The representation theory of simple Liegroups is the best and most important example.Every element v of the tangent space T e at the identity element e of G determines aunique left–invariant vector–field whose value at the element g of G is denoted by gv;the vector space underlying the Lie algebra g may therefore be identified with T e.Every vector–field v in the Lie algebra g determines a function c : R → G whosederivative everywhere is given by the corresponding left–invariant vector–field: c ′ (t) =T L c(t) v and which has the property: c(s + t) = c(s)c(t), (for all s and t) (theoperation on the r.h.s. is the group multiplication in G). The formal similarity of thisformula with the one valid for the elementary exponential function justifies the definition:exp(v) = c(1). This is called the exponential map, and it maps the Lie algebra g intothe Lie group G. It provides a diffeomorphism between a neighborhood of 0 in g anda neighborhood of e in G. This exponential map is a generalization of the exponentialfunction for real numbers (since R is the Lie algebra of the Lie group of positive realnumbers with multiplication), for complex numbers (since C is the Lie algebra of theLie group of non–zero complex numbers with multiplication) and for matrices (sinceM(n, R) with the regular commutator is the Lie algebra of the Lie group GL(n, R) of allinvertible matrices). As the exponential map is surjective on some neighborhood N of e,it is common to call elements of the Lie algebra infinitesimal generators of the group G.The exponential map and the Lie algebra determine the local group structure of everyconnected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists aneighborhood U of the zero element of the Lie algebra g, such that for u, v ∈ U we haveexp(u)exp(v) = exp(u + v + 1/2[u, v] + 1/12[[u, v], v] − 1/12[[u, v], u] − ...),where the omitted terms are known and involve Lie brackets of four or more elements.


30 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction1.1.7.1 Application: Physical Examples of Lie GroupsHere are a few examples of Lie groups and their relations to other areas ofmathematics and physics:(1) Euclidean space R n is an Abelian Lie group (with ordinary vector additionas the group operation).(2) The group GL n (R) of invertible matrices (under matrix multiplication)is a Lie group of dimension n 2 . It has a subgroup SL n (R) of matricesof determinant 1 which is also a Lie group.(3) The group O n (R) generated by all rotations and reflections of an nDvector space is a Lie group called the orthogonal group. It has a subgroupof elements of determinant 1, called the special orthogonal groupSO(n), which is the group of rotations in R n . 42(4) Spin groups are double covers of the special orthogonal groups (usede.g., for studying fermions in quantum field theory).(5) The group Sp 2n (R) of all matrices preserving a symplectic form is aLie group called the symplectic group.(6) The Lorentz group and the Poincaré group of isometries of space–timeare Lie groups of dimensions 6 and 10 that are used in special relativity.(7) The Heisenberg group is a Lie group of dimension 3, used in quantummechanics.(8) The unitary group U(n) is a compact group of dimension n 2 consistingof unitary matrices. It has a subgroup of elements of determinant 1,called the special unitary group SU(n).(9) The group U(1) × SU(2) × SU(3) is a Lie group of dimension 1 + 3 +8 = 12 that is the gauge group of the Standard Model of elementaryparticles, whose dimension corresponds to: 1 photon + 3 vector bosons+ 8 gluons.In case u and v commute, this formula reduces to the familiar exponential law:exp(u)exp(v) = exp(u + v).Every homomorphism f : G → H of Lie groups induces a homomorphism between thecorresponding Lie algebras g and h. The association G =⇒ g is called the Lie Functor.42 For example, in terms of orthogonal matrices, the rotations about the standardCartesian coordinate axes (x, y, z) in R 3 through an angle φ are given by:0101011 0 0cos φ 0 sin φcos φ − sin φ 0R x(φ) = @ 0 cos φ − sin φA , R y(φ) = @ 0 1 0 A , R z(φ) = @ sin φ cos φ 0A0 sin φ cos φ− sin φ 0 cos φ0 0 1


Introduction 311.1.8 Application: A Bird View on Modern Physics1.1.8.1 Three Pillars of 20th Century PhysicsIn this subsection we make small digression into the field of modern physics,which is the major customer for machinery of differential geometry. Arguably,the three most influential geniuses that shaped the world of the20th Century physics, and at the same time showed the pathway to thecurrent unified physical ‘theory of everything’, have been:(1) In the first third of the Century, it had been Albert Einstein.(2) In the second third of the Century, it was Richard Feynman.(3) At the end of the Century – and still today, it has been Edward Witten.It is well–known that Einstein had three periods of his scientific career:(1) Before 1905, when he formulated Special Relativity in a quick series ofpapers published in Annalen der Physik (the most prestigious physicsjournal of the time). This early period was dominated by his ‘thoughtexperiments’, i.e., ‘concrete physical images’, described in the languageof non–professional mathematics. You can say, it was almost purevisualization. This quick and powerful series of ground–braking papers(with just enough maths to be accepted by scientific community) gavehim a reputation of the leading physicist and scientist. 43(2) Although an original and brilliant theory, Special Relativity was notcomplete, which was obvious to Einstein. So, he embarked onto thegeneral relativity voyage, incorporating gravitation. Now, for this goal,his maths was not strong enough. He spent 10 years fighting withgravity, using the ‘hard’ Riemannian geometry, and talking to the leadingmathematician of the time, David Hilbert. At the end, they bothsubmitted the same gravitational equations of general relativity (onlyderived in different ways) to Annalen der Physik in November of 1915.(3) Although even today considered as the most elegant physical theory,General Relativity is still not complete: it cannot live together in thesame world with quantum mechanics. So, Einstein embarked onto thelast journey of his life, the search for unified field theory – and he‘failed’ 44 after 30 years of unsuccessful struggle with a task to big for43 Recall that the Nobel Prize was ‘in the air’ for Einstein for more than 15 years; atthe end he got it in 1921, for his discovery of the Photo–Electric Effect.44 Einstein ‘failed’ in the same way as Hilbert ‘failed’ with his Program of axiomaticformalization of all mathematical sciences. Their apparent ‘failure’ still influences developmentof physics and mathematics, apparently converging into superstring theory.


32 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionone man.Feynman’s story is very different. All his life he was a profoundly originalscientist, similar to the young Einstein. He refused to take anybody’sword for anything, which meant that he had to reinvent for himself almostthe whole of physics. It took him five years of concentrated work to reinventquantum mechanics. At the end, he got a new version of quantummechanics that he (and only he) could understand. In orthodox physics itwas said: Suppose an electron is in this state at a certain time, then youcalculate its future behavior by solving Schrodinger equation. Instead ofthis, Feynman said simply: “The electron does whatever it likes.” A historyof the electron is any possible path in space and time. The behaviorof the electron is just the result of adding together all histories accordingto some simple rules that Feynman worked out. His path–integral andrelated Feynman diagrams, for long defied rigorous mathematical foundation.However, it is still the most powerful calculation tool in quantum(and statistical) mechanics. Later, Feynman generalized it to encompassphysical fields – which led to his version of quantum electrodynamics (thefirst prototype of a quantum field theory) – and his Nobel Prize. All hiscareer he consistently distrusted official mathematics and invented his ownmaths underpinned with a direct physical intuition.If the story had ended here, we might have said that visual physicalintuition is leading the way of science. However, the story does not end here.The leading authority in contemporary physics is Ed Witten, a physicistwho did not get the Nobel Prize, but rather the Fields Medal – togetherwith his ‘superstring theory of everything’. 45 Witten works at the sameplace where Einstein spent the last 30 years of his life – at the PrincetonInstitute of Advanced Study. He is dreaming Einstein’s dream: a unifiedtheory of everything, using the most powerful maths possible. His prophecy,delivered at a turn of the Century, has been: “In the 21 fist Century,mathematics will be dominated by string theory.”When superstring theory arrived in physics in 1984 as a potential theoryof the universe, it was considered by mainstream physicists as littlebetter than religion in terms of constituting a viable, testable theory. Instring theory, the fundamental particles were string–like, rather than pointparticles; the universe had 10 or 11 dimensions, rather than four; and theTheir joined work on gravity is called the Einstein–Hilbert action.45 Witten joined the ‘old Green–Schwarz bosonic string community’ after he won hisFields Medal for topological quantum field theory (TQFT)


Introduction 33theory itself existed at an energy so far from earthly energies that it took aleap of enormous faith to imagine the day when an experiment could evertest it. Quite simply, string theory seemed an excessively esoteric pursuit,which it still is.1.1.8.2 String Theory in ‘Plain English’With modern (super)string theory, 46 scientists might be on the verge of fulfillingEinstein’s dream: formulating the sought for ‘theory of everything’,which would unite our understanding of the four fundamental forces of Nature47 into a single equation (like, e.g., Newton, or Einstein, or Schrödingerequation) and explaining the basic nature of matter and energy.Fig. 1.2 All particles and forces of Nature are supposed to be manifestations of differentresonances of tiny 1D strings vibrating in a 10D hyper–space: (a) An ordinary matter;(b) A molecule; (c) An atom (around ten billionths of a centimeter in diameter; (d) Asubatomic particle (e.g., proton – around 100.000 times smaller than an atom); (e) Asuper–string (around 10 20 times smaller than a proton).In simplest terms, string theory states that all particles and forces of Natureare manifestations of different resonances of tiny 1–dimensional strings(rather than the zero–dimensional points (particles) that are the basis ofthe Standard Model of particle physics), vibrating in 10 dimensions (see46 Recall that ‘superstring’ means ‘supersymmetric string’. The supersymmetry (oftenabbreviated SUSY) is a hypothetical symmetry that relates bosons (particles thattransmit forces) and fermions (particles of matter). In supersymmetric theories, everyfundamental fermion has a bosonic ‘superpartner’ and vice versa.47 Recall that the four fundamental forces are: (i) Gravity (it describes the attractiveforce of matter; it is the same force that holds planets and moons in their orbits and keepsour feet on the ground; it is the weakest force of the four by many orders of magnitude);(ii) Electromagnetism (it describes how electric and magnetic fields work together; italso makes objects solid; once believed to be two separate forces, could be described bya relatively simple set of Maxwell equations); (iii) Strong nuclear force (it is responsiblefor holding the nucleus of atoms together; without it, protons would repel one anotherso no elements other than hydrogen, which has only one proton, would be able to form);(iv) Weak nuclear force (it explains beta decay and the associated radioactivity; it alsodescribes how elementary particles can change into other particles with different energiesand masses).


34 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFigure 1.2).Recall that the Standard Model is a theory which describes the strong,weak, and electromagnetic fundamental forces, as well as the fundamentalparticles that make up all matter. Developed between 1970 and 1973, itis a quantum field theory, and consistent with both quantum mechanicsand special relativity. The Standard Model contains both fermionic andbosonic fundamental particles. Fermions are particles which possess half–integer spin, obey the Fermi–Dirac statistics and also the Pauli exclusionprinciple, which states that no fermions can share the same quantum state.On the other hand, bosons possess integer spin, obey the Bose–Einsteinstatistics, and do not obey the Pauli exclusion principle. In the StandardModel, the theory of the electro–weak interaction (which describes the weakand electromagnetic interactions) is combined with the theory of quantumchromodynamics. All of these theories are gauge theories, 48 meaning thatthey model the forces between fermions by coupling them to bosons which48 Recall that the familiar Maxwell gauge field theory(or, in the non–Abelian case,Yang–Mills gauge field theory) is defined in terms of the fundamental gauge field (whichgeometrically represents a connection) A µ = (A 0 , ⃗ A), that is µ = 0, 3. Here A 0 is thescalar potential and ⃗ A is the vector potential. The Maxwell LagrangianL M = − 1 4 FµνF µν − A µJ µ (1.1)is expressed in terms of the field strength tensor (curvature) F µν = ∂ µA ν − ∂ νA µ, and amatter current J µ that is conserved: ∂ µJ µ = 0. This Maxwell Lagrangian is manifestlyinvariant under the gauge transformation A µ → A µ + ∂ µΛ; and, correspondingly, theclassical Euler-Lagrange equations of motion∂ µF µν = J ν (1.2)are gauge invariant. Observe that current conservation ∂ νJ ν = 0 follows from theantisymmetry of F µν.Note that this Maxwell theory could easily be defined in any space–time dimension dsimply by taking the range of the space–time index µ on the gauge field A µ to be µ =0, 1, 2, . . . , (d − 1) in dD space–time. The field strength tensor is still the antisymmetrictensor F µν = ∂ µA ν − ∂ νA µ, and the Maxwell Lagrangian (1.1) and the field equationsof motion (1.2) do not change their form. The only real difference is that the numberof independent fields contained in the field strength tensor F µν is different in differentdimensions. (Since F µν can be regarded as a d × d antisymmetric matrix, the numberof fields is equal to 1 d(d − 1).) So at this level, planar (2 + 1)D Maxwell theory is quite2similar to the familiar (3 + 1)D Maxwell theory. The main difference is simply that themagnetic field is a (pseudo–) scalar B = ɛ ij ∂ i A j in (2 + 1)D, rather than a (pseudo–)vector B ⃗ = ∇ ⃗ × A ⃗ in (3 + 1)D. This is just because in (2 + 1)D the vector potential⃗A is a 2D vector, and the curl in 2D produces a scalar. On the other hand, the electricfield E ⃗ = −∇A ⃗ 0 − ˙⃗ A is a 2D vector. So the antisymmetric 3 × 3 field–strength tensorhas three nonzero field components: two for the electric field E ⃗ and one for the magneticfield B.


Introduction 35mediate the forces. The Lagrangian of each set of mediating bosons isinvariant under a transformation called a gauge transformation, so thesemediating bosons are referred to as gauge bosons. There are twelve different‘flavours’ of fermions in the Standard Model. The proton, neutron areThe real novelty of (2 + 1)D is that, instead of considering this ‘reduced’ form ofMaxwell theory, we can also define a completely different type of gauge theory: a Chern–Simons gauge theory. It satisfies the usual criteria for a sensible gauge theory: it isLorentz invariant, gauge invariant, and local. The Chern–Simons Lagrangian is (see,e.g., [Dunne (1999)])L CS = κ 2 ɛµνρ A µ∂ νA ρ − A µJ µ . (1.3)Two things are important about this Chern–Simons Lagrangian. First, it does not lookgauge invariant, because it involves the gauge field A µ itself, rather than just the (manifestlygauge invariant) field strength F µν. Nevertheless, under a gauge transformation,the Chern–Simons Lagrangian changes by a total space–time derivativeδL CS = κ 2 ∂µ (λ ɛµνρ ∂ νA ρ) . (1.4)Therefore, if we can neglect boundary terms then the corresponding Chern–Simons action,ZS CS = d 3 x L CS ,is gauge invariant. This is reflected in the fact that the classical Euler–Lagrange equationsκ2 ɛµνρ F νρ = J µ , or equivalently F µν = 1 κ ɛµνρJ ρ , (1.5)are clearly gauge invariant. Note that the Bianchi identity, ɛ µνρ ∂ µF νρ = 0, is compatiblewith the current conservation: ∂ µJ µ = 0, which follows from the NoetherTheorem. A second important feature of the Chern–Simons Lagrangian (1.3) is thatit is first–order in space–time derivatives. This makes the canonical structure of thesetheories significantly different from that of Maxwell theory. A related property is thatthe Chern–Simons Lagrangian is particular to (2 + 1)D, in the sense that we cannotwrite down such a term in (3 + 1)D – the indices simply do not match up. Actually, it ispossible to write down a ‘Chern–Simons theory’ in any odd space–time dimension (forexample, the Chern–Simons Lagrangian in 5D space–time is L = ɛ µνρστ A µ∂ νA ρ∂ σA τ ),but it is only in (2 + 1)D that the Lagrangian is quadratic in the gauge field.Recently, increasingly popular has become Seiberg–Witten gauge theory. It refers toa set of calculations that determine the low–energy physics, namely the moduli spaceand the masses of electrically and magnetically charged supersymmetric particles asa function of the moduli space. This is possible and nontrivial in gauge theory withN = 2 extended supersymmetry, by combining the fact that various parameters of theLagrangian are holomorphic functions (a consequence of supersymmetry) and the knownbehavior of the theory in the classical limit. The extended supersymmetry is supersymmetrywhose infinitesimal generators Q α i carry not only a spinor index α, but also anadditional index i = 1, 2... The more extended supersymmetry is, the more it constrainsphysical observables and parameters. Only the minimal (un–extended) supersymmetryis a realistic conjecture for particle physics, but extended supersymmetry is very importantfor analysis of mathematical properties of quantum field theory and superstringtheory.


36 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmade up of two of these: the up quark and down quark, bound togetherby the strong nuclear force. Together with the electron (bound to thenucleus in atoms by the electromagnetic force), those fermions constitutethe vast majority of everyday matter. To date, almost all experimentaltests of the three forces described by the Standard Model have agreed withits predictions. However, the Standard Model is not a complete theoryof fundamental interactions, primarily because it does not describe thegravitational force.For this reason, string theories are able to avoid problems associatedwith the presence of point–like particles in a physical theory. The basicidea is that the fundamental constituents of Nature are strings of energyof the Planck length (around 10 −35 m), which vibrate at specific resonantfrequencies (modes). Another key claim of the theory is that no measurabledifferences can be detected between strings that wrap around dimensionssmaller than themselves and those that move along larger dimensions (i.e.,physical processes in a dimension of size R match those in a dimension ofsize 1/R). Singularities are avoided because the observed consequences of‘big crunches’ never reach zero size. In fact, should the universe begin a‘big crunch’ sort of process, string theory dictates that the universe couldnever be smaller than the size of a string, at which point it would actuallybegin expanding.Recently, physicists have been exploring the possibility that the stringsare actually membranes, that is strings with 2 or more dimensions (membranesare refereed to as p−branes, where p is the number of dimensions,see Figure 1.3). Every p−brane sweeps out a (p + 1)−dimensional world–volume as it propagates through space–time. A special class of p−branes arethe so–called D–branes, named for the mathematician Johann Dirichlet. 49D–branes are typically classified by their dimension, which is indicated bya number written after the D: a D0–brane is a single point, a D1–brane isa line (sometimes called a ‘D-string’), a D2–brane is a plane, and a D25–brane fills the highest–dimensional space considered in old bosonic string49 Recall that Dirichlet boundary conditions have long been used in the study of fluidsand potential theory, where they involve specifying some quantity all along a boundary.In fluid dynamics, fixing a Dirichlet boundary condition could mean assigning a knownfluid velocity to all points on a surface; when studying electrostatics, one may establishDirichlet boundary conditions by fixing the voltage to known values at particular locations,like the surfaces of conductors. In either case, the locations at which values arespecified is called a D–brane. These constructions take on special importance in stringtheory, because open strings must have their endpoints attached to D–branes.


Introduction 37theory. 50 Fig. 1.3 Visualizing strings and p−branes.According to superstring theory, all the different types of elementaryparticles can be derived from only five types of interactions between justtwo different states of strings, open and closed: (i) an open string can splitto create two smaller open strings (see Figure 1.4); (ii) a closed string cansplit to create two smaller closed strings; (iii) an open string can form botha new open and a new closed string; (iv) two open strings can collide andcreate two new open strings; (v) an open string can join its ends to become aclosed string. All the forces and particles of Nature are just different modesof vibrating strings (somewhat like vibrating strings on string instrumentsto produce a music: different strings have different frequencies that soundas different notes and combining several strings gives chords). For example,gravity is caused by the lowest vibratory mode of a circular string. Higherfrequencies and different interactions of superstrings create different formsof matter and energy.50 The central idea of the so–called brane–world scenario is that our visible 3D universeis entirely restricted to a D3–brane embedded in a higher–dimensional space–time, calledthe bulk. The additional dimensions may be taken to be compact, in which case theobserved universe contains the extra dimensions, and then no reference to the bulk isappropriate in this context. In the bulk model, other branes may be moving throughthis bulk. Interactions with the bulk, and possibly with other branes, can influence ourbrane and thus introduce effects not seen in more standard cosmological models. Asone of its attractive features, the model can ‘explain’ the weakness of gravity relativeto the other fundamental forces of nature. In the brane picture, the other three forces(electromagnetism and the weak and strong nuclear forces) are localized on the brane,but gravity has no such constraint and so much of its attractive power ‘leaks’ into thebulk. As a consequence, the force of gravity should appear significantly stronger on small(sub–millimetre) scales, where less gravitational force has ‘leaked’. Various experimentsare currently underway to test this. For example, in a particle accelerator, if a gravitonwere to be discovered and then observed to suddenly disappear, it might be assumedthat the graviton ‘leaked’ into the bulk.


38 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 1.4 An elementary particle split (a) and string split (b). When a single elementaryparticle splits in two particles, it occurs at a definite moment in space–time. On theother hand, when a string splits into two strings, different observers will disagree aboutwhen and where this occurred. A relativistic observer who considers the dotted line tobe a surface of constant time believes the string broke at the space–time point P whileanother observer who considers the dashed line to be a surface of constant time believesthe string broke at Q.String theory is a possible solution of the core quantum gravity problem,and in addition to gravity it can naturally describe interactions similar toelectromagnetism and the other forces of nature. Superstring theories includefermions, the building blocks of matter, and incorporate the so–calledsupersymmetry. 51 It is not yet known whether string theory will be able51 In a world based on supersymmetry, when a particle moves in space, it also canvibrate in the new fermionic dimensions. This new kind of vibration produces a ‘cousin’or ‘superpartner’ for every elementary particle that has the same electric charge butdiffers in other properties such as spin. Supersymmetric theories make detailed predictionsabout how superpartners will behave. To confirm supersymmetry, scientistswould like to produce and study the new supersymmetric particles. The crucial stepis building a particle accelerator that achieves high enough energies. At present, the


Introduction 39to describe a universe with the precise collection of forces and matter thatis observed, nor how much freedom to choose those details that the theorywill allow. String theory as a whole has not yet made falsifiable predictionsthat would allow it to be experimentally tested, though various special cornersof the theory are accessible to planned observations and experiments.Work on string theory has led to advances in both mathematics (mainlyin differential and algebraic geometry) and physics (supersymmetric gaugetheories). 52Historically, string theory was originally invented to explain peculiaritieshighest–energy particle accelerator is the Tevatron at Fermilab near Chicago. There,protons and antiprotons collide with an energy nearly 2,000 times the rest energy ofan individual proton (given by Einsteins well–known formula E = mc 2 ). Earlier inthis decade, physicists capitalized on Tevatron’s unsurpassed energy in their discoveryof the top quark, the heaviest known elementary particle. After a shutdown of severalyears, the Tevatron resumed operation in 2001 with even more intense particle beams.In 2007, the available energies will make a ‘quantum jump’ when the European Laboratoryfor Particle Physics, or CERN (located near Geneva, Switzerland) turns on theLarge Hadron Collider (LHC). The LHC should reach energies 15,000 times the protonrest energy. The LHC is a multi–billion dollar international project, funded mainly byEuropean countries with substantial contributions from the United States, Japan, andother countries.52 Recall that gauge theories are a class of physical theories based on the idea thatsymmetry transformations can be performed locally as well as globally. Yang–Millstheory is a particular example of gauge theories with non–Abelian symmetry groupsspecified by the Yang–Mills action. For example, the Yang–Mills action for the O(n)gauge theory for a set of n non–interacting scalar fields ϕ i , with equal masses m isZ nX 1S = (2 ∂µϕ i ∂µ ϕ i − 1 2 m2 ϕ 2 i ) d4 x.i=1Other gauge theories with a non–Abelian gauge symmetry also exist, e.g., the Chern–Simons model. Most physical theories are described by Lagrangians which are invariantunder certain transformations, when the transformations are identically performed atevery space–time point-they have global symmetries. Gauge theory extends this idea byrequiring that the Lagrangians must possess local symmetries as well-it should be possibleto perform these symmetry transformations in a particular region of space–time withoutaffecting what happens in another region. This requirement is a generalized version ofthe equivalence principle of general relativity. Gauge symmetries reflect a redundancy inthe description of a system. The importance of gauge theories for physics stems from thetremendous success of the mathematical formalism in providing a unified framework todescribe the quantum field theories of electromagnetism, the weak force and the strongforce. This theory, known as the Standard Model (see footnote 5), accurately describesexperimental predictions regarding three of the four fundamental forces of nature, and isa gauge theory with the gauge group SU(3) × SU(2) × U(1). Modern theories like stringtheory, as well as some formulations of general relativity, are, in one way or another,gauge theories. Sometimes, the term gauge symmetry is used in a more general sense toinclude any local symmetry, like for example, diffeomorphisms.


40 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionof hadron (subatomic particle which experiences the strong nuclear force)behavior. In particle–accelerator experiments, physicists observed that thespin of a hadron is never larger than a certain multiple of the square ofits energy. No simple model of the hadron, such as picturing it as a setof smaller particles held together by spring–like forces, was able to explainthese relationships. In 1968, theoretical physicist Gabriele Veneziano wastrying to understand the strong nuclear force when he made a startling discovery.He found that a 200–year–old Euler beta function perfectly matchedmodern data on the strong force. Veneziano applied the Euler beta functionto the strong force, but no one could explain why it worked.In 1970, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskindpresented a physical explanation for Euler’s strictly theoretical formula. Byrepresenting nuclear forces as vibrating, 1D strings, these physicists showedhow Euler’s function accurately described those forces. But even afterphysicists understood the physical explanation for Veneziano’s insight, thestring description of the strong force made many predictions that directlycontradicted experimental findings. The scientific community soon lostinterest in string theory, and the Standard Model, with its particles andfields, remained un–threatened.Then, in 1974, John Schwarz and Joel Scherk studied the messenger–like patterns of string vibration and found that their properties exactlymatched those of the gravitational force’s hypothetical messenger particle- the graviton. They argued that string theory had failed to catch on becausephysicists had underestimated its scope. This led to the developmentof bosonic string theory, which is still the version first taught to many students.The original need for a viable theory of hadrons has been fulfilled byquantum chromodynamics (QCD), the theory of Gell–Mann’s quarks andtheir interactions. It is now hoped that string theory (or some descendantof it) will provide a fundamental understanding of the quarks themselves.Bosonic string theory is formulated in terms of the so–called Polyakovaction, a mathematical quantity which can be used to predict how stringsmove through space and time. By applying the ideas of quantum mechanicsto the Polyakov action - a procedure known as quantization - one candeduce that each string can vibrate in many different ways, and that eachvibrational state appears to be a different particle. The mass the particlehas, and the fashion with which it can interact, are determined by the waythe string vibrates - in essence, by the ‘note’ which the string sounds. Thescale of notes, each corresponding to a different kind of particle, is termedthe spectrum of the theory. These early models included both open strings,


Introduction 41which have two distinct endpoints, and closed strings, where the endpointsare joined to make a complete loop. The two types of string behave inslightly different ways, yielding two spectra. Not all modern string theoriesuse both types; some incorporate only the closed variety. However,the bosonic theory has problems. Most importantly, the theory has a fundamentalinstability, believed to result in the decay of space-time itself.Additionally, as the name implies, the spectrum of particles contains onlybosons, particles like the photon which obey particular rules of behavior.While bosons are a critical ingredient of the Universe, they are not its onlyconstituents. Investigating how a string theory may include fermions in itsspectrum led to supersymmetry, a mathematical relation between bosonsand fermions which is now an independent area of study. String theorieswhich include fermionic vibrations are now known as superstring theories;several different kinds have been described.Roughly between 1984 and 1986, physicists realized that string theorycould describe all elementary particles and interactions between them, andhundreds of them started to work on string theory as the most promisingidea to unify theories of physics. This so–called first superstring revolutionwas started by a discovery of anomaly cancellation in type I string theory byMichael Green and John Schwarz in 1984. The anomaly is cancelled due tothe Green–Schwarz mechanism. Several other ground–breaking discoveries,such as the heterotic string, were made in 1985.Note that in the type IIA and type IIB string theories closed stringsare allowed to move everywhere throughout the 10D space-time (called thebulk), while open strings have their ends attached to D–branes, which aremembranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9– in type IIA and even – 0,2,4,6 or 8 – in type IIB, including the timedirection).While understanding the details of string and superstring theories requiresconsiderable geometrical sophistication, some qualitative propertiesof quantum strings can be understood in a fairly intuitive fashion. Forexample, quantum strings have tension, much like regular strings made oftwine; this tension is considered a fundamental parameter of the theory.The tension of a quantum string is closely related to its size. Consider aclosed loop of string, left to move through space without external forces.Its tension will tend to contract it into a smaller and smaller loop. Classicalintuition suggests that it might shrink to a single point, but this wouldviolate Heisenberg’s uncertainty principle. The characteristic size of thestring loop will be a balance between the tension force, acting to make it


42 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionContemporary String TheoriesType Dim DetailsBosonic 26 Only bosons, no fermions means only forces, no matter,with both open and closed strings; major flaw:a particle with imaginary mass, called the tachyon,representing an instability in the theoryI 10 Supersymmetry between forces and matter, with bothopen and closed strings, no tachyon, group symmetryis SO(32)IIA 10 Supersymmetry between forces and matter, withclosed strings and open strings bound to D–branes, notachyon, massless fermions spin both ways (nonchiral)IIB 10 Supersymmetry between forces and matter, withclosed strings and open strings bound to D–branes, notachyon, massless fermions only spin one way (chiral)HO 10 Supersymmetry between forces and matter, withclosed strings only, no tachyon, heterotic, meaningright moving and left moving strings differ, groupsymmetry is SO(32)HE 10 Supersymmetry between forces and matter, withclosed strings only, no tachyon, heterotic, meaningright moving and left moving strings differ, groupsymmetry is E 8 × E 8small, and the uncertainty effect, which keeps it ‘stretched’. Consequently,the minimum size of a string must be related to the string tension.Before the 1990s, string theorists believed that there were five distinctsuperstring theories: type I, types IIA and IIB, and the two heterotic stringtheories (SO(32) and E 8 × E 8 ). The thinking was that out of these fivecandidate theories, only one was the actual correct theory of everything,and that theory was the theory whose low energy limit, with ten dimensionsspacetime compactified down to four, matched the physics observed in ourworld today. But now it is known that this naïve picture was wrong, andthat the five superstring theories are connected to one another as if they areeach a special case of some more fundamental theory, of which there is onlyone. These theories are related by transformations that are called dualities.If two theories are related by a duality transformation, it means that the


Introduction 43first theory can be transformed in some way so that it ends up looking justlike the second theory. The two theories are then said to be dual to oneanother under that kind of transformation. Put differently, the two theoriesare two different mathematical descriptions of the same phenomena. Thesedualities link quantities that were also thought to be separate. Large andsmall distance scales, strong and weak coupling strengths – these quantitieshave always marked very distinct limits of behavior of a physical system,in both classical field theory and quantum particle physics. But strings canobscure the difference between large and small, strong and weak, and thisis how these five very different theories end up being related.This type of duality is called T–duality. T–duality relates type IIAsuperstring theory to type IIB superstring theory. That means if we taketype IIA and Type IIB theory and ‘compactify’ them both on a circle, thenswitching the momentum and winding modes, and switching the distancescale, changes one theory into the other. The same is also true for the twoheterotic theories. T–duality also relates type I superstring theory to bothtype IIA and type IIB superstring theories with certain boundary conditions(termed ‘orientifold’). Formally, the location of the string on the circle isdescribed by two fields living on it, one which is left-moving and anotherwhich is right-moving. The movement of the string center (and hence itsmomentum) is related to the sum of the fields, while the string stretch(and hence its winding number) is related to their difference. T-dualitycan be formally described by taking the left-moving field to minus itself, sothat the sum and the difference are interchanged, leading to switching ofmomentum and winding.On the other hand, every force has a coupling constant, which is a measureof its strength, and determines the chances of one particle to emitor receive another particle. For electromagnetism, the coupling constant isproportional to the square of the electric charge. When physicists study thequantum behavior of electromagnetism, they can’t solve the whole theoryexactly, because every particle may emit and receive many other particles,which may also do the same, endlessly. So events of emission and receptionare considered as perturbations and are dealt with by a series of approximations,first assuming there is only one such event, then correcting theresult for allowing two such events, etc (this method is called Perturbationtheory. This is a reasonable approximation only if the coupling constant issmall, which is the case for electromagnetism. But if the coupling constantgets large, that method of calculation breaks down, and the little piecesbecome worthless as an approximation to the real physics. This can also


44 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionhappen in string theory. String theories have a string coupling constant.But unlike in particle theories, the string coupling constant is not just anumber, but depends on one of the oscillation modes of the string, calledthe dilaton. Exchanging the dilaton field with minus itself exchanges a verylarge coupling constant with a very small one. This symmetry is called S–duality. If two string theories are related by S–duality, then one theorywith a strong coupling constant is the same as the other theory with weakcoupling constant. The theory with strong coupling cannot be understoodby means of perturbation theory, but the theory with weak coupling can.So if the two theories are related by S-duality, then we just need to understandthe weak theory, and that is equivalent to understanding the strongtheory. Superstring theories related by S–duality are: type I superstringtheory with heterotic SO(32) superstring theory, and type IIB theory withitself.Around 1995, Edward Witten and others found strong evidence thatthe different superstring theories were different limits of a new 11D theorycalled M–theory. With the discovery of M–theory, an extra dimension appearedand the fundamental string of string theory became a 2-dimensionalmembrane called an M2–brane (or supermembrane). Its magnetic dual isan M5–brane. The various branes of string theory are thought to be relatedto these higher dimensional M–branes wrapped on various cycles. Thesediscoveries sparked the so–called second superstring revolution.One intriguing feature of string theory is that it predicts the numberof dimensions which the universe should possess. Nothing in Maxwell’stheory of electromagnetism, or Einstein’s theory of relativity, makes thiskind of prediction; these theories require physicists to insert the numberof dimensions ‘by hand’. The first person to add a fifth dimension to Einstein’sfour space–time dimensions was German mathematician TheodorKaluza in 1919. The reason for the un–observability of the fifth dimension(its compactness) was suggested by Swedish physicist Oskar Klein in 1926.Today, this is called the 5D Kaluza–Klein theory.Instead, string theory allows one to compute the number of spacetimedimensions from first principles. Technically, this happens because for adifferent number of dimensions, the theory has a gauge anomaly. Thiscan be understood by noting that in a consistent theory which includesa photon (technically, a particle carrying a force related to an unbrokengauge symmetry), it must be massless. The mass of the photon whichis predicted by string theory depends on the energy of the string modewhich represents the photon. This energy includes a contribution from


Introduction 45the Casimir effect, namely from quantum fluctuations in the string. Thesize of this contribution depends on the number of dimensions since for alarger number of dimensions, there are more possible fluctuations in thestring position. Therefore, the photon will be massless – and the theoryconsistent – only for a particular number of dimensions.The only problem is that when the calculation is done, the universe’sdimensionality is not four as one may expect (three axes of space and oneof time), but 26. More precisely, bosonic string theories are 26D, whilesuperstring and M–theories turn out to involve 10 and 11 dimensions, respectively.In bosonic string theories, the 26 dimensions come from thePolyakov equation. However, these results appear to contradict the observedfour dimensional space–time.Two different ways have been proposed to solve this apparent contradiction.The first is to compactify the extra dimensions; i.e., the 6 or 7 extradimensions are so small as to be undetectable in our phenomenal experience.The 6D model’s resolution is achieved with the so–called Calabi–Yaumanifolds (see Figure 1.5). In 7D, they are termed G 2 −manifolds. Essentiallythese extra dimensions are compactified by causing them to loopback upon themselves. A standard analogy for this is to consider multidimensionalspace as a garden hose. If the hose is viewed from a sufficientdistance, it appears to have only one dimension, its length. Indeed, thinkof a ball small enough to enter the hose but not too small. Throwing sucha ball inside the hose, the ball would move more or less in one dimension;in any experiment we make by throwing such balls in the hose, the onlyimportant movement will be one-dimensional, that is, along the hose. However,as one approaches the hose, one discovers that it contains a seconddimension, its circumference. Thus, a ant crawling inside it would move intwo dimensions (and a fly flying in it would move in three dimensions). This‘extra dimension’ is only visible within a relatively close range to the hose,or if one ‘throws in’ small enough objects. Similarly, the extra compactdimensions are only visible at extremely small distances, or by experimentingwith particles with extremely small wave lengths (of the order of thecompact dimension’s radius), which in quantum mechanics means very highenergies. Another possibility is that we are stuck in a 3+1 dimensional (i.e.,three spatial dimensions plus one time dimension) subspace of the full universe.This subspace is supposed to be a D–brane, hence this is known as abrane–world theory. In either case, gravity acting in the hidden dimensionsaffects other non–gravitational forces such as electromagnetism. In principle,therefore, it is possible to deduce the nature of those extra dimensions


46 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionby requiring consistency with the Standard Model, but this is not yet apractical possibility. It is also be possible to extract information regardingthe hidden dimensions by precision tests of gravity, but so far these haveonly put upper limitations on the size of such hidden dimensions.Fig. 1.5 Calabi–Yau manifold – a 3D projection created using Mathematica T M .For popular expose on string theory, see [Witten (2002); Greene (2000)],while the main textbook is still [Green et. al. (1987)].1.2 Application: Paradigm of <strong>Diff</strong>erential–<strong>Geom</strong>etricModelling of Dynamical SystemsIn this section we give a paradigm of differential–geometric modelling andanalysis of complex dynamical systems (see [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]for more background details). This is essentially a recipe how to develop acovariant formalism on smooth manifolds, given a certain physical, or bio–physical, or psycho–physical, or socio–physical system, here labelled by ageneric name: ‘physical situation’. We present this recipe in the form ofthe following five–step algorithm.(I) So let’s start: given a certain physical situation, the first stepin its predictive modelling and analysis, that is, in applying a powerfuldifferential–geometric machinery to it, is to associate with this situationtwo independent coordinate systems, constituting two independent smoothRiemannian manifolds. Let us denote these two coordinate systems andtheir respective manifolds as:• Internal coordinates: x i = x i (t), (i = 1, ..., m), constituting the mDinternal configuration manifold: M m ≡ {x i }; and


Introduction 47• External coordinates: y e = y e (t), (e = 1, ..., n), constituting the nDexternal configuration manifold: N n ≡ {y e }.The main example that we have in mind is a standard robotic or biodynamic(loco)motion system, in which x i denote internal joint coordinates,while y e denote external Cartesian coordinates of segmental centersof mass. However, we believe that such developed methodology can fit ageneric physical situation.Therefore, in this first, engineering step (I) of our differential–geometricmodelling, we associate to the given natural system, not one but two differentand independent smooth configuration manifolds, somewhat like viewingfrom two different satellites a certain place on Earth with a footballgame playing in it.(II) Once that we have precisely defined two smooth manifolds, astwo independent views on the given physical situation, we can apply ourdifferential–geometric modelling to it and give it a natural physical interpretation.More precisely, once we have two smooth Riemannian manifolds,M m ≡ {x i } and N n ≡ {y e }, we can formulate two smooth maps betweenthem: 53f : N → M, given by coordinate transformation: x i = f i (y e ), (1.6)andg : M → N, given by coordinate transformation: y e = g e (x i ). (1.7)If the Jacobian matrices of these two maps are nonsingular (regular), thatis if their Jacobian determinants are nonzero, then these two maps aremutually inverse, f = g −1 , and they represent standard forward and inversekinematics.(III) Although, maps f and g define some completely general nonlinearcoordinate (functional) transformations, which are even unknown at themoment, there is something linear and simple that we know about them(from calculus). Namely, the corresponding infinitesimal transformationsare linear and homogenous: from (1.6) we have (applying everywhere Einstein’ssummation convention over repeated indices)dx i = ∂f i∂y e dye , (1.8)53 This obviously means that we are working in the category of smooth manifolds.


48 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhile from (1.7) we havedy e = ∂ge∂x i dxi . (1.9)Furthermore, (1.8) implies the linear and homogenous transformation ofinternal velocities,v i ≡ ẋ i = ∂f i∂y e ẏe , (1.10)while (1.9) implies the linear and homogenous transformation of externalvelocities,u e ≡ ẏ e = ∂ge∂x i ẋi . (1.11)In this way, we have defined two velocity vector–fields, the internal one:v i = v i (x i , t) and the external one: u e = u e (y e , t), given respectively by thetwo nonlinear systems of ODEs, (1.10) and (1.11). 54(IV) The next step in our differential–geometrical modelling/analysisis to define second derivatives of the manifold maps f and g, that is thetwo acceleration vector–fields, which we will denote by a i = a i (x i , ẋ i , t)and w e = w e (y e , ẏ e , t), respectively. However, unlike simple physics inlinear Euclidean spaces, these two acceleration vector–fields on manifoldsM and N are not the simple time derivatives of the corresponding velocityvector–fields (a i ≠ ˙v i and w e ≠ ˙u e ), due to the existence of the Levi–Civitaconnections ∇ M and ∇ N on both M and N. Properly defined, these twoacceleration vector–fields respectively read:a i = ˙v i + Γ i jkv j v k = ẍ i + Γ i jkẋ j ẋ k , and (1.12)w e = ˙u e + Γ e hlu h u l = ÿ e + Γ e hlẏ h ẏ l , (1.13)where Γ i jkand Γe hldenote the (second–order) Christoffel symbols of theconnections ∇ M and ∇ N .Therefore, in the step (III) we gave the first–level model of our physicalsituation in the form of two ordinary vector–fields, the first–order vector–fields (1.10) and (1.11). For some simple situations (e.g., modelling ecologicalsystems), we could stop at this modelling level. Using physical terminologywe call them velocity vector–fields. Following this, in the step (IV)we have defined the two second–order vector–fields (1.12) and (1.13), as54 Although transformations of differentials and associated velocities are linear and homogeneous,the systems of ODE’s define nonlinear vector–fields, as they include Jacobian(functional) matrices.


Introduction 49a connection–base derivations of the previously defined first–order vector–fields. Using physical terminology, we call them ‘acceleration vector–fields’.(V) Finally, following our generic physical terminology, as a naturalnext step we would expect to define some kind of generic Newton–Maxwellforce–fields. And we can actually do this, with a little surprise that individualforces involved in the two force–fields will not be vectors, but ratherthe dual objects called 1–forms (or, 1D differential forms). Formally, wedefine the two covariant force–fields asF i = mg ij a j = mg ij ( ˙v j + Γ j ik vi v k ) = mg ij (ẍ j + Γ j ikẋi ẋ k ), and (1.14)G e = mg eh w h = mg eh ( ˙u h + Γ h elu e u l ) = mg eh (ÿ h + Γ h elẏ e ẏ l ), (1.15)where m is the mass of each single segment (unique, for symplicity), whileg ij = gij M and g eh = geh N are the two Riemannian metric tensors correspondingto the manifolds M and N. The two force–fields, F i defined by(1.14) and G e defined by (1.15), are generic force–fields corresponding tothe manifolds M and N, which represent the material cause for the givenphysical situation. Recall that they can be physical, bio–physical, psycho–physical or socio–physical force–fields. Physically speaking, they are thegenerators of the corresponding dynamics and kinematics.Main geometrical relations behind this fundamental paradigm, formingthe so–called covariant force functor, are depicted in Figure 1.6.Fig. 1.6 The covariant force functor, including the main relations used by differential–geometric modelling.


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Chapter 2Technical Preliminaries: Tensors,Actions and Functors2.1 Tensors: Local Machinery of <strong>Diff</strong>erential <strong>Geom</strong>etryPhysical and engineering laws must be independent of any particular coordinatesystems used in describing them mathematically, if they are to bevalid. In other words, all physical and engineering equations need to betensorial or covariant. Therefore, for the reference purpose, in this section,we give the basic formulas from the standard tensor calculus, which is usedthroughout the text. The basic notational convention used in tensor calculusis Einstein’s summation convention over repeated indices. More onthis subject can be found in any standard textbook on mathematical methodsfor scientists and engineers, or mathematical physics (we recommend[Misner et al. (1973)]).2.1.1 Transformation of Coordinates and ElementaryTensorsTo introduce tensors, consider a standard linear nD matrix system, Ax = b.It can be rewritten in the so–called covariant form asa ij x j = b i , (i, j = 1, ..., n). (2.1)Here, i is a free index and j is a dummy index to be summed upon, so theexpansion of (2.1) givesa 11 x 1 + a 12 x 2 + ... + a 1n x n = b 1 ,a 21 x 1 + a 22 x 2 + ... + a 2n x n = b 2 ,...a n1 x 1 + a n2 x 2 + ... + a nn x n = b n ,51


52 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionas expected from the original matrix form Ax = b. This indicial notationcan be more useful than the matrix one, like e.g., in computer science,where indices would represent loop variables. However, the full potentialof tensor analysis is to deal with nonlinear multivariate systems, whichare untractable by linear matrix algebra and analysis. The core of thisnonlinear multivariate analysis is general functional transformation.2.1.1.1 Transformation of CoordinatesSuppose that we have two sets of curvilinear coordinates that are single–valued, continuous and smooth functions of time, x j = x j (t), (j = 1, ..., m)and ¯x i = ¯x i (t), (i = 1, ..., n), respectively, representing trajectories of motionof some physical or engineering system. Then a general (m × n)Dtransformation (i.e., a nonlinear map) x j ↦→ ¯x i is defined by the set oftransformation equations¯x i = ¯x i (x j ), (i = 1, ..., n; j = 1, ..., m). (2.2)In case of the square transformation, m = n, we can freely exchange theindices, like e.g., in general relativity theory. On the other hand, in thegeneral case of rectangular transformation, m ≠ n, like e.g., in robotics,and we need to take care of these ‘free’ indices.Now, if the Jacobian determinant of this coordinate transformation isdifferent from zero,∣ ∂¯x i ∣∣∣∣∂x j ≠ 0,then the transformation (2.2) is reversible and the inverse transformation,x j = x j (¯x i ),exists as well. Finding the inverse transformation is the problem of matrixinverse: in case of the square matrix it is well defined, although the inversemight not exist if the matrix is singular. However, in case of the squarematrix, its proper inverse does not exist, and the only tool that we areleft with is the so–called Moore–Penrose pseudoinverse, which gives anoptimal solution (in the least–squares sense) of an overdetermined system ofequations. Every (overdetermined) rectangular coordinate transformationinduces a redundant system.For example, in Euclidean 3D space R 3 , transformation from Cartesiancoordinates y k = {x, y, z} into spherical coordinates x i = {ρ, θ, ϕ} is given


Technical Preliminaries: Tensors, Actions and Functors 53byy 1 = x 1 cos x 2 cos x 3 , y 2 = x 1 sin x 2 cos x 3 , y 3 = x 1 sin x 3 , (2.3)with the Jacobian matrix given by⎛( ) ∂yk cos x 2 cos x 3 −x 1 sin x 2 cos x 3 −x 1 cos x 2 sin x 3 ⎞∂x i = ⎝ sin x 2 cos x 3 x 1 cos x 2 cos x 3 −x 1 sin x 2 sin x 3 ⎠ (2.4)sin x 3 0 x 1 cos x 3 ∣ ∣∣and the corresponding Jacobian determinant, ∣ ∂yk∂x = (x 1 ) 2 cos x 3 .iAn inverse transform is given byx 1 = √ ( ) y(y 1 ) 2 + (y 2 ) 2 + (y 3 ) 2 , x 2 2= arctany 1 ,()∣x 3 y 3= arctan √ , with∂x i ∣∣∣ 1(y1 ) 2 + (y 2 ) 2 ∣∂y k =(x 1 ) 2 cos x 3 .As an important engineering (robotic) example, we have a rectangulartransformation from 6 DOF external, end–effector (e.g., hand) coordinates,into n DOF internal, joint–angle coordinates. In most cases this is a redundantmanipulator system, with infinite number of possible joint trajectories.2.1.1.2 Scalar InvariantsA scalar invariant (or, a zeroth order tensor) with respect to the transformation(2.2) is the quantity ϕ = ϕ(t) defined asϕ(x i ) = ¯ϕ(¯x i ),which does not change at all under the coordinate transformation. In otherwords, ϕ is invariant under (2.2). For example, biodynamic examples ofscalar invariants include various energies (kinetic, potential, biochemical,mental) with the corresponding kinds of work, as well as related thermodynamicquantities (free energy, temperature, entropy, etc.).2.1.1.3 Vectors and CovectorsAny geometrical object v i = v i (t) that under the coordinate transformation(2.2) transforms as¯v i = v j ∂¯xi∂x j ,(remember, summing upon j−index),


54 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionrepresents a vector, traditionally called a contravariant vector, or, a first–order contravariant tensor. Standard physical and engineering examplesinclude both translational and rotational velocities and accelerations.On the other hand, any geometrical object v i = v i (t) that under thecoordinate transformation (2.2) transforms as¯v i = v j∂x j∂¯x i ,represents a one–form or covector, traditionally called a covariant vector,or, a first–order covariant tensor. Standard physical and engineering examplesinclude both translational and rotational momenta, forces and torques.2.1.1.4 Second–Order TensorsAny geometrical object t ik = t ik (t) that under the coordinate transformation(2.2) transforms as¯t ik jl ∂¯xi ∂¯x k= t∂x j ∂x l ,(i, k = 1, ..., n; j, l = 1, ..., m),represents a second–order contravariant tensor. It can be get as an outerproduct of two contravariant vectors, t ik = u i v k .Any geometrical object t ik = t ik (t) that under the coordinate transformation(2.2) transforms as¯t ik = t jl∂x j∂¯x i ∂x l∂¯x k ,represents a second–order covariant tensor. It can be get as an outer productof two covariant vectors, t ik = u i v k .Any geometrical object t i k = ti k(t) that under the coordinate transformation(2.2) transforms as¯t i k = t j l∂¯x i ∂x l∂x j ∂¯x k ,represents a second–order mixed tensor. It can be get as an outer productof a covariant vector and a contravariant vector, t i k = ui v k .Standard physical and engineering examples examples include:(1) The fundamental (material) covariant metric tensor g ≡ g ik , i.e., inertiamatrix, given usually by the transformation from Cartesian coordinates


Technical Preliminaries: Tensors, Actions and Functors 55y j to curvilinear coordinates x i ,g ik = ∂yj ∂y j∂x i , (summing over j). (2.5)∂xk It is used in the quadratic metric form ds 2 of the space in consideration(e.g., a certain physical or engineering configuration space)ds 2 ≡ dy j dy j = g ik dx i dx k ,where the first term on the r.h.s denotes the Euclidean metrics, whilethe second term is the Riemannian metric of the space, respectively.(2) Its inverse g −1 ≡ g ik , given byg ik = (g ik ) −1 = G ik|g ik | , G ik is the cofactor of the matrix (g ik );(3) The Kronecker–delta symbol δ i k, given byδ i k ={ 1 if i = k0 if i ≠ k ,used to denote the metric tensor in Cartesian orthogonal coordinates.δ i k is a discrete version of the Dirac δ−function. The generalizedKronecker–delta symbol δ ijklmn(in 3D) is the product of Ricci antisymmetrictensors ε ijk and ε lmn ,δ ijklmn = εijk ε lmn =⎧⎨ 0 if at least two indices are equal+1 if both ijk and lmn are either even or odd .⎩−1 if one of ijk, lmn is even and the other is oddFor example, to derive components of the metric tensor g ≡ g ij instandard spherical coordinates, we use the relations (2.3–2.4) between thespherical coordinates x i = {ρ, θ, ϕ} and the Cartesian coordinates y k ={x, y, z}, and the definition, g ij = ∂yk ∂y k∂x i ∂x, to get the metric tensor (injmatrix form)⎛⎞ ⎛⎞1 0 0 1 0 0(g ij ) = ⎝ 0 (x 1 ) 2 cos 2 x 3 0 ⎠ = ⎝ 0 ρ 2 cos 2 ϕ 0 ⎠ , (2.6)0 0 (x 1 ) 2 0 0 ρ 2


56 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand the inverse metric tensor⎛1 0 0(g ij ⎜ 1) = ⎝ 0(x 1 ) 2 cos 2 x0 3 10 0(x 1 ) 2 ⎞⎟⎠ =⎛1 0 0⎜ 1⎝ 0ρ 2 cos 2 ϕ 010 0⎟⎠ . (2.7)ρ 2 ⎞Given a tensor, we can derive other tensors by raising and loweringits indices, by their multiplication with covariant and contravariant metrictensors. In this way, the so–called associated tensors to the given tensorare be formed. For example, v i and v i are associated tensors, related byv i = g ik v k and v i = g ik v k .Given two vectors, u ≡ u iproduct is given byand v ≡ v i , their inner (dot, or scalar)u · v ≡ g ij u i v j ,while their vector (cross) product (in 3D) is given by2.1.1.5 Higher–Order Tensorsu × v ≡ ε ijk u j v k .As a generalization of above tensors, consider a geometrical object Rkps i =Rkps i (t) that under the coordinate transformation (2.2) transforms as¯R kps i = R j ∂¯x i ∂x l ∂x q ∂x tlqt∂x j ∂¯x k ∂¯x p , (all indices = 1, ..., n). (2.8)∂¯xsClearly, R i kjl = Ri kjl(x, t) is a fourth order tensor, once contravariant andthree times covariant, representing the central tensor in Riemannian geometry,called the Riemann curvature tensor. As all physical and engineeringconfiguration spaces are Riemannian manifolds, they are all characterizedby curvature tensors. In case R i kjl= 0, the corresponding Riemannianmanifold reduces to the Euclidean space of the same dimension, in whichg ik = δ i k.If one contravariant and one covariant index of a tensor a set equal, theresulting sum is a tensor of rank two less than that of the original tensor.This process is called tensor contraction.If to each point of a region in an nD space there corresponds a definitetensor, we say that a tensor–field has been defined. In particular, this is avector–field or a scalar–field according as the tensor is of rank one or zero.It should be noted that a tensor or tensor–field is not just the set of its


Technical Preliminaries: Tensors, Actions and Functors 57components in one special coordinate system, but all the possible sets ofcomponents under any transformation of coordinates.2.1.1.6 Tensor SymmetryA tensor is called symmetric with respect to two indices of the same varianceif its components remain unaltered upon interchange of the indices;e.g., a ij = a ji , or a ij = a ji . A tensor is called skew–symmetric (or, antisymmetric)with respect to two indices of the same variance if its componentschange sign upon interchange of the indices; e.g., a ij = −a ji , or a ij = −a ji .Regarding tensor symmetry, in the following we will prove several usefulpropositions.(i) Every second–order tensor can be expressed as the sum of two tensors,one of which is symmetric and the other is skew–symmetric. For example, asecond–order tensor a ij , which is for i, j = 1, ..., n given by the n×n−matrixcan be rewritten as⎛⎞a 11 a 12 ... a 1na ij = ⎜ a 21 a 22 ... a n2⎟⎝ ... ... ... ... ⎠ ,a n1 a n2 ... a nna ij = 1 2 a ij + 1 2 a ij + 1 2 a ji − 1 2 a ji ,that can be rearranged as= 1 2 a ij + 1 2 a ji + 1 2 a ij − 1 2 a ji , which can be regrouped as= 1 2 (a ij + a ji ) + 1 2 (a ij − a ji ), which can be written as= a (ij) + a [ij] ,where a (ij) denotes its symmetric part, while a [ij] denotes its skew–symmetric part, as required.(ii) Every quadratic form can be made symmetric. For example, aquadratic form a ij x i x j , that (for i, j = 1, ..., n) expands asa ij x i x j = a 11 x 1 x 1 + a 12 x 1 x 2 + ... + a 1n x 1 x n ++ a 21 x 2 x 1 + a 22 x 2 x 2 + ... + a 2n x 2 x n +...+ a n1 x n x 1 + a n2 x n x 2 + ... + a nn x n x n ,with a non–symmetric second–order tensor a ij , can be made symmetric in


58 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe following way.a ij x i x j = 1 2 a ijx i x j + 1 2 a ijx i x j .If we swap indices in the second term, we get= 1 2 a ijx i x j + 1 2 a jix j x i , which is equal to= 1 2 (a ij + a ji ) x i x j .If we now use a substitution,12 (a ij + a ji ) ≡ b ij = b ji , we geta ij x i x j = b ij x i x j ,where a ij is non–symmetric and b ij is symmetric, as required.(iii) Every second–order tensor that is the sum a ij = u i v j + u j v i , or,a ij = u i v j + u j v i is symmetric. In both cases, if we swap the indices i andj, we get a ji = u j v i + u i v j , (resp. a ji = u j v i + u i v j ), which implies thatthe tensor a ij (resp. a ij ) is symmetric.(iv) Every second–order tensor that is the difference b ij = u i v j − u j v i ,or, b ij = u i v j − u j v i is skew–symmetric. In both cases, if we swap theindices i and j, we get b ji = −(u j v i − u i v j ), (resp. b ji = −(u j v i − u i v j )),which implies that the tensor b ij (resp. b ij ) is skew–symmetric.2.1.2 Euclidean Tensors2.1.2.1 Basis Vectors and the Metric Tensor in R nThe natural Cartesian coordinate basis in an nD Euclidean space R n isdefined as a set of nD unit vectors e i given bye 1 = [{1, 0, 0, ...} t , e 2 = {0, 1, 0, ...} t , e 3 = {0, 0, 1, ...} t , ..., e n = {0, 0, ..., 1} t ],(where index t denotes transpose) while its dual basis e i is given by:e 1 = [{1, 0, 0, ...}, e 2 = {0, 1, 0, ...}, e 3 = {0, 0, 1, ...}, ..., e n = {0, 0, ..., 1}],


Technical Preliminaries: Tensors, Actions and Functors 59(no transpose) where the definition of the dual basis is given by the Kronecker’sδ−symbol, i.e., the n × n identity matrix:⎡e i · e j = δ i j =⎢⎣1 0 0 ... 00 1 0 ... 00 0 1 ... 0... ... ... ... ...0 0 0 ... 1that is the metric tensor in Cartesian coordinates equals g = δ i j. In general,(i.e., curvilinear) coordinate system, the metric tensor g = g ij is defined asthe scalar product of the dual basis vectors, i.e., the n × n matrix:⎡⎤g 11 g 12 g 13 ... g 1ng 21 g 22 g 23 ... g 2ng ij = e i · e j =⎢ g 31 g 32 g 33 ... g 3n⎥⎣ ... ... ... ... ... ⎦ .g n1 g n2 g n3 ... g nn2.1.2.2 Tensor Products in R nLet u and v denote two vectors in R n , with their components given by⎤⎥⎦ ,u i = u · e i , and v j = v · e j ,where u = |u| and v = |v| are their respective norms (or, lengths). Thentheir inner product (i.e., scalar, or dot product) u · v is a scalar invariantS, defined asS = u i · v j = g ij u i v j .Besides the dot product of two vectors u, v ∈ R n , there is also their tensorproduct (i.e., generalized vector, or cross product), which is a second–order tensorT = u ⊗ v, in components, T ij = u i ⊗ v j .In the natural basis e i this tensor is expanded asT = T ij e i ⊗ e j ,while its components in the dual basis read:T ij = T (e i , e j ),


60 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere T = |T| is its norm. To get its components in curvilinear coordinates,we need first to substitute it in Cartesian basis:then to evaluate it on the slots:T ij = T mn (e m ⊗ e n )(e i , e j ),T ij = T mn e m · e i e n · e j ,and finally to calculate the other index configurations by lowering indices,by means of the metric tensor:T i j = g jm T im , T ij = g im g jn T mn .2.1.3 Covariant <strong>Diff</strong>erentiationIn this subsection, we need to consider some nD Riemannian manifold M(see section (3.10.1) below) with the metric form (i.e., line element) ds 2 =g ik dx i dx k , as a configuration space for a certain physical or engineeringsystem (e.g., robotic manipulator).2.1.3.1 Christoffel’s SymbolsPartial derivatives of the metric tensor g ik (2.5) form themselves specialsymbols that do not transform as tensors (with respect to the coordinatetransformation (2.2)), but nevertheless represent important quantities intensor analysis. They are called Christoffel symbols of the first kind, definedbyΓ ijk = 1 (2 (∂ x kg ij + ∂ x j g ki − ∂ x ig jk ), remember, ∂ x i ≡ ∂ )∂x iand Christoffel symbols of the second kind, defined byΓ k ij = g kl Γ ijl .(2.8) of the manifold M, can be ex-The Riemann curvature tensor Rijk lpressed in terms of the later asR l ijk = ∂ x j Γ l ik − ∂ x kΓ l ij + Γ l rjΓ r ik − Γ l rkΓ r ij.For example, in 3D spherical coordinates, x i = {ρ, θ, ϕ}, with the metrictensor and its inverse given by (2.6, 2.7), it can be shown that the only


Technical Preliminaries: Tensors, Actions and Functors 61nonzero Christoffel’s symbols are:2.1.3.2 GeodesicsΓ 2 12 = Γ 2 21 = Γ 3 13 = Γ 3 31 = 1 ρ , Γ3 23 = Γ 2 32 = − tan θ, (2.9)Γ 1 22 = −ρ, Γ 1 33 = −ρ cos 2 θ, Γ 2 33 = sin θ cos θ.From the Riemannian metric form ds 2 = g ik dx i dx k it follows that thedistance between two points t 1 and t 2 on a curve x i = x i (t) in M is givenbys =∫ t2t 1√gik ẋ i ẋ k dt.That curve x i = x i (t) in M which makes the distance s a minimum is calleda geodesic of the space M (e.g., in a sphere, the geodesics are arcs of greatcircles). Using the calculus of variations, the geodesics are found from thedifferential geodesic equation,ẍ i + Γ i jkẋ j ẋ k = 0, (2.10)where overdot means derivative upon the line parameter s.For example, in 3D spherical coordinates x i = {ρ, θ, ϕ}, using (2.9),geodesic equation (2.10) becomes a system of three scalar ODEs,¨ρ − ρ˙θ 2 − ρ cos 2 θ ˙ϕ 2 = 0, ¨θ +2ρ ˙ρ ˙ϕ + sin θ cos θ ˙ϕ2 = 0,¨ϕ + 2 ρ ˙ρ ˙ϕ − 2 tan θ ˙θ ˙ϕ = 0. (2.11)2.1.3.3 Covariant DerivativeOrdinary total and partial derivatives of vectors (covectors) do not transformas vectors (covectors) with respect to the coordinate transformation(2.2). For example, let y k be Cartesian coordinates and x i be general curvilinearcoordinates of a dynamical system (with i, k = 1, ..., n). We have:x i (t) = x i [y k (t)], which implies thatdx idt = ∂xi∂y k dy kdt , or equivalently, ẋi = ∂xi∂y k ẏk ,that is a transformation law for the contravariant vector, which means thatthe velocity v i ≡ ẋ i ≡ dxidtis a proper contravariant vector. However, if we


62 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionperform another time differentiation, we getd 2 x idt 2= ∂xi d 2 y k∂y k dt 2 + ∂2 x i dy k dy m∂y k ∂y m dt dt ,which means that d2 x id 2 x idt 2dt 2is not a proper vector.is an acceleration vector only in a special case when x i are an-∂ 2 x i∂y k ∂y mother Cartesian coordinates; then = 0, and therefore the originalcoordinate transformation is linear, x i = a i k yk + b i (where a i k and bi areconstant).Therefore, d2 x idtrepresents an acceleration vector only in terms of Newtonianmechanics in a Euclidean space R n , while it is not a proper acceleration2vector in terms of Lagrangian or Hamiltonian mechanics in general curvilinearcoordinates on a smooth manifold M n . And we know that Newtonianmechanics in R n is sufficient only for fairly simple mechanical systems.The above is true for any tensors. So we need to find another derivativeoperator to be able to preserve their tensor character. The solution to thisproblem is called the covariant derivative.The covariant derivative v;k i of a contravariant vector vi is defined asv i ;k = ∂ x kv i + Γ i jkv j .Similarly, the covariant derivative v i;k of a covariant vector v i is defined asv i;k = ∂ x kv i − Γ j ik v j.Generalization for the higher order tensors is straightforward; e.g., the covariantderivative t j kl;q of the third order tensor tj klis given byt j kl;q = ∂ x qtj kl + Γj qst s kl − Γ s kqt j sl − Γs lqt j ks .The covariant derivative is the most important tensor operator in generalrelativity (its zero defines parallel transport) as well as the basis fordefining other differential operators in mechanics and physics.2.1.3.4 Covariant Form of <strong>Diff</strong>erential OperatorsHere we give the covariant form of classical vector differential operators:gradient, divergence, curl and Laplacian.Gradient. If ϕ = ϕ(x i , t) is a scalar field, the gradient one–form grad(ϕ)is defined bygrad(ϕ) = ∇ϕ = ϕ ;i = ∂ x iϕ.


Technical Preliminaries: Tensors, Actions and Functors 63Divergence. The divergence div(v i ) of a vector–field v i = v i (x i , t) isdefined by contraction of its covariant derivative with respect to the coordinatesx i = x i (t), i.e., the contraction of v i ;k , namelydiv(v i ) = v i ;i = 1 √ g∂ x i( √ gv i ).Curl. The curl curl(θ i ) of a one–form θ i = θ i (x i , t) is a second–ordercovariant tensor defined ascurl(θ i ) = θ i;k − θ k;i = ∂ x kθ i − ∂ x iθ k .Laplacian. The Laplacian ∆ϕ of a scalar invariant ϕ = ϕ(x i , t) is thedivergence of grad(ϕ), or∆ϕ = ∇ 2 ϕ = div(grad(ϕ)) = div(ϕ ;i ) = 1 √ g∂ x i( √ gg ik ∂ x kϕ).2.1.3.5 Absolute DerivativeThe absolute derivative (or intrinsic, or Bianchi’s derivative) of a contravariantvector v i along a curve x k = x k (t) is denoted by ˙¯v i ≡ Dv i /dtand defined as the inner product of the covariant derivative of v i andẋ k ≡ dx k /dt, i.e., v ;kẋk i , and is given by˙¯v i = ˙v i + Γ i jkv j ẋ k .Similarly, the absolute derivative ˙¯v i of a covariant vector v i is defined as˙¯v i = ˙v i − Γ j ik v jẋ k .Generalization for the higher order tensors is straightforward; e.g., the absolutederivative ˙¯t j kl of the third order tensor tj klis given by˙¯t j kl = ṫj kl + Γj qst s klẋ q − Γ s kqt j slẋq − Γ s lqt j ksẋq .The absolute derivative is the most important differential operator inphysics and engineering, as it is the basis for the covariant form of bothLagrangian and Hamiltonian equations of motion of many physical andengineering systems.


64 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.1.3.6 3D Curve <strong>Geom</strong>etry: Frenet–Serret FormulaeGiven three unit vectors: tangent τ i , principal normal β i , and binormalν i , as well as two scalar invariants: curvature K and torsion T, of a curveγ(s) = γ[x i (s)], the so–called Frenet–Serret formulae are valid 1˙¯τ i ≡ ˙τ i + Γ i jkτ j ẋ k = Kβ i ,˙¯β i ≡ ˙β i + Γ i jkβ j ẋ k = −(Kτ i + T ν i ),˙¯ν i ≡ ˙ν i + Γ i jkν j ẋ k = T β i .2.1.3.7 Mechanical Acceleration and ForceIn modern analytical mechanics, the two fundamental notions of accelerationand force in general curvilinear coordinates are substantially differentfrom the corresponding terms in Cartesian coordinates as commonly usedin engineering mechanics. Namely, the acceleration vector is not an ordinarytime derivative of the velocity vector; ‘even worse’, the force, which isa paradigm of a vector in statics and engineering vector mechanics, is nota vector at all. Proper mathematical definition of the acceleration vectoris the absolute time derivative of the velocity vector, while the force is adifferential one–form.To give a brief look at these ‘weird mathematical beasts’, consider a materialdynamical system described by n curvilinear coordinates x i = x i (t).First, recall from section 2.1.3.3 above, that an ordinary time derivative ofthe velocity vector v i (t) = ẋ i (t) does not transform as a vector with respectto the general coordinate transformation (2.2). Therefore, a i ≠ ˙v i . So, weneed to use its absolute time derivative to define the acceleration vector(with i, j, k = 1, ..., n),a i = ˙¯v i ≡ Dvidt= v i ;kẋ k ≡ ˙v i + Γ i jkv j v k ≡ ẍ i + Γ i jkẋ j ẋ k , (2.12)which is equivalent to the l.h.s of the geodesic equation (2.10). Only inthe particular case of Cartesian coordinates, the general acceleration vector(2.12) reduces to the familiar engineering form of the Euclidean accelerationvector 2 , a = ˙v.1 In this paragraph, the overdot denotes the total derivative with respect to the lineparameter s (instead of time t).2 Any Euclidean space can be defined as a set of Cartesian coordinates, while anyRiemannian manifold can be defined as a set of curvilinear coordinates. Christoffel’ssymbols Γ i jk vanish in Euclidean spaces defined by Cartesian coordinates; however, theyare nonzero in Riemannian manifolds defined by curvilinear coordinates.


Technical Preliminaries: Tensors, Actions and Functors 65For example, in standard spherical coordinates x i = {ρ, θ, ϕ}, we havethe components of the acceleration vector given by (2.11), if we now reinterpretoverdot as the time derivative,a ρ = ¨ρ − ρ˙θ 2 − ρ cos 2 θ ˙ϕ 2 , a θ = ¨θ + 2 ρ ˙ρ ˙ϕ + sin θ cos θ ˙ϕ2 ,a ϕ = ¨ϕ + 2 ρ ˙ρ ˙ϕ − 2 tan θ ˙θ ˙ϕ.Now, using (2.12), the Newton’s fundamental equation of motion, thatis the basis of all science, F = m a, gets the following tensorial formF i = ma i = m ˙¯v i = m(v i ;kẋ k ) ≡ m( ˙v i + Γ i jkv j v k ) = m(ẍ i + Γ i jkẋ j ẋ k ),(2.13)which defines Newtonian force as a contravariant vector.However, modern Hamiltonian dynamics reminds us that: (i) Newton’sown force definition was not really F = m a, but rather F = ṗ, wherep is the system’s momentum, and (ii) the momentum p is not really avector, but rather a dual quantity, a differential one–form 3 . Consequently,the force, as its time derivative, is also a one–form (see Figure 2.1; also,compare with Figure Figure 5.2 above). This new force definition includesthe precise definition of the mass distribution within the system, by meansof its Riemannian metric tensor g ij . Thus, (2.13) has to be modified asF i = mg ij a j ≡ mg ij ( ˙v j + Γ j ik vi v k ) = mg ij (ẍ j + Γ j ikẋi ẋ k ), (2.14)where the quantity mg ij is called the material metric tensor, or inertiamatrix. Equation (2.14) generalizes the notion of the Newtonian force F,from Euclidean space R n to the Riemannian manifold M.2.1.4 Application: Covariant MechanicsRecall that a material system is regarded from the dynamical standpointas a collection of particles which are subject to interconnections and constraintsof various kinds (e.g., a rigid body is regarded as a number ofparticles rigidly connected together so as to remain at invariable distancesfrom each other). The number of independent coordinates which determinethe configuration of a dynamical system completely is called the number ofdegrees of freedom (DOF) of the system. In other words, this number, n,3 For example, in Dirac’s < bra|ket > formalism, kets are vectors, while bras areone–forms; in matrix notation, columns are vectors, while rows are one–forms.


66 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 2.1 A one–form θ (which is a family of parallel (hyper)surfaces, the so–calledGrassmann planes) pierced by the vector v to give a scalar product θ(v) ≡< θ, v >= 2.6(see text for explanation).is the dimension of the system’s configuration manifold. This viewpoint isthe core of our applied differential geometry.For simplicity, let us suppose that we have a dynamical system withthree DOF (e.g., a particle of mass M, or a rigid body of mass M withone point fixed); generalization to n DOF, with N included masses M α , isstraightforward. The configuration of our system at any time is then givenby three coordinates {q 1 , q 2 , q 3 }. As the coordinates change in value thedynamical system changes its configuration. Obviously, there is an infinitenumber of sets of independent coordinates which will determine the configurationof a dynamical system, but since the position of the system iscompletely given by any one set, these sets of coordinates must be functionallyrelated. Hence, if ¯q i is any other set of coordinates, these quantitiesmust be connected with q i by formulae of the type¯q i = ¯q i (q i ), (i = 1, ..., n(= 3)). (2.15)Relations (2.15) are the equations of transformation from one set of dynamicalcoordinates to another and, in a standard tensorial way (see [Misneret al. (1973)]), we can define tensors relative to this coordinate transformation.The generalized coordinates q i , (i = 1, ..., n) constitute the system’sconfiguration manifold.In particular, in our ordinary Euclidean 3−dimensional (3D) space R 3 ,the ordinary Cartesian axes are x i = {x, y, z}, and the system’s center ofmass (COM) is given byC i =M αx i α∑ Nα=1 M ,αwhere Greek subscript α labels the masses included in the system. If wehave a continuous distribution of matter V = V (M) rather than the dis-


Technical Preliminaries: Tensors, Actions and Functors 67crete system of masses M α , all the α−sums should be replaced by volumeintegrals, the element of mass dM taking the place of M α ,N∑∫∫∫M α ⇒ dM.α=1An important quantity related to the system’s COM is the double symmetriccontravariant tensorVI ij = M α x i αx j α, (2.16)called the inertia tensor, calculated relative to the origin O of the Cartesianaxes x i α = {x α , y α , z α }. If we are given a straight line through O, definedby its unit vector λ i , and perpendiculars p α are drawn from the differentparticles on the line λ i , the quantityI(λ i ) = M α p 2 αis called the moment of inertia around λ i . The moment of inertia I(λ i )can be expressed through inertia tensor (2.16) asI(λ i ) = (Ig ij − I ij )λ i λ j ,where g ij is the system’s Euclidean 3D metric tensor (as defined above),I = g ij I ij , and I ij = g rm g sn I mn is the covariant inertia tensor. If we nowconsider the quadric Q whose equation is(Ig ij − I ij )x i x j = 1, (2.17)we find that the moment of inertia around λ i is 1/R, where R is the radiusvector of Q in the direction of λ i . The quadric Q defined by relation (2.17)is called the ellipsoid of inertia at the origin O. It has always three principalaxes, which are called the principal axes of inertia at O, and the planescontaining them in pairs are called the principal planes of inertia at O.The principal axes of inertia are given by the equations(Ig ij − I ij )λ j = θλ i ,where θ is a root of the determinant equation|(I − θ)g ij − I ij | = 0.More generally, if we suppose that the points of our dynamical systemare referred to rectilinear Cartesian axes x i in a Euclidean n−dimensional


68 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(nD) space R n , then when we are given the time and a set of generalizedcoordinates q i we are also given all the points x i of the dynamical system,as the system is determined uniquely. Consequently, the x i are functionsof q i and possibly also of the time, that is,x i = x i (q i , t).If we restrict ourselves to the autonomous dynamical systems in which theseequations do not involve t, i.e.,x i = x i (q i ), (2.18)then differentiating (2.18) with respect to the time t givesẋ i = ∂xi∂q j ˙qj . (2.19)The quantities ˙q i , which form a vector with reference to coordinate transformations(2.15), we shall call the generalized velocity vector. We see from(2.19) that when the generalized velocity vector is given we know the velocityof each point of our system. Further, this gives us the system’s kineticenergy,E kin = 1 2 M αg mn ẋ m α ẋ n α = 1 2 M αg mn∂x m α∂q i∂x n α∂q j ˙qi ˙q j . (2.20)Now, if we use the Euclidean metric tensor g ij to define the materialmetric tensor G ij , including the distribution of all the masses M α of oursystem, asG ij = M α g mn∂x m α∂q i∂x n α∂q j , (2.21)the kinetic energy (2.20) becomes a homogenous quadratic form in the generalizedsystem’s velocities ˙q i ,E kin = 1 2 G ij ˙q i ˙q j . (2.22)From the transformation relation (2.21) we see that the material metrictensor G ij is symmetric in i and j. Also, since E kin is an invariant for alltransformations of generalized coordinates, from (2.22) we conclude thatG ij is a double symmetric tensor. Clearly, this is the central quantityin classical tensor system dynamics. We will see later, that G ij definesthe Riemannian geometry of the system dynamics. For simplicity reasons,


Technical Preliminaries: Tensors, Actions and Functors 69G ij is often denoted by purely geometrical symbol g ij , either assuming orneglecting the material properties of the system.Now, let us find the equations of motion of our system. According to theD’Alembert’s Principle of virtual displacements, the equations of motion inCartesian coordinates x i in R n are embodied in the single tensor equationg mn (M α ẍ m α − X m α )δx n α = 0, (2.23)where Xα i is the total force vector acting on the particle M α , while δx i α isthe associated virtual displacement vector, so that the product g ij Xαδx i j αis the virtual work of the system, and we can neglect in Xα i all the internalor external forces which do not work in the displacement δx i α. If we givethe system a small displacement compatible to with the constraints of thesystem, we see that this displacement may be effected by giving incrementsδq i to the generalized coordinates q i of the system, and these are relatedto the δx i in accordance with the transformation formulae δx i α = ∂xi α∂qδq j . jFurthermore, in this displacement the internal forces due to the constraintsof the system will do no work, since these constraints are preserved,and consequently only the external forces will appear in (2.23), so it becomes[ (d ∂xmg mn M ααdt ∂q jNow, using (2.20–2.22), we derive(d ∂xmM α g αmndt ∂q jAlso, if we put˙qj ) ∂xnα∂q i− Xm α) ∂xnα ˙qj ∂q i = d dt (G ij ˙q j )− 1 ∂G st2 ∂q i ˙q j ˙q k = d dtF i = g mn X m α∂x n α∂q i ,∂x n ]α∂q i δq i = 0. (2.24)we get( )∂Ekin∂ ˙q i − ∂E kin∂q i .F i δq i = g mn X m α δx n α = δW, (2.25)where δW is the virtual work done by the external forces in the smalldisplacement δq i , which shows that F i is the covariant vector, called thegeneralized force vector. Now (2.24) takes the form[ ( )d ∂Ekindt ∂ ˙q i − ∂E ]kin∂q i − F i δq i = 0.


70 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSince the coordinates q i are independent this equation is true for all variationsδq i and we get as a final result the covariant Lagrangian equations ofmotion,( )d ∂Ekindt ∂ ˙q i − ∂E kin∂q i = F i .If the force system is conservative and E pot is the system’s potential energygiven byF i = − ∂E pot∂q i ,then, using (2.25) the Lagrangian equations take the standard form( )d ∂Ldt ∂ ˙q i = ∂L∂q i , (2.26)where the Lagrangian function L = L(q, ˙q) of the system is given by L =E kin − E pot (since E pot does not contain ˙q i ).Now, the kinetic energy E kin of the system, given by quadratic form(2.22), is always positive except when ˙q i is zero in which case E kin vanishes.In other words, the quadratic form (2.22) is positive definite. Consequently,we can always find the line (or arc) element, defined byds 2 = G ij dq i dq j . (2.27)A manifold in which ds 2 is given by relation of the type of (2.27), geometricallywith g ij instead of G ij , is called a Riemannian manifold.2.1.4.1 Riemannian Curvature TensorEvery Riemannian manifold is characterized by the Riemann curvature tensor.In physical literature (see, e.g., [Misner et al. (1973)]) it is usuallyintroduced through the Jacobi equation of geodesic deviation, showing theacceleration of the relative separation of nearby geodesics (the shortest,straight lines on the manifold). For simplicity, consider a sphere of radiusa in R 3 . Here, Jacobi equation is pretty simple,d 2 ξ+ Rξ = 0,ds2 where ξ is the geodesic separation vector (the so–called Jacobi vector–field),s denotes the geodesic arc parameter given by (2.27) and R = 1/a 2 is theGaussian curvature of the surface.


Technical Preliminaries: Tensors, Actions and Functors 71In case of a higher–dimensional manifold M, the situation is naturallymore complex, but the main structure of the Jacobi equation remains similar,D 2 ξ+ R(u, ξ, u) = 0,ds2 where D denotes the covariant derivative and R(u, ξ, u) is the curvaturetensor, a three–slot linear machine. In components defined in a local coordinatechart (x i ) on M, this equation readsD 2 ξ ids 2+ dx j dx lRi jklds ξk ds = 0,where Rjkl i are the components of the Riemann curvature tensor.2.1.4.2 Exterior <strong>Diff</strong>erential FormsRecall that exterior differential forms are a special kind of antisymmetricalcovariant tensors (see, e.g., [De Rham (1984); Flanders (1963)]). Suchtensor–fields arise in many applications in physics, engineering, and differentialgeometry. The reason for this is the fact that the classical vectoroperations of grad, div, and curl as well as the theorems of Green, Gauss,and Stokes can all be expressed concisely in terms of differential forms andthe main operator acting on them, the exterior derivative d. <strong>Diff</strong>erentialforms inherit all geometrical properties of the general tensor calculus andadd to it their own powerful geometrical, algebraic and topological machinery(see Figures 2.2 and 2.3). <strong>Diff</strong>erential p−forms formally occur asintegrands under ordinary integral signs in R 3 :• a line integral ∫ P dx + Q dy + R dz has as its integrand the one–formω = P dx + Q dy + R dz;• a surface integral ∫∫ A dydz + B dzdx + C dxdy has as its integrand thetwo–form α = A dydz + B dzdx + C dxdy;• a volume integral ∫∫∫ K dxdydz has as its integrand the three–formλ = K dxdydz.By means of an exterior derivative d, a derivation that transformsp−forms into (p + 1)−forms, these geometrical objects generalize ordinaryvector differential operators in R 3 :• a scalar function f = f(x) is a zero–form;


72 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 2.2 Basis vectors and 1–forms in Euclidean R 3 −space: (a) Translational case; and(b) Rotational case.• its gradient df, is a one–form 4df = ∂f ∂f ∂fdx + dy +∂x ∂y ∂z dz;• a curl dω, of a one–form ω above, is a two–form( ∂Rdω =∂y − ∂Q ) ( ∂Pdydz +∂z∂z − ∂R ) ( ∂Qdzdx +∂x∂x − ∂P )dxdy;∂y4 We use the same symbol, d, to denote both ordinary and exterior derivation, inorder to avoid extensive use of the boldface symbols. It is clear from the context whichderivative (differential) is in place: exterior derivative operates only on differential forms,while the ordinary differential operates mostly on coordinates.


Technical Preliminaries: Tensors, Actions and Functors 73• a divergence dα, of the two–form α above, is a three–form( ∂Adα =∂x + ∂B∂y + ∂C )dxdydz.∂zFig. 2.3 Fundamental two–form and its flux in R 3 : (a) Translational case; (b) Rotationalcase. In both cases the flux through the plane u ∧ v is defined as R R u∧v c dp idq iand measured by the number of tubes crossed by the circulation oriented by u ∧ v.Now, although visually intuitive, our Euclidean 3D space R 3 is notsufficient for thorough physical or engineering analysis. The fundamentalconcept of a smooth manifold, locally topologically equivalent to theEuclidean nD space R n , is required (with or without Riemannian metrictensor defined on it). In general, a proper definition of exterior derivatived for a p−form β on a smooth manifold M, includes the Poincaré lemma:


74 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiond(dβ) = 0, and validates the general integral Stokes formula∫ ∫β = dβ,∂Mwhere M is a p−dimensional manifold with a boundary and ∂M is its (p −1)−dimensional boundary, while the integrals have appropriate dimensions.A p−form β is called closed if its exterior derivative is equal to zero,Mdβ = 0.From this condition one can see that the closed form (the kernel of theexterior derivative operator d) is conserved quantity. Therefore, closedp−forms possess certain invariant properties, physically corresponding tothe conservation laws.A p−form β that is an exterior derivative of some (p − 1)−form α,β = dα,is called exact (the image of the exterior derivative operator d).Poincaré lemma, exact forms prove to be closed automatically,Bydβ = d(dα) = 0.Similarly to the components of a 3D vector v defined above, a one–formθ defined on an nD manifold M can also be expressed in components, usingthe coordinate basis {dx i } along the local nD coordinate chart {x i } ∈ M,asθ = θ i dx i .Now, the components of the exterior derivative of θ are equal to the componentsof its commutator defined on M bydθ = ω ij dx i dx j ,where the components of the form commutator ω ij are given byω ij =( ∂θi∂x i− ∂θ i∂x j ).The space of all smooth p−forms on a smooth manifold M is denoted byΩ p (M). The wedge, or exterior product of two differential forms, a p−form


Technical Preliminaries: Tensors, Actions and Functors 75α ∈ Ω p (M) and a q−form β ∈ Ω q (M) is a (p+q)−form α∧β. For example,if θ = a i dx i , and η = b j dx j , their wedge product θ ∧ η is given byθ ∧ η = a i b j dx i dx j ,so that the coefficients a i b j of θ ∧ η are again smooth functions, beingpolynomials in the coefficients a i of θ and b j of η. The exterior product ∧is related to the exterior derivative d : Ω p (M) → Ω p+1 (M), byd(α ∧ β) = dα ∧ β + (−1) p α ∧ dβ.Another important linear operator is the Hodge star ∗ : Ω p (M) →Ω n−p (M), where n is the dimension of the manifold M. This operatordepends on the inner product (i.e., Riemannian metric) on M and alsodepends on the orientation (reversing orientation will change the sign). Forany p−forms α and β,∗ ∗ α = (−1) p(n−p) α, and α ∧ ∗β = β ∧ ∗α.Hodge star is generally used to define dual (n − p)−forms on nD smoothmanifolds.For example, in R 3 with the ordinary Euclidean metric, if f and g arefunctions then (compare with the 3D forms of gradient, curl and divergencedefined above)df = ∂f ∂f ∂fdx + dy +∂x ∂y ∂z dz,∗df = ∂f ∂f ∂fdydz + dzdx +df ∧ ∗dg =∂x( ∂f∂x∂y∂g∂x + ∂f∂y∂g∂y + ∂f∂z∂z dxdy,∂g∂z)dxdydz = ∆f dxdydz,where ∆f is the Laplacian on R 3 . Therefore the three–form df ∧ ∗dg is theLaplacian multiplied by the volume element, which is valid, more generally,in any local orthogonal coordinate system in any smooth domain U ∈ R 3 .The subspace of all closed p−forms on M we will denote by Z p (M) ⊂Ω p (M), and the sub-subspace of all exact p−forms on M we will denote byB p (M) ⊂ Z p (M). Now, the quotient spaceH p (M) = Zp (M)B p M = Ker ( d : Ω p (M) → Ω p+1 (M) )Im (d : Ω p−1 (M) → Ω p (M))


76 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis called the pth de Rham cohomology group (or vector space) of a manifoldM. Two p−forms α and β on M are equivalent, or belong to the samecohomology class [α] ∈ H p (M), if their difference equals α − β = dθ, whereθ is a (p − 1)−form on M.2.1.4.3 The Covariant Force LawObjective of this final tensor section is to generalize the fundamental Newtonian3D equation, F = ma, for a generic robotic/biodynamical system,consisting of a number of flexibly–coupled rigid segments (see Figures 3.7–3.8 above), and thus to formulate the covariant force law.To be able to apply the covariant formalism, we need to start with thesuitable coordinate transformation (2.2), in this case as a relation betweenthe 6 external SE(3) rigid–body coordinates, y e = y e (t) (e = 1, ..., 6), and2n internal joint coordinates, x i = x i (t) (i = 1, ..., 2n) (n angles, formingthe constrained n−torus T n , plus n very restricted translational coordinates,forming the hypercube I n ⊂ R n ). Once we have these two sets ofcoordinates, external–y e and internal–x i , we can perform the general functionaltransformation (2.2) between them,x i = x i (y e ). (2.28)Now, although the coordinate transformation (2.28) is nonlinear and evenunknown at this stage, there is something known and simple about it: thecorresponding transformation of differentials is linear and homogenous,dx i = ∂xi∂y e dye ,which implies the linear and homogenous transformation of velocities,ẋ i = ∂xi∂y e ẏe . (2.29)Our internal velocity vector–field is defined by the set of ODEs (2.29), ateach representative point x i = x i (t) of the system’s configuration manifoldM = T n × I n , as v i ≡ v i (x i , t) := ẋ i (x i , t).Note that in general, a vector–field represents a field of vectors defined atevery point x i within some region U (e.g., movable segments/joints only) ofthe total configuration manifold M (consisting of all the segments/joints).Analytically, vector–field is defined as a set of autonomous ODEs (in ourcase, the set (2.29)). Its solution gives the flow, consisting of integral curves


Technical Preliminaries: Tensors, Actions and Functors 77of the vector–field, such that all the vectors from the vector–field are tangentto integral curves at different representative points x i ∈ U. In this way,through every representative point x i ∈ U passes both a curve from the flowand its tangent vector from the vector–field. <strong>Geom</strong>etrically, vector–field isdefined as a cross–section of the tangent bundle T M, the so–called velocityphase–space. Its geometrical dual is the 1–form–field, which represents afield of one–forms (see Figure 2.1), defined at the same representative pointsx i ∈ U. Analytically, 1–form–field is defined as an exterior differentialsystem, an algebraic dual to the autonomous set of ODEs. <strong>Geom</strong>etrically,it is defined as a cross–section of the cotangent bundle T ∗ M, the so–calledmomentum phase–space. Together, the vector–field and its corresponding1–form–field define the scalar potential field (e.g., kinetic and/or potentialenergy) at the same movable region U ⊂ M.Next, we need to formulate the internal acceleration vector–field, a i ≡a i (x i , ẋ i , t), acting in all movable joints, and at the same time generalizingthe Newtonian 3D acceleration vector a.According to Newton, acceleration is a rate–of–change of velocity. But,from the previous subsections, we know that a i ≠ ˙v i . However,a i := ˙¯v i = ˙v i + Γ i jkv j v k = ẍ i + Γ i jkẋ j ẋ k . (2.30)Once we have the internal acceleration vector–field a i= a i (x i , ẋ i , t),defined by the set of ODEs (2.30) (including Levi–Civita connections Γ i jkof the Riemannian configuration manifold M), we can finally define theinternal force 1–form field, F i = F i (x i , ẋ i , t), as a family of force one–forms, half of them rotational and half translational, acting in all movablejoints,F i := mg ij a j = mg ij ( ˙v j + Γ j ik vi v k ) = mg ij (ẍ j + Γ j ikẋi ẋ k ), (2.31)where we have used the simplified material metric tensor, mg ij , for thesystem (considering, for simplicity, all segments to have equal mass m),defined by its Riemannian kinetic energy formT = 1 2 mg ijv i v j .Equation F i = mg ij a j , defined properly by (2.31) at every representativepoint x i of the system’s configuration manifold M, formulates thesought for covariant force law, that generalizes the fundamental Newtonianequation, F = ma, for the generic physical or engineering system. Its


78 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmeaning is:Force 1–form–field = Mass distribution×Acceleration vector–fieldIn other words, the field (or, family) of force one–forms F i , acting inall movable joints (with constrained rotations on T n and very restrictedtranslations on I n ), causes both rotational and translational accelerationsof all body segments, within the mass distribution mg ij 5 , along the flow–lines of the vector–field a j .2.1.5 Application: Nonlinear Fluid DynamicsIn this subsection we will derive the general form of the Navier–Stokesequations in nonlinear fluid dynamics.2.1.5.1 Continuity EquationRecall that the most important equation in fluid dynamics, as well as ingeneral continuum mechanics, is the celebrated equation of continuity, (weexplain the symbols in the following text)∂ t ρ + div(ρu) = 0. (2.32)As a warm–up for turbulence, we will derive the continuity equation(2.32), starting from the mass conservation principle. Let dm denote aninfinitesimal mass of a fluid particle. Then, using the absolute time derivativeoperator ( ˙ ) ≡ D dt, the mass conservation principle reads˙dm = 0. (2.33)If we further introduce the fluid density ρ = dm/dv, where dv is an infinitesimalvolume of a fluid particle, then the mass conservation principle(2.33) can be rewritten as˙ρdv = 0,5 More realistically, instead of the simplified metric mg ij we have the material metrictensor G ij (2.21), including all k segmental masses m χ, as well as the correspondingmoments and products of inertia,G ij (x, m) =kXχ=1m χδ rs∂y r∂x i ∂y s∂x j ,(r, s = 1, ..., 6; i, j = 1, ..., 2n).


Technical Preliminaries: Tensors, Actions and Functors 79which is the absolute derivative of a product, and therefore expands into˙ρdv + ρ ˙ dv = 0. (2.34)Now, as the fluid density ρ = ρ(x k , t) is a function of both time t andspatial coordinates x k , for k = 1, 2, 3, that is, a scalar–field, its total timederivative ˙ρ, figuring in (2.34), is defined by˙ρ = ∂ t ρ + ∂ x kρ ∂ t x k ≡ ∂ t ρ + ρ ;k u k , (2.35)or, in vector form ˙ρ = ∂ t ρ + grad(ρ) · u,where u k = u k (x k , t) ≡ u is the velocity vector–field of the fluid.Regarding dv, ˙ the other term figuring in (2.34), we start by expanding anelementary volume dv along the sides {dx i (p) , dxj (q) , dxk (r)} of an elementaryparallelepiped, asdv = 1 3! δpqr ijk dxi (p) dxj (q) dxk (r), (i, j, k, p, q, r = 1, 2, 3)so that its absolute derivative becomesdv˙= 1 ˙2! δpqr dx i (p)dx j (q) dxk (r)ijk= 1 2! ui ;lδ pqrijk dxl (p) dxj (q) dxk (r) (using ˙ dx i (p) = u i ;ldx l (p) ),which finally simplifies into˙dv = u k ;kdv ≡ div(u) dv. (2.36)Substituting (2.35) and (2.36) into (2.34) gives˙ρdv ≡ ( ∂ t ρ + ρ ;k u k) dv + ρu k ;kdv = 0. (2.37)As we are dealing with arbitrary fluid particles, dv ≠ 0, so from (2.37)follows∂ t ρ + ρ ;k u k + ρu k ;k ≡ ∂ t ρ + (ρu k ) ;k = 0. (2.38)Equation (2.38) is the covariant form of the continuity equation, which instandard vector notation becomes (2.32), i.e., ∂ t ρ + div(ρu) = 0.


80 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.1.5.2 Forces Acting on a FluidA fluid contained in a finite volume is subject to the action of both volumeforces F i and surface forces S i , which are respectively defined by∫∮F i = ρf i dv, and S i = σ ij da j . (2.39)vHere, f i is a force vector acting on an elementary mass dm, so that theelementary volume force is given bydF i = f i dm = ρf i dv,which is the integrand in the volume integral on l.h.s of (2.39). σ ij =σ ij (x k , t) is the stress tensor–field of the fluid, so that the elementary forceacting on the closed oriented surface a is given bydS i = σ ij da j ,where da j is an oriented element of the surface a; this is the integrand inthe surface integral on the r.h.s of (2.39).On the other hand, the elementary momentum dK i of a fluid particle(with elementary volume dv and elementary mass dm = ρdv) equals theproduct of dm with the particle’s velocity u i , i.e.,dK i = u i dm = ρu i dv,so that the total momentum of the finite fluid volume v is given by thevolume integral∫K i = ρu i dv. (2.40)vNow, the Newtonian–like force law for the fluid states that the timederivative of the fluid momentum equals the resulting force acting on it,˙K i = F i , where the resulting force F i is given by the sum of surface andvolume forces,∮F i = S i + F i =From (2.40), taking the time derivative and usingaa∫σ ij da j + ρf i dv. (2.41)v∫˙K i = ρ ˙u i dv,v˙ ρdv = 0, we get


Technical Preliminaries: Tensors, Actions and Functors 81where ˙u i = ˙u i (x k , t) ≡ ˙u is the acceleration vector–field of the fluid, so that(2.41) gives∮ ∫σ ij da j + ρ(f i − ˙u i )dv = 0. (2.42)avNow, assuming that the stress tensor σ ij = σ ij (x k , t) does not have anysingular points in the volume v bounded by the closed surface a, we cantransform the surface integral in (2.42) in the volume one, i.e.,∮a∫σ ij da j =vσ ij;j dv, (2.43)where σ ij;j denotes the divergence of the stress tensor. The expression (2.43)shows us that the resulting surface force acting on the closed surface a equalsthe flux of the stress tensor through the surface a. Using this expression,we can rewrite (2.42) in the form∫ (σ ij;j + ρf i − ρ ˙u i) dv = 0.vAs this equation needs to hold for an arbitrary fluid element dv ≠ 0, itimplies the dynamical equation of motion for the fluid particles, also calledthe first Cauchy law of motion,2.1.5.3 Constitutive and Dynamical Equationsσ ij;j + ρf i = ρ ˙u i . (2.44)Recall that, in case of a homogenous isotropic viscous fluid, the stress tensorσ ij depends on the strain–rate tensor–field e ij = e ij (x k , t) of the fluid insuch a way thatσ ij = −pg ij , when e ij = 0,where the scalar function p = p(x k , t) represents the pressure field. Therefore,pressure is independent on the strain–rate tensor e ij . Next, we introducethe viscosity tensor–field β ij = β ij (x k , t), asβ ij = σ ij + pg ij , (2.45)which depends exclusively on the strain–rate tensor (i.e., β ij = 0 whenevere ij = 0). A viscous fluid in which the viscosity tensor β ij can be expressed


82 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionas a function of the strain–rate tensor e ij in the formβ ij = α 1 (e I , e II , e III )g ij + α 2 (e I , e II , e III )e ij + α 3 (e I , e II , e III )e i ke kj ,(2.46)where α l = α l (e I , e II , e III ), (l = 1, 2, 3) are scalar functions of the basicinvariants (e I , e II , e III ) of the strain–rate tensor e ij , is called the Stokesfluid.If we take only the linear terms in (2.46), we get the constitutive equationfor the Newtonian fluid,β ij = α 1 e I g ij + α 2 e ij , (2.47)which is, therefore, a linear approximation of the constitutive equation(2.46) for the Stokes fluid.If we now put (2.47) into (2.45) we get the dynamical equation for theNewtonian fluid,σ ij = −pg ij + µe I g ij + 2ηe ij , (2.48)If we put µ = η V − 2 3 η, where η V is called the volume viscositycoefficient, while η is called the shear viscosity coefficient, we can rewrite(2.48) asσ ij = −pg ij +(η V − 2 )3 η e I g ij + 2ηe ij . (2.49)2.1.5.4 Navier–Stokes EquationsFrom the constitutive equation of the Newtonian viscous fluid (2.49), bytaking the divergence, we getσ ij;j = −p ;jg ij +(η V − 2 )3 η e I;j g ij + 2ηe ij;j .However, as e I;j = u k ;kjas well ase ij;j = 1 2 (ui;j + u j;i ) ;j = 1 2 (ui;j j+ u j;ij ) = 1 2 ∆ui + 1 2 uki ;k


Technical Preliminaries: Tensors, Actions and Functors 83we getor;j = −p ;jg ij +(η V − 2 )3 ησ ijσ ij;j = −p ;jg ij +u k ;kjg ij + η∆u i + ηu k ;kjg ij ,(η V − 1 3 η )u k ;kjg ij + η∆u i .If we now substitute this expression into (2.44) we getρ ˙u i = ρf i − p ;j g ij +(η V − 1 )3 η u k ;kjg ij + η∆u i , (2.50)that is a system of 3 scalar PDEs called the Navier–Stokes equations, whichin vector form readρ ˙u = ρf − grad p +(η V − 1 )3 η grad(div u) + η∆u. (2.51)In particular, for incompressible fluids, div u = 0, we have˙u = f − 1 ρ grad p + ν∆u, where ν = η ρ(2.52)is the coefficient of kinematic viscosity.2.2 Actions: The Core Machinery of Modern PhysicsIt is now well–known that a contemporary development of theoreticalphysics progresses according to the heuristic action paradigm 6 (see e.g.,[Ramond (1990); Feynman and Hibbs (1965); Siegel (2002)]), which followsthe common three essential steps (note that many technical detailsare omitted here for brevity):(1) In order to develop a new physical theory, we first define a new actionA[w], a functional in N system variables w i , as a time integral fromthe initial point t 0 to the final point t 1 ,A[w] =∫ t1t 0L[w] dt. (2.53)6 The action principle is a fundamental concept in physics (of as great importance assymmetry). It is very powerful for classical physics, allowing all field equations to bederived from a single function, and making symmetries simpler to check. In quantumphysics the dynamics is necessarily formulated in terms of an action (in the path–integralapproach), or an equivalent Hamiltonian (in the Heisenberg and Schrödinger approaches).


84 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow, the nature of the integrand L[w] in (6.230) depends on whether weare dealing with particles or fields. In case of particles, L[w] = L(q, ˙q) isan ordinary finite–dimensional mechanical Lagrangian (usually kineticminus potential energy), defined through mechanical (total system energy)Hamiltonian H = H(q, p) as L[w] = L(q, ˙q) = p i ˙q i − H(q, p),where q, ˙q, p are generalized coordinates, velocities and canonical momenta,respectively.In case of fields, the integrand L[w] is more involved, as fields justinfinite–dimensional particles. Thus,∫L[w] = d n q L(ϕ, ˙ϕ, ϕ q ),where the integral is taken over all n space coordinates 7 , while ϕ, ˙ϕ, ϕ qdenote field variables, their velocities and their coordinate (partial)derivatives, respectively. The subintegrand L = L(ϕ, ˙ϕ, ϕ q ) is the systemLagrangian density, defined through the system Hamiltonian densityH = H(ϕ, π, π q ) as L(ϕ, ˙ϕ, ϕ q ) = π i ˙ϕ i − H(ϕ, π, π q ), where π, π qare field (canonical) momenta, and their coordinate derivatives.(2) Variate the action A[w] using the extremal (least) action principleδA[w] = 0, (2.54)and using techniques from calculus of variations (see, e.g., [Arfken(1985); Fox (1988); Ramond (1990)]), derive classical field and motionequations, as Euler–Lagrangian equations, describing the extremalpath, or direct system path: t ↦→ w(t), from t 0 to t 1 .Again, we have two cases. The particle Euler–Lagrangian equationreads∂ t L ˙q i = L q i,and can be recast in Hamiltonian form, using the Poisson bracket (or,classical commutator) 8 , as a pair of canonical equations˙q i = [q i , H], ṗ i = [p i , H]. (2.55)7 In particular, n = 3 for the fields in Euclidean 3D space.8 Recall that for any two functions A = A(q k , p k , t) and B = B(q k , p k , t), their Poissonbracket is defined as„ ∂A ∂B[A, B] partcl =∂q k − ∂A «∂B∂p k ∂p k ∂q k .


Technical Preliminaries: Tensors, Actions and Functors 85The field Euler–Lagrangian equation∂ q L ∂qϕ i = L ϕ i,in Hamiltonian form gives a pair of field canonical equations 9˙ϕ i = [ϕ i , H], ˙π i = [π i , H]. (2.56)(3) Once we have a satisfactory description of fields and motions, we canperform the Feynman quantization 10 of classical equations [Feynmanand Hibbs (1965)], using the same action A[w] as given by (6.230),but now including all trajectories rather than just the extremal one.Namely, to get the probability amplitude 〈f|i〉 of the system transitionfrom initial state i(w(t 0 )) at time t 0 to final state f(w(t 1 )) at time t 1 ,9 Here the field Poisson brackets are slightly generalized in the sense that partialderivatives ∂ are replaced with the corresponding variational derivatives δ, i.e.,„ δA δB[A, B] field =δq k − δA «δBδp k δp k δq k .10 Recall that quantum systems have two modes of evolution in time. The first, governedby Schrödinger equation:i ∂ |ψ〉 = Ĥ |ψ〉 ,∂t(where Ĥ is the Hamiltonian (energy) operator, i = √ −1 and is Planck’s constantdivided by 2π (≡ 1 in natural units)), describes the time evolution of quantum systemswhen they are undisturbed by measurements. ‘Measurements’ are defined as interactionsof the system with its environment. As long as the system is sufficiently isolated fromthe environment, it follows Schrödinger equation. If an interaction with the environmenttakes place, i.e., a measurement is performed, the system abruptly decoheres i.e.,collapses or reduces to one of its classically allowed states.A time–dependent state of a quantum system is determined by a normalized, complex,wave psi–function ψ = ψ(t), that is a solution of the above Schrödinger equation. InDirac’s words, this is a unit ‘ket’ vector |ψ〉 (that makes a scalar product ‘bracket’ 〈, 〉with the dual, ‘bra’ vector 〈ψ|) , which is an element of the Hilbert space L 2 (ψ) witha coordinate basis (q i ). The state ket–vector |ψ(t)〉 is subject to action of the Hermitianoperators (or, self–adjoint operators), obtained by the procedure of quantization ofclassical mechanical quantities, and whose real eigen–values are being measured. Quantumsuperposition is a generalization of the algebraic principle of linear combination ofvectors.The (first) quantization can be performed in three different quantum evolution pictures,namely Schrödinger (S)–picture, in which the system state vector |ψ(t)〉 rotatesand the coordinate basis (q i ) is fixed; Heisenberg (H)–picture, in which the coordinatebasis rotates and the state vector is fixed; and Dirac interaction (I)–picture, in whichboth the state vector and the coordinate basis rotate.


86 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwe put the action A[w] into the path integral 11 (see, e.g., [Feynman andHibbs (1965); Schulman (1981); Siegel (2002)]), symbolically written as∫〈f|i〉 = D[w] e iA[w] , (2.57)Ωwhere i = √ −1 is the imaginary unit, Ω represents the space of allsystem paths w i (t) which contribute to the system transition with equalprobabilities, and the implicit Planck constant ħ is normalized to unity.This ‘functional integral’ is usually calculated by breaking up the timeinterval [t 0 , t 1 ] into discrete points and taking the continuum limit.The symbolic differential D[w] in the path integral (2.57) representinga (somewhat non–rigorous) path measure, defines a productD[w] =N∏dw i ,i=1which in case of quantum–mechanical particles readsD[w] =and in case of quantum fields readsD[w] =N∏dq i dp i ,i=1N∏dϕ i dπ i .i=1The path integral scheme (2.57) is commonly used for calculating thepropagator of an arbitrary quantum(gravity) system, as well as that11 The Feynman path integral is an expression for the system’s propagator in terms ofan integral over an infinite–dimensional space of paths within the system’s configurationspace. It constitutes a formulation of non–relativistic quantum mechanics which is alternativeto the usual Schrödinger equation. Whereas the Schrödinger equation is basedon Hamiltonians, the Feynman path integral is based on Lagrangians. In the last threedecades, path integrals have proven to be invaluable in quantum field theory, statisticalmechanics, condensed matter physics, and quantum gravity. The path integral is nowthe preferred method for quantizing gauge fields, as well as setting up perturbation expansionsin quantum field theory. It also leads very quickly to important conclusionsin certain problems. However, for most simple non–relativistic quantum problems, thepath integral is not as easy to use as the Schrödinger equation, and most of the resultsobtained with it can be obtained more easily by other means. Nevertheless, one cannothelp but be impressed with the elegance and beauty of the Feynman path integral, orrecognize that it is a result of fundamental scientific importance.


Technical Preliminaries: Tensors, Actions and Functors 87of a Markov–Gaussian stochastic systems described by either Langevinrate ODEs or Fokker–Planck PDEs.The most general quantization method, the path integral (2.57), canbe reduced to the most common Dirac quantization rule [Dirac (1982)],which uses modified particle equation (2.55) for quantum particles,˙ˆq i = i{Ĥ, ˆqi }, ˙ˆpi = i{Ĥ, ˆp i}, (2.58)and modified field equation (6.147) for quantum fields,˙ˆϕ i = i{Ĥ, ˆϕi }, ˙ˆπi = i{Ĥ, ˆπ i], (2.59)where coordinate and field variables (of ordinary Euclidean space) arereplaced by the corresponding Hermitian operators (i.e., self–adjointoperators) in the complex Hilbert space and the Poisson bracket [, ]is replaced by the quantum commutator {, } (multiplied by −i). Inaddition, the Dirac rule postulates the Heisenberg uncertainty relationsbetween the canonical pairs of coordinate and field variables, namely∆q i · ∆p i ≥ 1 2and ∆ϕ i · ∆π i ≥ 1 2 .The action paradigm, as outlined above, provides both classical 12 andquantum description for any new physical theory, even for those yet tobe discovered. It represents a heuristic tool in search for a unified forceof nature; the latest theory described in this way has been the celebratedsuperstring theory.2.3 Functors: Global Machinery of Modern MathematicsIn modern mathematical sciences whenever one defines a new class of mathematicalobjects, one proceeds almost in the next breath to say what kindsof maps between objects will be considered [Switzer (1975)]. A generalframework for dealing with situations where we have some objects and mapsbetween objects, like sets and functions, vector spaces and linear operators,points in a space and paths between points, etc. – gives the modern metalanguageof categories and functors. Categories are mathematical universesand functors are ‘projectors’ from one universe onto another. For this rea-12 For example, the Einstein–Hilbert action is used to derive the Einstein equation ofgeneral relativity (see below).


88 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionson, in this book we extensively use this language, mainly following itsfounder, S. MacLane [MacLane (1971)].2.3.1 Maps2.3.1.1 Notes from Set TheoryGiven a map (or, a function) f : A → B, the set A is called the domain off, and denoted Dom f. The set B is called the codomain of f, and denotedCod f. The codomain is not to be confused with the range of f(A), whichis in general only a subset of B.A map f : X → Y is called injective or 1–1 or an injection if forevery y in the codomain Y there is at most one x in the domain X withf(x) = y. Put another way, given x and x ′ in X, if f(x) = f(x ′ ), then itfollows that x = x ′ . A map f : X → Y is called surjective or onto or asurjection if for every y in the codomain Cod f there is at least one x inthe domain X with f(x) = y. Put another way, the range f(X) is equal tothe codomain Y . A map is bijective iff it is both injective and surjective.Injective functions are called the monomorphisms, and surjective functionsare called the epimorphisms in the category of sets (see below).Two main classes of maps (or, functions) that we will use int this bookare: (i) continuous maps (denoted as C 0 −class), and (ii) smooth or differentiablemaps (denoted as C k −class). The former class is the core oftopology, the letter of differential geometry. They are both used in the coreconcept of manifold.A relation is any subset of a Cartesian product (see below). By definition,an equivalence relation α on a set X is a relation which is reflexive,symmetrical and transitive, i.e., relation that satisfies the following threeconditions:(1) Reflexivity: each element x ∈ X is equivalent to itself, i.e., xαx,(2) Symmetry: for any two elements x, x ′ ∈ X, xαx ′ implies x ′ αx, and(3) Transitivity: a ≤ b and b ≤ c implies a ≤ c.Similarly, a relation ≤ defines a partial order on a set S if it has thefollowing properties:(1) Reflexivity: a ≤ a for all a ∈ S,(2) Antisymmetry: a ≤ b and b ≤ a implies a = b, and(3) Transitivity: a ≤ b and b ≤ c implies a ≤ c.


Technical Preliminaries: Tensors, Actions and Functors 89A partially ordered set (or poset) is a set taken together with a partialorder on it. Formally, a partially ordered set is defined as an ordered pairP = (X, ≤), where X is called the ground set of P and ≤ is the partialorder of P .2.3.1.2 Notes From Calculus2.3.1.3 MapsRecall that a map (or, function) f is a rule that assigns to each elementx in a set A exactly one element, called f(x), in a set B. A map could bethought of as a machine [[f]] with x−input (the domain of f is the set of allpossible inputs) and f(x)−output (the range of f is the set of all possibleoutputs) [Stuart (1999)]x → [[f]] → f(x)There are four possible ways to represent a function (or map): (i) verbally(by a description in words); (ii) numerically (by a table of values); (iii)visually (by a graph); and (iv) algebraically (by an explicit formula). Themost common method for visualizing a function is its graph. If f is afunction with domain A, then its graph is the set of ordered input–outputpairs{(x, f(x)) : x ∈ A}.A generalization of the graph concept is a concept of a cross–section of afibre bundle, which is one of the core geometrical objects for dynamics ofcomplex systems.2.3.1.4 Algebra of MapsLet f and g be maps with domains A and B. Then the maps f + g, f − g,fg, and f/g are defined as follows [Stuart (1999)](f + g)(x) = f(x) + g(x) domain = A ∩ B,(f − g)(x) = f(x) − g(x) domain = A ∩ B,(fg)(x) = f(x) g(x) domain = A ∩ B,( ) f(x) = f(x) domain = {x ∈ A ∩ B : g(x) ≠ 0}.g g(x)


90 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.3.1.5 Compositions of MapsGiven two maps f and g, the composite map f ◦ g (also called the compositionof f and g) is defined by(f ◦ g)(x) = f(g(x)).The (f ◦ g)−machine is composed of the g−machine (first) and then thef−machine [Stuart (1999)],x → [[g]] → g(x) → [[f]] → f(g(x))For example, suppose that y = f(u) = √ u and u = g(x) = x 2 + 1. Since yis a function of u and u is a function of x, it follows that y is ultimately afunction of x. We calculate this by substitutiony = f(u) = f ◦ g = f(g(x)) = f(x 2 + 1) = √ x 2 + 1.2.3.1.6 The Chain RuleIf f and g are both differentiable (or smooth, i.e., C k ) maps and h = f ◦ gis the composite map defined by h(x) = f(g(x)), then h is differentiableand h ′ is given by the product [Stuart (1999)]h ′ (x) = f ′ (g(x)) g ′ (x).In Leibniz notation, if y = f(u) and u = g(x) are both differentiable maps,thendydx = dy dudu dx .The reason for the name chain rule becomes clear if we add another linkto the chain. Suppose that we have one more differentiable map x = h(t).Then, to calculate the derivative of y with respect to t, we use the chainrule twice,dydt = dydududxdxdt .2.3.1.7 Integration and Change of Variables1–1 continuous (i.e., C 0 ) map T with a nonzero Jacobian∣∣ ∂(x,...)∂(u,...)∣ thatmaps a region S onto a region R, (see [Stuart (1999)]) we have the followingsubstitution formulas:


Technical Preliminaries: Tensors, Actions and Functors 911. for a single integral,∫2. for a double integral,∫∫R3. for a triple integral,∫∫∫RR∫∫f(x, y) dA =∫∫f(x, y, z) dV =∫f(x) dx =4. similarly for n−tuple integrals.SSSf(x(u)) ∂x∂u du,f(x(u, v), y(u, v))∂(x, y)∣∂(u, v) ∣ dudv,f(x(u, v, w), y(u, v, w), z(u, v, w))∂(x, y, z)∣∂(u, v, w) ∣ dudvdw2.3.1.8 Notes from General TopologyTopology is a kind of abstraction of Euclidean geometry, and also a naturalframework for the study of continuity. 13 Euclidean geometry is abstractedby regarding triangles, circles, and squares as being the same basic object.Continuity enters because in saying this one has in mind a continuous deformationof a triangle into a square or a circle, or any arbitrary shape. Onthe other hand, a disk with a hole in the center is topologically differentfrom a circle or a square because one cannot create or destroy holes by continuousdeformations. Thus using topological methods one does not expectto be able to identify a geometrical figure as being a triangle or a square.However, one does expect to be able to detect the presence of gross featuressuch as holes or the fact that the figure is made up of two disjoint piecesetc. In this way topology produces theorems that are usually qualitative innature – they may assert, for example, the existence or non–existence of anobject. They will not, in general, give the means for its construction [Nashand Sen (1983)].13 Intuitively speaking, a function f : R −→ R is continuous near a point x in its domainif its value does not jump there. That is, if we just take δx to be small enough, the twofunction values f(x) and f(x + δx) should approach each other arbitrarily closely. Inmore rigorous terms, this leads to the following definition: A function f : R −→ R iscontinuous at x ∈ R if for all ɛ > 0, there exists a δ > 0 such that for all y ∈ R with|y − x| < δ, we have that |f(y) − f(x)| < ɛ. The whole function is called continuous if itis continuous at every point x.


92 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.3.1.9 Topological SpaceStudy of topology starts with the fundamental notion of topological space.Let X be any set and Y = {X α } denote a collection, finite or infinite ofsubsets of X. Then X and Y form a topological space provided the X αand Y satisfy:(1) Any finite or infinite subcollection {Z α } ⊂ X α has the property that∪Z α ∈ Y , and(2) Any finite subcollection {Z α1 , ..., Z αn } ⊂ X α has the property that∩Z αi ∈ Y .The set X is then called a topological space and the X α are called opensets. The choice of Y satisfying (2) is said to give a topology to X.Given two topological spaces X and Y , a function (or, a map)f : X → Y is continuous if the inverse image of an open set in Y is an openset in X.The main general idea in topology is to study spaces which can be continuouslydeformed into one another, namely the idea of homeomorphism.If we have two topological spaces X and Y , then a map f : X → Y is calleda homeomorphism iff(1) f is continuous (C 0 ), and(2) There exists an inverse of f, denoted f −1 , which is also continuous.Definition (2) implies that if f is a homeomorphism then so is f −1 . Homeomorphismis the main topological example of reflexive, symmetrical andtransitive relation, i.e., equivalence relation. Homeomorphism divides alltopological spaces up into equivalence classes. In other words, a pair oftopological spaces, X and Y , belong to the same equivalence class if theyare homeomorphic.The second example of topological equivalence relation is homotopy.While homeomorphism generates equivalence classes whose members aretopological spaces, homotopy generates equivalence classes whose membersare continuous (C 0 ) maps. Consider two continuous maps f, g : X →Y between topological spaces X and Y . Then the map f is said to behomotopic to the map g if f can be continuously deformed into g (seebelow for the precise definition of homotopy). Homotopy is an equivalencerelation which divides the space of continuous maps between two topologicalspaces into equivalence classes [Nash and Sen (1983)].


Technical Preliminaries: Tensors, Actions and Functors 93Another important notions in topology are covering, compactness andconnectedness. Given a family of sets {X α } = X say, then X is a coveringof another set Y if ∪X α contains Y . If all the X α happen to be open setsthe covering is called an open covering. Now consider the set Y and all itspossible open coverings. The set Y is compact if for every open covering{X α } with ∪X α ⊃ Y there always exists a finite subcovering {X 1 , ..., X n }of Y with X 1 ∪ ... ∪ X n ⊃ Y . Again, we define a set Z to be connected ifit cannot be written as Z = Z 1 ∪ Z 2 , where Z 1 and Z 2 are both open andZ 1 ∩ Z 2 is an empty set.Let A 1 , A 2 , ..., A n be closed subspaces of a topological space X such thatX = ∪ n i=1 A i. Suppose f i : A i → Y is a function, 1 ≤ i ≤ n, ifff i |A i ∩ A j = f j |A i ∩ A j , 1 ≤ i, j ≤ n. (2.60)In this case f is continuous iff each f i is. Using this procedure we can definea C 0 −function f : X → Y by cutting up the space X into closed subsets A iand defining f on each A i separately in such a way that f|A i is obviouslycontinuous; we then have only to check that the different definitions agreeon the overlaps A i ∩ A j .The universal property of the Cartesian product: let p X : X × Y → X,and p Y : X × Y → Y be the projections onto the first and second factors,respectively. Given any pair of functions f : Z → X and g : Z → Y thereis a unique function h : Z → X × Y such that p X ◦ h = f, and p Y ◦ h = g.Function h is continuous iff both f and g are. This property characterizesX/α up to homeomorphism. In particular, to check that a given functionh : Z → X is continuous it will suffice to check that p X ◦ h and p Y ◦ h arecontinuous.The universal property of the quotient: let α be an equivalence relationon a topological space X, let X/α denote the space of equivalence classesand p α : X → X/α the natural projection. Given a function f : X → Y ,there is a function f ′ : X/α → Y with f ′ ◦ p α = f iff xαx ′ implies f(x) =f(x ′ ), for all x ∈ X. In this case f ′ is continuous iff f is. This propertycharacterizes X/α up to homeomorphism.2.3.1.10 HomotopyNow we return to the fundamental notion of homotopy. Let I be a compactunit interval I = [0, 1]. A homotopy from X to Y is a continuousfunction F : X × I → Y . For each t ∈ I one has F t : X → Y defined byF t (x) = F (x, t) for all x ∈ X. The functions F t are called the ‘stages’ of the


94 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionhomotopy. If f, g : X → Y are two continuous maps, we say f is homotopicto g, and write f ≃ g, if there is a homotopy F : X × I → Y such thatF 0 = f and F 1 = g. In other words, f can be continuously deformed intog through the stages F t . If A ⊂ X is a subspace, then F is a homotopyrelative to A if F (a, t) = F (a, 0), for all a ∈ A, t ∈ I.The homotopy relation ≃ is an equivalence relation. To prove thatwe have f ≃ f is obvious; take F (x, t = f(x), for all x ∈ X, t ∈ I. Iff ≃ g and F is a homotopy from f to g, then G : X × I → Y defined byG(x, t) = F (x, 1 − t), is a homotopy from g to f, i.e., g ≃ f. If f ≃ g withhomotopy F and g ≃ f with homotopy G, then f ≃ h with homotopy Hdefined byH(x, t) ={ F (x, t), 0 ≤ t ≤ 1/2G(x, 2t − 1), 1/2 ≤ t ≤ 1 .To show that H is continuous we use the relation (2.60).In this way, the set of all C 0 −functions f : X → Y between two topologicalspaces X and Y , called the function space and denoted by Y X , ispartitioned into equivalence classes under the relation ≃. The equivalenceclasses are called homotopy classes, the homotopy class of f is denoted by[f], and the set of all homotopy classes is denoted by [X; Y ].If α is an equivalence relation on a topological space X and F : X ×I →Y is a homotopy such that each stage F t factors through X/α, i.e., xαx ′implies F t (x) = F t (x ′ ), then F induces a homotopy F ′ : (X/α) × I → Ysuch that F ′ ◦ (p α × 1) = F .Homotopy theory has a range of applications of its own, outside topologyand geometry, as for example in proving Cauchy Theorem in complexvariable theory, or in solving nonlinear equations of artificial neural networks.A pointed set (S, s 0 ) is a set S together with a distinguished points 0 ∈ S. Similarly, a pointed topological space (X, x 0 ) is a space X togetherwith a distinguished point x 0 ∈ X. When we are concerned with pointedspaces (X, x 0 ), (Y, y 0 ), etc., we always require that all functions f : X →Y shell preserve base points, i.e., f(x 0 ) = y 0 , and that all homotopiesF : X × I → Y be relative to the base point, i.e., F (x 0 , t) = y 0 , for allt ∈ I. We denote the homotopy classes of base point–preserving functionsby [X, x 0 ; Y, y 0 ] (where homotopies are relative to x 0 ). [X, x 0 ; Y, y 0 ] is apointed set with base point f 0 , the constant function: f 0 (x) = y 0 , for allx ∈ X.A path γ(t) from x 0 to x 1 in a topological space X is a continuous map


Technical Preliminaries: Tensors, Actions and Functors 95γ : I → X with γ(0) = x 0 and γ(1) = x 1 . Thus X I is the space of all pathsin X with the compact–open topology. We introduce a relation ∼ on X bysaying x 0 ∼ x 1 iff there is a path γ : I → X from x 0 to x 1 . ∼ is clearly anequivalence relation, and the set of equivalence classes is denoted by π 0 (X).The elements of π 0 (X) are called the path components, or 0−componentsof X. If π 0 (X) contains just one element, then X is called path connected,or 0−connected. A closed path, or loop in X at the point x 0 is a path γ(t)for which γ(0) = γ(1) = x 0 . The inverse loop γ −1 (t) based at x 0 ∈ X isdefined by γ −1 (t) = γ(1 − t), for 0 ≤ t ≤ 1. The homotopy of loops is theparticular case of the above defined homotopy of continuous maps.If (X, x 0 ) is a pointed space, then we may regard π 0 (X) as a pointed setwith the 0−component of x 0 as a base point. We use the notation π 0 (X, x 0 )to denote p 0 (X, x 0 ) thought of as a pointed set. If f : X → Y is a map thenf sends 0−components of X into 0−components of Y and hence defines afunction π 0 (f) : π 0 (X) → π 0 (Y ). Similarly, a base point–preserving mapf : (X, x 0 ) → (Y, y 0 ) induces a map of pointed sets π 0 (f) : π 0 (X, x 0 ) →π 0 (Y, y 0 ). In this way defined π 0 represents a ‘functor’ from the ‘category’of topological (point) spaces to the underlying category of (point) sets (seethe next section).Combination of topology and calculus gives differential topology, or differentialgeometry.2.3.1.11 Commutative DiagramsThe category theory (see below) was born with an observation that manyproperties of mathematical systems can be unified and simplified by a presentationwith commutative diagrams of arrows [MacLane (1971)]. Eacharrow f : X → Y represents a function (i.e., a map, transformation, operator);that is, a source (domain) set X, a target (codomain) set Y , and arule x ↦→ f(x) which assigns to each element x ∈ X an element f(x) ∈ Y .A typical diagram of sets and functions isXf✲ YXf ✲ f(X)❅❅❅❅❘Zhg❄or❅h ❅❅❅❘g(f(X)) g❄This diagram is commutative iff h = g◦f, where g◦f is the usual compositefunction g ◦ f : X → Z, defined by x ↦→ g(f(x)).


96 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSimilar commutative diagrams apply in other mathematical, physicaland computing contexts; e.g., in the ‘category’ of all topological spaces, theletters X, Y, and Z represent topological spaces while f, g, and h stand forcontinuous maps. Again, in the category of all groups, X, Y, and Z standfor groups, f, g, and h for homomorphisms.Less formally, composing maps is like following directed paths from oneobject to another (e.g., from set to set). In general, a diagram is commutativeiff any two paths along arrows that start at the same point and finishat the same point yield the same ‘homomorphism’ via compositions alongsuccessive arrows. Commutativity of the whole diagram follows from commutativityof its triangular components (depicting a ‘commutative flow’, seeFigure 2.4). Study of commutative diagrams is popularly called ‘diagramchasing’, and provides a powerful tool for mathematical thought.Fig. 2.4 A commutative flow (denoted by curved arrows) on a triangulated digraph.Commutativity of the whole diagram follows from commutativity of its triangular components.As an example from linear algebra, consider an elementary diagrammaticdescription of matrices, using the following pull–back diagram [Barry(1993)]:Matrix Aentries ✲ List Ashapelength❄Nat × Nat❄✲ Natproductasserts that a matrix is determined by its shape, given by a pair of naturalnumbers representing the number of rows and columns, and its data, givenby the matrix entries listed in some specified order.


Technical Preliminaries: Tensors, Actions and Functors 97Many properties of mathematical constructions may be represented byuniversal properties of diagrams [MacLane (1971)]. Consider the Cartesianproduct X × Y of two sets, consisting as usual of all ordered pairs 〈x, y〉 ofelements x ∈ X and y ∈ Y . The projections 〈x, y〉 ↦→ x, 〈x, y〉 ↦→ y of theproduct on its ‘axes’ X and Y are functions p : X ×Y → X, q : X ×Y → Y .Any function h : W → X × Y from a third set W is uniquely determinedby its composites p ◦ h and q ◦ h. Conversely, given W and two functionsf and g as in the diagram below, there is a unique function h which makesthe following diagram commute:W ❅f h ❅ g❅❅❘✠ ❄X ✛ p X × Y ✲q YThis property describes the Cartesian product X × Y uniquely; the samediagram, read in the category of topological spaces or of groups, describesuniquely the Cartesian product of spaces or of the direct product of groups.The construction ‘Cartesian product’ is technically called a ‘functor’because it applies suitably both to the sets and to the functions betweenthem; two functions k : X → X ′ and l : Y → Y ′ have a function k × l astheir Cartesian product:k × l : X × Y → X ′ × Y ′ ,〈x, y〉 ↦→ 〈kx, ly〉.2.3.1.12 Groups and Related Algebraic StructuresAs already stated, the basic functional unit of lower biodynamics is thespecial Euclidean group SE(3) of rigid body motions. In general, a groupis a pointed set (G, e) with a multiplication µ : G × G → G and an inverseν : G → G such that the following diagrams commute [Switzer (1975)]:(1)G(e is a two–sided identity)(e, 1)✲G × G(1, e)✲ G❅1 ❅❅❅❘ µ 1❄ ✒G


98 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(2)G × G × Gµ × 1 ✲ G × G1 × µ❄G × Gµµ❄✲ G(3)(associativity)G(ν, 1)✲G × G(1, ν)✲ G❅e❅ µ e❅❅❘❄ ✒G(inverse).Here e : G → G is the constant map e(g) = e for all g ∈ G. (e, 1) meansthe map such that (e, 1)(g) = (e, g), etc. A group G is called commutativeor Abelian group if in addition the following diagram commutesG × GT❅µ ❅❅ ❅❘G✲ G × Gµ ✠where T : G × G → G × G is the switch map T (g 1 , g 2 ) = (g 1 , g 2 ), for all(g 1 , g 2 ) ∈ G × G.A group G acts (on the left) on a set A if there is a function α : G×A →A such that the following diagrams commute [Switzer (1975)]:(1)A(e, 1)✲G × A❅❅❅❅❘A1α❄


Technical Preliminaries: Tensors, Actions and Functors 99(2)G × G × A1 × α ✲ G × Aµ × 1❄G × Aαα❄✲ Awhere (e, 1)(x) = (e, x) for all x ∈ A. The orbits of the action are thesets Gx = {gx : g ∈ G} for all x ∈ A.Given two groups (G, ∗) and (H, ·), a group homomorphism from (G, ∗)to (H, ·) is a function h : G → H such that for all x and y in G it holdsthath(x ∗ y) = h(x) · h(y).From this property, one can deduce that h maps the identity element e Gof G to the identity element e H of H, and it also maps inverses to inversesin the sense that h(x −1 ) = h(x) −1 . Hence one can say that h is compatiblewith the group structure.The kernel Ker h of a group homomorphism h : G → H consists of allthose elements of G which are sent by h to the identity element e H of H,i.e.,Ker h = {x ∈ G : h(x) = e H }.The image Im h of a group homomorphism h : G → H consists of allelements of G which are sent by h to H, i.e.,Im h = {h(x) : x ∈ G}.The kernel is a normal subgroup of G and the image is a subgroup ofH. The homomorphism h is injective (and called a group monomorphism)iff Ker h = e G , i.e., iff the kernel of h consists of the identity element of Gonly.Similarly, a ring is a set S together with two binary operators + and ∗(commonly interpreted as addition and multiplication, respectively) satisfyingthe following conditions:(1) Additive associativity: For all a, b, c ∈ S, (a + b) + c = a + (b + c),(2) Additive commutativity: For all a, b ∈ S, a + b = b + a,


100 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(3) Additive identity: There exists an element 0 ∈ S such that for all a ∈ S,0 + a = a + 0 = a,(4) Additive inverse: For every a ∈ S there exists -a ∈ S such that a +(−a) = (−a) + a = 0,(5) Multiplicative associativity: For all a, b, c ∈ S, (a ∗ b) ∗ c = a ∗ (b ∗ c),(6) Left and right distributivity: For all a, b, c ∈ S, a∗(b+c) = (a∗b)+(a∗c)and (b + c) ∗ a = (b ∗ a) + (c ∗ a).A ring (the term introduced by David Hilbert) is therefore an Abeliangroup under addition and a semigroup under multiplication. A ring that iscommutative under multiplication, has a unit element, and has no divisorsof zero is called an integral domain. A ring which is also a commutativemultiplication group is called a field. The simplest rings are the integers Z,polynomials R[x] and R[x, y] in one and two variables, and square n × nreal matrices.An ideal is a subset I of elements in a ring R which forms an additivegroup and has the property that, whenever x belongs to R and y belongsto I, then xy and yx belong to I. For example, the set of even integers isan ideal in the ring of integers Z. Given an ideal I, it is possible to definea factor ring R/I.A ring is called left (respectively, right) Noetherian if it does not containan infinite ascending chain of left (respectively, right) ideals. In this case,the ring in question is said to satisfy the ascending chain condition on left(respectively, right) ideals. A ring is said to be Noetherian if it is bothleft and right Noetherian. If a ring R is Noetherian, then the following areequivalent:(1) R satisfies the ascending chain condition on ideals.(2) Every ideal of R is finitely generated.(3) Every set of ideals contains a maximal element.A module is a mathematical object in which things can be added togethercommutatively by multiplying coefficients and in which most of therules of manipulating vectors hold. A module is abstractly very similar to avector space, although in modules, coefficients are taken in rings which aremuch more general algebraic objects than the fields used in vector spaces.A module taking its coefficients in a ring R is called a module over R orR−module. Modules are the basic tool of homological algebra.Examples of modules include the set of integers Z, the cubic latticein d dimensions Z d , and the group ring of a group. Z is a module over


Technical Preliminaries: Tensors, Actions and Functors 101itself. It is closed under addition and subtraction. Numbers of the formnα for n ∈ Z and α a fixed integer form a submodule since, for (n, m) ∈ Z,nα ± mα = (n ± m)α and (n ± m) is still in Z. Also, given two integersa and b, the smallest module containing a and b is the module for theirgreatest common divisor, α = GCD(a, b).A module M is a Noetherian module if it obeys the ascending chaincondition with respect to inclusion, i.e., if every set of increasing sequencesof submodules eventually becomes constant. If a module M is Noetherian,then the following are equivalent:(1) M satisfies the ascending chain condition on submodules.(2) Every submodule of M is finitely generated.(3) Every set of submodules of M contains a maximal element.Let I be a partially ordered set. A direct system of R−modules overI is an ordered pair {M i , ϕ i j } consisting of an indexed family of modules{M i : i ∈ I} together with a family of homomorphisms {ϕ i j : M i → M j }for i ≤ j, such that ϕ i i = 1 M ifor all i and such that the following diagramcommutes whenever i ≤ j ≤ kM i❅ϕ j ❅k❅❅❘ϕ i kM j✲ϕ ✒ i jSimilarly, an inverse system of R−modules over I is an ordered pair{M i , ψ j i } consisting of an indexed family of modules {M i : i ∈ I} togetherwith a family of homomorphisms {ψ j i : M j → M i } for i ≤ j, such thatψ i i = 1 Mi for all i and such that the following diagram commutes wheneveri ≤ j ≤ kM kM kψ k i✲M i❅ψ k j ❅❅❅❘M jψ j i ✒


102 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.3.2 CategoriesA category is a generic mathematical structure consisting of a collectionof objects (sets with possibly additional structure), with a correspondingcollection of arrows, or morphisms, between objects (agreeing with thisadditional structure). A category K is defined as a pair (Ob(K), Mor(K)) ofgeneric objects A, B, . . . in Ob(K) and generic arrows f : A → B, g : B →C, . . . in Mor(K) between objects, with associative composition:Af ✲ Bg ✲ C = Af ◦g ✲ C,and identity (loop) arrow. (Note that in topological literature, Hom(K) orhom(K) is used instead of Mor(K); see [Switzer (1975)]).A category K is usually depicted as a commutative diagram (i.e., adiagram with a common initial object A and final object D):✬fA ✲ B✩h K g❄ ❄C ✲ Dk✫ ✪To make this more precise, we say that a category K is defined if we have:(1) A class of objects {A, B, C, ...} of K, denoted by Ob(K);(2) A set of morphisms, or arrows Mor K (A, B), with elements f : A → B,defined for any ordered pair (A, B) ∈ K, such that for two differentpairs (A, B) ≠ (C, D) in K, we have Mor K (A, B) ∩ Mor K (C, D) = ∅;(3) For any triplet (A, B, C) ∈ K with f : A → B and g : B → C, there isa composition of morphismsMor K (B, C) × Mor K (A, B) ∋ (g, f) → g ◦ f ∈ Mor K (A, C),written schematically asf : A → B, g : B → C.g ◦ f : A → CIf we have a morphism f ∈ Mor K (A, B), (otherwise written f : A → B,for A ✲ B), then A = dom(f) is a domain of f, and B = cod(f) is acodomain of f (of which range of f is a subset) and denoted B = ran(f).


Technical Preliminaries: Tensors, Actions and Functors 103To make K a category, it must also fulfill the following two properties:(1) Associativity of morphisms: for all f ∈ Mor K (A, B), g ∈ Mor K (B, C),and h ∈ Mor K (C, D), we have h ◦ (g ◦ f) = (h ◦ g) ◦ f; in other words,the following diagram is commutativefA❄Bh ◦ (g ◦ f) = (h ◦ g) ◦✲fgD✻h✲ C(2) Existence of identity morphism: for every object A ∈ Ob(K) exists aunique identity morphism 1 A ∈ Mor K (A, A); for any two morphismsf ∈ Mor K (A, B), and g ∈ Mor K (B, C), compositions with identity morphism1 B ∈ Mor K (B, B) give 1 B ◦ f = f and g ◦ 1 B = g, i.e., thefollowing diagram is commutative:fA ✲ gB ✲ C❅f❅1 B g❅ ❅❘ ❄ ✒BThe set of all morphisms of the category K is denoted⋃Mor(K) = Mor K (A, B).A,B∈Ob(K)If for two morphisms f ∈ Mor K (A, B) and g ∈ Mor K (B, A) the equalityg ◦ f = 1 A is valid, then the morphism g is said to be left inverse (orretraction), of f, and f right inverse (or section) of g. A morphism whichis both right and left inverse of f is said to be two–sided inverse of f.A morphism m : A → B is called monomorphism in K (i.e., 1–1, orinjection map), if for any two parallel morphisms f 1 , f 2 : C → A in K theequality m◦f 1 = m◦f 2 implies f 1 = f 2 ; in other words, m is monomorphismif it is left cancellable. Any morphism with a left inverse is monomorphism.A morphism e : A → B is called epimorphism in K (i.e., onto, orsurjection map), if for any two morphisms g 1 , g 2 : B → C in K the equalityg 1 ◦ e = g 2 ◦ e implies g 1 = g 2 ; in other words, e is epimorphism if it is rightcancellable. Any morphism with a right inverse is epimorphism.


104 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionA morphism f : A → B is called isomorphism in K (denoted as f : A ∼ =B) if there exists a morphism f −1 : B → A which is a two–sided inverse off in K. The relation of isomorphism is reflexive, symmetric, and transitive,i.e., equivalence relation.For example, an isomorphism in the category of sets is called a set–isomorphism, or a bijection, in the category of topological spaces is called atopological isomorphism, or a homeomorphism, in the category of differentiablemanifolds is called a differentiable isomorphism, or a diffeomorphism.A morphism f ∈ Mor K (A, B) is regular if there exists a morphismg : B → A in K such that f ◦ g ◦ f = f. Any morphism with either a leftor a right inverse is regular.An object T is a terminal object in K if to each object A ∈ Ob(K) thereis exactly one arrow A → T . An object S is an initial object in K if toeach object A ∈ Ob(K) there is exactly one arrow S → A. A null objectZ ∈ Ob(K) is an object which is both initial and terminal; it is unique up toisomorphism. For any two objects A, B ∈ Ob(K) there is a unique morphismA → Z → B (the composite through Z), called the zero morphism from Ato B.A notion of subcategory is analogous to the notion of subset. A subcategoryL of a category K is said to be a complete subcategory iff for anyobjects A, B ∈ L, every morphism A → B of L is in K.A groupoid is a category in which every morphism is invertible. Atypical groupoid is the fundamental groupoid Π 1 (X) of a topological spaceX. An object of Π 1 (X) is a point x ∈ X, and a morphism x → x ′ ofΠ 1 (X) is a homotopy class of paths f from x to x ′ . The composition ofpaths g : x ′ → x ′′ and f : x → x ′ is the path h which is ‘f followedby g’. Composition applies also to homotopy classes, and makes Π 1 (X) acategory and a groupoid (the inverse of any path is the same path tracedin the opposite direction).A group is a groupoid with one object, i.e., a category with one object inwhich all morphisms are isomorphisms. Therefore, if we try to generalizethe concept of a group, keeping associativity as an essential property, weget the notion of a category.A category is discrete if every morphism is an identity. A monoid is acategory with one object. A group is a category with one object in whichevery morphism has a two–sided inverse under composition.Homological algebra was the progenitor of category theory (see e.g.,[Dieudonne (1988)]). Generalizing L. Euler’s formula f + v = e + 2 for thefaces, vertices and edges of a convex polyhedron, E. Betti defined numerical


Technical Preliminaries: Tensors, Actions and Functors 105invariants of spaces by formal addition and subtraction of faces of variousdimensions; H. Poincaré formalized these and introduced homology. E.Noether stressed the fact that these calculations go on in Abelian groups,and that the operation ∂ n taking a face of dimension n to the alternatingsum of faces of dimension n−1 which form its boundary is a homomorphism,and it also satisfies ∂ n ◦ ∂ n+1 = 0. There are many ways of approximatinga given space by polyhedra, but the quotient H n = Ker ∂ n / Im ∂ n+1 is aninvariant, the homology group. Since Noether, the groups have been theobject of study instead of their dimensions, which are the Betti numbers.2.3.3 FunctorsIn algebraic topology, one attempts to assign to every topological spaceX some algebraic object F(X) in such a way that to every C 0 −functionf : X → Y there is assigned a homomorphism F(f) : F(X) −→ F(Y )(see [Switzer (1975); Dodson and Parker (1997)]). One advantage of thisprocedure is, e.g., that if one is trying to prove the non–existence of aC 0 −function f : X → Y with certain properties, one may find it relativelyeasy to prove the non–existence of the corresponding algebraic functionF(f) and hence deduce that f could not exist. In other words, F is to bea ‘homomorphism’ from one category (e.g., T ) to another (e.g., G or A).Formalization of this notion is a functor.A functor is a generic picture projecting one category into another.Let K = (Ob(K), Mor(K)) be a source (or domain) category and L =(Ob(L), Mor(L)) be a target (or codomain) category. A functor F =(F O , F M ) is defined as a pair of maps, F O : Ob(K) → Ob(L) and F M :Mor(K) → Mor(L), preserving categorical symmetry (i.e., commutativity ofall diagrams) of K in L.More precisely, a covariant functor, or simply a functor, F ∗ : K → Lis a picture in the target category L of (all objects and morphisms of) thesource category K:✬ ✩ ✬✩fA ✲F(f)BF(A) ✲ F(B)Fhg∗K✲❄ ❄C ✲ Dk✫ ✪F(h)❄F(C)✫L F(g)❄✲ F(D)F(k)✪Similarly, a contravariant functor, or a cofunctor, F ∗ : K → L is a dual


106 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionpicture with reversed arrows:✬ ✩ ✬✩fA ✲ BF(A) ✛ F(f) F(B)hg FK∗ ✲ ✻F(h) L✻ F(g)❄ ❄C ✲ DF(C) ✛ F(D)kF(k)✫ ✪ ✫✪In other words, a functor F : K → L from a source category K to atarget category L, is a pair F = (F O , F M ) of maps F O : Ob(K) → Ob(L),F M : Mor(K) → Mor(L), such that(1) If f ∈ Mor K (A, B) then F M (f) ∈ Mor L (F O (A), F O (B)) in case of thecovariant functor F ∗ , and F M (f) ∈ Mor L (F O (B), F O (A)) in case ofthe contravariant functor F ∗ ;(2) For all A ∈ Ob(K) : F M (1 A ) = 1 FO (A);(3) For all f, g ∈ Mor(K): if cod(f) = dom(g), thenF M (g ◦ f) = F M (g) ◦ F M (f) in case of the covariant functor F ∗ , andF M (g ◦ f) = F M (f) ◦ F M (g) in case of the contravariant functor F ∗ .Category theory originated in algebraic topology, which tried to assignalgebraic invariants to topological structures. The golden rule of such invariantsis that they should be functors. For example, the fundamentalgroup π 1 is a functor. Algebraic topology constructs a group called thefundamental group π 1 (X) from any topological space X, which keeps trackof how many holes the space X has. But also, any map between topologicalspaces determines a homomorphism φ : π 1 (X) → π 1 (Y ) of the fundamentalgroups. So the fundamental group is really a functor π 1 : T → G. Thisallows us to completely transpose any situation involving spaces and continuousmaps between them to a parallel situation involving groups andhomomorphisms between them, and thus reduce some topology problemsto algebra problems.Also, singular homology in a given dimension n assigns to each topologicalspace X an Abelian group H n (X), its nth homology group of X,and also to each continuous map f : X → Y of spaces a correspondinghomomorphism H n (f) : H n (X) → H n (Y ) of groups, and this in such away that H n (X) becomes a functor H n : T → A.The leading idea in the use of functors in topology is that H n or π ngives an algebraic picture or image not just of the topological spaces X, Ybut also of all the continuous maps f : X → Y between them.


Technical Preliminaries: Tensors, Actions and Functors 107Similarly, there is a functor Π 1 : T → G, called the ‘fundamentalgroupoid functor’, which plays a very basic role in algebraic topology. Here’show we get from any space X its ‘fundamental groupoid’ Π 1 (X). To saywhat the groupoid Π 1 (X) is, we need to say what its objects and morphismsare. The objects in Π 1 (X) are just the points of X and the morphisms arejust certain equivalence classes of paths in X. More precisely, a morphismf : x → y in Π 1 (X) is just an equivalence class of continuous paths fromx to y, where two paths from x to y are decreed equivalent if one can becontinuously deformed to the other while not moving the endpoints. (If thisequivalence relation holds we say the two paths are ‘homotopic’, and we callthe equivalence classes ‘homotopy classes of paths’ (see [MacLane (1971);Switzer (1975)]).Another examples are covariant forgetful functors:• From the category of topological spaces to the category of sets;it ‘forgets’ the topology–structure.• From the category of metric spaces to the category of topological spaceswith the topology induced by the metrics; it ‘forgets’ the metric.For each category K, the identity functor I K takes every K−object andevery K−morphism to itself.Given a category K and its subcategory L, we have an inclusion functorIn : K −→ K.Given a category K, a diagonal functor ∆ : K −→ K takes each objectA ∈ K to the object (A, A) in the product category K × K.Given a category K and a category of sets S, each object A ∈ K determinesa covariant Hom–functor K[A, ] : K → S, a contravariant Hom–functor K[ , A] : K −→ S, and a Hom–bifunctor K[ , ] : K op × K → S.A functor F : K → L is a faithful functor if for all A, B ∈ Ob(K) and forall f, g ∈ Mor K (A, B), F(f) = F(g) implies f = g; it is a full functor if forevery h ∈ Mor L (F(A), F(B)), there is g ∈ Mor K (A, B) such that h = F(g);it is a full embedding if it is both full and faithful.A representation of a group is a functor F : G → V.Similarly, we can define a representation of a category to be a functorF : K → V from the 2−category K (a ‘big’ category including all ordinary,or ‘small’ categories, see section (2.3.8.1) below) to the category of vectorspaces V. In this way, a category is a generalization of a group and grouprepresentations are a special case of category representations.


108 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.3.4 Natural TransformationsA natural transformation (i.e., a functor morphism) τ : F → ·G is a mapbetween two functors of the same variance, (F, G) : K ⇒ L, preservingcategorical symmetry:✬ ✩ ✬✩F F(f)✲ F(A) ✲ F(B)A✫f ✲ BKτ ⇓ τ A L τ B❄❄G ✲ G(A) ✲ G(B)G(f)✪ ✫✪More precisely, all functors of the same variance from a source category Kto a target category L form themselves objects of the functor category L K .Morphisms of L K , called natural transformations, are defined as follows.Let F : K → L and G : K → L be two functors of the same variancefrom a category K to a category L. Natural transformation F −→ τG is afamily of morphisms such that for all f ∈ Mor K (A, B) in the source categoryK, we have G(f) ◦ τ A = τ B ◦ F(f) in the target category L. Then we saythat the component τ A : F(A) → G(A) is natural in A.If we think of a functor F as giving a picture in the target category Lof (all the objects and morphisms of) the source category K, then a naturaltransformation τ represents a set of morphisms mapping the picture F toanother picture G, preserving the commutativity of all diagrams.An invertible natural transformation, such that all components τ A areisomorphisms) is called a natural equivalence (or, natural isomorphism). Inthis case, the inverses (τ A ) −1 in L are the components of a natural isomorphism(τ ) −1 : G −→ ∗F. Natural equivalences are among the most importantmetamathematical constructions in algebraic topology (see [Switzer(1975)]).For example, let B be the category of Banach spaces over R and boundedlinear maps. Define D : B → B by taking D(X) = X ∗ = Banach space ofbounded linear functionals on a space X and D(f) = f ∗ for f : X → Y abounded linear map. Then D is a cofunctor. D 2 = D ◦ D is also a functor.We also have the identity functor 1 : B → B. Define T : 1 → D ◦ D asfollows: for every X ∈ B let T (X) : X → D 2 X = X ∗∗ be the naturalinclusion – that is, for x ∈ X we have [T (X)(x)](f) = f(x) for everyf ∈ X ∗ . T is a natural transformation. On the subcategory of nD Banachspaces T is even a natural equivalence. The largest subcategory of B onwhich T is a natural equivalence is called the category of reflexive Banach


Technical Preliminaries: Tensors, Actions and Functors 109spaces [Switzer (1975)].As S. Eilenberg and S. MacLane first observed, ‘category’ has beendefined in order to define ‘functor’ and ‘functor’ has been defined in orderto define ‘natural transformation’ [MacLane (1971)]).2.3.4.1 Compositions of Natural TransformationsNatural transformations can be composed in two different ways. First, wehave an ‘ordinary’ composition: if F, G and H are three functors from thesource category A to the target category B, and then α : F → ·G, β : G → ·Hare two natural transformations, then the formula(β ◦ α) A= β A ◦ α A , for all A ∈ A, (2.61)defines a new natural transformation β ◦ α : F → ·H. This compositionlaw is clearly associative and possesses a unit 1 F at each functor F, whoseA−-component is 1 FA .Second, we have the Godement product of natural transformations, usuallydenoted by ∗. Let A, B and C be three categories, F, G, H and K befour functors such that (F, G) : A ⇒ B and (H, K) : B ⇒ C, and α : F → ·G,β : H → ·K be two natural transformations. Now, instead of (2.61), theGodement composition is given by(β ∗ α) A= β GA ◦ H (α A ) = K (α A ) ◦ β F A , for all A ∈ A, (2.62)which defines a new natural transformation β ∗ α : H ◦ F ·→ K ◦ G.Finally, the two compositions (2.61) and (2.61) of natural transformationscan be combined as(δ ∗ γ) ◦ (β ∗ α) = (δ ◦ β) ∗ (γ ◦ α) ,where A, B and C are three categories, F, G, H, K, L, M are six functors,and α : F → ·H, β : G → ·K, γ : H → ·L, δ : K → ·M are four naturaltransformations.2.3.4.2 Dinatural TransformationsDouble natural transformations are called dinatural transformations. Anend of a functor S : C op × C → X is a universal dinatural transformationfrom a constant e to S. In other words, an end of S is a pair 〈e, ω〉, wheree is an object of X and ω : e .. → S is a wedge (dinatural) transformationwith the property that to every wedge β : x .. → S there is a unique arrow


110 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionh : x → e of B with β c = ω c h for all a ∈ C. We call ω the ending wedgewith components ω c , while the object e itself, by abuse of language, is calledthe end of S and written with integral notation as ∫ S(c, c); thusc∫S(c, c) →ωcS(c, c) = e.cNote that the ‘variable of integration’ c appears twice under the integralsign (once contravariant, once covariant) and is ‘bound’ by the integralsign, in that the result no longer depends on c and so is unchanged if‘c’ is replaced by any other letter standing for an object of the categoryC. These properties are like those of the letter x under the usual integralsymbol ∫ f(x) dx of calculus.Every end is manifestly a limit (see below) – specifically, a limit of asuitable diagram in X made up of pieces like S(b, b) → S(b, c) → S(c, c).For each functor T : C → X there is an isomorphism∫ ∫S(c, c) = T c ∼ = Lim T,ccvalid when either the end of the limit exists, carrying the ending wedge tothe limiting cone; the indicated notation thus allows us to write any limitas an integral (an end) without explicitly mentioning the dummy variable(the first variable c of S).A functor H : X → Y is said to preserve the end of a functor S :C op ×C → X when ω : e → .. S an end of S in X implies that Hω : He → .. HSis an and for HS; in symbols∫ ∫H S(c, c) = HS(c, c).cSimilarly, H creates the end of S when to each end v : y .. → HS in Y thereis a unique wedge ω : e .. → S with Hω = v, and this wedge ω is an end of S.The definition of the coend of a functor S : C op ×C → X is dual to thatof an end. A coend of S is a pair 〈d, ζ〉, consisting of an object d ∈ X and awedge ζ : S .. → d. The object d (when it exists, unique up to isomorphism)will usually be written with an integral sign and with the bound variable cas superscript; thus∫ cS(c, c) ζ c→ S(c, c) = d.c


Technical Preliminaries: Tensors, Actions and Functors 111The formal properties of coends are dual to those of ends. Both are muchlike those for integrals in calculus (see [MacLane (1971)], for technical details).2.3.5 Limits and ColimitsIn abstract algebra constructions are often defined by an abstract propertywhich requires the existence of unique morphisms under certain conditions.These properties are called universal properties. The limit of a functorgeneralizes the notions of inverse limit and product used in various parts ofmathematics. The dual notion, colimit, generalizes direct limits and directsums. Limits and colimits are defined via universal properties and providemany examples of adjoint functors.A limit of a covariant functor F : J → C is an object L of C, togetherwith morphisms φ X : L → F(X) for every object X of J , such that forevery morphism f : X → Y in J , we have F(f)φ X = φ Y , and such that thefollowing universal property is satisfied: for any object N of C and any setof morphisms ψ X : N → F(X) such that for every morphism f : X → Y inJ , we have F(f)ψ X = ψ Y , there exists precisely one morphism u : N → Lsuch that φ X u = ψ X for all X. If F has a limit (which it need not), thenthe limit is defined up to a unique isomorphism, and is denoted by lim F.Analogously, a colimit of the functor F : J → C is an object L ofC, together with morphisms φ X : F(X) → L for every object X of J ,such that for every morphism f : X → Y in J , we have φ Y F(X) = φ X ,and such that the following universal property is satisfied: for any objectN of C and any set of morphisms ψ X : F(X) → N such that for everymorphism f : X → Y in J , we have ψ Y F(X) = ψ X , there exists preciselyone morphism u : L → N such that uφ X = ψ X for all X. The colimit ofF, unique up to unique isomorphism if it exists, is denoted by colim F.Limits and colimits are related as follows: A functor F : J → C hasa colimit iff for every object N of C, the functor X ↦−→ Mor C (F(X), N)(which is a covariant functor on the dual category J op ) has a limit. If thatis the case, then Mor C (colim F, N) = lim Mor C (F(−), N) for every objectN of C.2.3.6 AdjunctionThe most important functorial operation is adjunction; as S. MacLane oncesaid, “Adjoint functors arise everywhere” [MacLane (1971)].


112 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe adjunction ϕ : F ⊣ G between two functors (F, G) : K ⇆ L ofopposite variance [Kan (1958)], represents a weak functorial inversef : F(A) → Bϕ(f) : A → G(B)ϕforming a natural equivalence ϕ : Mor K (F(A), B) −→ Mor L (A, G(B)). Theadjunction isomorphism is given by a bijective correspondence (a 1–1 andonto map on objects) ϕ : Mor(K) ∋ f → ϕ(f) ∈ Mor(L) of isomorphisms inthe two categories, K (with a representative object A), and L (with a representativeobject B). It can be depicted as a (non–commutative) diagram✬F(A) ✛f K❄B✫✩FG✪✬A✩L ϕ(f)❄✲ G(B)✫ ✪In this case F is called left adjoint, while G is called right adjoint.In other words, an adjunction F ⊣ G between two functors (F, G) ofopposite variance, from a source category K to a target category L, isdenoted by (F, G, η, ε) : K ⇆ L. Here, F : L → K is the left (upper) adjointfunctor, G : L ← K is the right (lower) adjoint functor, η : 1 L → G ◦ F isthe unit natural transformation (or, front adjunction), and ε : F ◦ G → 1 Kis the counit natural transformation (or, back adjunction).For example, K = S is the category of sets and L = G is the categoryof groups. Then F turns any set into the free group on that set, whilethe ‘forgetful’ functor F ∗ turns any group into the underlying set of thatgroup. Similarly, all sorts of other ‘free’ and ‘underlying’ constructions arealso left and right adjoints, respectively.Right adjoints preserve limits, and left adjoints preserve colimits.The category C is called a cocomplete category if every functor F : J →C has a colimit. The following categories are cocomplete: S, G, A, T , andPT .The importance of adjoint functors lies in the fact that every functorwhich has a left adjoint (and therefore is a right adjoint) is continuous.In the category A of Abelian groups, this e.g., shows that the kernel of aproduct of homomorphisms is naturally identified with the product of thekernels. Also, limit functors themselves are continuous. A covariant functor


Technical Preliminaries: Tensors, Actions and Functors 113F : J → C is cocontinuous if it transforms colimits into colimits. Everyfunctor which has a right adjoint (and is a left adjoint) is cocontinuous.The analogy between adjoint functors and adjoint linear operators reliesupon a deeper analogy: just as in quantum theory the inner product 〈φ, ψ〉represents the amplitude to pass from φ to ψ, in category theory Mor(A, B)represents the set of ways to go from A to B. These are to Hilbert spacesas categories are to sets. The analogues of adjoint linear operators betweenHilbert spaces are certain adjoint functors between 2−Hilbert spaces [Baez(1997); Baez and Dolan (1998)]. Similarly, the adjoint representation ofa Lie group G is the linearized version of the action of G on itself byconjugation, i.e., for each g ∈ G, the inner automorphism x ↦→ gxg −1 givesa linear transformation Ad(g) : g → g, from the Lie algebra g of G to itself.2.3.7 Abelian Categorical AlgebraAn Abelian category is a certain kind of category in which morphisms andobjects can be added and in which kernels and cokernels exist and have theusual properties. The motivating prototype example of an Abelian categoryis the category of Abelian groups A. Abelian categories are the frameworkfor homological algebra (see [Dieudonne (1988)]).Given a homomorphism f : A → B between two objects A ≡ Dom fand B ≡ Cod f in an Abelian category A, then its kernel, image, cokerneland coimage in A are defined respectively as:Ker f = f −1 (e B ), Coker f = Cod f/ Im f,Im f = f(A), Coim f = Dom f/ Ker f,where e B is a unit of B [Dodson and Parker (1997)].In an Abelian category A a composable pair of arrows,f• ✲ Bg ✲ •is exact at B iff Im f ≡ Ker g (equivalence as subobjects of B) – or, equivalently,if Coker f ≡ Coim g [MacLane (1971)].For each arrow f in an Abelian category A the triangular identities readKer(Coker(Ker f)) = Ker f, Coker(Ker(Coker f)) = Coker f.The diagram (with 0 the null object)0 ✲ Af ✲ Bg ✲ C ✲ 0 (2.63)


114 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis a short exact sequence when it is exact at A, at B, and at C.Since 0 → a is the zero arrow, exactness at A means just that f ismonic (i.e., 1–1, or injective map); dually, exactness at C means that g isepic (i.e., onto, or surjective map). Therefore, (2.63) is equivalent tof = Ker g, g = Coker f.Similarly, the statement that h = Coker f becomes the statement that thesequenceAf ✲ Bg ✲ C ✲ 0is exact at B and at C. Classically, such a sequence was called a shortright exact sequence. Similarly, k = Ker f is expressed by a short left exactsequence0 ✲ Af ✲ Bg ✲ C.If A and A ′ are Abelian categories, an additive functor F : A → A ′ isa functor from A to A ′ withF(f + f ′ ) = Ff + Ff ′ ,for any parallel pair of arrows f, f ′ : b → c in A. It follows that F0 = 0.A functor F : A → A ′ between Abelian categories A and A ′ is, bydefinition, exact when it preserves all finite limits and all finite colimits. Inparticular, an exact functor preserves kernels and cokernels, which meansthatKer(Ff) = F(Ker f) and Coker(Ff) = F(Coker f);then F also preserves images, coimages, and carries exact sequences toexact sequences. By construction of limits from products and equalizersand dual constructions, F : A → A ′ is exact iff it is additive and preserveskernels and cokernels.A functor F is left exact when it preserves all finite limits. In otherwords, F is left exact iff it is additive and Ker(Ff) = F(Ker f) for all f:the last condition is equivalent to the requirement that F preserves shortleft exact sequences.Similarly, a functor F is right exact when it preserves all finite colimits.In other words, F is right exact iff it is additive and Coker(Ff) =


Technical Preliminaries: Tensors, Actions and Functors 115F(Coker f) for all f: the last condition is equivalent to the requirementthat F preserves short right exact sequences.In an Abelian category A, a chain complex is a sequence... ✲ cn+1∂ n+1✲ cn∂ n ✲ cn−1✲ ...of composable arrows, with ∂ n ∂ n+1 = 0 for all n. The sequence need not beexact at c n ; the deviation from exactness is measured by the nth homologyobjectH n c = Ker(∂ n : c n✲ cn−1 )/ Im(∂ n+1 : c n+1✲ cn ).Similarly, a cochain complex in an Abelian category A is a sequence... ✲ wn+1d n+1✲ wnd n ✲ wn−1✲ ...of composable arrows, with d n d n+1 = 0 for all n. The sequence neednot be exact at w n ; the deviation from exactness is measured by the nthcohomology objectH n w = Ker(d n+1 : w n✲ wn+1 )/ Im(d n : w n−1✲ wn ).A cycle is a chain C such that ∂C = 0. A boundary is a chain C suchthat C = ∂B, for any other chain B.A cocycle (a closed form) is a cochain ω such that dω = 0. A coboundary(an exact form) is a cochain ω such that ω = dθ, for any other cochain θ.


116 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.3.8 n−CategoriesIn this subsection we introduce the concept of modern n−categories.Intuitively, in describing dynamical systems (processes) by means ofn−categories, instead of classical starting with a set of things:we can now start with a category of things and processes between things:or, a 2−category of things, processes, and processes between processes:... and so on. For example, this n−categorical framework can be used forhigher gauge theory [Baez (2002)], which .


Technical Preliminaries: Tensors, Actions and Functors 1172.3.8.1 Generalization of ‘Small’ CategoriesIf we think of a point in geometrical space (either natural, or abstract)as an object (or, a 0−cell), and a path between two points as an arrow(or, a 1−morphism, or a 1−cell), we could think of a ‘path of paths’ as a2−arrow (or, a 2−morphism, or a 2−cell), and a ‘path of paths of paths’(or, a 3−morphism, or a 3−cell), etc. Here a ‘path of paths’ is just acontinuous 1–parameter family of paths from between source and targetpoints, which we can think of as tracing out a 2D surface, etc. In this waywe get a ‘skeleton’ of an n−category, where a 1−category operates with0−cells (objects) and 1−cells (arrows, causally connecting source objectswith target ones), a 2−category operates with all the cells up to 2−cells[Bénabou (1967)], a 3−category operates with all the cells up to 3−cells,etc. This skeleton clearly demonstrates the hierarchical self–similarity ofn–categories:0 − cell :x •1 − cell :x •2 − cell :x •f ✲ • yf∨ h❘ • y✒gf3 − cell :x •hj> gi❘• y✒where triple arrow goes in the third direction, perpendicular to both singleand double arrows. Categorical composition is defined by pasting arrows.Thus, a 1−category can be depicted as a commutative triangle:AF ✲ F (A)❅ G ◦ F ❅❅❅❘ G✠G(F (A))


118 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona 2−category is a commutative triangle:Af∨ α❘ FB✒g❅❅❅❅❅❅❅❘G ◦ FG(F (A))G(F (f))∨G(F (g))F (f)✲❘F (A) F (α) F (B)∨ ✒ F (g) G✠ ❘G(F (α))G(F (B))✒a 3−category is a commutative triangle:fF (f)Aαψ ❘> βFB ❅✒g ❅❅❅F ◦ G❅ ❅❅❅❅ ❅❘G(F (f))G(F (ψ))G(F (α))G(F (A))G(F (g))F (ψ)✲ F (A) F (α) > F (g)G✠ ❘> G(F (β))G(F (B)) ✒❘F (β) F (B)✒etc., up to n−categories.Many deep–sounding results in mathematical sciences are get by theprocess of categorification 14 of the high school mathematics [Crane andFrenkel (1994); Baez and Dolan (1998)].An n−category is a generic mathematical structure consisting of a collectionof objects, a collection of arrows between objects, a collection of2−arrows between arrows [Bénabou (1967)], a collection of 3−arrows be-14 Categorification means replacing sets with categories, functions with functors, andequations between functions by natural equivalences between functors. Iterating thisprocess requires a theory of n−categories.


Technical Preliminaries: Tensors, Actions and Functors 119tween 2−arrows, and so on up to n [Baez (1997); Baez and Dolan (1998);Leinster (2002); Leinster (2003); Leinster (2004)].More precisely, an n−category (for n ≥ 0) consists of:• 0−cells, or objects, A, B, . . .• 1−cells, or arrows, Af ✲ B, with a compositionAf ✲ Bg ✲ C = Ag◦f ✲ C• 2−cells, ‘arrows between arrows’, A∨ α❘ B, with vertical compositions(denoted by ◦) and horizontal compositions (denoted by ∗),✒grespectively given byfAAfαg ∨ ✲◆B = Aβ ✍∨hf∨ α❘ ✒gA ′ f ′f∨ β◦α❘ B✒hα ′❘ A ′′ = A∨ ✒g ′andf ′ ◦fα ′ ∗ α ❘ ∨ ✒g ′ ◦gA ′′• 3−cells, ‘arrows between arrowsfΓ ❘between arrows’, A α > β B (where the Γ−arrow goes in✒ ga direction perpendicular to f and α), with various kinds of vertical,horizontal and mixed compositions,• etc., up to n−cells.Calculus of n−categories has been developed as follows. First, thereis K 2 , the 2–category of all ordinary (or small) categories. K 2 has cate-


120 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongories K, L, ... as objects, functors F, G : K ⇒ L as arrows, and naturaltransformations, like τ : F → ·G as 2–arrows.In a similar way, the arrows in a 3–category K 3 are 2–functors F 2 , G 2 , ...sending objects in K 2 to objects in L 2 , arrows to arrows, and 2–arrows to2–arrows, strictly preserving all the structure of K 2Af∨ α❘ B✒gF 2 ✲ F2 (A)F 2 (f)❘F 2 (α) F 2 (B).∨ ✒F 2 (g)The 2–arrows in K 3 are 2–natural transformations, like τ 2 : F 22·⇒ G 2 between2–functors F 2 , G 2 : K 2 −→ L 2 that sends each object in K 2 to anarrow in L 2 and each arrow in K 2 to a 2–arrow in L 2 , and satisfies naturaltransformation–like conditions. We can visualize τ 2 as a prism goingfrom one functorial picture of K 2 in L 2 to another, built using commutativesquares:AfF 2 ✒F 2 (A)F 2 (f)❘F 2 (α) F 2 (B)∨ ✒F 2 (g)∨ α❘ B ⇓τ 2 (A)τ 2 (B)✒g❅G G 2 (f)2 ❅❅❘ ❄G 2 (A) G 2 (α) ❘ ❄K 2G 2 (B)∨ ✒G 2 (g) L 2Similarly, the arrows in a 4–category K 4 are 3–functors F 3 , G 3 , ... sendingobjects in K 3 to objects in L 3 , arrows to arrows, and 2–arrows to 2–arrows,strictly preserving all the structure of K 3fF 3 (f)Aαψ> gβ❘B✒F 3 ✲F 3 (ψ) ❘F 3 (A) F 3 (α) > F 3 (β) F 3 (B)✒ F 3 (g)


Technical Preliminaries: Tensors, Actions and Functors 121The 2–arrows in K 4 are 3–natural transformations, like τ 3 : F 3·⇒ G between3–functors F 3 , G 3 : K 3 → L 3 that sends each object in K 3 to aarrow in L 3 and each arrow in K 3 to a 2–arrow in L 3 , and satisfies naturaltransformation–like conditions. We can visualize τ 3 as a prism going fromone picture of K 3 in L 3 to another, built using commutative squares:F 3 (f)F 3 (ψ) ❘F 3 (A) F 3 (α) > F 3 (β) F 3 (B)fF ✒ ✒ 3 ψF❘ 3 (g)A α > β B ⇓ τ 3 (A)τ 3 (B)✒ ❅G 3 (f) gG 3 ❅❅ ❄ ❅❘G 3 (ψ) ❄K 3 ❘G 3 (A) G 3 (α) > G 3 (β) G 3 (B)✒ LG 3 (g)32.3.8.2 Topological Structure of n−CategoriesWe already emphasized the topological nature of ordinary category theory.This fact is even more obvious in the general case of n−categories (see[Leinster (2002); Leinster (2003); Leinster (2004)]).2.3.8.3 Homotopy Theory and Related n−CategoriesAny topological manifold M induces an n−category Π n (M) (its fundamentaln−groupoid), in which 0–cells are points in M; 1–cells are pathsin M (i.e., parameterized continuous maps f : [0, 1] → M); 2–cells arehomotopies (denoted by ≃) of paths relative to endpoints (i.e., parameterizedcontinuous maps h : [0, 1] × [0, 1] → M); 3–cells are homotopiesof homotopies of paths in M (i.e., parameterized continuous mapsj : [0, 1] × [0, 1] × [0, 1] → M); categorical composition is defined by pastingpaths and homotopies. In this way the following ‘homotopy skeleton’


122 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionemerges:0 − cell : x • x ∈ M;1 − cell : x •f ✲ • y f : x ≃ y ∈ M,f : [0, 1] → M, f : x ↦→ y, y = f(x), f(0) = x, f(1) = y;e.g., linear path: f(t) = (1 − t)x + ty;2 − cell : x •f∨ h❘ • y h : f ≃ g ∈ M,✒gh : [0, 1] × [0, 1] → M, h : f ↦→ g, g = h(f(x)),h(x, 0) = f(x), h(x, 1) = g(x), h(0, t) = x, h(1, t) = ye.g., linear homotopy: h(x, t) = (1 − t)f(x) + tg(x);3 − cell : x •hfj> gi❘• y j : h ≃ i ∈ M,✒j : [0, 1] × [0, 1] × [0, 1] → M, j : h ↦→ i, i = j(h(f(x)))j(x, t, 0) = h(f(x)), j(x, t, 1) = i(f(x)),j(x, 0, s) = f(x), j(x, 1, s) = g(x),j(0, t, s) = x, j(1, t, s) = ye.g., linear composite homotopy: j(x, t, s) = (1 − t) h(f(x)) + t i(f(x)).If M is a smooth manifold, then all included paths and homotopies needto be smooth. Recall that a groupoid is a category in which every morphismis invertible; its special case with only one object is a group.Category T TTopological n−category T T has:• 0–cells: topological spaces X• 1–cells: continuous maps Xf ✲ Y


Technical Preliminaries: Tensors, Actions and Functors 123• 2–cells: homotopies h between f and g : X∨ h❘ Y✒gi.e., continuous maps h : X × [0, 1] → Y , such that ∀x ∈ X, h(x, 0) =f(x) and h(x, 1) = g(x)ff• 3–cells: homotopies between homotopies : Xi.e., continuous maps j : X × [0, 1] × [0, 1] → Y .hj> gi❘Y✒Category CKConsider an n−category CK, which has:• 0–cells: chain complexes A (of Abelian groups, say)• 1–cells: chain maps Af ✲ Bf• 2–cells: chain homotopies A∨ α❘ B,✒gi.e., maps α : A → B of degree 1fΓ ❘• 3–cells A α > β B: homotopies between homotopies,✒ gi.e., maps Γ : A → B of degree 2 such that dΓ − Γd = β − α.There ought to be some kind of map CC : T T ⇒ CK (see [Leinster (2002);Leinster (2003); Leinster (2004)]).2.3.8.4 CategorificationCategorification is the process of finding category–theoretic analogs of set–theoretic concepts by replacing sets with categories, functions with functors,


124 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand equations between functions by natural isomorphisms between functors,which in turn should satisfy certain equations of their own, called ‘coherencelaws’. Iterating this process requires a theory of n−categories.Categorification uses the following analogy between set theory and categorytheory [Crane and Frenkel (1994); Baez and Dolan (1998)]:Set Theoryelementsequationsbetween elementssetsfunctionsequationsbetween functionsCategory Theoryobjectsisomorphismsbetween objectscategoriesfunctorsnatural isomorphismsbetween functorsJust as sets have elements, categories have objects. Just as there arefunctions between sets, there are functors between categories. Now, theproper analog of an equation between elements is not an equation betweenobjects, but an isomorphism. Similarly, the analog of an equation betweenfunctions is a natural isomorphism between functors.2.3.9 Application: n−Categorical Framework for HigherGauge FieldsRecall that in the 19th Century, J.C. Maxwell unified Faraday’s electric andmagnetic fields. Maxwell’s theory led to Einstein’s special relativity wherethis unification becomes a spin–off of the unification of space end time inthe form of the Faraday tensor [Misner et al. (1973)]F = E ∧ dt + B,where F is electromagnetic 2−form on space–time, E is electric 1−form onspace, and B is magnetic 2−form on space. Gauge theory considers F assecondary object to a connection–potential 1−form A. This makes half ofthe Maxwell equations into tautologies [Baez (2002)], i.e.,F = dA =⇒ dF = 0 the Bianchi relation,but does not imply the dual Bianchi relation, which is a second half ofMaxwell’s equations,∗d ∗ F = J,


Technical Preliminaries: Tensors, Actions and Functors 125where ∗ is the dual Hodge star operator and J is current 1−form.To understand the deeper meaning of the connection–potential 1−formγA, we can integrate it along a path γ in space–time, x ✲ y. Classically,the integral ∫ A represents an action for a charged point particleγto move along the path γ. Quantum–mechanically, exp(i ∫ )γ A representsa phase (within the unitary group U(1)) by which the particle’s wave–function changes as it moves along the path γ, so A is a U(1)−connection.The only thing that matters here is the difference α between two pathsγ 1 and γ 2 in the action ∫ A [Baez (2002)], which is a two–morphismγγ 1xα ❘ y∨ ✒γ 2To generalize this construction, consider any compact Lie group G. Aconnection A on a trivial G−bundle is a γ−valued 1−form. A assignsa holonomy P exp(i ∫ )γ A γ∈ G along any path x ✲ y and has acurvature F given byF = dA + A ∧ A.The curvature F implies the extended Bianchi relationdF + A ∧ F = 0,


126 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbut does not imply the dual Bianchi relation, i.e., Yang–Mills relation 15∗(d ∗ F + A ∧ ∗F ) = J.15 Recall that the Yang–Mills (YM) Lagrangian density L YM is a functional of thevector potential fields A i µ, where the internal index i ranges over {1, · · · , n}, where n isthe dimension of the gauge group, and µ is a space–time index (µ = 0, · · · , 3). The fieldtensor derived from these potential fields is (see, e.g., [Pons et. al. (2000)])F i αβ = Ai β,α − Ai α,β − Ci jk Aj αA k β ,where Cjk i are the structure constants of the gauge group Lie.density is consequently given byThe YM LagrangianL YM = − 1 4p|g|Fiµν F j αβ gµα g νβ C ij ,where C ij is a nonsingular, symmetric group metric and g is the determinant of thespace–time metric tensor (in a semi–simple Lie group, C ij is usually taken to be Cit s Ct js ;in an Abelian Lie group, one usually takes C ij = δ ij ).The derivatives of L YM with respect to the velocities of the configuration–space variables,Ȧ i α give the tangent–space functions ˆP iα corresponding to the phase–space conjugatemomenta:ˆP iα = ∂L YM= p |g|Fµνg j αµ g 0ν C ij .∂Ȧi αThe Legendre map FL is defined by mapping ˆP α i to P α i in the phase–space. Because ofthe antisymmetry of the field tensor, the primary constraints are0 = ˆP i 0 = ∂L YM∂Ȧi 0= p |g|F j µνg 0µ g 0ν C ij .A generator of a projectable gauge transformation thus must be independent of Ȧ i 0 .An infinitesimal YM gauge transformation is defined by an array of gauge fields Λ iand transforms the potential byWe denote this transformation byδ R [Λ]A i µ = −Λ i ,µ − C i jk Λj A k µ.δ R A i µ = −(D µΛ) j ,where D µ is the Yang–Mills covariant derivative (in its action on space–time scalars andYM vectors). Under this transformation, the field transforms asδ R F i µν = −Ci jk Λj F k µν ,where we work to first order in Λ i and use the Jacobi identityC i jl Cl mn + Ci ml Cl nj + Ci nl Cl jm = 0.The YM Lagrangian L YM is invariant under this transformation provided that the groupmetric is symmetric,C k li C kj = −C k lj C ki (which is if C ij = C s it Ct js ).


Technical Preliminaries: Tensors, Actions and Functors 127Further generalization is performed with string theory. Just as pointparticles naturally interact with a 1−form A, strings naturally interactwith a 2−form B, such that [Baez (2002)]∫( ∫ )action = B, and phase = exp i B .ΣΣThis 2−form connection B has a 3−form curvature G = dB, which satisfiesMaxwell–like equations, i.e., implies Bianchi–like relation dG = 0, but doesnot imply the dual, current relation ∗d ∗ G = J, with the current 2−formJ.In this way, the higher Yang–Mills theory assigns holonomies to pathsand also to paths of paths, so that we have a 3−morphismγ 1xα 1γ 2 ✲◆y∨α 2 ✍∨γ 3allowing us to ask not merely whether holonomies along paths are equal,but whether and how they are isomorphic.This generalization actually proposes categorification of the basic geometricalconcepts of manifold, group, Lie group and Lie algebra [Baez(2002)]. Replacing the words set and function by smooth manifold andsmooth map we get the concept of smooth category. Replacing themby group and homomorphism we get the concept of 2−group. Replacingthem by Lie group and smooth homomorphism we get the concept of Lie2−group. Replacing them by Lie algebra and Lie algebra homomorphismwe get the concept of Lie 2−algebra. Examples of the smooth categoriesare the following:(1) A smooth category with only identity morphisms is a smooth manifold.(2) A smooth category with one object and all morphisms invertible is aLie group.(3) Any Lie groupoid gives a smooth category with all morphisms invertible.(4) A generalization of a vector bundle (E, M, π), where E and M aresmooth manifolds and projection π : E → M is a smooth map, gives avector 2−bundle (E, M, π) where E and M are smooth categories andprojection π : E → M is a smooth functor.


128 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2.3.10 Application: Natural <strong>Geom</strong>etrical StructuresClosely related to the higher–dimensional automata are various natural geometricalstructures, most of which are commonly called tangles.For example, a 2D flow–chart–like complex 1D−structure could be adiagram of the form [Leinster (2002); Leinster (2003)]:Its 3D–generalization is a surface diagram with the same information–flow:Moreover, if we allow crossings, as in a braid:then we start getting pictures that look like knots which are again related


Technical Preliminaries: Tensors, Actions and Functors 129to higher categorical structures [Leinster (2002); Baez (1997)].A category C with only one object is a monoid (= semigroup with unit)M. A 2−category C with only one 0−cell is a monoidal category M. Abraided monoidal category is a monoidal category equipped with a mapcalled braidingA ⊗ B β A,B ✲ B ⊗ A ,for each pair A, B of objects.The canonical example of a braided monoidal category is BR [Leinster(2003)]. This has:(1) Objects: natural numbers 0, 1, . . .;(2) Morphisms: braids, e.g.,(taken up to deformation);there are no morphisms m ✲ n when m ≠ n;(3) Tensor product: placing side–by–side (which on objects means addition);and(4) Braiding: right over left, e.g.,Knots, links and braids are all special cases of tangles (see [Reshetikhinand Turaev (1990)]). The mysterious relationships between topology, algebraand physics amount in large part to the existence of interesting functorsfrom various topologically defined categories to Hilbert, the categoryof Hilbert spaces. These topologically defined categories are always∗−categories, and the really interesting functors from them to Hilbert arealways ∗−functors, which preserve the ∗−structure. Physically, the ∗ operationcorresponds to reversing the direction of time. For example, there isa ∗−category whose objects are collections of points and whose morphisms


130 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionare tangles (see [Baez (1997); Baez and Dolan (1998)]):We can think of this morphism f : x → y as representing the trajectories ofa collection of particles and antiparticles, where particles and antiparticlescan be created or annihilated in pairs. Reversing the direction of time, weget the ‘dual’ morphism f ∗ : y → x:..This morphism is not the inverse of f, since the composite f ◦ f ∗nontrivial tangle:is aIndeed, any groupoid becomes a ∗−category if we set f ∗ = f −1 for every.


Technical Preliminaries: Tensors, Actions and Functors 131morphism f.The above example involves 1D curves in 3D space. More generally,topological quantum field theory studies nD manifolds embedded in (n+k)Dspace–time, which in the k → ∞ limit appear as ‘abstract’ nD manifolds.It appears that these are best described using certain ‘n−categories withduals’, meaning n−categories in which every j−morphism f has a dual f ∗ .Therefore, a tangle is a box in 3D space with knotted and linked stringembedded within it and a certain number of strands of that string emanatingfrom the surface of the box. There are no open ends of string insidethe box. We usually think of some subset of the strands as inputs to thetangle and the remaining strands as the outputs from the tangle. Usuallythe inputs are arranged to be drawn vertically and so that they enter tanglefrom below, while the outputs leave the tangle from above. The tangle itself(within the box) is arranged as nicely as possible with respect to a verticaldirection. This means that a definite vertical direction is chosen, and thatthe tangle intersects planes perpendicular to this direction transversely exceptfor a finite collection of critical points. These basic critical points arelocal maxima and local minima for the space curves inside the tangle. Twotangles configured with respect to the same box are ambient isotopic if thereis an isotopy in three space carrying one to the other that fixes the inputand output strands of each tangle. We can compose two tangles A and Bwhere the number of output strand of A is equal to the number of inputstrands of B. Composition is accomplished by joining each output strandof A to a corresponding input strand of B [Kauffman and Radford (1995);Kauffman and Radford (1999); Kauffman (1994)].A tangle diagram is a box in the plane, arranged parallel to a chosen verticaldirection with a left–right ordered sequence of input strands enteringthe bottom of the box, and a left–right ordered sequence of output strandsemanating from the top of the box. Inside the box is a diagram of thetangle represented with crossings (broken arc indicating the undercrossingline) in the usual way for knot and links. We assume, as above, that thetangle is represented so that it is transverse to lines perpendicular to thevertical except for a finite number of points in the vertical direction alongthe tangle. It is said that the tangle is well arranged, or Morse with respectto the vertical direction when these transversality conditions are met.At the critical points we will see a local maximum, a local minimum or acrossing in the diagram. Tangle composition is well–defined (for matchinginput/output counts) since the input and output strands have an ordering(from left to right for the reader facing the plane on which the tangle di-


132 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionagram is drawn). Note that the cardinality of the set of input strands oroutput strands can be equal to zero. If they are both zero, then the tangleis simply a knot or link diagram arranged well with respect to the verticaldirection [Kauffman and Radford (1995); Kauffman and Radford (1999);Kauffman (1994)].The Reidemeister moves are a set of moves on diagrams that combinatoriallygenerate isotopy for knots, links and and tangles [Reidemeister(1948)]. If two tangles are equivalent in 3D space, then corresponding diagramsof these tangles can be obtained one from another, by a sequenceof Reidemeister moves. Each move is confined to the tangle box and keepsthe input and output strands of the tangle diagram fixed.Two (tangle) diagrams are said to be regularly isotopic if one can beobtained from the other by a sequence of Reidemeister moves of type 0,2,3(move number 1 is not used in regular isotopy).If A and B are given tangles, we denote the composition of A and B byAB where the diagram of A is placed below the diagram of B and the outputstrands of A are connected to the input strands of B. If the cardinalitiesof the sets of input and output strands are zero, then we simple place onetangle below the other to form the product [Kauffman and Radford (1995);Kauffman and Radford (1999); Kauffman (1994)].Along with tangle composition, as defined in the previous paragraph, wealso have an operation of product or juxtaposition of tangles. To juxtaposetwo tangles A and B simply place their diagrams side by side with A tothe left of B and regard this new diagram as a new tangle whose inputsare the inputs of A followed by the inputs of B, and whose outputs are theoutputs of A followed by the outputs of B. We denote the tangle productof A and B by A ⊗ B.It remains to describe the equivalence relation on tangles that makesthem represent regular isotopy classes of embedded string. Every tangle isa composition of elementary tangles where an elementary tangle is one ofthe following list: a cup (a single minimum – zero inputs, two outputs),a cap (a single maximum – two inputs, zero outputs), a crossing (a singlelocal crossing diagram – two inputs and two outputs).2.3.11 Ultimate Conceptual Machines:Weak n−CategoriesAs traditionally conceived, an n−category is an algebraic structurehaving objects or 0−morphisms, 1−morphisms between 0−morphisms,


Technical Preliminaries: Tensors, Actions and Functors 1332−morphisms between 1−morphisms, and so on up to n−morphisms.There should be various ways of composing j−morphisms, and these compositionoperations should satisfy various laws, such as associativity laws.In the so–called strict n−categories, these laws are equations. While well–understood and tractable, strict n−categories are insufficiently generalfor many applications: what one usually encounters in nature are weakn−categories, in which composition operations satisfy the appropriate lawsonly up to equivalence. Here the idea is that n−morphisms are equivalentprecisely when they are equal, while for j < n an equivalence betweenj−morphisms is recursively defined as a (j + 1)−morphism from one to theother that is invertible up to equivalence [Baez and Dolan (1998)].Now, what makes it difficult to define weak n−categories is that lawsformulated as equivalences should satisfy laws of their own – the so–calledcoherence laws – so that one can manipulate them with some of the samefacility as equations. Moreover, these coherence laws should also be equivalencessatisfying their own coherence laws, again up to equivalence, andso on [Baez and Dolan (1998)].For example, a weak 1−category is just an ordinary category. In acategory, composition of 1−morphisms is associative:(fg)h = f(gh).Weak 2−categories first appeared in the work of Bénabou [Bénabou(1967)], under the name of bicategories. In a bicategory, composition of1−morphisms is associative only up to an invertible 2−morphism, the ‘associator’:A f,g,h : (fg)h → f(gh).The associator allows one to re–bracket parenthesized composites of arbitrarilymany 1−morphisms, but there may be many ways to use it to gofrom one parenthesization to another. For all these to be equal, the associatormust satisfy a coherence law, the pentagon identity, which says thatthe following diagram commutes:((fg)h)i❄(f(gh))i(fg)(hi)✲ f(g(hi))✻✲ f((gh)i)


134 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere all the arrows are 2−morphisms built using the associator. Weak3−categories or tricategories were defined by [Gordon et. al. (1995)].In a tricategory, the pentagon identity holds only up to an invertible3−morphism, which satisfies a further coherence law of its own.When one explicitly lists the coherence laws this way, the definitionof weak n−category tends to grow ever more complicated with increasingn. To get around this, one must carefully study the origin of these coherencelaws. So far, most of our insight into coherence laws has been wonthrough homotopy theory, where it is common to impose equations onlyup to homotopy, with these homotopies satisfying coherence laws, again upto homotopy, and so on. For example, the pentagon identity and highercoherence laws for associativity first appeared in Stasheff’s work on thestructure inherited by a space equipped with a homotopy equivalence to aspace with an associative product [Stasheff (1963)]. Subsequent work ledto a systematic treatment of coherence laws in homotopy theory throughthe formalism of topological operads [Adams (1978)].Underlying the connection between homotopy theory and n−categorytheory is a hypothesis made quite explicit by Grothendieck [Grothendieck(1983)]: to any topological space one should be able to associatean n−category having points as objects, paths between points as1−morphisms, certain paths of paths as 2−morphisms, and so on, withcertain homotopy classes of n−fold paths as n−morphisms. This shouldbe a special sort of weak n−category called a weak n−groupoid, in whichall j−morphisms (0 < j ≤ n) are equivalences. Moreover, the processof assigning to each space its fundamental n−groupoid, as Grothendieckcalled it, should set up a complete correspondence between the theory ofhomotopy n−types (spaces whose homotopy groups vanish above the nth)and the theory of weak n−groupoids. This hypothesis explains why all thecoherence laws for weak n−groupoids should be deducible from homotopytheory. It also suggests that weak n−categories will have features not foundin homotopy theory, owing to the presence of j−morphisms that are notequivalences [Baez and Dolan (1998)].Homotopy theory also makes it clear that when setting up a theoryof n−categories, there is some choice involved in the shapes of onesj−morphisms – or in the language of topology, j−cells. The traditionalapproach to n−cate-gories is globular. This means that for j > 0, eachj−cell f : x → y has two (j − 1)−cells called its source, sf = x, and target,


Technical Preliminaries: Tensors, Actions and Functors 135tf = y, which for j > 1 satisfys(sf) = s(tf),t(sf) = t(tf)).Thus a j−cell can be visualized as a globe, a jD ball whose boundaryis divided into two (j − 1)D hemispheres corresponding to its source andtarget. However, in homotopy theory, the simplicial approach is much morepopular. In a simplicial set, each j−cell f is shaped like a jD simplex,and has j + 1 faces, certain (j − 1)−cells d 0 f, . . . , d n f. In addition tothese there are (j + 1)−cells i 0 f, . . . , i n+1 f called degeneracies, and theface and degeneracy maps satisfy certain well–known relations [Baez andDolan (1998)].


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Chapter 3<strong>Applied</strong> Manifold <strong>Geom</strong>etry3.1 IntroductionAlbert Einstein once said: “Nature is simple only when analyzed locally.Why? Because, locally any system appears to be linear, and therefore fullypredictable and controllable. <strong>Geom</strong>etrical elaboration of this fundamentalidea has produced the concept of a manifold, a topological space whichlocally looks like Euclidean R n −spaces, but globally can be totally different.In addition, to be able to use calculus on our manifolds, in much thesame way as in ordinary R n −spaces, the manifolds need to be smooth (i.e.,differentiable as many times as required, technically denoted by C k ).Consider a classical example, comparing a surface of an apple with aEuclidean plane. A small neighborhood of every point on the surface ofan apple (excluding its stem) looks like a Euclidean plane (denoted byR 2 ), with its local geodesics appearing like straight lines. In other words,a smooth surface is locally topologically equivalent to the Euclidean plane.This same concept of nonlinear geometry holds in any dimension. If dimensionis high, we are dealing with complex systems. Therefore, whilecontinuous–time linear systems live in Euclidean R n −spaces, continuous–time complex systems live in nD smooth manifolds, usually denoted byM.Finally, note that there are two dynamical paradigms of smooth manifolds:(i) Einstein’s 4D space–time manifold, historically the first one, and(ii) nD configuration manifold, which is the modern geometrical concept.As the Einstein space–time manifold is both simpler to comprehend andconsequently much more elaborated, we start our geometrical machinerywith it, keeping in mind that the same fundamental dynamics holds for all137


138 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsmooth manifolds, regardless of their dimension.Throughout the book we will try to follow the Hilbertian pedagogicalmethod of development: (i) intuitively introduce a new geometrical concept;(ii) rigorously define it; (iii) apply it to solve a real–world problem.3.1.1 Intuition Behind Einstein’s <strong>Geom</strong>etrodynamicsBriefly, Einstein–Wheeler geometrodynamics can be summarized as[Wheeler (1961); Wheeler (1962)]:(1) Gravity is not a Newtonian force, but an aspect of the geometry ofspace–time.(2) Space is not an absolute invariant entity, but is influenced by the distributionof mass and energy in the Universe. The fundamental <strong>Geom</strong>etrodynamicsPrinciple states:Space tells matter how to move, while matter tells space how to curve.(3) Large masses introduce a strong curvature in space–time. Light andmatter are forced to move according to this metric. Since all the matteris in motion, the geometry of space is constantly changing.The celebrated Einstein equation relates the curvature of space–time to themass/energy density. It reads (in the so–called ‘normal’ units: c = 8πG =1):G = T, or, in components, G αβ = T αβ , (3.1)where G = G αβ is the Einstein curvature tensor, representing space–time geometry, while T = T αβ is the stress–energy–momentum tensor,the ‘mystical’ SEM–tensor, representing matter; the 4D indices α, β =(0, 1, 2, 3) label respectively the four space–time directions: (t, x, y, z).To grasp the intuition behind the Einstein equation (3.1), we need toconsider a ball filled with test particles that are all initially at rest relativeto each other. Let V = V (t) be the volume of the ball after a proper timet has elapsed, as measured by the particle at the center of the ball. Thenthe Einstein equation says:⎛⎞flow of t−momentum in t − direction +¨V∣ = − 1 ⎜ flow of x−momentum in x − direction +⎟V t=0 2 ⎝ flow of y−momentum in y − direction + ⎠ ,flow of z−momentum in z − direction


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 139where these flows are measured at the center of the ball at time t = 0,using local inertial coordinates. These flows are the diagonal componentsof the SEM–tensor T. Its components T αβ tell us how much momentumin the α−direction is flowing in the β−direction through a given point ofspace–time. The flow of t−momentum in the t−direction is just the energydensity, T 00 = ρ. The flow of x−momentum in the x−direction is the‘pressure in the x−direction’, T 11 = P 1 ≡ P x , and similarly for y and z.In any event, we may summarize the Einstein equation (3.1) as¨V∣ = − 1 V t=0 2 (ρ + P x + P y + P z ) ≡ − 1 2 (T 00 + T 11 + T 22 + T 33 ). (3.2)This new equation tells us that positive energy density and positive pressurecurve space–time in a way that makes a freely falling ball of point particlestend to shrink. Since E = mc 2 and we are working in normal units, ordinarymass density counts as a form of energy density. Thus a massive object willmake a swarm of freely falling particles at rest around it start to shrink.In short, (3.2) tells us that gravity attracts (see e.g., [Misner et al. (1973);Baez (2001)]).To see why equation (3.2) is equivalent to the Einstein equation (3.1),we need to understand the Riemann curvature tensor and the geodesic deviationequation. Namely, when space–time is curved, the result of paralleltransport depends on the path taken. To quantify this notion, pick twovectors u and v at a point p in space–time. In the limit where ɛ −→ 0, wecan approximately speak of a ‘parallelogram’ with sides ɛu and ɛv. Takeanother vector w at p and parallel transport it first along ɛv and then alongɛu to the opposite corner of this parallelogram. The result is some vectorw 1 . Alternatively, parallel transport w first along ɛu and then along ɛv.The result is a slightly different vector, w 2 . The limitw 2 − w 1limɛ−→0 ɛ 2 = R(u, v)w (3.3)is well–defined, and it measures the curvature of space–time at the point p.In local coordinates, we can write it asR(u, v)w = R α βγδu β v γ w δ .The quantity Rβγδ α is called the Riemann curvature tensor. We can use thistensor to calculate the relative acceleration of nearby particles in free fall ifthey are initially at rest relative to one another. Consider two freely fallingparticles at nearby points p and q. Let v be the velocity of the particle at


140 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionp, and let ɛu be the vector from p to q. Since the two particles start outat rest relative to one other, the velocity of the particle at q is obtained byparallel transporting v along ɛu.Now let us wait a short while. Both particles trace out geodesics as timepasses, and at time ɛ they will be at new points, say p ′ and q ′ . The pointp ′ is displaced from p by an amount ɛv, so we get a little parallelogram,exactly as in the definition of the Riemann curvature:Next let us calculate the new relative velocity of the two particles. Tocompare vectors we must carry one to another using parallel transport. Letv 1 be the vector we get by taking the velocity vector of the particle at p ′and parallel transporting it to q ′ along the top edge of our parallelogram.Let v 2 be the velocity of the particle at q ′ . The difference v 2 − v 1 is thenew relative velocity. It follows that over this passage of time, the averagerelative acceleration of the two particles is a = (v 2 − v 1 )/ɛ. By equation(3.3),v 2 − v 1alimɛ→0 ɛ 2 = R(u, v)v, therefore lim = R(u, v)v.ɛ→0 ɛThis is the simplified form of the geodesic deviation equation. From the definitionof the Riemann curvature it is easy to see that R(u, v)w = −R(v, u)w,so we can also write this equation asa αlimɛ−→0 ɛ = −Rα βγδv β u γ v δ . (3.4)Using geodesic deviation equation (3.4) we can work out the secondtime derivative of the volume V (t) of a small ball of test particles thatstart out at rest relative to each other. For this we must let u range over anorthonormal basis of tangent vectors, and sum the ‘outwards’ componentof acceleration for each one of these. By equation (3.4) this gives¨Vlim ∣ = −R αV −→0 V t=0βαδv β v δ .In terms of the so–called Ricci tensor, which is a contracted Riemann tensor,R βδ = R α βαδ ,we may write the above expression as¨Vlim ∣ = −R βδ v β v δ .V −→0 V t=0


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 141In local inertial coordinates, where the ball starts out at rest, we havev = (1, 0, 0, 0), so¨Vlim ∣ = −R 00 . (3.5)V −→0 V t=0In short, the Ricci tensor says how our ball of freely falling test particlesstarts changing in volume. The Ricci tensor only captures some of theinformation in the Riemann curvature tensor. The rest is captured bythe so–called the Weyl tensor (see e.g., [Penrose (1989); Penrose (1994);Penrose (1997)]), which says how any such ball starts changing in shape.The Weyl tensor describes tidal forces, gravitational waves and the like.Now, the Einstein equation in its usual form saysG αβ = T αβ .Here the right side is the stress-energy tensor, while the left side, the ‘Einsteintensor’, is just an abbreviation for a quantity constructed from theRicci tensor:G αβ = R αβ − 1 2 g αβR γ γ.Thus the Einstein equation really saysThis impliesbut g α α = 4, soR αβ − 1 2 g αβR γ γ = T αβ . (3.6)R α α − 1 2 gα αR γ γ = T α α ,−R α α = T α α .Substituting this into equation (3.6), we getR αβ = T αβ − 1 2 g αβT γ γ . (3.7)This is an equivalent version of the Einstein equation, but with the rolesof R and T switched [Baez (2001)]. This is a formula for the Ricci tensor,which has a simple geometrical meaning.Equation (3.7) will be true if any one component holds in all localinertial coordinate systems. This is a bit like the observation that all ofMaxwell’s equations are contained in Gauss’s law and and ∇ · B = 0.


142 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionClearly, this is only true if we know how the fields transform under changeof coordinates. Here we assume that the transformation laws are known.Given this, the Einstein equation (3.1) is equivalent to the fact thatR 00 = T 00 − 1 2 g 00T γ γ (3.8)in every local inertial coordinate system about every point. In such coordinateswe have⎛ ⎞−1 0 0 0g = ⎜ 0 1 0 0⎟⎝ 0 0 1 0 ⎠ (3.9)0 0 0 1so g 00 = −1, as well asT γ γ = −T 00 + T 11 + T 22 + T 33 .Equation (3.8) thus says thatR 00 = 1 2 (T 00 + T 11 + T 22 + T 33 ).By equation (3.5), this is equivalent to the required¨Vlim ∣ = − 1V →0 V t=0 2 (T 00 + T 11 + T 22 + T 33 ).3.1.2 Einstein’s <strong>Geom</strong>etrodynamics in BriefAs a final introductory motivation, we give an ‘express–flight bird–view’on derivation of the Einstein equation from the Hilbert action principle,starting from the Einstein space–time manifold M. For all technical details,see [Misner et al. (1973)], which is still, after 33 years, the core textbook


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 143on the subject.M ... the space–time manifold Mg ij = g ij (x i ) ∈ T x M ... metric tensor on Mg ij = (g ij ) −1 ... inverse metric tensor on MΓ ijk = 1 2 (∂ x kg ij + ∂ x j g ki − ∂ x ig jk )... 1–order Christoffel symbolsΓ k ij = g kl Γ ijl ... 2–order Christoffel symbols (Levi–Civita connection)Rijk l = ∂ x j Γ l ik − ∂ x kΓ l ij + Γ l rjΓ r ik − Γ l rkΓ r ij ... Riemann curvature tensorR ij = Rijll ... Ricci tensor is the trace of RiemannR = g ij R ij ... scalar curvature is the trace of RicciG ij = R ij − 1 2 Rg ij... Einstein tensor is the trace–reversed RicciT ij = −2 δL Hilbδg ij + g ij L Hilb ... stress–energy–momentum (SEM) tensorL Hilb = 116π gij R ij (−g) 1/2∫... is derived from the Hilbert LagrangianδS = δ L Hilb (−g) 1/2 d 4 x = 0 ... the Hilbert action principle givesG ij = 8πT ij... the Einstein equation.We will continue Einstein’s geometrodynamics in section 6.4 below.3.2 Intuition Behind the Manifold ConceptAs we have already got the initial feeling, in the heart of applied differentialgeometry is the concept of a manifold. As a warm–up, to get some dynamicalintuition behind this concept, let us consider a simple 3DOF mechanicalsystem determined by three generalized coordinates, q i = {q 1 , q 2 , q 3 }.There is a unique way to represent this system as a 3D manifold, such thatto each point of the manifold there corresponds a definite configuration ofthe mechanical system with coordinates q i ; therefore, we have a geometricalrepresentation of the configurations of our mechanical system, calledthe configuration manifold. If the mechanical system moves in any way,its coordinates are given as the functions of the time. Thus, the motionis given by equations of the form: q i = q i (t). As t varies (i.e., t ∈ R), weobserve that the system’s representative point in the configuration manifolddescribes a curve and q i = q i (t) are the equations of this curve.


144 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 3.1An intuitive geometrical picture behind the manifold concept (see text).On the other hand, to get some geometrical intuition behind the conceptof a manifold, consider a set M (see Figure 3.1) which is a candidate fora manifold. Any point x ∈ M 1 has its Euclidean chart, given by a 1–1and onto map ϕ i : M → R n , with its Euclidean image V i = ϕ i (U i ). Moreprecisely, a chart ϕ i is defined byϕ i : M ⊃ U i ∋ x ↦→ ϕ i (x) ∈ V i ⊂ R n ,where U i ⊂ M and V i ⊂ R n are open sets (see [Boothby (1986); Arnold(1978); De Rham (1984)]).Clearly, any point x ∈ M can have several different charts (see Figure3.1). Consider a case of two charts, ϕ i , ϕ j : M → R n , having in theirimages two open sets, V ij = ϕ i (U i ∩ U j ) and V ji = ϕ j (U i ∩ U j ). Then wehave transition functions ϕ ij between them,ϕ ij = ϕ j ◦ ϕ −1i : V ij → V ji , locally given by ϕ ij (x) = ϕ j (ϕ −1i (x)).If transition functions ϕ ij exist, then we say that two charts, ϕ i and ϕ j arecompatible. Transition functions represent a general (nonlinear) transformationsof coordinates, which are the core of classical tensor calculus.A set of compatible charts ϕ i : M → R n , such that each point x ∈ Mhas its Euclidean image in at least one chart, is called an atlas. Two atlasesare equivalent iff all their charts are compatible (i.e., transition functions1 Note that sometimes we will denote the point in a manifold M by m, and sometimesby x (thus implicitly assuming the existence of coordinates x = (x i )).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 145exist between them), so their union is also an atlas. A manifold structureis a class of equivalent atlases.Finally, as charts ϕ i : M → R n were supposed to be 1-1 and onto maps,they can be either homeomorphisms, in which case we have a topological(C 0 ) manifold, or diffeomorphisms, in which case we have a smooth (C k )manifold.Slightly more precisely, a topological (respectively smooth) manifold isa separable space M which is locally homeomorphic (resp. diffeomorphic)to Euclidean space R n , having the following properties (reflected in Figure3.1):(1) M is a Hausdorff space: For every pair of points x 1 , x 2 ∈ M, there aredisjoint open subsets U 1 , U 2 ⊂ M such that x 1 ∈ U 1 and x 2 ∈ U 2 .(2) M is second–countable space: There exists a countable basis for thetopology of M.(3) M is locally Euclidean of dimension n: Every point of M has a neighborhoodthat is homeomorphic (resp. diffeomorphic) to an open subsetof R n .This implies that for any point x ∈ M there is a homeomorphism (resp.diffeomorphism) ϕ : U → ϕ(U) ⊆ R n , where U is an open neighborhoodof x in M and ϕ(U) is an open subset in R n . The pair (U, ϕ) is called acoordinate chart at a point x ∈ M, etc.3.3 Definition of a <strong>Diff</strong>erentiable ManifoldGiven a chart (U, ϕ), we call the set U a coordinate domain, or a coordinateneighborhood of each of its points. If in addition ϕ(U) is an open ballin R n , then U is called a coordinate ball. The map ϕ is called a (local)coordinate map, and the component functions (x 1 , ..., x n ) of ϕ, defined byϕ(m) = (x 1 (m), ..., x n (m)), are called local coordinates on U.Two charts (U 1 , ϕ 1 ) and (U 2 , ϕ 2 ) such that U 1 ∩ U 2 ≠ ∅ are calledcompatible if ϕ 1 (U 1 ∩U 2 ) and ϕ 2 (U 2 ∩U 1 ) are open subsets of R n . A family(U α , ϕ α ) α∈A of compatible charts on M such that the U α form a covering ofM is called an atlas. The maps ϕ αβ = ϕ β ◦ ϕ −1α : ϕ α (U αβ ) → ϕ β (U αβ ) arecalled the transition maps, for the atlas (U α , ϕ α ) α∈A , where U αβ = U α ∩U β ,


146 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionso that we have a commutative triangle:U αβ ⊆ M ❅ϕ α ❅❅❅❘ ϕ β✠ϕ α (U αβ ) ✲ ϕ β (U αβ )ϕ αβAn atlas (U α , ϕ α ) α∈A for a manifold M is said to be a C k −atlas, ifall transition maps ϕ αβ : ϕ α (U αβ ) → ϕ β (U αβ ) are of class C k . Two C katlases are called C k −equivalent, if their union is again a C k −atlas for M.An equivalence class of C k −atlases is called a C k −structure on M. Inother words, a smooth structure on M is a maximal smooth atlas on M,i.e., such an atlas that is not contained in any strictly larger smooth atlas.By a C k −manifold M, we mean a topological manifold together with aC k −structure and a chart on M will be a chart belonging to some atlas ofthe C k −structure. Smooth manifold means C ∞ −manifold, and the word‘smooth’ is used synonymously with C ∞ [De Rham (1984)].Sometimes the terms ‘local coordinate system’ or ‘parametrization’ areused instead of charts. That M is not defined with any particular atlas, butwith an equivalence class of atlases, is a mathematical formulation of thegeneral covariance principle. Every suitable coordinate system is equallygood. A Euclidean chart may well suffice for an open subset of R n , but thiscoordinate system is not to be preferred to the others, which may requiremany charts (as with polar coordinates), but are more convenient in otherrespects.For example, the atlas of an n−sphere S n has two charts. If N =(1, 0, ..., 0) and S = (−1, ..., 0, 0) are the north and south poles of S n respectively,then the two charts are given by the stereographic projectionsfrom N and S:ϕ 1 : S n \{N} → R n , ϕ 1 (x 1 , ..., x n+1 ) = (x 2 /(1 − x 1 ), . . . , x n+1 /(1 − x 1 )), andϕ 2 : S n \{S} → R n , ϕ 2 (x 1 , ..., x n+1 ) = (x 2 /(1 + x 1 ), . . . , x n+1 /(1 + x 1 )),while the overlap map ϕ 2 ◦ ϕ −11 : R n \{0} → R n \{0} is given by the diffeomorphism(ϕ 2 ◦ ϕ −11 )(z) = z/||z||2 , for z in R n \{0}, from R n \{0} toitself.Various additional structures can be imposed on R n , and the correspondingmanifold M will inherit them through its covering by charts. Forexample, if a covering by charts takes their values in a Banach space E,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 147then E is called the model space and M is referred to as a C k −Banachmanifold modelled on E. Similarly, if a covering by charts takes their valuesin a Hilbert space H, then H is called the model space and M is referredto as a C k −Hilbert manifold modelled on H. If not otherwise specified, wewill consider M to be an Euclidean manifold, with its covering by chartstaking their values in R n .For a Hausdorff C k −manifold the following properties are equivalent[Kolar et al. (1993)]: (i) it is paracompact; (ii) it is metrizable; (iii) itadmits a Riemannian metric; 2 (iv) each connected component is separable.3.4 Smooth Maps Between Smooth ManifoldsA map ϕ : M → N between two manifolds M and N, with M ∋ m ↦→ϕ(m) ∈ N, is called a smooth map, or C k −map, if we have the followingcharting:2 Recall the corresponding properties of a Euclidean metric d. For any three pointsx, y, z ∈ R n , the following axioms are valid:M 1 : d(x, y) > 0, for x ≠ y; and d(x, y) = 0, for x = y;M 2 : d(x, y) = d(y, x); M 3 : d(x, y) ≤ d(x, z) + d(z, y).


148 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction✬★Um⊙✧✫✥M✦✩✪ϕ✬✩★ ✥N V✲ ⊙ϕ(m)✧ ✦✫✪φψ✻R m✬ ✩✬ ✩✻φ(U) ❄❄ψ(V )⊙✲ ⊙φ(m)ψ ◦ ϕ ◦ φ −1ψ(ϕ(m))✫✪✲✛✫✪This means that for each m ∈ M and each chart (V, ψ) on N with ϕ (m) ∈ Vthere is a chart (U, φ) on M with m ∈ U, ϕ (U) ⊆ V , and Φ = ψ ◦ ϕ ◦ φ −1is C k , that is, the following diagram commutes:M ⊇ Uφϕ✲ V ⊆ N❄❄φ(U) ✲ ψ(V )ΦLet M and N be smooth manifolds and let ϕ : M → N be a smoothmap. The map ϕ is called a covering, or equivalently, M is said to coverN, if ϕ is surjective and each point n ∈ N admits an open neighborhood Vsuch that ϕ −1 (V ) is a union of disjoint open sets, each diffeomorphic via ϕto V .A C k −map ϕ : M → N is called a C k −diffeomorphism if ϕ is a bijection,ϕ −1 : N → M exists and is also C k . Two manifolds are calleddiffeomorphic if there exists a diffeomorphism between them. All smoothmanifolds and smooth maps between them form the category M.ψR n3.4.1 Intuition Behind Topological Invariants of ManifoldsNow, restricting to the topology of nD compact (i.e., closed and bounded)and connected manifolds, the only cases in which we have a complete understandingof topology are n = 0, 1, 2. The only compact and connected


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 1490D manifold is a point. A 1D compact and connected manifold can eitherbe a line element or a circle, and it is intuitively clear (and can easily beproven) that these two spaces are topologically different. In 2D, there isalready an infinite number of different topologies: a 2D compact and connectedsurface can have an arbitrary number of handles and boundaries,and can either be orientable or non–orientable (see figure 3.2). Again, it isintuitively quite clear that two surfaces are not homeomorphic if they differin one of these respects. On the other hand, it can be proven that any twosurfaces for which these data are the same can be continuously mapped toone another, and hence this gives a complete classification of the possibletopologies of such surfaces.Fig. 3.2 Three examples of 2D manifolds: (a) The sphere S 2 is an orientable manifoldwithout handles or boundaries. (b) An orientable manifold with one boundary and onehandle. (c) The Möbius strip is an unorientable manifold with one boundary and nohandles.A quantity such as the number of boundaries of a surface is called atopological invariant. A topological invariant is a number, or more generallyany type of structure, which one can associate to a topological space, andwhich does not change under continuous mappings. Topological invariantscan be used to distinguish between topological spaces: if two surfaces havea different number of boundaries, they can certainly not be topologicallyequivalent. On the other hand, the knowledge of a topological invariantis in general not enough to decide whether two spaces are homeomorphic:a torus and a sphere have the same number of boundaries (zero), but areclearly not homeomorphic. Only when one has some complete set of topologicalinvariants, such as the number of handles and boundaries in the 2Dcase, is it possible to determine whether or not two topological spaces arehomeomorphic. In more than 2D, many topological invariants are known,but for no dimension larger than two has a complete set of topological invariantsbeen found. In 3D, it is generally believed that a finite number ofcountable invariants would suffice for compact manifolds, but this is not rig-


150 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionorously proven, and in particular there is at present no generally acceptedconstruction of a complete set. A very interesting and intimately relatedproblem is the famous Poincaré conjecture, stating that if a 3D manifoldhas a certain set of topological invariants called its ‘homotopy groups’ equalto those of the 3–sphere S 3 , it is actually homeomorphic to the three-sphere.In four or more dimensions, a complete set of topological invariants wouldconsist of an uncountably infinite number of invariants! A general classificationof topologies is therefore very hard to get, but even without such ageneral classification, each new invariant that can be constructed gives usa lot of interesting new information. For this reason, the construction oftopological invariants of manifolds is one of the most important issues intopology.3.5 (Co)Tangent Bundles of Smooth Manifolds3.5.1 Tangent Bundle and Lagrangian Dynamics3.5.1.1 Intuition Behind a Tangent BundleIn mechanics, to each nD configuration manifold M there is associated its2nD velocity phase–space manifold, denoted by T M and called the tangentbundle of M (see Figure 3.3). The original smooth manifold M is calledthe base of T M. There is an onto map π : T M −→ M, called the projection.Above each point x ∈ M there is a tangent space T x M = π −1 (x) to M at x,which is called a fibre. The fibre T x M ⊂ T M is the subset of T M, such thatthe total tangent bundle, T M =⊔T x M, is a disjoint union of tangentm∈Mspaces T x M to M for all points x ∈ M. From dynamical perspective,the most important quantity in the tangent bundle concept is the smoothmap v : M −→ T M, which is an inverse to the projection π, i.e, π ◦ v =Id M , π(v(x)) = x. It is called the velocity vector–field. Its graph (x, v(x))represents the cross–section of the tangent bundle T M. This explains thedynamical term velocity phase–space, given to the tangent bundle T M ofthe manifold M.3.5.1.2 Definition of a Tangent BundleRecall that if [a, b] is a closed interval, a C 0 −map γ : [a, b] → M is saidto be differentiable at the endpoint a if there is a chart (U, φ) at γ(a) such


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 151Fig. 3.3 A sketch of a tangent bundle T M of a smooth manifold M (see text forexplanation).that the following limit exists and is finite [Abraham et al. (1988)]:ddt (φ ◦ γ)(a) ≡ (φ ◦ (φ ◦ γ)(t) − (φ ◦ γ)(a)γ)′ (a) = lim. (3.10)t→a t − aGeneralizing (3.10), we get the notion of the curve on a manifold. Fora smooth manifold M and a point m ∈ M a curve at m is a C 0 −mapγ : I → M from an interval I ⊂ R into M with 0 ∈ I and γ(0) = m.Two curves γ 1 and γ 2 passing though a point m ∈ U are tangent at mwith respect to the chart (U, φ) if (φ ◦ γ 1 ) ′ (0) = (φ ◦ γ 2 ) ′ (0). Thus, twocurves are tangent if they have identical tangent vectors (same directionand speed) in a local chart on a manifold.For a smooth manifold M and a point m ∈ M, the tangent space T m Mto M at m is the set of equivalence classes of curves at m:T m M = {[γ] m : γ is a curve at a point m ∈ M}.A C k −map ϕ : M ∋ m ↦→ ϕ(m) ∈ N between two manifolds M and Ninduces a linear map T m ϕ : T m M → T ϕ(m) N for each point m ∈ M, calleda tangent map, if we have:


152 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionT m (M)T MT (N)T ϕ(m) (N)⊙T (ϕ)✲⊙π Mπ N★M✧❄⊙m✥✦ϕ★✧✲❄⊙ Nϕ(m)✥✦i.e., the following diagram commutes:T m MT m ϕ✲ T ϕ(m) Nπ M❄M ∋ mϕπ N❄✲ ϕ(m) ∈ Nwith the natural projection π M : T M → M, given by π M (T m M) = m,that takes a tangent vector v to the point m ∈ M at which the vector v isattached i.e., v ∈ T m M.For an nD smooth manifold M, its nD tangent bundle T M is the disjointunion of all its tangent spaces T m M at all points m ∈ M, T M =⊔T m M.m∈MTo define the smooth structure on T M, we need to specify how toconstruct local coordinates on T M. To do this, let (x 1 (m), ..., x n (m)) belocal coordinates of a point m on M and let (v 1 (m), ..., v n (m)) be componentsof a tangent vector in this coordinate system. Then the 2n numbers(x 1 (m), ..., x n (m), v 1 (m), ..., v n (m)) give a local coordinate system on T M.T M =⊔T m M defines a family of vector spaces parameterized by M.m∈MThe inverse image π −1M(m) of a point m ∈ M under the natural projectionπ M is the tangent space T m M. This space is called the fibre of the tangentbundle over the point m ∈ M [Steenrod (1951)].A C k −map ϕ : M → N between two manifolds M and N induces alinear tangent map T ϕ : T M → T N between their tangent bundles, i.e.,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 153the following diagram commutes:T MT ϕ✲ T Nπ M❄Mϕπ N❄✲ NAll tangent bundles and their tangent maps form the category T B. Thecategory T B is the natural framework for Lagrangian dynamics.Now, we can formulate the global version of the chain rule. If ϕ : M →N and ψ : N → P are two smooth maps, then we have T (ψ ◦ ϕ) = T ψ ◦ T ϕ(see [Kolar et al. (1993)]). In other words, we have a functor T : M ⇒ T Bfrom the category M of smooth manifolds to the category T B of theirtangent bundles:Nϕ ✠MψT M❅ ❅❅T(ψ ◦ ϕ) =⇒ T ϕ ❅ T (ψ ◦ ϕ)❅ ❅ ❅❘ ✠❅❘✲ PT N ✲ T PT ψ3.5.2 Cotangent Bundle and Hamiltonian Dynamics3.5.2.1 Definition of a Cotangent BundleA dual notion to the tangent space T m M to a smooth manifold M at apoint m is its cotangent space TmM ∗ at the same point m. Similarly tothe tangent bundle, for a smooth manifold M of dimension n, its cotangentbundle T ∗ M is the disjoint union of all its cotangent spaces TmM ∗ at allpoints m ∈ M, i.e., T ∗ M =⊔TmM. ∗ Therefore, the cotangent bundlem∈Mof an n−manifold M is the vector bundle T ∗ M = (T M) ∗ , the (real) dualof the tangent bundle T M.If M is an n−manifold, then T ∗ M is a 2n−manifold. To definethe smooth structure on T ∗ M, we need to specify how to constructlocal coordinates on T ∗ M. To do this, let (x 1 (m), ..., x n (m)) be localcoordinates of a point m on M and let (p 1 (m), ..., p n (m)) be componentsof a covector in this coordinate system. Then the 2n numbers(x 1 (m), ..., x n (m), p 1 (m), ..., p n (m)) give a local coordinate system onT ∗ M. This is the basic idea one uses to prove that indeed T ∗ M is a


154 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2n−manifold.T ∗ M =⊔m∈MT ∗ mM defines a family of vector spaces parameterized byM, with the conatural projection, π ∗ M : T ∗ M → M, given by π ∗ M (T mM) ∗ =m, that takes a covector p to the point m ∈ M at which the covector pis attached i.e., p ∈ TmM. ∗ The inverse image π −1M(m) of a point m ∈ Munder the conatural projection π ∗ M is the cotangent space T mM. ∗ This spaceis called the fibre of the cotangent bundle over the point m ∈ M.In a similar way, a C k −map ϕ : M → N between two manifolds Mand N induces a linear cotangent map T ∗ ϕ : T ∗ M → T ∗ N between theircotangent bundles, i.e., the following diagram commutes:T ∗ MT ∗ ϕ✲ T ∗ Nπ ∗ M❄Mϕπ ∗ N❄✲ NAll cotangent bundles and their cotangent maps form the category T ∗ B.The category T ∗ B is the natural stage for Hamiltonian dynamics.Now, we can formulate the dual version of the global chain rule. Ifϕ : M → N and ψ : N → P are two smooth maps, then we have T ∗ (ψ◦ϕ) =T ∗ ψ ◦ T ∗ ϕ. In other words, we have a cofunctor T ∗ : M ⇒ T ∗ B from thecategory M of smooth manifolds to the category T ∗ B of their cotangentbundles:Nϕ✠MT ∗ M❅❅T(ψ ◦ ϕ) =⇒∗T ∗ ϕ❅❅❘ ✒ ❅■❅T ∗ (ψ ◦ ϕ)❅❅✲ PTψ∗ N ✛T ∗ PT ∗ ψ3.5.3 Application: Command/Control in Human–RobotInteractionsSuppose that we have a human–robot team, consisting of m robots and n humans.To be able to put the modelling of the fully controlled human–robotteam performance into the rigorous geometrical settings, we suppose thatall possible behaviors of m robots can be described by a set of continuousand smooth, time–dependent robot configuration coordinates x r = x r (t),while all robot–related behaviors of n humans can be described by a set of


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 155continuous and smooth, time–dependent human configuration coordinatesq h = q h (t). In other words, all robot coordinates, x r = x r (t), constitutethe smooth Riemannian manifold M r g (such that r = 1, ..., dim(M r g )), withthe positive–definite metric formg ↦→ ds 2 = g rs (x)dx r dx s (3.11)similarly, all human coordinates q h = q h (t), constitute a smooth Riemannianmanifold N h a (such that h = 1, ..., dim(N h a )) , with the positive–definitemetric forma ↦→ dσ 2 = a hk (q)dq h dq k . (3.12)In this Riemannian geometry settings, the feedforward command/controlaction of humans upon robots is defined by a smooth map,C : N h a → M r g ,which is in local coordinates given by a general (nonlinear) functional transformationx r = x r (q h ), (r = 1, ..., dim(M r g ); h = 1, ..., dim(N h a )), (3.13)while its inverse, the feedback map from robots to humans is defined by asmooth map,F = C −1 : M r g → N h a ,which is in local coordinates given by an inverse functional transformationq h = q h (x r ), (h = 1, ..., dim(N h a ); r = 1, ..., dim(M r g )). (3.14)Now, although the coordinate transformations (3.13) and (3.14) arecompletely general, nonlinear and even unknown at this stage, there issomething known and simple about them: the corresponding transformationsof differentials are linear and homogenous, namelydx r = ∂xr∂q h dqh , and dq h = ∂qh∂x r dxr ,which imply linear and homogenous transformations of robot and humanvelocities,ẋ r = ∂xr∂q h ˙qh , and ˙q h = ∂qh∂x r ẋr . (3.15)


156 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRelation (3.15), representing two autonomous dynamical systems, givenby two sets of ordinary differential equations (ODEs), geometrically definestwo velocity vector–fields: (i) robot velocity vector–field, v r ≡ v r (x r , t) :=ẋ r (x r , t); and human velocity vector–field, u h ≡ u h (q h , t) := ˙q h (q h , t). Recallthat a vector–field defines a single vector at each point x r (in somedomain U) of a manifold in case. Its solution gives the flow, consisting ofintegral curves of the vector–field, such that all the vectors from the vector–field are tangent to integral curves at different points x i ∈ U. <strong>Geom</strong>etrically,a velocity vector–field is defined as a cross–section of the tangent bundleof the manifold. In our case, the robot velocity vector–field v r = ẋ r (x r , t)represents a cross–section of the robot tangent bundle T M r g , while the humanvelocity vector–field u h = ˙q h (q h , t) represents a cross–section of thehuman tangent bundle T N h a . In this way, two local velocity vector–fields,v r and u h , give local representations for the following two global tangentmaps,T C : T N h a → T M r g , and T F : T M r g → T N h a .To be able to proceed along the geometrodynamical line, we neednext to formulate the two corresponding acceleration vector–fields, a r ≡a r (x r , ẋ r , t) and w h ≡ w h (q h , ˙q h , t), as time rates of change of the two velocityvector–fields v r and u h . Now, recall that the acceleration vector–fieldis defined as the absolute time derivative, ˙¯v r = D dt vr , of the velocity vector–field. In our case, we have the robotic acceleration vector–field a r := ˙¯v rdefined on M r g bya r := ˙¯v r = ˙v r + Γ r stv s v t = ẍ r + Γ r stẋ s ẋ t , (3.16)and the human acceleration vector–field w h := ˙ū h defined on N h abyw h := ˙ū h = ˙u h + Γ h jku j u k = ẍ r + Γ h jk ˙q j ˙q k , (3.17)<strong>Geom</strong>etrically, an acceleration vector–field is defined as a cross–section ofthe second tangent bundle of the manifold. In our case, the robot accelerationvector–field a r = ˙¯v r (x r , ẋ r , t), given by the ODEs (3.16), represents across–section of the second robot tangent bundle T T M r g , while the humanacceleration vector–field w h = ˙ū h (q h , ˙q h , t), given by the ODEs (3.17), representsa cross–section of the second human tangent bundle T T N h a . In thisway, two local acceleration vector–fields, a r and w h , give local representationsfor the following two second tangent maps,T T C : T T N h a → T T M r g , and T T F : T T M r g → T T N h a .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 157In other words, we have the feedforward command/control commutativediagram:T T Nah✻T T C✲ T T Mgr✻T Nah✻T C✲ T Mgr✻N h aC✲ Mgras well as the feedback commutative diagram:T T Nah✻✛T T FT T Mgr✻T Nah✻✛T FT Mgr✻N h a✛FM r gThese two commutative diagrams formally define the global feedforwardand feedback human–robot interactions at the positional, velocity, and accelerationlevels of command and control.3.5.4 Application: Generalized Bidirectional AssociativeMemorySystem ArchitectureHere we present a covariant model of generalized bidirectional associativememory (GBAM), a neurodynamical classification machine generaliz-


158 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioning Kosko BAM and RBAM systems (see [Kosko (1992)]). Mathematically,the GBAM is a tensor–field system (q, p, W ) defined on a manifold M calledthe GBAM manifold. The system (q, p, W ) includes two nonlinearly coupled(yet non–chaotic and stable) subsystems (see Figure 3.4): (i) activation(q, p)−-dynamics, where q and p represent neuronal 1D tensor–fields, and(ii) self–organized learning W −-dynamics, where W is a symmetric synaptic2D tensor–field.Fig. 3.4Architecture of the GBAM neurodynamical classifier.GBAM Activation DynamicsThe GBAM–manifold M can be viewed as a Banach space with aC ∞ −smooth structure on it, so that in each local chart U open in M,an nD smooth coordinate system U α exists.GBAM–activation (q, p)−-dynamics, is defined as a system of two coupled,first–order oscillator tensor–fields, dual to each other, in a local Banachchart U α , α = 1, ..., n on M:1. An excitatory neural vector–field q α = q α (t) : M → T M, being across–section of the tangent bundle T M; and2. An inhibitory neural 1–form p α = p α (t) : M → T ∗ M, being across–section of the cotangent bundle T ∗ M.To start with conservative linear (q, p)–system, we postulate the GBAM


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 159scalar activation–potential V to be a negative bilinear form:V = − 1 2 ω αβq α q β − 1 2 ωαβ p α p β + q α p α , (α, β = 1, ..., n), (3.18)where n is the number of neurons in each neural field, while ω αβ and ω αβrepresent respectively inhibitory–covariant and excitatory–contravariantcomponents of the symmetric (with zero–trace) coupling GBAM synaptictensor W .The Lyapunov–stable, conservative, linear (q, p)−-dynamics is given asa bidirectional (excitatory–inhibitory) gradient system:˙q α = − ∂V∂p α= ω αβ p β − q α ,ṗ α = − ∂V∂q α = ω αβq β − p α . (3.19)As W is a symmetric and zero–trace synaptic coupling tensor, the conservativelinear dynamics (3.19) is equivalent to the rule that the state ofeach neuron (in both excitatory and inhibitory neural fields) is changedin time if and only if the scalar action potential V , defined by relation(3.18), is lowered. Therefore, the scalar action potential V is a monotonicallynon–increasing Lyapunov function ˙V ≤ 0 for the conservative linear(q, p)−-dynamics (3.19), which converges to a local minimum or groundstate of V .Applying the inputs I α and J α , we get the non–conservative linear(q, p)−-system equations:˙q α = I α + ω αβ p β − q α , ṗ α = J α + ω αβ q β − p α . (3.20)Further, applying the sigmoid GBAM activation functions S α (·) andS α (·) to the synaptic product–terms, we get the non-conservative nonlinear(q, p)–system equations, which generalize the transient RC–circuit neurodynamicalmodel:˙q α = I α + ω αβ S β (p β ) − q α , ṗ α = J α + ω αβ S β (q β ) − p α . (3.21)The equations in (3.21) represent a 2–input system that can be appliede.g., to classification of two–feature data. The generalization to an N–inputsystem working in a ND feature–space is given by˙q α e = I α e + ω αβe S β (p β ) − q α e , ṗ o α = I o α + ω o αβS β (q β ) − p o α, (3.22)where e(= 2, 4, ..., N) and o(= 1, 3, ..., N − 1) denote respectively even andodd partitions of the total sample of N features.The GBAM model (3.22) gives a generalization of four well–known recurrentNN models:


160 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction1. Continuous Hopfield amplifier–circuit model [Hopfield (1984)]C j ˙v j = I j − v jR j+ T ij u i ,(i, j = 1, ..., N),where v j = v j (t) represent the activation potentials in the jth processingunit, C j and R j denote input capacitances and leakage resistances, u i =f i [v j (t)] are output functions from processing elements, and T ij = w ij isthe inverse of the resistors connection–matrix; and the functions f i aresigmoidal.2. Cohen–Grossberg general ART–system [Cohen and Grossberg(1983)],˙v j = −a j (v j )[b j (v j ) − f k (v k )m jk ],(j = 1, ..., N),with proved asymptotical stability.3. Hecht–Nielsen counter–propagation network [Hecht-Nielsen (1987)],˙v j = −Av j + (B − v j )I j − v j I k ,where A, B are positive constants and I j are input values for each processingunit.4. Kosko’s BAM (ABAM and RABAM) bidirectional models [Kosko(1992)]˙v j = −a j (v j )[b j (v j ) − f k (v k )m jk ],˙u k = −a k (u k )[b k (u k ) − f j (u j )m jk ],which is globally stable for the cases of signal and random–signal Hebbianlearning.GBAM Self–Organized Learning DynamicsThe continuous (and at least C 1 −-differentiable) unsupervised updatelaw for the coupling synaptic GBAM tensor–field W can be viewed both asan inhibitory–covariant Hebbian learning scheme, generalized from [Kosko(1992)]:˙ω αβ = −ω αβ + Φ αβ (q α , p α ), (α, β = 1, ..., n), (3.23)and, as an excitatory–contravariant Hebbian learning scheme:˙ω αβ = −ω αβ + Φ αβ (q α , p α ), (3.24)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 161where the three terms from the left to the right denote respectively thenew–update value, the old value and the innovation of the synaptic tensorW . In this case the nonlinear (usually sigmoid) innovation functions Φ αβand Φ αβ are defined by one of following four Hebbian models:Signal Hebbian learning, with innovation in both variance–forms:Φ αβ = S α (q α ) S β (p β ),Φ αβ = S α (q α ) S β (p β ); (3.25)<strong>Diff</strong>erential Hebbian learning, with innovation in both variance–forms:Φ αβ = S α (q α )S β (p β ) + Ṡα(q α )Ṡβ(p β ),Φ αβ = S α (q α )S β (p β ) + Ṡα (q α )Ṡβ (p β ), (3.26)where Ṡ−terms denote the so–called ‘signal velocities’ (for details see[Kosko (1992)]).Random signal Hebbian learning, with innovation in both variance–forms:Φ αβ = S α (q α ) S β (p β ) + n αβ ,Φ αβ = S α (q α ) S β (p β ) + n αβ , (3.27)where n αβ = {n αβ (t)}, n αβ = {n αβ (t)} respectively denote covariant andcontravariant additive, zero–mean, Gaussian white–noise processes independentof the main innovation signal; andRandom differential signal Hebbian learning, with innovation in bothvariance–forms:Φ αβ = S α (q α )S β (p β ) + Ṡα(q α )Ṡβ(p β ) + n αβ ,Φ αβ = S α (q α )S β (p β ) + Ṡα (q α )Ṡβ (p β ) + n αβ . (3.28)Total GBAM (q, p, W )−neurodynamics and biological interpretationTotal GBAM tensorial neurodynamics is defined as a union of the neuraloscillatory activation (q, p)−-dynamics (3.22) and the synaptic learning


162 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionW −-dynamics (3.28), namely˙q eα = Ie α + ω αβe S β (p β ) − qe α , (3.29)ṗ o α = Iα o + ω o αβS β (q β ) − p o α,= −ω αβe + Φ αβe (q α , p α ),˙ω o αβ = −ω o αβ + Φ o αβ(q α , p α ),˙ω αβe(α, β = 1, ..., n),where the tensorial innovation Φ−-functions are given by one of Hebbianmodels (3.25–3.28), α(= 1, ..., n) is the number of continuous–time (or,graded–response) neurons in each neural–activation field, e(= 2, 4, ..., N)and o(= 1, 3, ..., N − 1) denote respectively even and odd partitions of thetotal sample of N features.Artificial neural networks are generally inspired by biological neuralsystems, but in fact, some important features of biological systems are notpresent in most artificial neural networks. In particular, unidirectional neuralnetworks, which include all associative neural networks except the BAMmodel introduced by [Kosko (1992)], do not resemble oscillatory biologicalneural systems. GBAM is a generalization of Kosko’s ABAM and RABAMneural systems and inherits their oscillatory (excitatory/inhibitory) neurosynapticbehavior. Such oscillatory behavior is a basic characteristic of anumber of biological systems. Examples of similar oscillatory neural ensemblesin the human nervous system are:– Motoneurons and Renshaw interneurons in the spinal cord;– Pyramidal and basket cells in the hippocampus;– Mitral and granule cells in the olfactory bulb;– Pyramidal cells and thalamic inter–neurons in cortico–thalamic system;– Interacting excitatory and inhibitory populations of neurons found inthe cerebellum, olfactory cortex, and neocortex, all representing the basicmechanisms for the generation of oscillating (EEG–monitored) activity inthe brain.Therefore, GBAM can be considered as a model for any of above–mentioned oscillatory biological neural systems.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 1633.6 Tensor Fields on Smooth Manifolds3.6.1 Tensor BundleA tensor bundle T associated to a smooth n−manifold M is defined as atensor product of tangent and cotangent bundles:T =q⊗T ∗ M ⊗p⊗T M ={ }} {p timesT M ⊗ ... ⊗ T M ⊗{ }} {q timesT ∗ M ⊗ ... ⊗ T ∗ M.Tensor bundles are special case of more general fibre bundles (see section4.1 below).A tensor–field of type (p, q) on a smooth n−manifold M is defined as asmooth section τ : M −→ T of the tensor bundle T . The coefficients of thetensor–field τ are smooth (C k ) functions with p indices up and q indicesdown. The classical position of indices can be explained in modern termsas follows. If (U, φ) is a chart at a point m ∈ M with local coordinates(x 1 , ..., x n ), we have the holonomous frame field∂ x i 1 ⊗ ∂ x i 2 ⊗ ... ⊗ ∂ x ip ⊗ dx j1 ⊗ dx j2 ... ⊗ dx jq ,for i ∈ {1, ..., n} p , j = {1, ..., n} q , over U of this tensor bundle, and for any(p, q)−tensor–field τ we haveτ|U = τ i1...ipj 1...j q∂ x i 1 ⊗ ∂ x i 2 ⊗ ... ⊗ ∂ x ip ⊗ dx j1 ⊗ dx j2 ... ⊗ dx jq .For such tensor–fields the Lie derivative along any vector–field is defined(see section 3.7 below), and it is a derivation (i.e., both linearity and Leibnizrules hold) with respect to the tensor product. Tensor bundle T admitsmany natural transformations (see [Kolar et al. (1993)]). For example, a‘contraction’ like the trace T ∗ M ⊗ T M = L (T M, T M) → M × R, butapplied just to one specified factor of type T ∗ M and another one of typeT M, is a natural transformation. And any ‘permutation of the same kindof factors’ is a natural transformation.The tangent bundle π M : T M → M of a manifold M (introduced above)is a special tensor bundle over M such that, given an atlas {(U α , ϕ α )} ofM, T M has the holonomic atlasΨ = {(U α , ϕ α = T ϕ α )}.The associated linear bundle coordinates are the induced coordinates (ẋ λ )at a point m ∈ M with respect to the holonomic frames {∂ λ } in tangent


164 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionspaces T m M. Their transition functions readẋ ′λ = ∂x′λ∂x µ ẋµ .Technically, the tangent bundle T M is a tensor bundle with the structureLie group GL(dim M, R) (see section 3.8 below).Recall that the cotangent bundle of M is the dual T ∗ M of T M. It isequipped with the induced coordinates (ẋ λ ) at a point m ∈ M with respectto holonomic coframes {dx λ } dual of {∂ λ }. Their transition functions readẋ ′ λ = ∂x′µ∂x λ ẋµ.3.6.1.1 Pull–Back and Push–ForwardIn this section we define two important operations, following [Abrahamet al. (1988)], which will be used in the further text.Let ϕ : M → N be a C k map of manifolds and f ∈ C k (N, R). Definethe pull–back of f by ϕ byϕ ∗ f = f ◦ ϕ ∈ C k (M, R).If f is a C k diffeomorphism and X ∈ X k (M), the push–forward of Xby ϕ is defined byϕ ∗ X = T ϕ ◦ X ◦ ϕ −1 ∈ X k (N).If x i are local coordinates on M and y j local coordinates on N, thepreceding formula gives the components of ϕ ∗ X by(ϕ ∗ X) j (y) = ∂ϕj∂x i (x) Xi (x), where y = ϕ(x).We can interchange pull–back and push–forward by changing ϕ to ϕ −1 ,that is, defining ϕ ∗ (resp. ϕ ∗ ) by ϕ ∗ = (ϕ −1 ) ∗ (resp. ϕ ∗ = (ϕ −1 ) ∗ ). Thusthe push–forward of a function f on M is ϕ ∗ f = f ◦ ϕ −1 and the pull–backof a vector–field Y on N is ϕ ∗ Y = (T ϕ) −1 ◦ Y ◦ ϕ.Notice that ϕ must be a diffeomorphism in order that the pull–back andpush–forward operations make sense, the only exception being pull–back offunctions. Thus vector–fields can only be pulled back and pushed forwardby diffeomorphisms. However, even when ϕ is not a diffeomorphism we cantalk about ϕ−related vector–fields as follows.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 165Let ϕ : M → N be a C k map of manifolds. The vector–fields X ∈X k−1 (M) and Y∈ X k−1 (N) are called ϕ−related, denoted X ∼ ϕ Y , ifT ϕ ◦ X = Y ◦ ϕ.Note that if ϕ is diffeomorphism and X and Y are ϕ−related, then Y =ϕ ∗ X. However, in general, X can be ϕ−related to more than one vector–field on N. ϕ−relatedness means that the following diagram commutes:T M✻XMT ϕϕ✲ T N✻Y✲ NThe behavior of flows under these operations is as follows: Let ϕ : M →N be a C k −map of manifolds, X ∈ X k (M) and Y ∈ X k (N). Let F t and G tdenote the flows of X and Y respectively. Then X ∼ ϕ Y iff ϕ ◦ F t = G t ◦ ϕ.In particular, if ϕ is a diffeomorphism, then the equality Y = ϕ ∗ X holds iffthe flow of Y is ϕ◦F t ◦ϕ −1 (This is called the push–forward of F t by ϕ sinceit is the natural way to construct a diffeomorphism on N out of one on M).In particular, (F t ) ∗X = X. Therefore, the flow of the push–forward of avector–field is the push–forward of its flow.3.6.1.2 Dynamical Evolution and FlowAs a motivational example, consider a mechanical system that is capableof assuming various states described by points in a set U. For example, Umight be R 3 × R 3 and a state might be the positions and momenta (x i , p i )of a particle moving under the influence of the central force field, withi = 1, 2, 3. As time passes, the state evolves. If the state is γ 0 ∈ U at times and this changes to γ at a later time t, we setF t,s (γ 0 ) = γ,and call F t,s the evolution operator; it maps a state at time s to what thestate would be at time t; that is, after time t−s. has elapsed. Determinismis expressed by the Chapman–Kolmogorov law [Abraham et al. (1988)]:F τ,t ◦ F t,s = F τ,s, F t,t = identity. (3.30)The evolution laws are called time independent, or autonomous, whenF t,s depends only on t − s. In this case the preceding law (3.30) becomes


166 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe group property:F t ◦ F s = F t+s, F 0 = identity. (3.31)We call such an F t a flow and F t,s a time–dependent flow, or an evolutionoperator. If the system is irreversible, that is, defined only for t ≥ s, wespeak of a semi–flow [Abraham et al. (1988)].Usually, instead of F t,s the laws of motion are given in the form of ODEsthat we must solve to find the flow. These equations of motion have theform:˙γ = X(γ), γ(0) = γ 0 ,where X is a (possibly time–dependent) vector–field on U.As a continuation of the previous example, consider the motion of a particleof mass m under the influence of the central force field (like gravity, orCoulombic potential) F i (i = 1, 2, 3), described by the Newtonian equationof motion:mẍ i = F i (x). (3.32)By introducing momenta p i = mẋ i , equation (3.32) splits into two Hamiltonianequations:ẋ i = p i /m, ṗ i = F i (x). (3.33)Note that in Euclidean space we can freely interchange subscripts and superscripts.However, in general case of a Riemannian manifold, p i = mg ij ẋ jand (3.33) properly readsẋ i = g ij p j /m, ṗ i = F i (x). (3.34)The phase–space here is the Riemannian manifold (R 3 \{0}) × R 3 , thatis, the cotangent bundle of R 3 \{0}, which is itself a smooth manifold forthe central force field. The r.h.s of equations (3.34) define a Hamiltonianvector–field on this 6D manifold byX(x, p) = ( (x i , p i ), (p i /m, F i (x)) ) . (3.35)Integration of equations (3.34) produces trajectories (in this particularcase, planar conic sections). These trajectories comprise the flow F t ofthe vector–field X(x, p) defined in (3.35).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 1673.6.1.3 Vector–Fields and Their Flows3.6.1.4 Vector–Fields on MA vector–field X on U, where U is an open chart in n−manifold M, is asmooth function from U to M assigning to each point m ∈ U a vector atthat point, i.e., X(m) = (m, X(m)). If X(m) is tangent to M for eachm ∈ M, X is said to be a tangent vector–field on M. If X(m) is orthogonalto M (i.e., X(p) ∈ Mm) ⊥ for each X(m) ∈ M, X is said to be a normalvector–field on M.In other words, let M be a C k −manifold. A C k −vector–field on M isa C k −section of the tangent bundle T M of M. Thus a vector–field X ona manifold M is a C k −map X : M → T M such that X(m) ∈ T m M forall points m ∈ M,and π M ◦ X = Id M . Therefore, a vector–field assigns toeach point m of M a vector based (i.e., bound) at that point. The set ofall C k vector–fields on M is denoted by X k (M).A vector–field X ∈ X k (M) represents a field of direction indicators[Thirring (1979)]: to every point m of M it assigns a vector in the tangentspace T m M at that point. If X is a vector–field on M and (U, φ) is a charton M and m ∈ U, then we have X(m) = X(m) φ i ∂ . Following [Kolar∂φ iet al. (1993)], we write X| U = X φ i ∂ .∂φ iLet M be a connected n−manifold, and let f : U → R (U an open setin M) and c ∈ R be such that M = f −1 (c) (i.e., M is the level set of thefunction f at height c) and ∇f(m) ≠ 0 for all m ∈ M. Then there existon M exactly two smooth unit normal vector–fields N 1,2 (m) = ± ∇f(m)|∇f(m)|(here |X| = (X · X) 1/2 denotes the norm or length of a vector X, and (·)denotes the scalar product on M) for all m ∈ M, called orientations on M.Let ϕ : M → N be a smooth map. Recall that two vector–fields X ∈X k (M) and Y ∈ X (N) are called ϕ−related, if T ϕ ◦ X = Y ◦ ϕ holds, i.e.,if the following diagram commutes:T M✻XMT ϕϕ✲ T N✻Y✲ NIn particular, a diffeomorphism ϕ : M → N induces a linear map betweenvector–fields on two manifolds, ϕ ∗ : X k (M) → X (N), such that ϕ ∗ X =


168 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionT ϕ ◦ X ◦ ϕ −1 : N → T N, i.e., the following diagram commutes:T M✻XMT ϕϕ✲ T N✻ϕ ∗ X✲ NThe correspondences M → T M and ϕ → T ϕ obviously define a functorT : M ⇒ M from the category of smooth manifolds to itself. T is anotherspecial case of the vector bundle functor (4.3.2), and closely related to thetangent bundle functor (3.5).A C k time–dependent vector–field is a C k −map X : R×M → T M suchthat X(t, m) ∈ T m M for all (t, m) ∈ R × M, i.e., X t (m) = X(t, m).3.6.1.5 Integral Curves as Dynamical TrajectoriesRecall (3.5) that a curve γ at a point m of an n−manifold M is a C 0 −mapfrom an open interval I ⊂ R into M such that 0 ∈ I and γ(0) = m. Forsuch a curve we may assign a tangent vector at each point γ(t), t ∈ I, by˙γ(t) = T t γ(1).Let X be a smooth tangent vector–field on the smooth n−manifold M,and let m ∈ M. Then there exists an open interval I ⊂ R containing 0 anda parameterized curve γ : I → M such that:(1) γ(0) = m;(2) ˙γ(t) = X(γ(t)) for all t ∈ I; and(3) If β : Ĩ → M is any other parameterized curve in M satisfying (1) and(2), then Ĩ ⊂ I and β(t) = γ(t) for all t ∈ Ĩ.A parameterized curve γ : I → M satisfying condition (2) is calledan integral curve of the tangent vector–field X. The unique γ satisfyingconditions (1)–(3) is the maximal integral curve of X through m ∈ M.In other words, let γ : I → M, t ↦→ γ (t) be a smooth curve in a manifoldM defined on an interval I ⊆ R. ˙γ(t) = d dtγ(t) defines a smooth vector–fieldalong γ since we have π M ◦ ˙γ = γ. Curve γ is called an integral curve orflow line of a vector–field X ∈ X k (M) if the tangent vector determined byγ equals X at every point m ∈ M, i.e.,˙γ = X ◦ γ,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 169or, if the following diagram commutes:T I✻1I;T u✲ T M✻˙γ X ✒✲γ MOn a chart (U, φ) with coordinates φ(m) = ( x 1 (m), ..., x n (m) ) , forwhich ϕ ◦ γ : t ↦→ γ i (t) and T ϕ ◦ X ◦ ϕ −1 : x i ↦→ ( x i , X i (m) ) , this iswritten˙γ i (t) = X i (γ (t)) , for all t ∈ I ⊆ R, (3.36)which is an ordinary differential equation of first–order in n dimensions.The velocity ˙γ of the parameterized curve γ (t) is a vector–field along γdefined by˙γ(t) = (γ(t), ẋ 1 (t), . . . ẋ n (t)).Its length | ˙γ| : I → R, defined by | ˙γ|(t) = | ˙γ(t)| for all t ∈ I, is a functionalong α. | ˙γ| is called speed of γ [Arnold (1989)].Each vector–field X alongγ is of the form X(t) = (γ(t), X 1 (t), . . . , X n (t)), where each componentX i is a function along γ. X is smooth if each X i : I → M is smooth. Thederivative of a smooth vector–field X along a curve γ(t) is the vector–fieldẊ along γ defined byẊ(t) = (γ(t), Ẋ 1 (t), . . . Ẋ n (t)).Ẋ(t) measures the rate of change of the vector part (X 1 (t), . . . X n (t)) ofX(t) along γ. Thus, the acceleration ¨γ(t) of a parameterized curve γ(t) isthe vector–field along γ get by differentiating the velocity field ˙γ(t).<strong>Diff</strong>erentiation of vector–fields along parameterized curves has the followingproperties. For X and Y smooth vector–fields on M along theparameterized curve γ : I → M and f a smooth function along γ, we have:(1) d dt (X + Y ) = Ẋ + Ẏ ;(2) d dt (fX) = fX ˙ + fẊ; and(X · Y ) = ẊY + XẎ .(3) d dt


170 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionA geodesic in M is a parameterized curve γ : I → M whose acceleration¨γ is everywhere orthogonal to M; that is, ¨γ(t) ∈ Mα(t) ⊥ for all t ∈ I ⊂ R.Thus a geodesic is a curve in M which always goes ‘straight ahead’ in thesurface. Its acceleration serves only to keep it in the surface. It has nocomponent of acceleration tangent to the surface. Therefore, it also has aconstant speed ˙γ(t).Let v ∈ M m be a vector on M. Then there exists an open interval I ⊂ Rcontaining 0 and a geodesic γ : I → M such that:(1) γ(0) = m and ˙γ(0) = v; and(2) If β : Ĩ → M is any other geodesic in M with β(0) = m and ˙β(0) = v,then Ĩ ⊂ I and β(t) = γ(t) for all t ∈ Ĩ.The geodesic γ is now called the maximal geodesic in M passing throughm with initial velocity v.By definition, a parameterized curve γ : I → M is a geodesic of M iffits acceleration is everywhere perpendicular to M, i.e., iff ¨γ(t) is a multipleof the orientation N(γ(t)) for all t ∈ I, i.e., ¨γ(t) = g(t) N(γ(t)), whereg : I → R. Taking the scalar product of both sides of this equation withN(γ(t)) we find g = − ˙γṄ(γ(t)). Thus γ : I → M is geodesic iff it satisfiesthe differential equation¨γ(t) + Ṅ(γ(t)) N(γ(t)) = 0.This vector equation represents the system of second–order componentODEsẍ i + N i (x + 1, . . . , x n ) ∂N j∂x k (x + 1, . . . , xn ) ẋ j ẋ k = 0.The substitution u i = ẋ i reduces this second–order differential system (inn variables x i ) to the first–order differential systemẋ i = u i ,˙u i = −N i (x + 1, . . . , x n ) ∂N j∂x k (x + 1, . . . , xn ) ẋ j ẋ k(in 2n variables x i and u i ). This first–order system is just the differentialequation for the integral curves of the vector–field X in U × R (U openchart in M), in which case X is called a geodesic spray.Now, when an integral curve γ(t) is the path a mechanical system Ξfollows, i.e., the solution of the equations of motion, it is called a trajectory.In this case the parameter t represents time, so that (3.36) describes motionof the system Ξ on its configuration manifold M.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 171If X i (m) is C 0 the existence of a local solution is guaranteed, and aLipschitz condition would imply that it is unique. Therefore, exactly oneintegral curve passes through every point, and different integral curves cannever cross. As X ∈ X k (M) is C k , the following statement about thesolution with arbitrary initial conditions holds [Thirring (1979); Arnold(1989)]:Theorem. Given a vector–field X ∈ X (M), for all points p ∈ M, thereexist(η > 0, a neighborhood V of p, and a function γ : (−η, η) × V → M,t, x i (0) ) ↦→ γ ( t, x i (0) ) such that˙γ = X ◦ γ, γ ( 0, x i (0) ) = x i (0) for all x i (0) ∈ V ⊆ M.For all |t| < η, the map x i (0) ↦→ γ ( t, x i (0) ) is a diffeomorphism ftX betweenV and some open set of M. For proof, see [Dieudonne (1969)], I,10.7.4 and 10.8.This Theorem states that trajectories that are near neighbors cannotsuddenly be separated. There is a well–known estimate (see [Dieudonne(1969)], I, 10.5) according to which points cannot diverge faster than exponentiallyin time if the derivative of X is uniformly bounded.An integral curve γ (t) is said to be maximal if it is not a restriction ofan integral curve defined on a larger interval I ⊆ R. It follows from theexistence and uniqueness theorems for ODEs with smooth r.h.s and fromelementary properties of Hausdorff spaces that for any point m ∈ M thereexists a maximal integral curve γ m of X, passing for t = 0 through pointm, i.e., γ(0) = m.Theorem (Local Existence, Uniqueness, and Smoothness) [Abrahamet al. (1988)]. Let E be a Banach space, U ⊂ E be open, and supposeX : U ⊂ E → E is of class C k , k ≥ 1. Then1. For each x 0 ∈ U, there is a curve γ : I → U at x 0 such that˙γ(t) = X (γ(t)) for all t ∈ I.2. Any two such curves are equal on the intersection of their domains.3. There is a neighborhood U 0 of the point x 0 ∈ U, a real number a > 0,and a C k map F : U 0 × I → E, where I is the open interval ] − a, a[ , suchthat the curve γ u : I → E, defined by γ u (t) = F (u, t) is a curve at u ∈ Esatisfying the ODEs ˙γ u (t) = X (γ u (t)) for all t ∈ I.Proposition (Global Uniqueness). Suppose γ 1 and γ 2 are two integralcurves of a vector–field X at a point m ∈ M. Then γ 1 = γ 2 on theintersection of their domains [Abraham et al. (1988)].If for every point m ∈ M the curve γ m is defined on the entire real axis


172 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionR, then the vector–field X is said to be complete.The support of a vector–field X defined on a manifold M is defined to bethe closure of the set {m ∈ M|X(m) = 0}. A C k vector–field with compactsupport on a manifold M is complete. In particular, a C k vector–field on acompact manifold is complete. Completeness corresponds to well–defineddynamics persisting eternally.Now, following [Abraham et al. (1988)], for the derivative of a C kfunction f : E → R in the direction X we use the notation X[f] = df · X, where df stands for the derivative map. In standard coordinates on R nthis is a standard gradientdf(x) = ∇f = (∂ x 1f, ..., ∂ x nf), and X[f] = X i ∂ x if.Let F t be the flow of X. Then f (F t (x)) = f (F s (x)) if t ≥ s.For example, Newtonian equations for a moving particle of mass min a potential field V in R n are given by ¨q i (t) = −(1/m)∇V ( q i (t) ) , fora smooth function V : R n → R. If there are constants a, b ∈ R, b ≥0 such that (1/m)V (q i ) ≥ a − b ∥ ∥q i∥ ∥ 2 , then every solution exists for alltime. To show this, rewrite the second–order equations as a first–ordersystem ˙q i = (1/m) p i , ṗ i = −V (q i ) and note that the energy E(q i , p i ) =((1/2m) ‖ p i ‖ 2 +V (q) is a first integral of the motion. Thus, for any solutionq i (t), p i (t) ) we have E ( q i (t), p i (t) ) = E ( q i (0), p i (0) ) = V (q(0)).Let X t be a C k time–dependent vector–field on an n−manifold M,k ≥ 1, and let m 0 be an equilibrium of X t , that is, X t (m 0 ) = 0 for all t.Then for any T there exists a neighborhood V of m 0 such that any m ∈ Vhas integral curve existing for time t ∈ [−T, T ].3.6.1.6 Dynamical Flows on MRecall (6.289) that the flow F t of a C k vector–field X ∈ X k (M) is the one–parameter group of diffeomorphisms F t : M → M such that t ↦→ F t (m)is the integral curve of X with initial condition m for all m ∈ M andt ∈ I ⊆ R. The flow F t (m) is C k by induction on k. It is defined as[Abraham et al. (1988)]:ddt F t(x) = X(F t (x)).Existence and uniqueness theorems for ODEs guarantee that F tsmooth in m and t. From uniqueness, we get the flow property:isF t+s = F t ◦ F s


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 173along with the initial conditions F 0 = identity. The flow property generalizesthe situation where M = V is a linear space, X(x) = A x for a(bounded) linear operator A, and where F t (x) = e tA x – to the nonlinearcase. Therefore, the flow F t (m) can be defined as a formal exponentialF t (m) = exp(t X) = (I + t X + t2 2 X2 + ...) =∞∑ X k t k.k!recall that a time–dependent vector–field is a map X : M × R →T Msuch that X(m, t) ∈ T m M for each point m ∈ M and t ∈ R. An integralcurve of X is a curve γ(t) in M such thatk=0˙γ(t) = X (γ (t) , t) , for all t ∈ I ⊆ R.In this case, the flow is the one–parameter group of diffeomorphismsF t,s : M → M such that t ↦→ F t,s (m) is the integral curve γ(t) with initialcondition γ(s) = m at t = s. Again, the existence and uniqueness Theoremfrom ODE–theory applies here, and in particular, uniqueness gives thetime–dependent flow property, i.e., the Chapman–Kolmogorov lawF t,r = F t,s ◦ F s,r .If X happens to be time independent, the two notions of flows are relatedby F t,s = F t−s (see [Marsden and Ratiu (1999)]).3.6.1.7 Categories of ODEsOrdinary differential equations are naturally organized into their categories(see [Kock (1981)]). First order ODEs are organized into a category ODE 1 .A first–order ODE on a manifold–like object M is a vector–field X : M →T M, and a morphism of vector–fields (M 1 , X 1 ) → (M 2 , X 2 ) is a map f :M 1 → M 2 such that the following diagram commutesT M 1T f ✲ T M 2✻✻X 2X 1M 1✲ Mf2A global solution of the differential equation (M, X), or a flow line of avector–field X, is a morphism from ( R, ∂∂x)to (M, X).


174 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSimilarly, second–order ODEs are organized into a category ODE 2 . Asecond–order ODE on M is usually constructed as a vector–field on T M,ξ : T M → T T M, and a morphism of vector–fields (M 1 , ξ 1 ) → (M 2 , ξ 2 ) is amap f : M 1 → M 2 such that the following diagram commutesT T M 1T T f ✲ T T M 2✻✻ξ 1ξ 2T M 1✲ T MT f2Unlike solutions for first–order ODEs, solutions for second–order ODEs arenot in general homomorphisms from R, unless the second–order ODE is aspray [Kock and Reyes (2003)].3.6.2 <strong>Diff</strong>erential Forms on Smooth ManifoldsRecall (see section 2.1.4.2 above) that exterior differential forms are a specialkind of antisymmetrical covariant tensors, that formally occur as integrandsunder ordinary integral signs in R 3 . To give a more precise exposition,here we start with 1−forms, which are dual to vector–fields, and afterthat introduce general k−forms.3.6.2.1 1−Forms on MDual to the notion of a C k vector–field X on an n−manifold M is a C kcovector–field, or a C k 1−form α, which is defined as a C k −section of thecotangent bundle T ∗ M, i.e., α : M → T ∗ M is smooth and π ∗ M ◦ X =Id M . We denote the set of all C k 1−forms by Ω 1 (M). A basic exampleof a 1−form is the differential df of a real–valued function f ∈ C k (M, R).With point wise addition and scalar multiplication Ω 1 (M) becomes a vectorspace.In other words, a C k 1−form α on a C k manifold M is a real–valuedfunction on the set of all tangent vectors to M, i.e., α : T M → R with thefollowing properties:(1) α is linear on the tangent space T m M for each m ∈ M;(2) For any C k vector–field X ∈ X k (M), the function f : M → R is C k .Given a 1−form α, for each point m ∈ M the map α(m) : T m M → R isan element of the dual space T ∗ mM. Therefore, the space of 1−forms Ω 1 (M)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 175is dual to the space of vector–fields X k (M).In particular, the coordinate 1−forms dx 1 , ..., dx n are locally definedat any point m ∈ M by the property that for any vector–field X =(X 1 , ..., X n) ∈ X k (M),dx i (X) = X i .The dx i ’s form a basis for the 1−forms at any point m ∈ M, with localcoordinates ( x 1 , ..., x n) , so any 1−form α may be expressed in the formα = f i (m) dx i .If a vector–field X on M has the form X(m) = ( X 1 (m), ..., X n (m) ) ,then at any point m ∈ M,α m (X) = f i (m) X i (m),where f ∈ C k (M, R).Suppose we have a 1D closed curve γ = γ(t) inside a smooth manifoldM. Using a simplified ‘physical’ notation, a 1–form α(x) defined at a pointx ∈ M, given byα(x) = α i (x) dx i , (3.37)can be unambiguously integrated over a curve γ ∈ M, as follows. Parameterizeγ by a parameter t, so that its coordinates are given by x i (t). Attime t, the velocity ẋ = ẋ(t) is a tangent vector to M at x(t). One caninsert this tangent vector into the linear map α(x) to get a real number.By definition, inserting the vector ẋ(t) into the linear map dx i gives thecomponent ẋ i = ẋ i (t). Doing this for every t, we can then integrate over t,∫ (αi(x(t))ẋ i) dt. (3.38)Note that this expression is independent of the parametrization in termsof t. Moreover, from the way that tangent vectors transform, one candeduce how the linear maps dx i should transform, and from this how thecoefficients α i (x) should transform. Doing this, one sees that the aboveexpression is also invariant under changes of coordinates on M. Therefore,a 1–form can be unambiguously integrated over a curve in M. We writesuch an integral as∫∫α i (x) dx i , or, even shorter, as α.γγ


176 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionClearly, when M is itself a 1D manifold, (3.38) gives precisely the ordinaryintegration of a function α(x) over x, so the above notation is indeednatural.The 1−forms on M are part of an algebra, called the exterior algebra,or Grassmann algebra on M. The multiplication ∧ in this algebra is calledwedge product (see (3.40) below), and it is skew–symmetric,dx i ∧ dx j = −dx j ∧ dx i .One consequence of this is that dx i ∧ dx i = 0.3.6.2.2 k−Forms on MA differential form, or an exterior form α of degree k, or a k−form for short,is a section of the vector bundle Λ k T ∗ M, i.e., α : M → Λ k T ∗ M. In otherwords, α(m) : T m M × ... × T m M → R (with k factors T m M) is a functionthat assigns to each point m ∈ M a skew–symmetric k−multilinear map onthe tangent space T m M to M at m. Without the skew–symmetry assumption,α would be called a (0, k)−tensor–field. The space of all k−forms isdenoted by Ω k (M). It may also be viewed as the space of all skew symmetric(0, k)−tensor–fields, the space of all mapsΦ : X k (M) × ... × X k (M) → C k (M, R),which are k−linear and skew–symmetric (see (3.40) below). We putΩ k (M) = C k (M, R).In particular, a 2−form ω on an n−manifold M is a section of the vectorbundle(Λ 2 T ∗ M. If (U, φ) is a chart at a point m ∈ M with local coordinatesx 1 , ..., x n) let {e 1 , ..., e n } = {∂ x 1, ..., ∂ x n} – be the corresponding basis forT m M, and let { e 1 , ..., e n} = { dx 1 , ..., dx n} – be the dual basis for TmM.∗Then at each point m ∈ M, we can write a 2−form ω asω m (v, u) = ω ij (m) v i u j , where ω ij (m) = ω m (∂ x i, ∂ x j ).Similarly to the case of a 1–form α (3.37), one would like to define a2–form ω as something which can naturally be integrated over a 2D surfaceΣ within a smooth manifold M. At a specific point x ∈ M, the tangentplane to such a surface is spanned by a pair of tangent vectors, (ẋ 1 , ẋ 2 ). So,to generalize the construction of a 1–form, we should give a bilinear mapfrom such a pair to R. The most general form of such a map isω ij (x) dx i ⊗ dx j , (3.39)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 177where the tensor product of two cotangent vectors acts on a pair of vectorsas,dx i ⊗ dx j (ẋ 1 , ẋ 2 ) = dx i (ẋ 1 ) dx j (ẋ 2 ).On the r.h.s. of this equation, one multiplies two ordinary numbers got byletting the linear map dx i act on ẋ 1 , and dx j on ẋ 2 .However, the bilinear map (3.39) is slightly too general to give a goodintegration procedure. The reason is that we would like the integral tochange sign if we change the orientation of integration, just like in the 1Dcase. In 2D, changing the orientation means exchanging ẋ 1 and ẋ 2 , so wewant our bilinear map to be antisymmetric under this exchange. This isachieved by defining a 2–form to beω = ω ij (x) ( dx i ⊗ dx j − dx j ⊗ dx i) ≡ ω ij (x) dx i ∧ dx jWe now see why a2–form corresponds to an antisymmetric tensor–field: thesymmetric part of ω ij would give a vanishing contribution to ω. Now, parameterizinga surface Σ in M with two coordinates t 1 and t 2 , and reasoningexactly like we did in the case of a 1–form, one can show that the integrationof a 2–form over such a surface is indeed well–defined, and independentof the parametrization of both Σ and M.If each summand of a differential form α ∈ Ω k (M) contains k basis1−forms dx i ’s, the form is called a k−form. Functions f ∈ C k (M, R) areconsidered to be 0−forms, and any form on an n−manifold M of degreek > n must be zero due to the skew–symmetry.Any k−form α ∈ Ω k (M) may be expressed in the formα = f I dx i1 ∧ ... ∧ dx i k= f I dx I ,where I is a multiindex I = (i 1 , ..., i k ) of length k, and ∧ is the wedgeproduct which is associative, bilinear and anticommutative.Just as 1−forms act on vector–fields to give real–valued functions, sok−forms act on k−tuples of vector–fields to give real–valued functions.The wedge product of two differential forms, a k−form α ∈ Ω k (M) andan l−form β ∈ Ω l (M) is a (k + l)−form α ∧ β defined as:α ∧ β =(k + l)!A(α ⊗ β), (3.40)k!l!∑σ∈S k(sign σ) τ(e σ(1) , ...,where A : Ω k (M) → Ω k (M), Aτ(e 1 , ..., e k ) = 1 k!e σ(k) ), where S k is the permutation group on k elements consisting of allbijections σ : {1, ..., k} → {1, ..., k}.


178 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFor any k−form α ∈ Ω k (M) and l−form β ∈ Ω l (M), the wedge productis defined fiberwise, i.e., (α ∧ β) m= α x ∧ β m for each point m ∈ M. It isalso associative, i.e., (α ∧ β) ∧ γ = α ∧ (β ∧ γ), and graded commutative,i.e., α∧β = (−1) kl β∧α. These properties are proved in multilinear algebra.So M =⇒ Ω k (M) is a contravariant functor from the category M into thecategory of real graded commutative algebras [Kolar et al. (1993)].Let M be an n−manifold, X ∈ X k (M), and α ∈ Ω k+1 (M). The interiorproduct, or contraction, i X α = X⌋α ∈ Ω k (M) of X and α (with insertionoperator i X ) is defined asi X α(X 1 , ..., X k ) = α(X, X 1 , ..., X k ).Insertion operator i X of a vector–field X ∈ X k (M) is natural withrespect to the pull–back F ∗ of a diffeomorphism F : M → N between twomanifolds, i.e., the following diagram commutes:Ω k (N)F ∗✲ Ω k (M)i X❄Ω k−1 (N)F ∗i F ∗ X❄✲ Ω k−1 (M)Similarly, insertion operator i X of a vector–field X ∈ Y k (M) is naturalwith respect to the push–forward F ∗ of a diffeomorphism F : M → N, i.e.,the following diagram commutes:Ω k (M)F ∗✲ Ω k (N)i Y❄Ω k−1 (M)F ∗i F∗Y❄✲ Ω k−1 (N)In case of Riemannian manifolds there is another exterior operation.Let M be a smooth n−manifold with Riemannian metric g = 〈, 〉 and thecorresponding volume element µ. The Hodge star operator ∗ : Ω k (M) →Ω n−k (M) on M is defined asα ∧ ∗β = 〈α, β〉 µfor α, β ∈ Ω k (M).The Hodge star operator satisfies the following properties for α, β ∈ Ω k (M)[Abraham et al. (1988)]:


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 179(1) α ∧ ∗β = 〈α, β〉 µ = β ∧ ∗α;(2) ∗1 = µ, ∗µ = (−1) Ind(g) ;(3) ∗ ∗ α = (−1) Ind(g) (−1) k(n−k) α;(4) 〈α, β〉 = (−1) Ind(g) 〈∗α, ∗β〉, where Ind(g) is the index of the metric g.3.6.2.3 Exterior <strong>Diff</strong>erential SystemsHere we give an informal introduction to exterior differential systems (EDS,for short), which are expressions involving differential forms related to anymanifold M. Later, when we fully develop the necessary differential geometricalas well as variational machinery (see (5.8) below), we will give amore precise definition of EDS.Central in the language of EDS is the notion of coframing, which is a realfinite–dimensional smooth manifold M with a given global cobasis and coordinates,but without requirement for a proper topological and differentialstructures. For example, M = R 3 is a coframing with cobasis {dx, dy, dz}and coordinates {x, y, z}. In addition to the cobasis and coordinates, acoframing can be given structure equations (3.10.2.4) and restrictions. Forexample, M = R 2 \{0} is a coframing with cobasis {e 1 , e 2 }, a single coordinate{r}, structure equations {dr = e 1 , de 1 = 0, de 2 = e 1 ∧ e 2 /r} andrestrictions {r ≠ 0}.A system S on M in EDS terminology is a list of expressions includingdifferential forms (e.g., S = {dz − ydx}).Now, a simple EDS is a triple (S, Ω, M), where S is a system on M,and Ω is an independence condition: either a decomposable k−form or asystem of k−forms on M. An EDS is a list of simple EDS objects wherethe various coframings are all disjoint.An integral element of an exterior system (S, Ω, M) is a subspace P ⊂T m M of the tangent space at some point m ∈ M such that all forms in Svanish when evaluated on vectors from P . Alternatively, an integral elementP ⊂ T m M can be represented by its annihilator P ⊥ ⊂ TmM, ∗ comprisingthose 1−forms at m which annul every vector in P . For example, withM = R 3 = {(x, y, z)}, S = {dx ∧ dz} and Ω = {dx, dz}, the integralelement P = {∂ x + ∂ z , ∂ y } is equally determined by its annihilator P ⊥ ={dz − dx}. Again, for S = {dz − ydx} and Ω = {dx}, the integral elementP = {∂ x + y∂ z } can be specified as {dy}.


180 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.6.3 Exterior Derivative and (Co)HomologyThe exterior derivative is an operation that takes k−forms to (k+1)−formson a smooth manifold M. It defines a unique family of maps d : Ω k (U) →Ω k+1 (U), U open in M, such that (see [Abraham et al. (1988)]):(1) d is a ∧−antiderivation; that is, d is R−linear and for two forms α ∈Ω k (U), β ∈ Ω l (U),d(α ∧ β) = dα ∧ β + (−1) k α ∧ dβ.(2) If f ∈ C k (U, R) is a function on M, then df = ∂f∂x i dx i : M → T ∗ M is thedifferential of f, such that df(X) = i X df = L X f −di X f = L X f = X[f]for any X ∈ X k (M).(3) d 2 = d ◦ d = 0 (that is, d k+1 (U) ◦ d k (U) = 0).(4) d is natural with respect to restrictions |U; that is, if U ⊂ V ⊂ M areopen and α ∈ Ω k (V ), then d(α|U) = (dα)|U, or the following diagramcommutes:Ω k (V )d❄Ω k+1 (V )|U|U✲ Ω k (U)d❄✲ Ω k+1 (U)(5) d is natural with respect to the Lie derivative L X (4.3.2) along anyvector–field X ∈ X k (M); that is, for ω ∈ Ω k (M) we have L X ω ∈Ω k (M) and dL X ω = L X dω, or the following diagram commutes:Ω k (M)d❄Ω k+1 (M)L XL X✲ Ω k (M)d❄✲ Ω k+1 (M)(6) Let ϕ : M → N be a C k map of manifolds. Then ϕ ∗ : Ω k (N) → Ω k (M)is a homomorphism of differential algebras (with ∧ and d) and d isnatural with respect to ϕ ∗ = F ∗ ; that is, ϕ ∗ dω = dϕ ∗ ω, or the following


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 181diagram commutes:Ω k (N)d❄Ω k+1 (N)ϕ ∗ϕ ∗✲ Ω k (M)d❄✲ Ω k+1 (M)(7) Analogously, d is natural with respect to diffeomorphism ϕ ∗ = (F ∗ ) −1 ;that is, ϕ ∗ dω = dϕ ∗ ω, or the following diagram commutes:Ω k (N)d❄Ω k+1 (N)ϕ ∗ϕ ∗✲ Ω k (M)d❄✲ Ω k+1 (M)(8) L X = i X ◦ d + d ◦ i X for any X ∈ X k (M) (the Cartan ‘magic’ formula).(9) L X ◦ d = d ◦ L X , i.e., [L X , d] = 0 for any X ∈ X k (M).(10) [L X , i Y ] = i [x,y] ; in particular, i X ◦L X = L X ◦i X for all X, Y ∈ X k (M).Given a k−form α = f I dx I ∈ Ω k (M), the exterior derivative is definedin local coordinates ( x 1 , ..., x n) of a point m ∈ M asdα = d ( f I dx I) = ∂f I∂x i dx i k∧ dx I = df I ∧ dx i1 ∧ ... ∧ dx i k.kIn particular, the exterior derivative of a function f ∈ C k (M, R) is a1−form df ∈ Ω 1 (M), with the property that for any m ∈ M, and X ∈X k (M),df m (X) = X(f),i.e., df m (X) is a Lie derivative of f at m in the direction of X. Therefore,in local coordinates ( x 1 , ..., x n) of a point m ∈ M we havedf = ∂f∂x i dxi .For any two functions f, g ∈ C k (M, R), exterior derivative obeys theLeibniz rule:d(fg) = g df + f dg,


182 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand the chain rule:d (g(f)) = g ′ (f) df.A k−form α ∈ Ω k (M) is called closed form if dα = 0, and it is calledexact form if there exists a (k − 1)−form β ∈ Ω k−1 (M) such that α = dβ.Since d 2 = 0, every exact form is closed. The converse is only partially true(Poincaré Lemma): every closed form is locally exact. This means that givena closed k−form α ∈ Ω k (M) on an open set U ⊂ M, any point m ∈ U hasa neighborhood on which there exists a (k − 1)−form β ∈ Ω k−1 (U) suchthat dβ = α| U .The Poincaré lemma is a generalization and unification of two well–known facts in vector calculus:(1) If curl F = 0, then locally F = grad f;(2) If div F = 0, then locally F = curl G.Poincaré lemma for contractible manifolds:smoothly contractible manifold is exact.Any closed form on a3.6.3.1 Intuition Behind CohomologyThe simple formula d 2 = 0 leads to the important topological notion ofcohomology. Let us try to solve the equation dω = 0 for a p−form ω. Atrivial solution is ω = 0. From the above formula, we can actually find amuch larger class of trivial solutions: ω = dα for a (p − 1)−form α. Moregenerally, if ω is any solution to dω = 0, then so is ω + dα. We want toconsider these two solutions as equivalent:ω ∼ ω + ω ′ if ω ′ ∈ Im d,where Im d is the image of d, that is, the collection of all p−forms ofthe form dα. (To be precise, the image of d contains q−forms for any0 < q ≤ n, so we should restrict this image to the p−forms for the p weare interested in.) The set of all p−forms which satisfy dω = 0 is called thekernel of d, denoted Ker d, so we are interested in Ker d up to the equivalenceclasses defined by adding elements of Im d. (Again, strictly speaking, Ker dconsists of q−forms for several values of q, so we should restrict it to thep−forms for our particular choice of p.) This set of equivalence classes iscalled H p (M), the p−th de Rham cohomology group of M,H p (M) = Ker dIm d .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 183Clearly, Ker d is a group under addition: if two forms ω (1) and ω (2) satisfydω (1) = dω (2) = 0, then so does ω (1) + ω (2) . Moreover, if we changeω (i) by adding some dα (i) , the result of the addition will still be in thesame cohomology class, since it differs from ω (1) + ω (2) by d(α (1) + α (2) ).Therefore, we can view this addition really as an addition of cohomologyclasses: H p (M) is itself an additive group. Also note that if ω (3) and ω (4)are in the same cohomology class (that is, their difference is of the formdα (3) ), then so are cω (3) and cω (4) for any constant factor c. In otherwords, we can multiply a cohomology class by a constant to get anothercohomology class: cohomology classes actually form a vector space.3.6.3.2 Intuition Behind HomologyAnother operator similar to the exterior derivative d is the boundary operatorδ, which maps compact submanifolds of a smooth manifold M totheir boundary. Here, δC = 0 means that a submanifold C of M has noboundary, and C = δU means that C is itself the boundary of some submanifoldU. It is intuitively clear, and not very hard to prove, that δ 2 = 0:the boundary of a compact submanifold does not have a boundary itself.That the objects on which δ acts are independent of its coordinates is alsoclear. So is the grading of the objects: the degree p is the dimension of thesubmanifold C. 3 What is less clear is that the collection of submanifoldsactually forms a vector space, but one can always define this vector spaceto consist of formal linear combinations of submanifolds, and this is preciselyhow one proceeds. The pD elements of this vector space are calledp−chains. One should think of -C as C with its orientation reversed, andof the sum of two disjoint sets, C 1 + C 2 , as their union. The equivalenceclasses constructed from δ are called homology classes.For example, in Figure 3.5, C 1 and C 2 both satisfy δC = 0, so theyare elements of Ker δ. Moreover, it is clear that neither of them separatelycan be viewed as the boundary of another submanifold, so they are not inthe trivial homology class Im δ. However, the boundary of U is C 1 − C 2 .(The minus sign in front of C 2 is a result of the fact that C 2 itself actuallyhas the wrong orientation to be considered a boundary of U.) This can bewritten as C 1 − C 2 = δU, or equivalently C 1 = C 2 + δU, showing that C 1and C 2 are in the same homology class.The cohomology groups for the δ−operator are called homology groups,3 Note that here we have an example of an operator that maps objects of degree p toobjects of degree p − 1 instead of p + 1.


184 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 3.5 The 1D submanifolds S 1 and S 2 represent the same homology class, since theirdifference is the boundary of U.and denoted by H p (M), with a lower index. 4 The p−chains C that satisfyδC = 0 are called p−cycles. Again, the H p (M) only exist for 0 ≤ p ≤ n.There is an interesting relation between cohomology and homologygroups. Note that we can construct a bilinear map from H p (M)×H p (M) →R by∫([ω], [C]) ↦→ ω, (3.41)where [ω] denotes the cohomology class of a p−form ω, and [Σ] the homologyclass of a p−cycle Σ. Using Stokes’ Theorem, it can be seen that theresult does not depend on the representatives for either ω or C∫∫ ∫ ∫ω + dα = ω + dα + ω + dαC+δUC C δU∫ ∫ ∫∫= ω + α + d(ω + dα) = ω,CδCwhere we used that by the definition of (co)homology classes, δC = 0 anddω = 0. As a result, the above map is indeed well–defined on homologyand cohomology classes. A very important Theorem by de Rham says thatthis map is nondegenerate [De Rham (1984)]. This means that if we takesome [ω] and we know the result of the map (3.41) for all [C], this uniquelydetermines [ω], and similarly if we start by picking an [C]. This in particularmeans that the vector space H p (M) is the dual vector space of H p (M).4 Historically, as can be seen from the terminology, homology came first and cohomologywas related to it in the way we will discuss below. However, since the cohomologygroups have a more natural additive structure, it is the name ‘cohomology’ which isactually used for generalizations.CUC


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 1853.6.3.3 De Rham Complex and Homotopy OperatorsAfter an intuitive introduction of (co)homology ideas, we now turn to theirproper definitions. Given a smooth manifold M, let Ω p (M) denote thespace of all smooth p−forms on M. The differential d, mapping p−formsto (p + 1)−forms, serves to define the de Rham complex on M0 → Ω 0 (M)d 0 ✲ Ω 1 (M)d 1 ✲ ...d n−1 ✲ Ω n (M) → 0. (3.42)Recall (from section 6.200 above) that in general, a complex is definedas a sequence of vector spaces, and linear maps between successive spaces,with the property that the composition of any pair of successive maps isidentically 0. In the case of the de Rham complex (3.42), this requirement isa restatement of the closure property for the exterior differential: d ◦ d = 0.In particular, for n = 3, the de Rham complex on a manifold M reads0 → Ω 0 (M)d 0 ✲ Ω 1 (M)If ω ≡ f(x, y, z) ∈ Ω 0 (M), thend 0 ω ≡ d 0 f = ∂f∂xdx +∂f∂yd 1 ✲ Ω 2 (M)If ω ≡ fdx + gdy + hdz ∈ Ω 1 (M), then( ∂gd 1 ω ≡∂x − ∂f ) ( ∂hdx∧dy+∂y∂y − ∂g )dy∧dz+∂z∂fdy + dz = grad ω.∂zd 2 ✲ Ω 3 (M) → 0.(3.43)( ∂f∂z − ∂h∂xIf ω ≡ F dy ∧ dz + Gdz ∧ dx + Hdx ∧ dy ∈ Ω 2 (M), thend 2 ω ≡ ∂F∂x + ∂G∂y + ∂H∂z= div ω.Therefore, the de Rham complex (3.43) can be written as)dz∧dx = curl ω.0 → Ω 0 (M)grad ✲ →Ω 1 (M)curl✲ Ω 2 (M)div ✲ Ω 3 (M) → 0.Using the closure property for the exterior differential, d ◦ d = 0, we get thestandard identities from vector calculuscurl · grad = 0 and div · curl = 0.The definition of the complex requires that the kernel of one of thelinear maps contains the image of the preceding map. The complex is


186 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionexact if this containment is equality. In the case of the de Rham complex(3.42), exactness means that a closed p−form ω, meaning that dω = 0, isnecessarily an exact p−form, meaning that there exists a (p − 1)−form θsuch that ω = dθ. (For p = 0, it says that a smooth function f is closed,df = 0, iff it is constant). Clearly, any exact form is closed, but the converseneed not hold. Thus the de Rham complex is not in general exact. Thecelebrated de Rham Theorem states that the extent to which this complexfails to be exact measures purely topological information about the manifoldM, its cohomology group.On the local side, for special types of domains in Euclidean space R m ,there is only trivial topology and we do have exactness of the de Rhamcomplex (3.42). This result, known as the Poincaré lemma, holds for star–shaped domains M ⊂ R m : Let M ⊂ R m be a star–shaped domain. Thenthe de Rham complex over M is exact.The key to the proof of exactness of the de Rham complex lies in theconstruction of suitable homotopy operators. By definition, these are linearoperators h : Ω p → Ω p−1 , taking differential p−forms into (p − 1)−forms,and satisfying the basic identity [Olver (1986)]ω = dh(ω) + h(dω), (3.44)for all p−forms ω ∈ Ω p . The discovery of such a set of operators immediatelyimplies exactness of the complex. For if ω is closed, dω = 0, then(3.44) reduces to ω = dθ where θ = h(ω), so ω is exact.3.6.3.4 Stokes Theorem and de Rham CohomologyStokes Theorem states that if α is an (n − 1)−form on an orientablen−manifold M, then the integral of dα over M equals the integral of αover ∂M, the boundary of M. The classical theorems of Gauss, Green, andStokes are special cases of this result.A manifold with boundary is a set M together with an atlas of charts(U, φ) with boundary on M. Define (see [Abraham et al. (1988)]) theinterior and boundary of M respectively asInt M = ⋃ Uφ −1 (Int (φ(U))) , and ∂M = ⋃ Uφ −1 (∂ (φ(U))) .If M is a manifold with boundary, then its interior Int M and its boundary∂M are smooth manifolds without boundary. Moreover, if f : M → Nis a diffeomorphism, N being another manifold with boundary, then f in-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 187duces, by restriction, two diffeomorphismsInt f : Int M → Int N, and ∂f : ∂M → ∂N.If n = dim M, then dim(Int M) = n and dim(∂M) = n − 1.To integrate a differential n−form over an n−manifold M, M must beoriented. If Int M is oriented, we want to choose an orientation on ∂Mcompatible with it. As for manifolds without boundary a volume form onan n−manifold with boundary M is a nowhere vanishing n−form on M.Fix an orientation on R n . Then a chart (U, φ) is called positively orientedif the map T m φ : T m M → R n is orientation preserving for all m ∈ U.Let M be a compact, oriented kD smooth manifold with boundary ∂M.Let α be a smooth (k − 1)−form on M. Then the classical Stokes formulaholds∫ ∫dα = α.If ∂M =Ø then ∫ dα = 0.MThe quotient spaceM∂MH k (M) = Ker ( d : Ω k (M) → Ω k+1 (M) )Im (d : Ω k−1 (M) → Ω k (M))represents the kth de Rham cohomology group of a manifold M. recall thatthe de Rham Theorem states that these Abelian groups are isomorphic tothe so–called singular cohomology groups of M defined in algebraic topologyin terms of simplices and that depend only on the topological structure ofM and not on its differentiable structure. The isomorphism is providedby integration; the fact that the integration map drops to the precedingquotient is guaranteed by Stokes’ Theorem.The exterior derivative commutes with the pull–back of differentialforms. That means that the vector bundle Λ k T ∗ M is in fact the valueof a functor, which associates a bundle over M to each manifold M anda vector bundle homomorphism over ϕ to each (local) diffeomorphism ϕbetween manifolds of the same dimension. This is a simple example ofthe concept of a natural bundle. The fact that the exterior derivative dtransforms sections of Λ k T ∗ M into sections of Λ k+1 T ∗ M for every manifoldM can be expressed by saying that d is an operator from Λ k T ∗ Minto Λ k+1 T ∗ M. That the exterior derivative d commutes with (local) diffeomorphismsnow means, that d is a natural operator from the functorΛ k T ∗ into functor Λ k+1 T ∗ . If k > 0, one can show that d is the unique


188 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionnatural operator between these two natural bundles up to a constant. Soeven linearity is a consequence of naturality [Kolar et al. (1993)].3.6.3.5 Euler–Poincaré Characteristics of MThe Euler–Poincaré characteristics of a manifold M equals the sum of itsBetti numbersn∑χ(M) = (−1) p b p .p=0In case of 2nD oriented compact Riemannian manifold M (Gauss–Bonnet Theorem) its Euler–Poincaré characteristics is equal∫χ(M) = γ,where γ is a closed 2n form on M, given byγ =M(−1)n(4π) n n! ɛ1...2n i 1...i 2nΩ i1i 2∧ Ω i2n−1i 2n,where Ω i j is the curvature 2−form of a Riemannian connection on M.Poincaré–Hopf Theorem: The Euler–Poincaré characteristics χ(M) of acompact manifold M equals the sum of indices of zeros of any vector–fieldon M which has only isolated zeros.3.6.3.6 Duality of Chains and Forms on MIn topology of finite–dimensional smooth (i.e., C p+1 with p ≥ 0) manifolds,a fundamental notion is the duality between p−chains C and p−forms(i.e., p−cochains) ω on the smooth manifold M, or domains of integrationand integrands – as an integral on M represents a bilinear functional (see[Choquet-Bruhat and DeWitt-Morete (1982); Dodson and Parker (1997)])∫ω ≡ 〈C, ω〉 , (3.45)Cwhere the integral is called the period of ω. Period depends only on thecohomology class of ω and the homology class of C. A closed form (cocycle)is exact (coboundary) if all its periods vanish, i.e., dω = 0 implies ω = dθ.The duality (3.45) is based on the classical Stokes formula∫ ∫dω = ω.C∂C


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 189This is written in terms of scalar products on M as〈C, dω〉 = 〈∂C, ω〉 ,where ∂C is the boundary of the p−chain C oriented coherently with C.While the boundary operator ∂ is a global operator, the coboundary operator,that is, the exterior derivative d, is local, and thus more suitable forapplications. The main property of the exterior differential,d 2 = 0 implies ∂ 2 = 0,can be easily proved by the use of Stokes’ formula〈∂ 2 C, ω 〉 = 〈∂C, dω〉 = 〈 C, d 2 ω 〉 = 0.The analysis of p−-chains and p−-forms on the finite–dimensionalsmooth manifold M is usually performed in (co)homology categories (see[Dodson and Parker (1997); Dieudonne (1988)]) related to M.Let M • denote the category of cochains, (i.e., p–forms) on the smoothmanifold M. When C = M • , we have the category S • (M • ) of generalizedcochain complexes A • in M • , and if A ′ = 0 for n < 0 we have a subcategoryS • DR (M• ) of the de Rham differential complexes in M •A • DR : 0 → Ω 0 (M)d ✲ Ω 1 (M)d ✲ Ω 2 (M) · · · (3.46)· · ·d ✲ Ω n (M)d ✲ · · · .Here A ′ = Ω n (M) is the vector space over R of all p−-forms ω on M (forp = 0 the smooth functions on M) and d n = d : Ω n−1 (M) → Ω n (M) is theexterior differential. A form ω ∈ Ω n (M) such that dω = 0 is a closed form orn–cocycle. A form ω ∈ Ω n (M) such that ω = dθ, where θ ∈ Ω n−1 (M), is anexact form or n–coboundary. Let Z n (M) = Ker(d) (resp. B n (M) = Im(d))denote a real vector space of cocycles (resp. coboundaries) of degree n.Since d n+1 d n = d 2 = 0, we have B n (M) ⊂ Z n (M). The quotient vectorspaceH n DR(M) = Ker(d)/ Im(d) = Z n (M)/B n (M)is the de Rham cohomology group. The elements of HDR n (M) representequivalence sets of cocycles. Two cocycles ω 1 , ω 2 belong to the sameequivalence set, or are cohomologous (written ω 1 ∼ ω 2 ) iff they differ by acoboundary ω 1 − ω 2 = dθ. The de Rham cohomology class of any formω ∈ Ω n (M) is [ω] ∈ HDR n (M). The de Rham differential complex (3.46) can


190 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbe considered as a system of second–order ODEs d 2 θ = 0, θ ∈ Ω n−1 (M)having a solution represented by Z n (M) = Ker(d).Analogously let M • denote the category of chains on the smooth manifoldM. When C = M • , we have the category S • (M • ) of generalizedchain complexes A • in M • , and if A n = 0 for n < 0 we have a subcategoryS• C (M • ) of chain complexes in M •A • : 0 ← C 0 (M)∂←− C 1 (M)∂←− C 2 (M) · · ·∂←− C n (M)∂←− · · · .Here A n = C n (M) is the vector space over R of all finite chains C on themanifold M and ∂ n = ∂ : C n+1 (M) → C n (M). A finite chain C suchthat ∂C = 0 is an n−cycle. A finite chain C such that C = ∂B is ann−boundary. Let Z n (M) = Ker(∂) (resp. B n (M) = Im(∂)) denote areal vector space of cycles (resp. boundaries) of degree n. Since ∂ n+1 ∂ n =∂ 2 = 0, we have B n (M) ⊂ Z n (M). The quotient vector spaceH C n (M) = Ker(∂)/ Im(∂) = Z n (M)/B n (M)is the n−homology group. The elements of H C n (M) are equivalence setsof cycles. Two cycles C 1 , C 2 belong to the same equivalence set, or arehomologous (written C 1 ∼ C 2 ), iff they differ by a boundary C 1 − C 2 =∂B). The homology class of a finite chain C ∈ C n (M) is [C] ∈ H C n (M).The dimension of the n−cohomology (resp. n−homology) group equalsthe nth Betti number b n (resp. b n ) of the manifold M. Poincaré lemmasays that on an open set U ∈ M diffeomorphic to R N , all closed forms(cycles) of degree p ≥ 1 are exact (boundaries). That is, the Betti numberssatisfy b p = 0 (resp. b p = 0) for p = 1, . . . , n.The de Rham Theorem states the following. The map Φ: H n ×H n → Rgiven by ([C], [ω]) → 〈C, ω〉 for C ∈ Z n , ω ∈ Z n is a bilinear nondegeneratemap which establishes the duality of the groups (vector spaces) H n and H nand the equality b n = b n .3.6.3.7 Hodge Star Operator and Harmonic FormsAs the configuration manifold M is an oriented ND Riemannian manifold,we may select an orientation on all tangent spaces T m M and all cotangentspaces T ∗ mM, with the local coordinates x i = (q i , p i ) at a point m ∈ M, in aconsistent manner. The simplest way to do that is to choose the Euclideanorthonormal basis ∂ 1 , ..., ∂ N of R N as being positive.Since the manifold M carries a Riemannian structure g = 〈, 〉, we havea scalar product on each T ∗ mM. So, we can define (as above) the linear


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 191Hodge star operator∗ : Λ p (T ∗ mM) → Λ N−p (T ∗ mM),which is a base point preserving operator∗ : Ω p (M) → Ω N−p (M),(Ω p (M) = Γ(Λ p (M)))(here Λ p (V ) denotes the p−fold exterior product of any vector space V ,Ω p (M) is a space of all p−forms on M, and Γ(E) denotes the space ofsections of the vector bundle E). Also,∗∗ = (−1) p(N−p) : Λ p (T ∗ x M) → Λ p (T ∗ mM).As the metric on TmM ∗ is given by g ij (x) = (g ij (x)) −1 , we have thevolume form defined in local coordinates as√∫∗(1) = det(g ij )dx 1 ∧ ... ∧ dx n , and Vol(M) = ∗(1).For any to p−forms α, β ∈ Ω p (M) with compact support, we define the(bilinear and positive definite) L 2 −product as∫∫(α, β) = 〈α, β〉 ∗ (1) = α ∧ ∗β.MWe can extend the product (·, ·) to L 2 (Ω p (M)); it remains bilinear andpositive definite, because as usual, in the definition of L 2 , functions thatdiffer only on a set of measure zero are identified.Using the Hodge star operator ∗, we can introduce the codifferentialoperator δ, which is formally adjoint to the exterior derivative d :Ω p (M) → Ω p+1 (M) on ⊕ N p=0Ω p (M) w.r.t. (·, ·). This means that forα ∈ Ω p−1 (M), β ∈ Ω p (M)(dα, β) = (α, δβ).Therefore, we have δ : Ω p (M) → Ω p−1 (M) andMδ = (−1) N(p+1)+1 ∗ d ∗ .Now, the Laplace–Beltrami operator (or, Hodge Laplacian, see [Griffiths(1983b); Voisin (2002)] as well as section (3.13.5.3) below), ∆ on Ω p (M),is defined by relation similar to (3.44) above∆ = dδ + δd : Ω p (M) → Ω p (M) (3.47)and an exterior differential form α ∈ Ω p (M) is called harmonic if ∆α = 0.M


192 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet M be a compact, oriented Riemannian manifold, E a vector bundlewith a bundle metric 〈·, ·〉 over M,D = d + A : Ω p−1 (Ad E ) → Ω p (Ad E ), with A ∈ Ω 1 (Ad E )– a tensorial and R−linear metric connection on E with curvature F D ∈Ω 2 (Ad E ) (Here by Ω p (Ad E ) we denote the space of those elements ofΩ p (End E ) for which the endomorphism of each fibre is skew symmetric;End E denotes the space of linear endomorphisms of the fibers of E).3.7 Lie Derivatives on Smooth ManifoldsLie derivative is popularly called ‘fisherman’s derivative’. In continuum mechanicsit is called Liouville operator. This is a central differential operatorin modern differential geometry and its physical and control applications.3.7.1 Lie Derivative Operating on FunctionsTo define how vector–fields operate on functions on an m−manifold M, wewill use the Lie derivative. Let f : M → R so T f : T M → T R = R × R.Following [Abraham et al. (1988)] we write T f acting on a vector v ∈ T m Min the formT f · v = (f(m), df(m) · v) .This defines, for each point m ∈ M, the element df(m) ∈ TmM.∗ Thusdf is a section of the cotangent bundle T ∗ M, i.e., a 1−form. The 1−formdf : M → T ∗ M defined this way is called the differential of f. If f is C k ,then df is C k−1 .If φ : U ⊂ M → V ⊂ E is a local chart for M, then the local representativeof f ∈ C k (M, R) is the map f : V → R defined by f = f ◦ φ −1 . Thelocal representative of T f is the tangent map for local manifolds,T f(x, v) = (f(x), Df(x) · v) .Thus the local representative of df is the derivative of the local representativeof f. In particular, if (x 1 , ..., x n ) are local coordinates on M, then thelocal components of df are(df) i = ∂ x if.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 193The introduction of df leads to the following definition of the Lie derivative.The directional or Lie derivative L X : C k (M, R) → C k−1 (M, R) of afunction f ∈ C k (M, R) along a vector–field X is defined byL X f(m) = X[f](m) = df(m) · X(m),for any m ∈ M. Denote by X[f] = df(X) the map M ∋ m ↦→ X[f](m) ∈ R.If f is F −valued, the same definition is used, but now X[f] is F −valued.If a local chart (U, φ) on an n−manifold M has local coordinates(x 1 , ..., x n ), the local representative of X[f] is given by the functionL X f = X[f] = X i ∂ x if.Evidently if f is C k and X is C k−1 then X[f] is C k−1 .Let ϕ : M → N be a diffeomorphism. Then L X is natural with respectto push–forward by ϕ. That is, for each f ∈ C k (M, R),L ϕ∗ X(ϕ ∗ f) = ϕ ∗ L X f,i.e., the following diagram commutes:C k ϕ(M, R) ∗ ✲ C k (N, R)L X❄❄C k (M, R) ✲ C k (N, R)ϕ ∗L ϕ∗ XAlso, L X is natural with respect to restrictions. That is, for U open inM and f ∈ C k (M, R),L X|U (f|U) = (L X f)|U,where |U : C k (M, R) → C k (U, R) denotes restriction to U, i.e., the followingdiagram commutes:C k |U(M, R) ✲ C k (U, R)L XL X|U❄❄C k (M, R) ✲ C k (U, R)|USince ϕ ∗ = (ϕ −1 ) ∗ the Lie derivative is also natural with respect topull–back by ϕ. This has a generalization to ϕ−related vector–fields as


194 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfollows: Let ϕ : M → N be a C k −map, X ∈ X k−1 (M) and Y ∈ X k−1 (N),k ≥ 1. If X ∼ ϕ Y , thenL X (ϕ ∗ f) = ϕ ∗ L Y ffor all f ∈ C k (N, R), i.e., the following diagram commutes:C k ϕ ∗(N, R) ✲ C k (M, R)L Y❄❄C k (N, R) ✲ C k (M, R)ϕ ∗The Lie derivative map L X : C k (M, R) → C k−1 (M, R) is a derivation,i.e., for two functions f, g ∈ C k (M, R) the Leibniz rule is satisfiedL XL X (fg) = gL X f + fL X g;Also, Lie derivative of a constant function is zero, L X (const) = 0.The connection between the Lie derivative L X f of a function f ∈C k (M, R) and the flow F t of a vector–field X ∈ X k−1 (M) is given as:ddt (F ∗ t f) = F ∗ t (L X f) .3.7.2 Lie Derivative of Vector FieldsIf X, Y ∈ X k (M), k ≥ 1 are two vector–fields on M, then[L X , L Y ] = L X ◦ L Y − L Y ◦ L Xis a derivation map from C k+1 (M, R) to C k−1 (M, R). Then there is a uniquevector–field, [X, Y ] ∈ X k (M) of X and Y such that L [X,Y ] = [L X , L Y ] and[X, Y ](f) = X (Y (f)) − Y (X(f)) holds for all functions f ∈ C k (M, R).This vector–field is also denoted L X Y and is called the Lie derivative ofY with respect to X, or the Lie bracket of X and Y . In a local chart(U, φ) at a point m ∈ M with coordinates (x 1 , ..., x n ), for X| U = X i ∂ x iand Y | U = Y i ∂ x i we have[X i ∂ x i, Y j ] (∂ x j = Xi ( ∂ x iY j) − Y i ( ∂ x iX j)) ∂ x j ,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 195since second partials commute. If, also X has flow F t , then [Abraham et al.(1988)]ddt (F ∗ t Y ) = F ∗ t (L X Y ) .In particular, if t = 0, this formula becomesddt | t=0 (F ∗ t Y ) = L X Y.Then the unique C k−1 vector–field L X Y = [X, Y ] on M defined by[X, Y ] = d dt | t=0 (F ∗ t Y ) ,is called the Lie derivative of Y with respect to X, or the Lie bracket of Xand Y, and can be interpreted as the leading order term that results fromthe sequence of flowsF −Yt◦ F −Xt ◦ F Y t ◦ F −Xt (m) = ɛ 2 [X, Y ](m) + O(ɛ 3 ), (3.48)for some real ɛ > 0. Therefore a Lie bracket can be interpreted as a ‘newdirection’ in which the system can flow, by executing the sequence of flows(3.48).Lie bracket satisfies the following property:[X, Y ][f] = X[Y [f]] − Y [X[f]],for all f ∈ C k+1 (U, R), where U is open in M.An important relationship between flows of vector–fields is given by theCampbell–Baker–Hausdorff formula:Ft Y ◦ Ft X = F X+Y + 1 2 [X,Y ]+ 1 12 ([X,[X,Y ]]−[Y,[X,Y ]])+...t (3.49)Essentially, if given the composition of multiple flows along multiple vector–fields, this formula gives the one flow along one vector–field which resultsin the same net flow. One way to prove the Campbell–Baker–Hausdorffformula (3.49) is to expand the product of two formal exponentials andequate terms in the resulting formal power series.Lie bracket is the R−bilinear map [, ] : X k (M)×X k (M) → X k (M) withthe following properties:(1) [X, Y ] = −[Y, X], i.e., L X Y = −L Y X for all X, Y ∈ X k (M) – skew–symmetry;(2) [X, X] = 0 for all X ∈ X k (M);


196 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(3) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 for all X, Y, Z ∈ X k (M) – theJacobi identity;(4) [fX, Y ] = f[X, Y ] − (Y f)X, i.e., L fX (Y ) = f(L X Y ) − (L Y f)X for allX, Y ∈ X k (M) and f ∈ C k (M, R);(5) [X, fY ] = f[X, Y ] + (Xf)Y , i.e., L X (fY ) = f(L X Y ) + (L X f)Y for allX, Y ∈ X k (M) and f ∈ C k (M, R);(6) [L X , L Y ] = L [x,y] for all X, Y ∈ X k (M).The pair (X k (M), [, ]) is the prototype of a Lie algebra [Kolar et al.(1993)]. In more general case of a general linear Lie algebra gl(n), which isthe Lie algebra associated to the Lie group GL(n), Lie bracket is given bya matrix commutator[A, B] = AB − BA,for any two matrices A, B ∈ gl(n).Let ϕ : M → N be a diffeomorphism. Then L X : X k (M) → X k (M) isnatural with respect to push–forward by ϕ. That is, for each f ∈ C k (M, R),L ϕ∗ X(ϕ ∗ Y ) = ϕ ∗ L X Y,i.e., the following diagram commutes:X k (M)ϕ ∗✲ X k (N)L X❄X k (M)ϕ ∗L ϕ∗ X❄✲ X k (N)Also, L X is natural with respect to restrictions. That is, for U open inM and f ∈ C k (M, R),[X|U, Y |U] = [X, Y ]|U,where U : C k (M, R) → C k (U, R) denotes restriction to U, i.e., the followingdiagram commutes [Abraham et al. (1988)]:X k (M)|U✲ X k (U)L X❄X k (M)|UL X|U❄✲ X k (U)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 197If a local chart (U, φ) on an n−manifold M has local coordinates(x 1 , ..., x n ), then the local components of a Lie bracket are[X, Y ] j = X i ∂ x iY j − Y i ∂ x iX j ,that is, [X, Y ] = (X · ∇)Y − (Y · ∇)X.Let ϕ : M → N be a C k −map, X ∈ X k−1 (M) and Y ∈ X k−1 (N),k ≥ 1. Then X ∼ ϕ Y , iff(Y [f]) ◦ ϕ = X[f ◦ ϕ]for all f ∈ C k (V, R), where V is open in N.For every X ∈ X k (M), the operator L X is a derivation on(C k (M, R), X k (M) ) , i.e., L X is R−linear.For any two vector–fields X ∈ X k (M) and Y ∈ X k (N), k ≥ 1 withflows F t and G t , respectively, if [X, Y ] = 0 then F ∗ t Y = Y and G ∗ t X = X.3.7.3 Time Derivative of the Evolution OperatorRecall that the time–dependent flow or evolution operator F t,s of a vector–field X ∈ X k (M) is defined by the requirement that t ↦→ F t,s (m) be theintegral curve of X starting at a point m ∈ M at time t = s, i.e.,ddt F t,s(m) = X (t, F t,s (m)) and F t,t (m) = m.By uniqueness of integral curves we have F t,s ◦ F s,r = F t,r (replacing theflow property F t+s = F t + F s ) and F t,t = identity.Let X t ∈ X k (M), k ≥ 1 for each t and suppose X(t, m) is continuous in(t, m) ∈ R×M. Then F t,s is of class C k and for f ∈ C k+1 (M, R) [Abrahamet al. (1988)], and Y ∈ X k (M), we have(1) d dt F t,s ∗ f = Ft,s ∗ (L Xt f) , and(2) d dt F t,s ∗ f = Ft,s([X ∗ t , Y ]) = Ft,s ∗ (L Xt Y ).From the above Theorem, the following identity holds:ddt F ∗ t,s f = −X t[F∗t,s f ] .3.7.4 Lie Derivative of <strong>Diff</strong>erential FormsSince F : M =⇒ Λ k T ∗ M is a vector bundle functor on M, the Lie derivative(4.3.2) of a k−form α ∈ Ω k (M) along a vector–field X ∈ X k (M) is


198 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondefined byIt has the following properties:L X α = d dt | t=0 F ∗ t α.(1) L X (α ∧ β) = L X α ∧ β + α ∧ L X β, so L X is a derivation.(2) [L X , L Y ] α = L [X,Y ] α.(3) d dt F ∗ t α = F ∗ t L X α = L X (F ∗ t α).Formula (3) holds also for time–dependent vector–fields in the sensethatd dt F (t,sα ∗ = Ft,sL ∗ X α = L X F∗t,s α ) and in the expression L X α thevector–field X is evaluated at time t.The famous Cartan magic formula (see [Marsden and Ratiu (1999)])states: the Lie derivative of a k−form α ∈ Ω k (M) along a vector–fieldX ∈ X k (M) on a smooth manifold M is defined asL X α = di X α + i X dα = d(X⌋α) + X⌋dα.Also, the following identities hold [Marsden and Ratiu (1999); Kolaret al. (1993)]:(1) L fX α = fL X α + df ∧ i x α.(2) L [X,Y ] α = L X L Y α − L Y L X α.(3) i [X,Y ] α = L X i Y α − i Y L X α.(4) L X dα = dL X α, i.e., [L X , d] = 0.(5) L X i X α = i X L X α, i.e., [L X , i X ] = 0.(6) L X (α ∧ β) = L X α ∧ β + α ∧ L X β.3.7.5 Lie Derivative of Various Tensor FieldsIn this section, we use local coordinates x i (i = 1, ..., n) on a biodynamicaln−manifold M, to calculate the Lie derivative L X i with respect to a genericvector–field X i . (As always, ∂ x i ≡ ∂∂x i ).Lie Derivative of a Scalar FieldGiven the scalar field φ, its Lie derivative L X iφ is given asL X iφ = X i ∂ x iφ = X 1 ∂ x 1φ + X 2 ∂ x 2φ + ... + X n ∂ x nφ.Lie Derivative of Vector and Covector–Fields


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 199asGiven a contravariant vector–field V i , its Lie derivative L X iV i is givenL X iV i = X k ∂ x kV i − V k ∂ x kX i ≡ [X i , V i ] − the Lie bracket.Given a covariant vector–field (i.e., a one–form) ω i , its Lie derivative L X iω iis given asL X iω i = X k ∂ x kω i + ω k ∂ x iX k .Lie Derivative of a Second–Order Tensor–FieldGiven a (2, 0) tensor–field S ij , its Lie derivative L X iS ij is given asL X iS ij = X i ∂ x iS ij − S ij ∂ x iX i − S ii ∂ x iX j .Given a (1, 1) tensor–field S i j , its Lie derivative L X iSi jis given asL X iS i j = X i ∂ x iS i j − S i j∂ x iX i + S i i∂ x j X i .Given a (0, 2) tensor–field S ij , its Lie derivative L X iS ij is given asL X iS ij = X i ∂ x iS ij + S ij ∂ x iX i + S ii ∂ x j X i .Lie Derivative of a Third–Order Tensor–FieldGiven a (3, 0) tensor–field T ijk , its Lie derivative L X iT ijk is given asL X iT ijk = X i ∂ x iT ijk − T ijk ∂ x iX i − T iik ∂ x iX j − T iji ∂ x iX k .Given a (2, 1) tensor–field T ijk , its Lie derivative L ijXiTkis given asL X iT ijk= Xi ∂ x iT ijk− T ijk ∂ x iXi + T iji∂ x kXi − T iik ∂ x iX j .Given a (1, 2) tensor–field Tjk i , its Lie derivative L X iT jk i is given asL X iT i jk = X i ∂ x iT i jk − T i jk∂ x iX i + T i ik∂ x j X i + T i ji∂ x kX i .Given a (0, 3) tensor–field T ijk , its Lie derivative L X iT ijk is given asL X iT ijk = X i ∂ x iT ijk + T ijk ∂ x iX i + T iik ∂ x j X i + T iji ∂ x kX i .Lie Derivative of a Fourth–Order Tensor–Field


200 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionGiven a (4, 0) tensor–field R ijkl , its Lie derivative L X iR ijkl is given asL X iR ijkl = X i ∂ x iR ijkl −R ijkl ∂ x iX i −R iikl ∂ x iX j −R ijil ∂ x iX k −R ijki ∂ x iX l .Given a (3, 1) tensor–field R ijkl, its Lie derivative L X iR ijklis given asL X iR ijkl= X i ∂ x iR ijkl− R ijkl∂ x iX i + R ijki∂ x lX i − Rliik ∂ x iX j − R ijil∂ x iX k .Given a (2, 2) tensor–field R ijkl , its Lie derivative L X iRij klis given asL X iR ijkl = Xi ∂ x iR ijkl − Rij kl ∂ x iXi + R ijil ∂ x kXi + R ijki ∂ x lXi − R iikl∂ x iX j .Given a (1, 3) tensor–field Rjkl i , its Lie derivative L X iRi jklis given asL X iR i jkl = X i ∂ x iR i jkl − R i jkl∂ x iX i + R i ikl∂ x j X i + R i jil∂ x kX i + R i jki∂ x lX i .Given a (0, 4) tensor–field R ijkl , its Lie derivative L X iR ijkl is given asL X iR ijkl = X i ∂ x iR ijkl +R ijkl ∂ x iX i +R iikl ∂ x j X i +R ijil ∂ x kX i +R ijki ∂ x lX i .Finally, recall that a spinor is a two–component complex column vector.Physically, spinors can describe both bosons and fermions, while tensors candescribe only bosons. The Lie derivative of a spinor φ is defined by¯φt (x) − φ(x)L X φ(x) = lim,t→0 twhere ¯φ t is the image of φ by a one–parameter group of isometries with Xits generator. For a vector–field X a and a covariant derivative ∇ a , the Liederivative of φ is given explicitly byL X φ = X a ∇ a φ − 1 8 (∇ aX b − ∇ b X a ) γ a γ b φ,where γ a and γ b are Dirac matrices (see, e.g., [Choquet-Bruhat andDeWitt-Morete (2000)]).3.7.6 Application: Lie–Derivative NeurodynamicsA Lie–derivative neuro–classifier is a self–organized, associative–memorymachine, represented by oscillatory (excitatory/inhibitory) tensor–field–system(x, v, ω) on the Banach manifold M. It consists of continual neural activation(x, y)−-dynamics and self–organizing synaptic learning ω−-dynamics.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 201The continual activation (x, y)−dynamics, is defined as a system of twocoupled, first–order oscillator tensor–fields, dual to each other, in a localBanach chart U α , (α = 1, ..., n) on M:1) an excitatory neural vector–field x i = x i (t) : M → T M, representinga cross–section of the tangent bundle T M; and2) an inhibitory neural one–form y i = y i (t) : M → T ∗ M, representinga cross–section of the cotangent bundle T ∗ M.The self–organized learning ω−dynamics is performed on a second–ordersymmetrical synaptic tensor–field ω = ω(t), given by its covariant componentsω ij = ω ij (t) and its contravariant components ω ij = ω ij (t), wherei, j = 1, ..., n.Starting with the Lyapunov–stable, negative scalar neural action potential:U = − 1 2 (ω ijx i x j + ω ij y i y j ),(i, j = 1, ..., n),the (x, y)−-dynamics is given in two versions, which are compared andcontrasted:(1) the Lie–linear neurodynamics with first–order Lie derivativesẋ i = J i + L X U, ẏ i = J i + L Y U,and(2) the Lie–quadratic neurodynamics with both first and second–orderLie derivativesẋ i = J i + L X U + L X L X U, ẏ i = J i + L Y U + L Y L Y U,where X = S i (x i ), Y = S i (y i ), S i represent sigmoid activation functions,while L X L X , L Y L Y : F (M) → F (M) denote the second–order (iterated)Lie derivatives.Self–organized learning ω−-dynamics is presented in the form of differentialHebbian learning scheme in both covariant and contravariant forms˙ω ij = −ω ij + S i (x i )S j (y j ) + Ṡi(x i )Ṡj(y j ),˙ω ij = −ω ij + S i (x i )S j (y j ) + Ṡi(x i )Ṡj(y j ),and(i, j = 1, ..., n),respectively.


202 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.7.7 Lie AlgebrasRecall from Introduction that an algebra A is a vector space with a product.The product must have the property thata(uv) = (au)v = u(av),for every a ∈ R and u, v ∈ A. A map φ : A → A ′ between algebras is calledan algebra homomorphism if φ(u · v) = φ(u) · φ(v). A vector subspace Iof an algebra A is called a left ideal (resp. right ideal) if it is closed underalgebra multiplication and if u ∈ A and i ∈ I implies that ui ∈ I (resp.iu ∈ I). A subspace I is said to be a two–sided ideal if it is both a left andright ideal. An ideal may not be an algebra itself, but the quotient of analgebra by a two–sided ideal inherits an algebra structure from A.A Lie algebra is an algebra A where the multiplication, i.e., the Liebracket (u, v) ↦→ [u, v], has the following properties:LA 1. [u, u] = 0 for every u ∈ A, andLA 2. [u, [v, w]] + [w, [u, v]] + [v, w, u]] = 0 for all u, v, w ∈ A.The condition LA 2 is usually called Jacobi identity. A subspace E ⊂ Aof a Lie algebra is called a Lie subalgebra if [u, v] ∈ E for every u, v ∈ E. Amap φ : A → A ′ between Lie algebras is called a Lie algebra homomorphismif φ([u, v]) = [φ(u), φ(v)] for each u, v ∈ A.All Lie algebras (over a given field K) and all smooth homomorphismsbetween them form the category LAL, which is itself a complete subcategoryof the category AL of all algebras and their homomorphisms.3.8 Lie Groups and Associated Lie AlgebrasIn the middle of the 19th Century S. Lie made a far reaching discoverythat techniques designed to solve particular unrelated types of ODEs, suchas separable, homogeneous and exact equations, were in fact all specialcases of a general form of integration procedure based on the invariance ofthe differential equation under a continuous group of symmetries. Roughlyspeaking a symmetry group of a system of differential equations is a groupthat transforms solutions of the system to other solutions. Once the symmetrygroup has been identified a number of techniques to solve and classifythese differential equations becomes possible. In the classical framework ofLie, these groups were local groups and arose locally as groups of transformationson some Euclidean space. The passage from the local Lie group tothe present day definition using manifolds was accomplished by E. Cartan


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 203at the end of the 19th Century, whose work is a striking synthesis of Lietheory, classical geometry, differential geometry and topology.These continuous groups, which originally appeared as symmetry groupsof differential equations, have over the years had a profound impact ondiverse areas such as algebraic topology, differential geometry, numericalanalysis, control theory, classical mechanics, quantum mechanics etc. Theyare now universally known as Lie groups.3.8.1 Definition of a Lie GroupA Lie group is a smooth (Banach) manifold M that has at the same time agroup G−structure consistent with its manifold M−structure in the sensethat group multiplicationand the group inversionµ : G × G → G, (g, h) ↦→ gh (3.50)ν : G → G, g ↦→ g −1 (3.51)are C k −maps [Chevalley (1955); Abraham et al. (1988); Marsden and Ratiu(1999); Puta (1993)]. A point e ∈ G is called the group identity element.For example, any nD Banach vector space V is an Abelian Lie groupwith group operations µ : V × V → V , µ(x, y) = x + y, and ν : V → V ,ν(x) = −x. The identity is just the zero vector. We call such a Lie groupa vector group.Let G and H be two Lie groups. A map G → H is said to be a morphismof Lie groups (or their smooth homomorphism) if it is their homomorphismas abstract groups and their smooth map as manifolds [Postnikov (1986)].All Lie groups and all their morphisms form the category LG (moreprecisely, there is a countable family of categories LG depending onC k −smoothness of the corresponding manifolds).Similarly, a group G which is at the same time a topological spaceis said to be a topological group if maps (3.50–3.51) are continuous, i.e.,C 0 −maps for it. The homomorphism G → H of topological groups is saidto be continuous if it is a continuous map. Topological groups and theircontinuous homomorphisms form the category T G.A topological group (as well as a smooth manifold) is not necessarilyHausdorff. A topological group G is Hausdorff iff its identity is closed. Asa corollary we have that every Lie group is a Hausdorff topological group(see [Postnikov (1986)]).


204 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFor every g in a Lie group G, the two maps,L g : G → G, h ↦→ gh, andR h : G → G,g ↦→ gh,are called left and right translation maps. Since L g ◦ L h = L gh , and R g ◦R h = R gh , it follows that (L g ) −1 = L g −1 and (R g ) −1 = R g −1, so both L gand R g are diffeomorphisms. Moreover L g ◦ R h = R h ◦ L g , i.e., left andright translation commute.A vector–field X on G is called left–invariant vector–field if for everyg ∈ G, L ∗ gX = X, that is, if (T h L g )X(h) = X(gh) for all h ∈ G, i.e., thefollowing diagram commutes:T G✻XT L g✲ T G✻XGL g✲ GThe correspondences G → T G and L g → T L g obviously define a functorF : LG ⇒ LG from the category G of Lie groups to itself. F is a specialcase of the vector bundle functor (see (4.3.2) below).Let X L (G) denote the set of left–invariant vector–fields on G; it is aLie subalgebra of X (G), the set of all vector–fields on G, since L ∗ g[X, Y ] =[L ∗ gX, L ∗ gY ] = [X, Y ], so the Lie bracket [X, Y ] ∈ X L (G).Let e be the identity element of G. Then for each ξ on the tangent spaceT e G we define a vector–field X ξ on G byX ξ (g) = T e L g (ξ).X L (G) and T e G are isomorphic as vector spaces. Define the Lie bracket onT e G by[ξ, η] = [X ξ , X η ] (e),for all ξ, η ∈ T e G. This makes T e G into a Lie algebra. Also, by construction,we have[X ξ , X η ] = X [ξ,η] ,this defines a bracket in T e G via left extension. The vector space T e G withthe above algebra structure is called the Lie algebra of the Lie group G andis denoted g.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 205For example, let V be a nD vector space. Then T e V ≃ V and theleft–invariant vector–field defined by ξ ∈ T e V is the constant vector–fieldX ξ (η) = ξ, for all η ∈ V . The Lie algebra of V is V itself.Since any two elements of an Abelian Lie group G commute, it followsthat all adjoint operators Ad g , g ∈ G, equal the identity. Therefore, theLie algebra g is Abelian; that is, [ξ, η] = 0 for all ξ, η ∈ g [Marsden andRatiu (1999)].Recall (3.7.7) that Lie algebras and their smooth homomorphisms formthe category LAL. We can now introduce the fundamental Lie functor,F : LG ⇒ LAL, from the category of Lie groups to the category of Liealgebras [Postnikov (1986)].Let X ξ be a left–invariant vector–field on G corresponding to ξ in g.Then there is a unique integral curve γ ξ : R → G of X ξ starting at e, i.e.,˙γ ξ (t) = X ξ(γξ (t) ) , γ ξ (0) = e.γ ξ (t) is a smooth one–parameter subgroup of G, i.e.,γ ξ (t + s) = γ ξ (t) · γ ξ (s),since, as functions of t both sides equal γ ξ (s) at t = 0 and both satisfydifferential equation˙γ(t) = X ξ(γξ (t) )by left invariance of X ξ , so they are equal. Left invariance can be also usedto show that γ ξ (t) is defined for all t ∈ R. Moreover, if φ : R → G is aone–parameter subgroup of G, i.e., a smooth homomorphism of the additivegroup R into G, then φ = γ ξ with ξ = ˙φ(0), since taking derivative at s = 0in the relationφ(t + s) = φ(t) · φ(s) gives ˙φ(t) = X ˙φ(0)(φ(t)) ,so φ = γ ξ since both equal e at t = 0. Therefore, all one–parametersubgroups of G are of the form γ ξ (t) for some ξ ∈ g.The map exp : g → G, given byexp(ξ) = γ ξ (1), exp(0) = e, (3.52)is called the exponential map of the Lie algebra g of G into G. exp is aC k −-map, similar to the projection π of tangent and cotangent bundles;exp is locally a diffeomorphism from a neighborhood of zero in g onto a


206 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionneighborhood of e in G; if f : G → H is a smooth homomorphism of Liegroups, thenf ◦ exp G = exp H ◦T e f .Also, in this case (see [Chevalley (1955); Marsden and Ratiu (1999);Postnikov (1986)])exp(sξ) = γ ξ (s).Indeed, for fixed s ∈ R, the curve t ↦→ γ ξ (ts), which at t = 0 passes throughe, satisfies the differential equationddt γ ξ(ts) = sX ξ(γξ (ts) ) = X sξ(γξ (ts) ) .Since γ sξ (t) satisfies the same differential equation and passes through e att = 0, it follows that γ sξ (t) = γ ξ (st). Putting t = 1 induces exp(sξ) = γ ξ (s)[Marsden and Ratiu (1999)].Hence exp maps the line sξ in g onto the one–parameter subgroup γ ξ (s)of G, which is tangent to ξ at e. It follows from left invariance that theflow F ξ t of X satisfies F ξ t (g) = g exp(sξ).Globally, the exponential map exp, as given by (3.52), is a naturaloperation, i.e., for any morphism ϕ : G → H of Lie groups G and H and aLie functor F, the following diagram commutes [Postnikov (1986)]:F(G)exp❄GF(ϕ)ϕ✲ F(H)exp❄✲ HLet G 1 and G 2 be Lie groups with Lie algebras g 1 and g 2 . Then G 1 ×G 2is a Lie group with Lie algebra g 1 × g 2 , and the exponential map is givenby [Marsden and Ratiu (1999)].exp : g 1 × g 2 → G 1 × G 2 , (ξ 1 , ξ 2 ) ↦→ (exp 1 (ξ 1 ), exp 2 (ξ 2 )) .For example, in case of a nD vector space, or infinite–dimensional Banachspace, the exponential map is the identity.The unit circle in the complex plane S 1 = {z ∈ C : |z| = 1} is an AbelianLie group under multiplication. The tangent space T e S 1 is the imaginary


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 207axis, and we identify R with T e S 1 by t ↦→ 2πit. With this identification,the exponential map exp : R → S 1 is given by exp(t) = e 2πit .The nD torus T n = S 1 ×···×S 1 (n times) is an Abelian Lie group. Theexponential map exp : R n → T n is given bySince S 1 = R/Z, it follows thatexp(t 1 , ..., t n ) = (e 2πit1 , ..., e 2πitn ).T n = R n /Z n ,the projection R n → T n being given by the exp map (see [Marsden andRatiu (1999); Postnikov (1986)]).For every g ∈ G, the mapAd g = T e(Rg −1 ◦ L g): g → gis called the adjoint map (or operator) associated with g.For each ξ ∈ g and g ∈ G we haveexp (Ad g ξ) = g (exp ξ) g −1 .The relation between the adjoint map and the Lie bracket is the following:For all ξ, η ∈ g we haveddt∣ Ad exp(tξ) η = [ξ, η].t=0A Lie subgroup H of G is a subgroup H of G which is also a submanifoldof G. Then h is a Lie subalgebra of g and moreover h = {ξ ∈ g| exp(tξ) ∈ H,for all t ∈ R}.Recall that one can characterize Lebesgue measure up to a multiplicativeconstant on R n by its invariance under translations. Similarly, on a locallycompact group there is a unique (up to a nonzero multiplicative constant)left–invariant measure, called Haar measure. For Lie groups the existenceof such measures is especially simple [Marsden and Ratiu (1999)]: Let Gbe a Lie group. Then there is a volume form Ub5, unique up to nonzeromultiplicative constants, that is left–invariant. If G is compact, Ub5 is rightinvariant as well.3.8.2 Actions of Lie Groups on Smooth ManifoldsLet M be a smooth manifold. An action of a Lie group G (with the unitelement e) on M is a smooth map φ : G × M → M, such that for all x ∈ M


208 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand g, h ∈ G, (i) φ(e, x) = x and (ii) φ (g, φ(h, x)) = φ(gh, x). In otherwords, letting φ g : x ∈ M ↦→ φ g (x) = φ(g, x) ∈ M, we have (i’) φ e = id Mand (ii’) φ g ◦ φ h = φ gh . φ g is a diffeomorphism, since (φ g ) −1 = φ g −1. Wesay that the map g ∈ G ↦→ φ g ∈ <strong>Diff</strong>(M) is a homomorphism of G into thegroup of diffeomorphisms of M. In case that M is a vector space and eachφ g is a linear operator, the function of G on M is called a representationof G on M [Puta (1993)]An action φ of G on M is said to be transitive group action, if for everyx, y ∈ M, there is g ∈ G such that φ(g, x) = y; effective group action, ifφ g = id M implies g = e, that is g ↦→ φ g is 1–1; and free group action, if foreach x ∈ M, g ↦→ φ g (x) is 1–1.For example,(1) G = R acts on M = R by translations; explicitly,φ : G × M → M, φ(s, x) = x + s.Then for x ∈ R, O x = R. Hence M/G is a single point, and the actionis transitive and free.(2) A complete flow φ t of a vector–field X on M gives an action of R onM, namely(t, x) ∈ R × M ↦→ φ t (x) ∈ M.(3) Left translation L g : G → G defines an effective action of G on itself.It is also transitive.(4) The coadjoint action of G on g ∗ is given byAd ∗ : (g, α) ∈ G × g ∗ ↦→ Ad ∗ g −1(α) = ( T e (R g −1 ◦ L g ) ) ∗α ∈ g ∗ .Let φ be an action of G on M. For x ∈ M the orbit of x is defined byO x = {φ g (x)|g ∈ G} ⊂ Mand the isotropy group of φ at x is given byG x = {g ∈ G|φ(g, x) = x} ⊂ G.An action φ of G on a manifold M defines an equivalence relation onM by the relation belonging to the same orbit; explicitly, for x, y ∈ M, wewrite x ∼ y if there exists a g ∈ G such that φ(g, x) = y, that is, if y ∈ O x .The set of all orbits M/G is called the group orbit space.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 209For example, let M = R 2 \{0}, G = SO(2), the group of rotations inplane, and the action of G on M given by([ ] )cos θ − sin θ, (x, y) ↦−→ (x cos θ − y sin θ, x sin θ + y cos θ).sin θ cos θThe action is always free and effective, and the orbits are concentric circles,thus the orbit space is M/G ≃ R ∗ +.A crucial concept in mechanics is the infinitesimal description of anaction. Let φ : G × M → M be an action of a Lie group G on a smoothmanifold M. For each ξ ∈ g,φ ξ : R × M → M, φ ξ (t, x) = φ (exp(tξ), x)is an R−-action on M. Therefore, φ exp(tξ) : M → M is a flow on M; thecorresponding vector–field on M, given byξ M (x) = d dt∣ φ exp(tξ) (x)t=0is called the infinitesimal generator of the action, corresponding to ξ in g.The tangent space at x to an orbit O x is given byT x O x = {ξ M (x)|ξ ∈ g}.Let φ : G×M → M be a smooth G−−action. For all g ∈ G, all ξ, η ∈ gand all α, β ∈ R, we have:(Ad g ξ) M= φ ∗ g −1ξ M , [ξ M , η M ] = − [ξ, η] M, and (αξ + βη) M = αξ M +βη M .Let M be a smooth manifold, G a Lie group and φ : G × M → M aG−action on M. We say that a smooth map f : M → M is with respectto this action if for all g ∈ G,f ◦ φ g = φ g ◦ f.Let f : M → M be an equivariant smooth map. Then for any ξ ∈ g wehaveT f ◦ ξ M = ξ M ◦ f.


210 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.8.3 Basic Dynamical Lie GroupsHere we give the first two examples of Lie groups, namely Galilei groupand general linear group. Further examples will be given in associationwith particular dynamical systems.3.8.3.1 Galilei GroupThe Galilei group is the group of transformations in space and time thatconnect those Cartesian systems that are termed ‘inertial frames’ in Newtonianmechanics. The most general relationship between two such framesis the following. The origin of the time scale in the inertial frame S ′ maybe shifted compared with that in S; the orientation of the Cartesian axes inS ′ may be different from that in S; the origin O of the Cartesian frame inS ′ may be moving relative to the origin O in S at a uniform velocity. Thetransition from S to S ′ involves ten parameters; thus the Galilei group is aten parameter group. The basic assumption inherent in Galilei–Newtonianrelativity is that there is an absolute time scale, so that the only way inwhich the time variables used by two different ‘inertial observers’ could possiblydiffer is that the zero of time for one of them may be shifted relativeto the zero of time for the other.Galilei space–time structure involves the following three elements:(1) World, as a 4D affine space A 4 . The points of A 4 are called world pointsor events. The parallel transitions of the world A 4 form a linear (i.e.,Euclidean) space R 4 .(2) Time, as a linear map t : R 4 → R of the linear space of the world paralleltransitions onto the real ‘time axes’. Time interval from the event a ∈A 4 to b ∈ A 4 is called the number t(b−a); if t(b−a) = 0 then the eventsa and b are called synchronous. The set of all mutually synchronousevents consists a 3D affine space A 3 , being a subspace of the world A 4 .The kernel of the mapping t consists of the parallel transitions of A 4translating arbitrary (and every) event to the synchronous one; it is alinear 3D subspace R 3 of the space R 4 .(3) Distance (metric) between the synchronous events,ρ(a, b) =‖ a − b ‖, for all a, b ∈ A 3 ,given by the scalar product in R 3 . The distance transforms arbitraryspace of synchronous events into the well known 3D Euclidean spaceE 3 .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 211The space A 4 , with the Galilei space–time structure on it, is calledGalilei space. Galilei group is the group of all possible transformationsof the Galilei space, preserving its structure. The elements of the Galileigroup are called Galilei transformations. Therefore, Galilei transformationsare affine transformations of the world A 4 preserving the time intervals anddistances between the synchronous events.The direct product R × R 3 , of the time axes with the 3D linear spaceR3 with a fixed Euclidean structure, has a natural Galilei structure. It iscalled Galilei coordinate system.3.8.3.2 General Linear GroupThe group of linear isomorphisms of R n to R n is a Lie group of dimensionn 2 , called the general linear group and denoted Gl(n, R). It is a smoothmanifold, since it is a subset of the vector space L(R n , R n ) of all linear mapsof R n to R n , as Gl(n, R) is the inverse image of R\{0} under the continuousmap A ↦→ det A of L(R n , R n ) to R. The group operation is compositionand the inverse map is(A, B) ∈ Gl(n, R) × Gl(n, R) ↦→ A ◦ B ∈ Gl(n, R)A ∈ Gl(n, R) ↦→ A −1 ∈ Gl(n, R).If we choose a basis in R n , we can represent each element A ∈ Gl(n, R)by an invertible (n × n)−-matrix. The group operation is then matrixmultiplication and the inversion is matrix inversion. The identity is theidentity matrix I n . The group operations are smooth since the formulas forthe product and inverse of matrices are smooth in the matrix components.The Lie algebra of Gl(n, R) is gl(n), the vector space L(R n , R n ) of alllinear transformations of R n , with the commutator bracketFor every A ∈ L(R n , R n ),[A, B] = AB − BA.γ A : t ∈ R ↦→γ A (t) =∞∑i=0is a one–parameter subgroup of Gl(n, R), becauseγ A (0) = I, and ˙γ A (t) =t ii! Ai ∈ Gl(n, R)∞∑i=0t i−1(i − 1)! Ai = γ A (t) A.


212 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionHence γ A is an integral curve of the left–invariant vector–field X A . Therefore,the exponential map is given byexp : A ∈ L(R n , R n ) ↦→ exp(A) ≡ e A = γ A (1) =∞∑i=0For each A ∈ Gl(n, R) the corresponding adjoint mapA ii!∈ Gl(n, R).Ad A : L(R n , R n ) → L(R n , R n )is given byAd A B = A · B · A −1 .3.8.4 Application: Lie Groups in Biodynamics3.8.4.1 Lie Groups of Joint RotationsRecall (see [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]) that local kinematics at eachrotational robot or (synovial) human joint, is defined as a group action ofan nD constrained rotational Lie group SO(n) on the Euclidean space R n .In particular, there is an action of SO(2)−-group in uniaxial human joints(cylindrical, or hinge joints, like knee and elbow) and an action of SO(3)−groupin three–axial human joints (spherical, or ball–and–socket joints, likehip, shoulder, neck, wrist and ankle). In both cases, SO(n) acts, with itsoperators of rotation, on the vector x = {x µ }, (i = 1, 2, 3) of external,Cartesian coordinates of the parent body–segment, depending, at the sametime, on the vector q = {q s }, (s = 1, · · · , n) on n group–parameters, i.e.,joint angles.Each joint rotation R ∈ SO(n) defines a mapR : x µ ↦→ ẋ µ , R(x µ , q s ) = R q sx µ ,where R q s ∈ SO(n) are joint group operators. The vector v = {v s }, (s =1, · · · , n) of n infinitesimal generators of these rotations, i.e., joint angularvelocities, given byv s = −[ ∂R(xµ , q s )∂q s ] q=0∂∂x µ ,constitute an nD Lie algebra so(n) corresponding to the joint rotation groupSO(n). Conversely, each joint group operator R q s, representing a one–parameter subgroup of SO(n), is defined as the exponential map of the


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 213corresponding joint group generator v sR q s = exp(q s v s ). (3.53)The exponential map (3.53) represents a solution of the joint operator differentialequation in the joint group–parameter space {q s }dR q sdq s = v s R q s.Uniaxial Group of Joint RotationsThe uniaxial joint rotation in a single Cartesian plane around a perpendicularaxis, e.g., xy−plane about the z axis, by an internal joint angle θ,leads to the following transformation of the joint coordinatesẋ = x cos θ − y sin θ, ẏ = x sin θ + y cos θ.In this way, the joint SO(2)−group, given by{ ( ) }cos θ − sin θSO(2) = R θ =|θ ∈ [0, 2π] ,sin θ cos θacts in a canonical way on the Euclidean plane R 2 by(( ) ( ))cos θ − sin θ xSO(2) =, ↦−→sin θ cos θ yIts associated Lie algebra so(2) is given by{( ) }0 −tso(2) = |t ∈ R ,t 0( x cos θ −y sin θx sin θ y cos θsince the curve γ θ ∈ SO(2) given by( )cos tθ − sin tθγ θ : t ∈ R ↦−→ γ θ (t) =∈ SO(2),sin tθ cos tθpasses through the identity I 2 =ddt∣ γ θ (t) =t=0( ) 1 0and then0 1( ) 0 −θ,θ 0so that I 2 is a basis of so(2), since dim (SO(2)) = 1.).


214 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe exponential map exp : so(2) → SO(2) is given by( ) ( )0 −θcos tθ − sin tθexp = γθ 0 θ (1) =.sin tθ cos tθThe infinitesimal generator of the action of SO(2) on R 2 , i.e., jointangular velocity v, is given byv = −y ∂∂x + x ∂ ∂y ,sincev R 2 (x, y) = d dt∣ exp(tv) (x, y) = d ( ) ( )cos tv − sin tv xt=0dt∣ .t=0sin tv cos tv yThe momentum map (see subsection 3.12.3.5 below) J : T ∗ R 2associated to the lifted action of SO(2) on T ∗ R 2 ≃ R 4 is given by→ RJ (x, y, p 1 , p 2 ) = xp y − yp x ,sinceJ (x, y, p x , p y ) (ξ) = (p x dx + p y dy)(v R 2) = −vp x y + −vp y x.The Lie group SO(2) acts on the symplectic manifold (R 4 , ω = dp x ∧dx + dp y ∧ dx) by(( ))cos θ − sin θφ, (x, y, p x , p y )sin θ cos θ= (x cos θ − y sin θ, x sin θ + y cos θ, p x cos θ − p y sin θ, p x sin θ + p y cos θ) .Three–Axial Group of Joint RotationsThe three–axial SO(3)−group of human–like joint rotations depends onthree parameters, Euler joint angles q i = (ϕ, ψ, θ), defining the rotationsabout the Cartesian coordinate triedar (x, y, z) placed at the joint pivotpoint. Each of the Euler angles are defined in the constrained range (−π, π),so the joint group space is a constrained sphere of radius π.Let G = SO(3) = {A ∈ M 3×3 (R) : A t A = I 3 , det(A) = 1} be thegroup of rotations in R 3 . It is a Lie group and dim(G) = 3. Let us isolateits one–parameter joint subgroups, i.e., consider the three operators of thefinite joint rotations R ϕ , R ψ , R θ ∈ SO(3), given by⎡⎤ ⎡⎤ ⎡⎤1 0 0cos ψ 0 sin ψcos θ − sin θ 0R ϕ = ⎣0 cos ϕ − sin ϕ0 sin ϕ cos ϕ⎦ , R ψ = ⎣0 1 0− sin ψ 0 cos ψ⎦ , R θ = ⎣sin θ cos θ 00 0 1⎦


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 215corresponding respectively to rotations about x−axis by an angle ϕ, abouty−axis by an angle ψ, and about z−axis by an angle θ.The total three–axial joint rotation A is defined as the product of aboveone–parameter rotations R ϕ , R ψ , R θ , i.e., A = R ϕ · R ψ · R θ is equal⎡⎤cos ψ cos ϕ − cos θ sin ϕ sin ψ cos ψ cos ϕ + cos θ cos ϕ sin ψ sin θ sin ψA = ⎣ − sin ψ cos ϕ − cos θ sin ϕ sin ψ − sin ψ sin ϕ + cos θ cos ϕ cos ψ sin θ cos ψ ⎦ .sin θ sin ϕ − sin θ cos ϕ cos θHowever, the order of these matrix products matters: different order productsgive different results, as the matrix product is noncommutative product.This is the reason why Hamilton’s quaternions 5 are today commonly usedto parameterize the SO(3)−group, especially in the field of 3D computergraphics.The one–parameter rotations R ϕ , R ψ , R θ define curves in SO(3) startingfrom I 3 = 0 1 0 . Their derivatives in ϕ = 0, ψ = 0 and θ = 0 belong( ) 1 0 00 0 1to the associated tangent Lie algebra so(3). That is the correspondinginfinitesimal generators of joint rotations – joint angular velocitiesv ϕ , v ψ , v θ ∈ so(3) – are respectively given byv ϕ =v θ =[ ] 0 0 00 0 −10 1 0[ ] 0 −1 01 1 00 0 0= −y ∂ ∂z + z ∂ ∂y , v ψ == −x ∂ ∂y + y ∂∂x .Moreover, the elements are linearly independent and so⎧⎡⎤ ⎫⎨ 0 −a b⎬so(3) = ⎣ a 0 −γ ⎦ |a, b, γ ∈ R⎩⎭ .−b γ 0[ ] 0 0 10 0 0 = −z ∂−1 0 0 ∂x + x ∂ ∂z ,The Lie algebra so(3) is identified with R 3 by associating to each v =5 Recall that the set of Hamilton’s quaternions H represents an extension of the setof complex numbers C. We can compute a rotation about the unit vector, u by an angleθ. The quaternion q that computes this rotation is„q = cos θ 2 , u sin θ «.2


216 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction[ ]0 −a b(v ϕ , v ψ , v θ ) ∈ R 3 the matrix v ∈ so(3) given by v = a 0 −γ . Then we−b γ 0have the following identities:(1) û × v = [û, v]; and(2) u · v = − 1 2Tr(û · v).The exponential map exp : so(3) → SO(3) is given by Rodrigues relationexp(v) = I +where the norm ‖v‖ is given bysin ‖v‖‖v‖ v + 1 2( )sin‖v‖ 22v 2 ,‖v‖2‖v‖ = √ (v 1 ) 2 + (v 2 ) 2 + (v 3 ) 2 .The the dual, cotangent Lie algebra so(3) ∗ , includes the three jointangular momenta p ϕ , p ψ , p θ ∈ so(3) ∗ , derived from the joint velocities v bymultiplying them with corresponding moments of inertia.Note that the parameterization of SO(3)−rotations is the subjectof continuous research and development in many theoretical and appliedfields of mechanics, such as rigid body, structural, and multibodydynamics, robotics, spacecraft attitude dynamics, navigation, imageprocessing, etc. For a complete discussion on the classical attituderepresentations see [Friberg (1988); Mladenova (1991); Shuster (1993);Schaub (1995)]. In addition, a modern vectorial parameterization of finiterotations, encompassing the mentioned earlier developments as wellas Gibbs, Wiener, and Milenkovic parameterizations [Mladenova (1999);Bauchau and Trainelli (2003)].3.8.4.2 Euclidean Groups of Total Joint MotionsBiodynamically realistic joint movement is predominantly rotational, plusrestricted translational (translational motion in human joints is observedafter reaching the limit of rotational amplitude). Gross translation in anyhuman joint means joint dislocation, which is a severe injury. Obviousmodels for uniaxial and triaxial joint motions are special Euclidean groupsof rigid body motions, SE(2) and SE(3), respectively.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 217Special Euclidean Group in the PlaneThe motion in uniaxial human joints is naturally modelled by the specialEuclidean group in the plane, SE(2). It consists of all transformations ofR 2 of the form Az + a, where z, a ∈ R 2 , andA ∈ SO(2) ={matrices of the form( cos θ − sin θsin θ cos θ)}.In other words [Marsden and Ratiu (1999)], group SE(2) consists of matricesof the ( form: )Rθ a(R θ , a) = , where a ∈ R0 I2 and R θ is the rotation matrix:( )cos θ − sin θR θ =, while I is the 3 × 3 identity matrix. The inversesin θ cos θ(R θ , a) −1 is given by(R θ , a) −1 =( ) −1Rθ a=0 I( )R−θ −R −θ a.0 IThe Lie algebra se(2) of SE(2) consists of 3 × 3 block matrices of the form( )( )−ξJ v0 1, where J = , (J T = J −1 = −J),0 0−1 0with the usual commutator bracket. If we identify se(2) with R 3 by theisomorphism( ) −ξJ v∈ se(2) ↦−→ (ξ, v) ∈ R 3 ,0 0then the expression for the Lie algebra bracket becomes[(ξ, v 1 , v 2 ), (ζ, w 1 , w 2 )] = (0, ζv 2 − ξw 2 , ξw 1 − ζv1) = (0, ξJ T w − ζJ T v),where v = (v 1 , v 2 ) and w = (w 1 , w 2 ).The adjoint group action of( )Rθ a(R θ , a)on (ξ, v) =0 I( ) −ξJ v0 0is given by the group conjugation,( ) ( ) ( ) ( )Rθ a −ξJ v R−θ −R −θ a −ξJ ξJa + Rθ v=,0 I 0 0 0 I0 0


218 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionor, in coordinates [Marsden and Ratiu (1999)],Ad (Rθ ,a)(ξ, v) = (ξ, ξJa + R θ v). (3.54)In proving (3.54) we used the identity(R θ J = JR θ . Identify the dualµalgebra, se(2) ∗ , with matrices of the form2 J 0 ), via the nondegenerateα 0pairing given by the trace of the product. Thus, se(2) ∗ is isomorphic to R 3via( µ2 J 0 )∈ se(2) ∗ ↦−→ (µ, α) ∈ R 3 ,α 0so that in these coordinates, the pairing between se(2) ∗ and se(2) becomes〈(µ, α), (ξ, v)〉 = µξ + α · v,that is, the usual dot product in R 3 . The coadjoint group action is thusgiven byAd ∗ (R θ ,a) −1(µ, α) = (µ − R θα · Ja + R θ α). (3.55)Formula (3.55) shows that the coadjoint orbits are the cylinders T ∗ S 1 α ={(µ, α)| ‖α‖ = const} if α ≠ 0 together with the points are on the µ−axis.The canonical cotangent bundle projection π : T ∗ S 1 α → S 1 α is defined asπ(µ, α) = α.Special Euclidean Group in the 3D SpaceThe most common group structure in human–like biodynamics is thespecial Euclidean group in 3D space, SE(3). It is defined as a semidirect(noncommutative) product of 3D rotations and 3D translations, SO(3)✄R 3 .The Heavy TopAs a starting point consider a rigid body (see (3.12.3.2) below) movingwith a fixed point but under the influence of gravity. This problem still has aconfiguration space SO(3), but the symmetry group is only the circle groupS 1 , consisting of rotations about the direction of gravity. One says thatgravity has broken the symmetry from SO(3) to S 1 . This time, eliminatingthe S 1 symmetry mysteriously leads one to the larger Euclidean groupSE(3) of rigid motion of R 3 . Conversely, we can start with SE(3) as


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 219the configuration space for the rigid–body and ‘reduce out’ translations toarrive at SO(3) as the configuration space (see [Marsden and Ratiu (1999)]).The equations of motion for a rigid body with a fixed point in a gravitationalfield give an interesting example of a system that is Hamiltonian (see(3.12.3.2)) relative to a Lie–Poisson bracket (see (3.13.2)). The underlyingLie algebra consists of the algebra of infinitesimal Euclidean motions in R 3 .The basic phase–space we start with is again T ∗ SO(3), parameterized byEuler angles and their conjugate momenta. In these variables, the equationsare in canonical Hamiltonian form. However, the presence of gravity breaksthe symmetry, and the system is no longer SO(3) invariant, so it cannotbe written entirely in terms of the body angular momentum p. One alsoneeds to keep track of Γ, the ‘direction of gravity’ as seen from the body.This is defined by Γ = A −1 k, where k points upward and A is the elementof SO(3) describing the current configuration of the body. The equationsof motion areṗ 1 = I 2 − I 3I 2 I 3p 2 p 3 + Mgl(Γ 2 χ 3 − Γ 3 χ 2 ),ṗ 2 = I 3 − I 1I 3 I 1p 3 p 1 + Mgl(Γ 3 χ 1 − Γ 1 χ 3 ),ṗ 3 = I 1 − I 2I 1 I 2p 1 p 2 + Mgl(Γ 1 χ 2 − Γ 2 χ 1 ),and ˙Γ = Γ × Ω,where Ω is the body’s angular velocity vector, I 1 , I 2 , I 3 are the body’s principalmoments of inertia, M is the body’s mass, g is the acceleration ofgravity, χ is the body fixed unit vector on the line segment connectingthe fixed point with the body’s center of mass, and l is the length of thissegment.The Euclidean Group and Its Lie AlgebraAn element of SE(3) is a pair (A, a) where A ∈ SO(3) and a ∈ R 3 .The action of SE(3) on R 3 is the rotation A followed by translation by thevector a and has the expression(A, a) · x = Ax + a.Using this formula, one sees that multiplication and inversion in SE(3) aregiven by(A, a)(B, b) = (AB, Ab + a) and (A, a) −1 = (A −1 , −A −1 a),


220 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor A, B ∈ SO(3) and a, b ∈ R 3 . The identity element is (l, 0).The Lie algebra of the Euclidean group SE(3) is se(3) = R 3 × R 3 withthe Lie bracket[(ξ, u), (η, v)] = (ξ × η, ξ × v − η × u). (3.56)The Lie algebra of the Euclidean group has a structure that is a specialcase of what is called a semidirect product. Here it is the product of the groupof rotations with the corresponding group of translations. It turns out thatsemidirect products occur under rather general circumstances when thesymmetry in T ∗ G is broken.The dual Lie algebra of the Euclidean group SE(3) is se(3) ∗ = R 3 × R 3with the same Lie bracket (3.56). For the further details on adjoint orbitsin se(3) as well as coadjoint orbits in se(3) ∗ see [Marsden and Ratiu (1999)].Symplectic Group in Hamiltonian Mechanics( ) 0 ILet J = , with I the n × n identity matrix. Now, A ∈−I 0L(R 2n , R 2n ) is called a symplectic matrix if A T J A = J. Let Sp(2n, R)be the set of 2n × 2n symplectic matrices. Taking determinants of thecondition A T J A = J gives det A = ±1, and so A ∈ GL(2n, R). Furthermore,if A, B ∈ Sp(2n, R), then (AB) T J(AB) = B T A T JAB = J. Hence,AB ∈ Sp(2n, R), and if A T J A = J, then JA = (A T ) −1 J = (A −1 ) T J,so J = (A − 1) T JA −1 , or A −1 ∈ Sp(2n, R). Thus, Sp(2n, R) is a group[Marsden and Ratiu (1999)].The symplectic Lie groupSp(2n, R) = { A ∈ GL(2n, R) : A T J A = J }is a noncompact, connected Lie group of dimension 2n 2 + n. Its Lie algebrasp(2n, R) = { A ∈ L(R 2n , R 2n ) : A T J A = J = 0 } ,called the symplectic Lie algebra, consists of the 2n × 2n matrices A satisfyingA T J A = 0 [Marsden and Ratiu (1999)].Consider a particle of mass m moving in a potential V (q), where q i =(q 1 , q 2 , q 3 ) ∈ R 3 . Newtonian second law states that the particle moves alonga curve q(t) in R 3 in such a way that m¨q i = − grad V (q i ). Introduce the


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 221momentum p i = m ˙q i , and the energyThenH(q, p) = 12m3∑p 2 i + V (q).i=1∂H∂q i = ∂V∂q i = −m¨qi = −ṗ i , and∂H= 1 ∂p i m p i = ˙q i , (i = 1, 2, 3),and hence Newtonian law F = m¨q i is equivalent to Hamiltonian equations˙q i = ∂H∂p i,ṗ i = − ∂H∂q i .Now, writing z = (q i , p i ) [Marsden and Ratiu (1999)],J grad H(z) =so Hamiltonian equations read( ) ( )0 I∂H∂q i∂H = ( ˙q i ), ṗ i = ż,−I 0∂p iż = J grad H(z). (3.57)Now let f : R 3 × R 3 → R 3 × R 3 and write w = f(z). If z(t) satisfiesHamiltonian equations (3.57) then w(t) = f(z(t)) satisfies ẇ = A T ż, whereA T = [∂w i /∂z j ] is the Jacobian matrix of f. By the chain rule,ẇ = A T J gradzH(z) = A T J A grad H(z(w)).wThus, the equations for w(t) have the form of Hamiltonian equations withenergy K(w) = H(z(w)) iff A T J A = J, that is, iff A is symplectic. Anonlinear transformation f is canonical iff its Jacobian matrix is symplectic.Sp(2n, R) is the linear invariance group of classical mechanics [Marsden andRatiu (1999)].


222 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.8.4.3 Group Structure of Biodynamical ManifoldPurely Rotational Biodynamical ManifoldKinematics of an n−-segment human–body chain (like arm, leg orspine) is usually defined as a map between external coordinates (usually,end–effector coordinates) x r (r = 1, . . . , n) and internal joint coordinatesq i (i = 1, . . . , N) (see [<strong>Ivancevic</strong> and Snoswell (2001); <strong>Ivancevic</strong> (2002);<strong>Ivancevic</strong> and Pearce (2001b); <strong>Ivancevic</strong> and Pearce (2001b); <strong>Ivancevic</strong>(2005)]). The forward kinematics are defined as a nonlinear map x r =x r (q i ) with a corresponding linear vector functions dx r = ∂x r /∂q i dq i ofdifferentials: and ẋ r = ∂x r /∂q i ˙q i of velocities. When the rank of theconfiguration–dependent Jacobian matrix J ≡ ∂x r /∂q i is less than n thekinematic singularities occur; the onset of this condition could be detectedby the manipulability measure. The inverse kinematics are defined converselyby a nonlinear map q i = q i (x r ) with a corresponding linear vectorfunctions dq i = ∂q i /∂x r dx r of differentials and ˙q i = ∂q i /∂x r ẋ r of velocities.Again, in the case of redundancy (n < N), the inverse kinematicproblem admits infinite solutions; often the pseudo–inverse configuration–control is used instead: ˙q i = J ∗ ẋ r , where J ∗ = J T (J J T ) −1 denotes theMoore–Penrose pseudo–inverse of the Jacobian matrix J.Humanoid joints, that is, internal coordinates q i (i = 1, . . . , N), constitutea smooth configuration manifold M, described as follows. Uniaxial,‘hinge’ joints represent constrained, rotational Lie groups SO(2) i cnstr,parameterized by constrained angles qcnstr i ≡ q i ∈ [qmin i , qi max]. Three–axial, ‘ball–and–socket’ joints represent constrained rotational Lie groupsSO(3) i cnstr, parameterized by constrained Euler angles q i = q φ icnstr (in thefollowing text, the subscript ‘cnstr’ will be omitted, for the sake of simplicity,and always assumed in relation to internal coordinates q i ).All SO(n)−-joints are Hausdorff C ∞ −-manifolds with atlases (U α , u α );in other words, they are paracompact and metrizable smooth manifolds,admitting Riemannian metric.Let A and B be two smooth manifolds described by smooth atlases(U α , u α ) and (V β , v β ), respectively. Then the family (U α × V β , u α × v β :U α × V β → R m × R n ) ( α, β) ∈ A × B is a smooth atlas for the directproduct A × B. Now, if A and B are two Lie groups (say, SO(n)), thentheir direct product G = A × B is at the same time their direct productas smooth manifolds and their direct product as algebraic groups, with the


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 223product law(a 1 , b 1 )(a 2 , b 2 ) = (a 1 a 2 , b 1 b 2 ), (a 1,2 ∈ A, b 1,2 ∈ B).Generalizing the direct product to N rotational joint groups, we candraw an anthropomorphic product–tree (see Figure 3.6) using a line segment‘–’ to represent direct products of human SO(n)−-joints. This is our basicmodel of the biodynamical configuration manifold M.Fig. 3.6 Purely rotational, whole–body biodynamical manifold, with a singleSO(3)−joint representing the whole spinal movability.Let T q M be a tangent space to M at the point q. The tangent bundleT M represents a union ∪ q∈M T q M, together with the standard topologyon T M and a natural smooth manifold structure, the dimension of whichis twice the dimension of M. A vector–field X on M represents a sectionX : M → T M of the tangent bundle T M.Analogously let Tq ∗ M be a cotangent space to M at q, the dual toits tangent space T q M. The cotangent bundle T ∗ M represents a union∪ q∈M Tq ∗ M, together with the standard topology on T ∗ M and a naturalsmooth manifold structure, the dimension of which is twice the dimensionof M. A 1−form θ on M represents a section θ : M → T ∗ M of thecotangent bundle T ∗ M.We refer to the tangent bundle T M of biodynamical configuration manifoldM as the velocity phase–space manifold, and to its cotangent bundleT ∗ M as the momentum phase–space manifold.


224 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionReduction of the Rotational Biodynamical ManifoldThe biodynamical configuration manifold M (Figure 3.6) can be (forthe sake of the brain–like motor control) reduced to N−-torus T N , in threesteps, as follows.First, a single three–axial SO(3)−joint can be reduced to the directproduct of three uniaxial SO(2)−joints, in the sense that three hinge jointscan produce any orientation in space, just as a ball–joint can. Algebraically,this means reduction (using symbol ‘’) of each of the three SO(3) rotationmatrices to the corresponding SO(2) rotation matrices⎛⎞1 0 0 ( )⎝ 0 cos φ − sin φ ⎠ cos φ − sin φsin φ cos φ0 sin φ cos φ⎛⎞cos ψ 0 sin ψ (⎝ 0 1 0 ⎠cos ψ− sin ψ cos ψ− sin ψ 0 cos ψ)sin ψ⎛⎞cos θ − sin θ 0( )⎝ sin θ cos θ 0 ⎠ cos θ − sin θsin θ cos θ0 0 1In this way we can set the reduction equivalence relation SO(3) SO(2) ✄ SO(2) ✄ SO(2), where ‘✄’ denotes the noncommutative semidirectproduct (see (3.8.4.2) above).Second, we have a homeomorphism: SO(2) ∼ S 1 , where S 1 denotes theconstrained unit circle in the complex plane, which is an Abelian Lie group.Third, let I N be the unit cube [0, 1] N in R N and ‘∼’ an equivalencerelation on R N get by ‘gluing’ together the opposite sides of I N , preservingtheir orientation. The manifold of human–body configurations (Figure 3.6)can be represented as the quotient space of R N by the space of the integrallattice points in R N , that is a constrained ND torus T N (3.221),R N /Z N = I N / ∼ ∼ =N∏Si 1 ≡ {(q i , i = 1, . . . , N) : mod 2π} = T N . (3.58)i=1Since S 1 is an Abelian Lie group, its N−-fold tensor product T N is also anAbelian Lie group, the toral group, of all nondegenerate diagonal N × Nmatrices. As a Lie group, the biodynamical configuration space M ≡ T N


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 225has a natural Banach manifold structure with local internal coordinatesq i ∈ U, U being an open set (chart) in T N .Conversely by ‘ungluing’ the configuration space we get the primary unitcube. Let ‘∼ ∗ ’ denote an equivalent decomposition or ‘ungluing’ relation.By the Tychonoff product–topology Theorem, for every such quotient spacethere exists a ‘selector’ such that their quotient models are homeomorphic,that is, T N / ∼ ∗ ≈ A N / ∼ ∗ . Therefore I N represents a ‘selector’ for theconfiguration torus T N and can be used as an N−-directional ‘command–space’ for the topological control of human motion. Any subset of DOFon the configuration torus T N representing the joints included in humanmotion has its simple, rectangular image in the command space – selectorI N . Operationally, this resembles what the brain–motor–controller, thecerebellum, actually performs on the highest level of human motor control.The Complete Biodynamical ManifoldThe full kinematics of a whole human–like body can be split down intofive kinematic chains: one for each leg and arm, plus one for spine with thehead. In all five chains internal joint coordinates, namely n 1 constrainedrotations x k rt together with n 2 of even more constrained translations x j tr(see Figure 3.7), constitute a smooth nD anthropomorphic configurationmanifold M, with local coordinates x i , (i = 1, . . . , n). That is, the motionspace in each joint is defined as a semidirect (noncommutative) product ofthe Lie group SO(n) of constrained rotations and a corresponding Lie groupR n of even more restricted translations. More precisely, in each movablehuman–like joint we have an action of the constrained special EuclideanSE(3) group (see (3.8.4.2) above). The joints themselves are linked bydirect (commutative) products.Realistic Human Spine ManifoldThe high–resolution human spine manifold is a dynamical chain consistingof 25 constrained SE(3)− joints. Each movable spinal joint has6 DOF: 3 dominant rotations, (performed first in any free spinal movement),restricted to about 7 angular degrees and 3 secondary translations(performed after reaching the limit of rotational amplitude), restricted toabout 5 mm (see Figure 3.8).Now, SE(3) = SO(3) ✄ R 3 is a non–compact group, so there is no anynatural metric given by the kinetic energy on SE(3), and consequently, no


226 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 3.7 A medium–resolution, whole–body biodynamical manifold, with just a singleSE(3)−joint representing the spinal movability.Fig. 3.8 The high–resolution human spine manifold is a dynamical chain consisting of25 constrained SE(3)−joints.natural controls in the sense of geodesics on SE(3). However, both of itssubgroups, SO(3) and R 3 , are compact with quadratic metric forms definedby standard line element g ij dq i dq j , and therefore admit optimal muscular–like controls in the sense of geodesics (see section 3.10.1.2 below).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2273.8.5 Application: Dynamical Games on SE(n)−GroupsIn this section we propose a general approach to modelling conflict resolutionmanoeuvres for land, sea and airborne vehicles, using dynamicalgames on Lie groups. We use the generic name ‘vehicle’ to representall planar vehicles, namely land and sea vehicles, as well asfixed altitude motion of aircrafts (see, e.g., [Lygeros et. al. (1998);Tomlin et. al. (1998)]). First, we elaborate on the two–vehicle conflictresolution manoeuvres, and after that discuss the multi–vehicle manoeuvres.We explore special features of the dynamical games solution when theunderlying dynamics correspond to left–invariant control systems on Liegroups. We show that the 2D (i.e., planar) motion of a vehicle may bemodelled as a control system on the Lie group SE(2). The proposed algorithmsurrounds each vehicle with a circular protected zone, while thesimplification in the derivation of saddle and Nash strategies follows fromthe use of symplectic reduction techniques [Marsden and Ratiu (1999)]. Tomodel the two–vehicle conflict resolution, we construct the safe subset ofthe state–space for one of the vehicles using zero–sum non–cooperative dynamicgame theory [Basar and Olsder (1995)] which we specialize to theSE(2) group. If the underlying continuous dynamics are left–invariant controlsystems, reduction techniques can be used in the computation of safesets.3.8.5.1 Configuration Models for Planar VehiclesThe configuration of each individual vehicle is described by an element ofthe Lie group SE(2) of rigid–body motions in R 2 . Let g i ∈ SE(2) denotethe configurations of vehicles labelled i, with i = 1, 2. The motion of eachvehicle may be modelled as a left–invariant vector–field on SE(2):ġ i = g i X i , (3.59)where the vector–fields X i belong to the vector space se(2), the Lie algebraassociated with the group SE(2).Each g i ∈ SE(2) can be represented in standard local coordinates(x i , y i , θ i ) as⎡⎤cos θ i − sin θ i x ig i = ⎣ sin θ i cos θ i y i⎦ ,0 0 1


228 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere x i , y i is the position of vehicle i and θ i is its orientation, or heading.The associated Lie algebra is se(2), with X i ∈ se(2) represented as⎡ ⎤0 −ω i v iX i = ⎣ ω i 0 0 ⎦ ,0 0 0where v i and ω i represent the translational (linear) and rotational (angular)velocities, respectively.Now, to determine dynamics of the relative configuration of two vehicles,we perform a change (transformation) of coordinates, to place the identityelement of the group SE(2) on vehicle 1. If g rel ∈ SE(2) denotes therelative configuration of vehicle 2 with respect to vehicle 1, theng 2 = g 1 g rel =⇒ g rel = g −11 g 2.<strong>Diff</strong>erentiation with respect to time yields the dynamics of the relativeconfiguration:which expands into:ġ rel = g rel X 2 − X 1 g rel ,ẋ rel = −v 1 + v 2 cos θ rel + ω 1 y rel ,ẏ rel = v 2 sin θ rel − ω 1 x rel ,˙θ rel = ω 2 − ω 1 .3.8.5.2 Two–Vehicles Conflict Resolution ManoeuvresNext, we seek control strategies for each vehicle, which are safe under (possible)uncertainty in the actions of neighbouring vehicle. For this, we expandthe dynamics of two vehicles (3.59),and write it in the matrix form asġ 1 = g 1 X 1 , ġ 2 = g 2 X 2 ,ġ = gX, (3.60)withg =[ ] [ ]g1 0X1 0, X = ,0 g 2 0 X 2


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 229in which g is an element in the configuration manifold M = SE(2)×SE(2),while the vector–fields X i ∈ se(2) × se(2) are linearly parameterised byvelocity inputs (ω 1 , v 1 ) ∈ R 2 and (ω 2 , v 2 ) ∈ R 2 .The goal of each vehicle is to maintain safe operation, meaning that(i) the vehicles remain outside of a specified target set T with boundary∂T , defined byT = {g ∈ M|l(g) < 0},where l(g) is a differentiable circular function,l(g) = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2 − ρ 2(with ρ denoting the radius of a circular protected zone) defines the minimumallowable lateral separation between vehicles; and(ii)dl(g) ≠ 0 on ∂T = {g ∈ M|l(g) = 0},where d represents the exterior derivative (a unique generalization of thegradient, divergence and curl).Now, due to possible uncertainty in the actions of vehicle 2, the safestpossible strategy of vehicle 1 is to drive along a trajectory which guaranteesthat the minimum allowable separation with vehicle 2 is maintainedregardless of the actions of vehicle 2. We therefore formulate this problemas a zero–sum dynamical game with two players: control vs. disturbance.The control is the action of vehicle 1,u = (ω 1 , v 1 ) ∈ U,and the disturbance is the action of vehicle 2,d = (ω 2 , v 2 ) ∈ D.Here the control and disturbance sets, U and D, are defined asU = ([ω min1 , ω max1 ], [v1 min , v1 max ]),D = ([ω min2 , ω max2 ], [v2 min , v2 max ])and the corresponding control and disturbance functional spaces, U and Dare defined as:U = {u(·) ∈ P C 0 (R 2 )|u(t) ∈ U, t ∈ R},D = {d(·) ∈ P C 0 (R 2 )|d(t) ∈ U, t ∈ R},


230 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere P C 0 (R 2 ) is the space of piecewise continuous functions over R 2 .We define the cost of a trajectory g(t) which starts at state g at initialtime t ≤ 0, evolves according to (3.60) with input (u(·), d(·)), and ends atthe final state g(0) as:J(g, u(·), d(·), t) : SE(2) × SE(2) × U × D × R − → R,such that J(g, u(·), d(·), t) = l(g(0)), (3.61)where 0 is the final time (without loss of generality). Thus the cost dependsonly on the final state g(0) (the Lagrangian, or running cost, is identicallyzero). The game is won by vehicle 1 if the terminal state g(0) is eitheroutside T or on ∂T (i.e., J(g, 0) ≥ 0), and is won by vehicle 2 otherwise.This two–player zero–sum dynamical game on SE(2) is defined as follows.Consider the matrix system (3.60), ġ = gX, over the time interval[t, 0] where t < 0 with the cost function J(g, u(·), d(·), t) defined by (3.61)As vehicle 1 attempts to maximize this cost assuming that vehicle 2 is actingblindly, the optimal control action and worst disturbance actions arecalculated asu ∗ = arg maxminu∈U d∈DJ(g, u(·), d(·), t), d∗= arg minmaxd∈D u∈UJ(g, u(·), d(·), t).The game is said to have a saddle solution (u ∗ ,d ∗ ) if the resulting optimalcost J ∗ (g, t) does not depend on the order of play, i.e., on the order in whichthe maximization and minimization is performed:J ∗ (g, t) = maxminu∈U d∈DJ(g, u(·), d(·), t) = minmaxd∈D u∈UJ(g, u(·), d(·), t).Using this saddle solution we calculate the ‘losing states’ for vehicle 1, calledthe predecessor P re t (T ) of the target set T ,P re t (T ) = {g ∈ M|J(g, u ∗ (·), d(·), t) < 0}.3.8.5.3 Symplectic Reduction and Dynamical Games on SE(2)Since vehicles 1 and 2 have dynamics given by left–invariant control systemson the Lie group SE(2), we haveX 1 = ξ 1 ω 1 + ξ 2 v 1 , X 2 = ξ 1 ω 2 + ξ 2 v 2 ,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 231with ξ 1 , ξ 2 being two of the three basis elements for the tangent Lie algebrase(2) given by⎡ ⎤0 −1 0ξ 1 = ⎣ 1 0 0 ⎦ ,⎡ ⎤0 0 1ξ 2 = ⎣ 1 0 0 ⎦ ,⎡ ⎤0 0 0ξ 3 = ⎣ 1 0 1 ⎦ .0 0 00 0 00 0 0If p 1 (resp. p 2 ) is a cotangent vector–field to SE(2) at g 1 (resp. g 2 ), belongingto the cotangent (dual) Lie algebra se(2) ∗ , we can define the momentumfunctions for both vehicles:P 1 1 = < p 1 , g 1 ξ 1 >, P 2 1 =< p 1 , g 1 ξ 2 >, P 3 1 =< p 1 , g 1 ξ 3 >,P 1 2 = < p 2 , g 2 ξ 1 >, P 2 2 =< p 2 , g 2 ξ 2 >, P 3 2 =< p 2 , g 2 ξ 3 >,which can be compactly written asP j i =< p i, g i ξ j > .Defining p = (p 1 , p 2 ) ∈ se(2) ∗ × se(2) ∗ , the optimal cost for the two-player,zero-sum dynamical game is given byJ ∗ (g, t) = maxminu∈U d∈DThe Hamiltonian H(g, p, u, d) is given byJ(g, u(·), d(·), t) = maxu∈U mind∈D l(g(0)).H(g, p, u, d) = P 1 1 ω 1 + P 2 1 v 1 + P 1 2 ω 1 + P 2 2 v 1for control and disturbance inputs (ω 1 , v 1 ) ∈ U and (ω 2 , v 2 ) ∈ D as definedabove. It follows that the optimal Hamiltonian H ∗ (g, p), defined on thecotangent bundle T ∗ SE(2), is given byH ∗ (g, p) = P 1 1ω max1 + ω min1+ P 1 ω max22− |P1 1 | ωmax 2 − ω min2+ P12 2+ |P1 2 | vmax 1 − v1min− |P1 2 | vmax2and the saddle solution (u ∗ , d ∗ ) is given byu ∗ = arg maxminu∈U d∈DH(g, p, u, d), d∗2 + ω min22v1 max + v1min+ P22 22 − v2min2= arg minmaxd∈D u∈U+ |P1 1 | ωmax 1 − ω min12v2 max + v2min2H(g, p, u, d). (3.62)Note that H(g, p, u, d) and H ∗ (g, p) do not depend on the state g andcostate p directly, rather through the momentum functions P j 1 , P j 2 . Thisis because the dynamics are determined by left–invariant vector–fields on


232 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe Lie group and the Lagrangian is state independent [Marsden and Ratiu(1999)].The optimal Hamiltonian H ∗ (g, p) determines a 12D Hamiltonianvector–field X H ∗ on the symplectic manifold T ∗ M = SE(2) × SE(2) ×se(2) ∗ × se(2) ∗ (which is the cotangent bundle of the configuration manifoldM), defined by Hamiltonian equationsX H ∗ : ġ = ∂H∗ (g, p)∂p, ṗ = − ∂H∗ (g, p),∂gwith initial condition at time t being g(t) = g and final condition at time 0being p(0) = dl(g(0)). In general, to solve for the saddle solution (3.62), oneneeds to solve the ODE system for all states. However since the original systemon M = SE(2)×SE(2) is left–invariant, it induces generic symmetriesin the Hamiltonian dynamics on T ∗ M , referred to as Marsden–Weinsteinreduction of Hamiltonian systems on symplectic manifolds, see [Marsdenand Ratiu (1999)]. In general for such systems one only needs to solvean ODE system with half of the dimensions of the underlying symplecticmanifold.For the two-vehicle case we only need to solve an ODE system with 6states. That is exactly given by the dynamics of the 6 momentum functions˙ P j i = L X H ∗ P j i = {P j i , H∗ (g, p)}, (3.63)for i, j = 1, 2, which is the Lie derivative of P j i with respect to the Hamiltonianvector–field X H ∗. In the equation (3.63), the bracket {·, ·} is thePoisson bracket [<strong>Ivancevic</strong> and Pearce (2001a)], giving the commutationrelations:{P 1 1 , P 2 1 } = P 3 1 , {P 2 1 , P 3 1 } = 0, {P 3 1 , P 1 1 } = P 2 1 ,{P 1 2 , P 2 2 } = P 3 2 , {P 2 2 , P 3 2 } = 0, {P 3 2 , P 1 2 } = P 2 2 .Using these commutation relations, equation (3.63) can be written explic-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 233itly:(P˙v1 1 = P13 max1 + v1min+ sign(P1 2 ) vmax 1 + v1min22P˙1 2 = P1(− 3 ωmax 1 + ω min1− sign(P1 1 ) ωmax2(P˙ω1 3 = P12 max1 + ω min1+ sign(P1 1 ) ωmax2),1 − ω min121 − ω min12(P˙v2 1 = P23 max2 + v2min+ sign(P2 2 ) vmax 2 + v2min22P˙2 2 = P2(− 3 ωmax 2 + ω min2− sign(P2 1 ) ωmax2(P˙ω2 3 = P22 max2 + ω min2+ sign(P2 1 ) ωmax2),),2 − ω min222 − ω min22).),),The final conditions for the variables P j 1 (t) and P j 2 (t) are get from theboundary of the safe set asP j 1 (0) =< d 1l(g), g 1 ξ j >, P j 2 (0) =< d 2l(g), g 2 ξ j >,where d 1 is the derivative of l taken with respect to its first argument g 1 only(and similarly for d 2 ). In this way, P j 1 (t) and P j 2 (t) are get for t ≤ 0. Oncethis has been calculated, the optimal input u ∗ (t) and the worst disturbanced ∗ (t) are given respectively as⎧ { ω⎪⎨ ω ∗ max1 if P1 1 (t) > 01(t) =u ∗ ω min1 if P1 1 (t) < 0(t) = { v ⎪⎩ v1(t) ∗ max1 if P1 2 (t) > 0=v1 min if P1 2 (t) < 0⎧ { ω⎪⎨ ω ∗ max2 if P2 1 (t) > 02(t) =d ∗ ω min2 if P2 1 (t) < 0(t) = { v ⎪⎩ v2(t) ∗ max2 if P2 2 (t) > 0=v2 min if P2 2 (t) < 03.8.5.4 Nash Solutions for Multi–Vehicle ManoeuvresThe methodology introduced in the previous sections can be generalized tofind conflict–resolutions for multi–vehicle manoeuvres. Consider the three–.


234 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionvehicle dynamics:ġ = gX, (3.64)with⎡ ⎤g 1 0 0g = ⎣ 0 g 2 0 ⎦ , X =0 0 g 3⎡⎣⎤X 1 0 00 X 2 0 ⎦ ,0 0 X 3where g is an element in the configuration space M = SE(2) × SE(2) ×SE(2) and X ∈ se(2) × se(2) × se(2) is linearly parameterised by inputs(ω 1 , v 1 ), (ω 2 , v 2 ) and (ω 3 , v 3 ).Now, the target set T is defined aswhereT = {g ∈ M|l 1 (g) < 0 ∨ l 2 (g) < 0 ∨ l 3 (g) < 0},l 1 (g) = min{(x 2 − x 1 ) 2 + (y 2 − y 1 ) 2 − ρ 2 , (x 3 − x 1 ) 2 + (y 3 − y 1 ) 2 − ρ 2 },l 2 (g) = min{(x 3 − x 2 ) 2 + (y 3 − y 2 ) 2 − ρ 2 , (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 − ρ 2 },l 3 (g) = min{(x 2 − x 3 ) 2 + (y 2 − y 3 ) 2 − ρ 2 , (x 1 − x 3 ) 2 + (y 1 − y 3 ) 2 − ρ 2 }.The control inputs u = (u 1 , u 2 , u 3 ) are the actions of vehicle 1, 2 and 3:where U i are defined asU i = ([ω miniu i = (ω i , v i ) ∈ U i ,, ω maxi ], [vimin , vi max ]).Clearly, this can be generalized to N vehicles.The cost functions J i (g, {u i (·)}, t) are defined asN∏N∏J i (g, {u i (·)}, t) : SE i (2) × U i × R − → R,i=1such that J i (g, {u i (·)}, t) = l i (g(0)).The simplest non–cooperative solution strategy is a so–called non–coopera-tive Nash equilibrium (see e.g., [Basar and Olsder (1995)]). A setof controls u ∗ i , (i = 1, ..., N) is said to be a Nash strategy, if for eachplayer modification of that strategy under the assumption that the othersi=1


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 235play their Nash strategies results in a decrease in his payoff, that is fori = 1, ..., N, and ∀u i (·),J i (u 1 , ..., u i , ..., u N ) ≤ J i (u ∗ 1, ..., u ∗ i , ..., u ∗ N ), (u ≠ u ∗ ).(Note that Nash equilibria may not be unique. It is also easy to see thatfor the two–player zero–sum game, a Nash equilibrium is a saddle solutionwith J = J 1 = −J 2 .)For N vehicles, the momentum functions are defined as in the two–vehicle case:P j i =< p i, g i ξ j >,with p i ∈ se(2) ∗ for i = 1, ..., N and ξ j defined as above.Then the Hamiltonian H(g, p, u 1 , ...u N ) is given byH(g, p, u 1 , ...u N ) = P 1 i ω i + P 2 i v i .The first case we consider is one in which all the vehicles are cooperating,meaning that each tries to avoid conflict assuming the others are doing thesame. In this case, the optimal Hamiltonian H ∗ (g, p) isH ∗ (g, p) = maxu i∈U iH(g, p, u 1 , ...u N ).For example, if N = 3, one may solve for (u ∗ 1, u ∗ 2, u ∗ 3), on the 9D quotientspace T ∗ M/M, so that the optimal control inputs are given as⎧ { ω⎪⎨ ω ∗ maxu ∗ i (t) = i if Pi 1(t) > 0ω mini if Pi (t) = { i 1(t) < 0v ⎪⎩ vi ∗ max(t) = i if Pi 2(t) > 0 .vi min if Pi 2(t) < 0One possibility for the optimal Hamiltonian corresponding to the non–cooperative case isH ∗ (g, p) = maxu 1∈U 1 u 2∈U 23.8.6 Classical Lie Theorymax max H(g, p, u 1 , u 2 , u 3 ).u 3∈U 3In this section we present the basics of classical theory of Lie groups andtheir Lie algebras, as developed mainly by Sophus Lie, Elie Cartan, FelixKlein, Wilhelm Killing and Hermann Weyl. For more comprehensivetreatment see e.g., [Chevalley (1955); Helgason (2001)].


236 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.8.6.1 Basic Tables of Lie Groups and Their Lie AlgebrasOne classifies Lie groups regarding their algebraic properties (simple,semisimple, solvable, nilpotent, Abelian), their connectedness (connectedor simply connected) and their compactness (see Tables A.1–A.3). Thisis the content of the Hilbert 5th problem (see, e.g., [Weisstein (2004);Wikipedia (2005)]).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 237Some real Lie groups and their Lie algebras:LiegroupDescription Remarks Liealgb.R n Euclidean spacewith additionR × nonzero realnumbers withmultiplicationR >0 positive realnumbers withmultiplicationS 1 = complex numbersR/Z of absolute value1, with multiplicationH × non–zero quaternionswith multiplicationS 3 quaternions ofabsolute value 1,with multiplication;a 3−sphereGL(n, R) general lineargroup: invertiblen−by-n real matricesGL + (n, R) n−by-n real matriceswith positivedeterminantAbelian, simplyconnected, notcompactAbelian, not connected,not compactAbelian, simplyconnected, notcompactAbelian, connected,not simplyconnected,compactsimply connected,not compactsimplyconnected, compact,simple andsemi–simple, isomorphicto SU(2), SO(3)and to Spin(3)not connected,not compactsimply connected,not compactR nRRRDescriptionthe Lie bracket iszerothe Lie bracket iszerothe Lie bracket iszerothe Lie bracket iszeroH quaternions, withLie bracket thecommutatorR 3 real 3−vectors,with Lie bracketthe cross product;isomorphicto su(2) and toso(3)M(n, R) n−by-n matrices,with Lie bracketthe commutatorM(n, R) n−by-n matrices,with Lie bracketthe commutatordim/Rn11143n 2n 2


238 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionClassical real Lie groups and their Lie algebras:LiegroupDescription Remarks Liealgb.SL(n, R) speciallinear group: realmatrices with determinant1O(n, R) orthogonalgroup: realorthogonal matricessimply connected,not compactif n > 1not connected,compactSO(n, R) special orthogonalconnected, com-group: real pact, for n ≥ 2:orthogonal matricesnot simply connected,with determi-nant 1for n = 3and n ≥ 5: simpleand semisimpleSpin(n) spinor group simply connected,compact,for n = 3 andn ≥ 5: simpleand semisimpleU(n) unitary group: isomorphic to S 1complex unitary for n =n−by-n matrices 1, not simply connected,compactSU(n) special unitarygroup: complexunitaryn−by-n matriceswith determinant1simply connected,compact,for n ≥ 2: simpleand semisimpleDescriptionsl(n, R) square matriceswith trace 0,with Lie bracketthe commutatorso(n, R) skew–symmetricsquare real matrices,with Liebracket the commutator;so(3, R)isisomorphicto su(2)and to R 3 withthe cross productso(n, R) skew–symmetricsquare real matrices,with Liebracket the commutatorso(n, R) skew–symmetricsquare real matrices,with Liebracket the commutatoru(n) square complexmatrices A satisfyingA = −A ∗ ,with Lie bracketthe commutatorsu(n) square complexmatrices Awith trace 0 satisfyingA = −A ∗ ,with Lie bracketthe commutatordim/Rn 2 −1n(n−1)/2n(n−1)/2n(n−1)/2n 2n 2 −1


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 239Basic complex Lie groups and their Lie algebras: 6LiegroupDescription Remarks Liealgb.C n group operationis additionAbelian, simplyconnected, notcompactAbelian, not simplyconnected,not compactsimply connected,not compact,for n = 1:isomorphic to C ×simple, semisimple,simply connected,for n ≥ 2:not compactnot connected,for n ≥ 2:not compactC × nonzero complexnumbers withmultiplicationGL(n, C) general lineargroup: invertiblen−by-n complexmatricesSL(n, C) special lineargroup: complexmatrices with determinant1O(n, C) orthogonalgroup: complexorthogonal matricessl(n, C) square matriceswith trace 0,with Lie bracketthe commutatorso(n, C) skew–symmetric squarecomplex matrices,with Liebracket the commutatorso(n, C) skew–symmetric squarecomplex matrices,with Liebracket the commutatorSO(n, C) special orthogonalgroup:complex orthogonalmatrices withdeterminant 1for n ≥ 2: notcompact,not simply connected,for n = 3and n ≥ 5: simpleand semisimpleC nCDescriptionthe Lie bracket iszerothe Lie bracket iszeroM(n, C) n−by-n matrices,with Lie bracketthe commutatordim/Cn1n 2n 2 −1n(n−1)/2n(n−1)/23.8.6.2 Representations of Lie groupsThe idea of a representation of a Lie group plays an important role in thestudy of continuous symmetry (see, e.g., [Helgason (2001)]). A great dealis known about such representations, a basic tool in their study being theuse of the corresponding ’infinitesimal’ representations of Lie algebras.Formally, a representation of a Lie group G on a vector space V (overa field K) is a group homomorphism G → Aut(V ) from G to the automorphismgroup of V . If a basis for the vector space V is chosen, therepresentation can be expressed as a homomorphism into GL(n, K). Thisis known as a matrix representation.6 The dimensions given are dimensions over C. Note that every complex Liegroup/algebra can also be viewed as a real Lie group/algebra of twice the dimension.


240 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOn the Lie algebra level, there is a corresponding linear map from theLie algebra of G to End(V ) preserving the Lie bracket [·, ·].If the homomorphism is in fact an monomorphism, the representationis said to be faithful.A unitary representation is defined in the same way, except that Gmaps to unitary matrices; the Lie algebra will then map to skew–Hermitianmatrices.Now, if G is a semisimple group, its finite–dimensional representationscan be decomposed as direct sums of irreducible representations. The irreduciblesare indexed by highest weight; the allowable (dominant) highestweights satisfy a suitable positivity condition. In particular, there exists aset of fundamental weights, indexed by the vertices of the Dynkin diagram ofG (see below), such that dominant weights are simply non–negative integerlinear combinations of the fundamental weights.If G is a commutative compact Lie group, then its irreducible representationsare simply the continuous characters of G. A quotient representationis a quotient module of the group ring.3.8.6.3 Root Systems and Dynkin DiagramsA root system is a special configuration in Euclidean space that has turnedout to be fundamental in Lie theory as well as in its applications. Also, theclassification scheme for root systems, by Dynkin diagrams, occurs in partsof mathematics with no overt connection to Lie groups (such as singularitytheory, see e.g., [Helgason (2001); Weisstein (2004); Wikipedia (2005)]).DefinitionsFormally, a root system is a finite set Φ of non–zero vectors (roots)spanning a finite–dimensional Euclidean space V and satisfying the followingproperties:(1) The only scalar multiples of a root α in V which belong to Φ are αitself and -α.(2) For every root α in V , the set Φ is symmetric under reflection throughthe hyperplane of vectors perpendicular to α.(3) If α and β are vectors in Φ, the projection of 2β onto the line throughα is an integer multiple of α.The rank of a root system Φ is the dimension of V . Two root systemsmay be combined by regarding the Euclidean spaces they span as mutually


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 241orthogonal subspaces of a common Euclidean space. A root system whichdoes not arise from such a combination, such as the systems A 2 , B 2 , andG 2 in Figure 3.9, is said to be irreducible.Two irreducible root systems (V 1 , Φ 1 ) and (V 2 , Φ 2 ) are considered tobe the same if there is an invertible linear transformation V 1 → V 2 whichpreserves distance up to a scale factor and which sends Φ 1 to Φ 2 .The group of isometries of V generated by reflections through hyperplanesassociated to the roots of Φ is called the Weyl group of Φ as it actsfaithfully on the finite set Φ, the Weyl group is always finite.ClassificationIt is not too difficult to classify the root systems of rank 2 (see Figure3.9).Fig. 3.9 Classification of root systems of rank 2.Whenever Φ is a root system in V and W is a subspace of V spannedby Ψ = Φ ∩ W , then Ψ is a root system in W . Thus, our exhaustive list ofroot systems of rank 2 shows the geometric possibilities for any two rootsin a root system. In particular, two such roots meet at an angle of 0, 30,45, 60, 90, 120, 135, 150, or 180 degrees.In general, irreducible root systems are specified by a family (indicatedby a letter A to G) and the rank (indicated by a subscript n). There arefour infinite families:• A n (n ≥ 1), which corresponds to the special unitary group, SU(n+1);• B n (n ≥ 2), which corresponds to the special orthogonal group,SO(2n + 1);• C n (n ≥ 3), which corresponds to the symplectic group, Sp(2n);• D n (n ≥ 4), which corresponds to the special orthogonal group,SO(2n),as well as five exceptional cases: E 6 , E 7 , E 8 , F 4 , G 2 .


242 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionDynkin DiagramsA Dynkin diagram is a graph with a few different kinds of possible edges(see Figure 3.10). The connected components of the graph correspond tothe irreducible subalgebras of g. So a simple Lie algebra’s Dynkin diagramhas only one component. The rules are restrictive. In fact, there are onlycertain possibilities for each component, corresponding to the classificationof semi–simple Lie algebras (see, e.g., [Conway et al. (1985)]).Fig. 3.10 The problem of classifying irreducible root systems reduces to the problem ofclassifying connected Dynkin diagrams.The roots of a complex Lie algebra form a lattice of rank k in a Cartansubalgebra h ⊂ g, where k is the Lie algebra rank of g. Hence, the rootlattice can be considered a lattice in R k . A vertex, or node, in the Dynkindiagram is drawn for each Lie algebra simple root, which corresponds toa generator of the root lattice. Between two nodes α and β, an edge isdrawn if the simple roots are not perpendicular. One line is drawn if theangle between them is 2π/3, two lines if the angle is 3π/4, and three linesare drawn if the angle is 5π/6. There are no other possible angles betweenLie algebra simple roots. Alternatively, the number of lines N between thesimple roots α and β is given byN = A αβ A βα =2 〈α, β〉|α| 2 2 〈β, α〉|β| 2 = 4 cos 2 θ,where A αβ = 2〈α,β〉|α|is an entry in the Cartan matrix (A 2αβ ) (for details onCartan matrix see, e.g., [Helgason (2001); Weisstein (2004)]). In a Dynkindiagram, an arrow is drawn from the longer root to the shorter root (whenthe angle is 3π/4 or 5π/6).Here are some properties of admissible Dynkin diagrams:(1) A diagram obtained by removing a node from an admissible diagram


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 243is admissible.(2) An admissible diagram has no loops.(3) No node has more than three lines attached to it.(4) A sequence of nodes with only two single lines can be collapsed to givean admissible diagram.(5) The only connected diagram with a triple line has two nodes.A Coxeter–Dynkin diagram, also called a Coxeter graph, is the same asa Dynkin diagram, but without the arrows. The Coxeter diagram is sufficientto characterize the algebra, as can be seen by enumerating connecteddiagrams.The simplest way to recover a simple Lie algebra from its Dynkin diagramis to first reconstruct its Cartan matrix (A ij ). The ith node andjth node are connected by A ij A ji lines. Since A ij = 0 iff A ji = 0, andotherwise A ij ∈ {−3, −2, −1}, it is easy to find A ij and A ji , up to order,from their product. The arrow in the diagram indicates which is larger.For example, if node 1 and node 2 have two lines between them, from node1 to node 2, then A 12 = −1 and A 21 = −2.However, it is worth pointing out that each simple Lie algebra can beconstructed concretely. For instance, the infinite families A n , B n , C n , andD n correspond to the special linear Lie algebra gl(n+1, C), the odd orthogonalLie algebra so(2n + 1, C), the symplectic Lie algebra sp(2n, C), andthe even orthogonal Lie algebra so(2n, C). The other simple Lie algebrasare called exceptional Lie algebras, and have constructions related to theoctonions.To prove this classification Theorem, one uses the angles between pairsof roots to encode the root system in a much simpler combinatorial object,the Dynkin diagram. The Dynkin diagrams can then be classified accordingto the scheme given above.To every root system is associated a corresponding Dynkin diagram.Otherwise, the Dynkin diagram can be extracted from the root system bychoosing a base, that is a subset ∆ of Φ which is a basis of V with thespecial property that every vector in Φ when written in the basis ∆ haseither all coefficients ≥ 0 or else all ≤ 0.The vertices of the Dynkin diagram correspond to vectors in ∆. Anedge is drawn between each non–orthogonal pair of vectors; it is a doubleedge if they make an angle of 135 degrees, and a triple edge if they makean angle of 150 degrees. In addition, double and triple edges are markedwith an angle sign pointing toward the shorter vector.


244 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAlthough a given root system has more than one base, the Weyl groupacts transitively on the set of bases. Therefore, the root system determinesthe Dynkin diagram. Given two root systems with the same Dynkin diagram,we can match up roots, starting with the roots in the base, and showthat the systems are in fact the same.Thus the problem of classifying root systems reduces to the problem ofclassifying possible Dynkin diagrams, and the problem of classifying irreducibleroot systems reduces to the problem of classifying connected Dynkindiagrams. Dynkin diagrams encode the inner product on E in terms of thebasis ∆, and the condition that this inner product must be positive definiteturns out to be all that is needed to get the desired classification (see Figure3.10).In detail, the individual root systems can be realized case–by–case, asin the following paragraphs:A n . Let V be the subspace of R n+1 for which the coordinates sumto 0, and let Φ be the set of vectors in V of length √ 2 and with integercoordinates in R n+1 . Such a vector must have all but two coordinates equalto 0, one coordinate equal to 1, and one equal to -1, so there are n 2 + nroots in all.B n . Let V = R n , and let Φ consist of all integer vectors in V of length1 or √ 2. The total number of roots is 2n 2 .C n : Let V = R n , and let Φ consist of all integer vectors in V of √ 2together with all vectors of the form 2λ, where λ is an integer vector oflength 1. The total number of roots is 2n 2 . The total number of roots is2n 2 .√D n . Let V = R n , and let Φ consist of all integer vectors in V of length2. The total number of roots is 2n(n − 1).E n . For V 8 , let V = R 8 , and let E 8 denote the set of vectors α oflength √ 2 such that the coordinates of 2α are all integers and are eitherall even or all odd. Then E 7 can be constructed as the intersection of E 8with the hyperplane of vectors perpendicular to a fixed root α in E 8 , andE 6 can be constructed as the intersection of E 8 with two such hyperplanescorresponding to roots α and β which are neither orthogonal to one anothernor scalar multiples of one another. The root systems E 6 , E 7 , and E 8 have72, 126, and 240 roots respectively.F 4 . For F 4 , let V = R 4 , and let Φ denote the set of vectors α of length1 or √ 2 such that the coordinates of 2α are all integers and are either alleven or all odd. There are 48 roots in this system.G 2 . There are 12 roots in G 2 , which form the vertices of a hexagram.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 245Root Systems and Lie TheoryIrreducible root systems classify a number of related objects in Lie theory,notably:(1) Simple complex Lie algebras;(2) Simple complex Lie groups;(3) Simply connected complex Lie groups which are simple modulo centers;and(4) Simple compact Lie groups.In each case, the roots are non–zero weights of the adjoint representation.A root system can also be said to describe a plant’s root and associatedsystems.3.8.6.4 Simple and Semisimple Lie Groups and AlgebrasA simple Lie group is a Lie group which is also a simple group. Thesegroups, and groups closely related to them, include many of the so–calledclassical groups of geometry, which lie behind projective geometry and othergeometries derived from it by the Erlangen programme of Felix Klein. Theyalso include some exceptional groups, that were first discovered by thosepursuing the classification of simple Lie groups. The exceptional groupsaccount for many special examples and configurations in other branchesof mathematics. In particular the classification of finite simple groups dependedon a thorough prior knowledge of the ‘exceptional’ possibilities.The complete listing of the simple Lie groups is the basis for the theoryof the semisimple Lie groups and reductive groups, and their representationtheory. This has turned out not only to be a major extension of the theoryof compact Lie groups (and their representation theory), but to be of basicsignificance in mathematical physics.Such groups are classified using the prior classification of the complexsimple Lie algebras. It has been shown that a simple Lie group has a simpleLie algebra that will occur on the list given there, once it is complexified(that is, made into a complex vector space rather than a real one). Thisreduces the classification to two further matters.The groups SO(p, q, R) and SO(p + q, R), for example, give rise to differentreal Lie algebras, but having the same Dynkin diagram. In generalthere may be different real forms of the same complex Lie algebra.Secondly, the Lie algebra only determines uniquely the simply connected


246 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(universal) cover G ∗ of the component containing the identity of a Lie groupG. It may well happen that G ∗ is not actually a simple group, for examplehaving a non–trivial center. We have therefore to worry about the globaltopology, by computing the fundamental group of G (an Abelian group: aLie group is an H−space). This was done by Elie Cartan.For an example, take the special orthogonal groups in even dimension.With −I a scalar matrix in the center, these are not actually simple groups;and having a two–fold spin cover, they aren’t simply–connected either.They lie ‘between’ G ∗ and G, in the notation above.Recall that a semisimple module is a module in which each submodule isa direct summand. In particular, a semisimple representation is completelyreducible, i.e., is a direct sum of irreducible representations (under a descendingchain condition). Similarly, one speaks of an Abelian category asbeing semisimple when every object has the corresponding property. Also,a semisimple ring is one that is semisimple as a module over itself.A semisimple matrix is diagonalizable over any algebraically closed fieldcontaining its entries. In practice this means that it has a diagonal matrixas its Jordan normal form.A Lie algebra g is called semisimple when it is a direct sum of simpleLie algebras, i.e., non–trivial Lie algebras L whose only ideals are {0} andL itself. An equivalent condition is that the Killing formB(X, Y ) = Tr(Ad(X) Ad(Y ))is non–degenerate [Schafer (1996)]. The following properties can be provedequivalent for a finite–dimensional algebra L over a field of characteristic0:1. L is semisimple.2. L has no nonzero Abelian ideal.3. L has zero radical (the radical is the biggest solvable ideal).4. Every representation of L is fully reducible, i.e., is a sum of irreduciblerepresentations.5. L is a (finite) direct product of simple Lie algebras (a Lie algebra iscalled simple if it is not Abelian and has no nonzero ideal ).A connected Lie group is called semisimple when its Lie algebra issemisimple; and the same holds for algebraic groups. Every finite dimensionalrepresentation of a semisimple Lie algebra, Lie group, or algebraicgroup in characteristic 0 is semisimple, i.e., completely reducible, but theconverse is not true. Moreover, in characteristic p > 0, semisimple Lie


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 247groups and Lie algebras have finite dimensional representations which arenot semisimple. An element of a semisimple Lie group or Lie algebra isitself semisimple if its image in every finite–dimensional representation issemisimple in the sense of matrices.Every semisimple Lie algebra g can be classified by its Dynkin diagram[Helgason (2001)].3.9 Lie Symmetries and Prolongations on ManifoldsIn this section we continue our expose on Lie groups of symmetry, as a linkto modern jet machinery, developed below.3.9.1 Lie Symmetry Groups3.9.1.1 Exponentiation of Vector Fields on MLet x = (x 1 , ..., x r ) be local coordinates at a point m on a smoothn−manifold M. Recall that the flow generated by the vector–fieldis a solution of the system of ODEsv = ξ i (x) ∂ x i ∈ M,dx idε = ξi (x 1 , ..., x m ),(i = 1, ..., r).The computation of the flow, or one–parameter group of diffeomorphisms,generated by a given vector–field v (i.e., solving the system of ODEs) isoften referred to as exponentiation of a vector–field, denoted by exp(εv) x(see [Olver (1986)]).If v, w ∈ M are two vectors defined bythenv = ξ i (x) ∂ x i and w = η i (x) ∂ x i,exp(εv) exp(θw) x = exp(θw) exp(εv) x,for all ɛ, θ ∈ R,x ∈ M, such that both sides are defined, iff they commute,i.e., [v, w] = 0 everywhere [Olver (1986)].A system of vector–fields {v 1 , ..., v r } on a smooth manifold M is ininvolution if there exist smooth real–valued functions h k ij (x), x ∈ M,


248 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioni, j, k = 1, ..., r, such that for each i, j,[v i , v j ] = h k ij · v k .Let v ≠ 0 be a right–invariant vector–field on a Lie group G. Then theflow generated by v through the identity e, namelyg ε = exp(εv) e ≡ exp(εv),is defined for all ε ∈ R and forms a one–parameter subgroup of G, withg ε+δ = g ε · g δ , g 0 = e, g −1ε = g −ε ,isomorphic to either R itself or the circle group SO(2). Conversely, anyconnected 1D subgroup of G is generated by such a right–invariant vector–field in the above manner [Olver (1986)].For example, let G = GL(n) with Lie algebra gl(n), the space of alln × n matrices with commutator as the Lie bracket. If A ∈ gl(n), then thecorresponding right–invariant vector–field v A on GL(n) has the expression[Olver (1986)]v A = a i kx k j ∂ x ij.The one–parameter subgroup exp(εv A ) e is found by integrating the systemof n 2 ordinary differential equationsdx i jdε = ai kx k j , x i j(0) = δ i j, (i, j = 1, ..., n),involving matrix entries of A. The solution is just the matrix exponentialX(ε) = e εA , which is the one–parameter subgroup of GL(n) generated bya matrix A in gl(n).Recall that the exponential map exp : g → G is get by setting ε = 1 inthe one–parameter subgroup generated by vector–field v :Its differential at 0,is the identity map.exp(v) ≡ exp(v) e.d exp : T g| 0 ≃ g → T G| e ≃ g


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2493.9.1.2 Lie Symmetry Groups and General DEsConsider a system S of general differential equations (DEs, to be distinguishedfrom ODEs) involving p independent variables x = (x 1 , ..., x p ), andq dependent variables u = (u 1 , ..., u q ). The solution of the system will be ofthe form u = f(x), or, in components, u α = f α (x 1 , ..., x p ), α = 1, ..., q (sothat Latin indices refer to independent variables while Greek indices referto dependent variables). Let X = R p , with coordinates x = (x 1 , ..., x p ), bethe space representing the independent variables, and let U = R q , with coordinatesu = (u 1 , ..., u q ), represent dependent variables. A Lie symmetrygroup G of the system S will be a local group of transformations acting onsome open subset M ⊂ X × U in such way that G transforms solutions ofS to other solutions of S [Olver (1986)].More precisely, we need to explain exactly how a given transformationg ∈ G, where G is a Lie group, transforms a function u = f(x). We firstlyidentify the function u = f(x) with its graphΓ f ≡ {(x, f(x)) : x ∈ dom f ≡ Ω} ⊂ X × U,where Γ f is a submanifold of X×U. If Γ f ⊂ M g ≡ dom g, then the transformof Γ f by g is defined asg · Γ f = {(˜x, ũ) = g · (x, u) : (x, u) ∈ Γ f } .We write ˜f = g · f and call the function ˜f the transform of f by g.For example, let p = 1 and q = 1, so X = R with a single independentvariable x, and U = R with a single dependent variable u, so we have asingle ODE involving a single function u = f(x). Let G = SO(2) be therotation group acting on X × U ≃ R 2 . The transformations in G are givenby(˜x, ũ) = θ · (x, u) = (x cos θ − u sin θ, x sin θ + u cos θ).Let u = f(x) be a function whose graph is a subset Γ f ⊂ X × U. Thegroup SO(2) acts on f by rotating its graph.In general, the procedure for finding the transformed function ˜f = g · fis given by [Olver (1986)]:g · f = [Φ g ◦ (1 × f)] ◦ [Ξ g ◦ (1 × f)] −1 , (3.65)where Ξ g = Ξ g (x, u), Φ g = Φ g (x, u) are smooth functions such that(˜x, ũ) = g · (x, u) = (Ξ g (x, u), Φ g (x, u)) ,


250 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhile 1 denotes the identity function of X, so 1(x) = x. Formula (3.65)holds whenever the second factor is invertible.Let S be a system of DEs. A symmetry group of the system S is a localLie group of transformations G acting on an open subset M ⊂ X ×U of thespace X × U of independent and dependent variables of the system withthe property that whenever u = f(x) is a solution of S, and whenever g · fis defined for g ∈ G, then u = g · f(x) is also a solution of the system.For example, in the case of the ODE u xx = 0, the rotation group SO(2)is obviously a symmetry group, since the solutions are all linear functionsand SO(2) takes any linear function to another linear function. Anothereasy example is given by the classical heat equation u t = u xx . Here thegroup of translations(x, t, u) ↦→ (x + εa, t + εb, u), ε ∈ R,is a symmetry group since u = f(x − εa, t − εb) is a solution to the heatequation whenever u = f(x, t) is.3.9.2 Prolongations3.9.2.1 Prolongations of FunctionsGiven a smooth real–valued function u = f(x) = f(x 1 , ..., x p ) of p independentvariables, there is an induced function u (n) = pr (n) f(x), called thenth prolongation of f [Olver (1986)], which is defined by the equations∂ k f(x)u J = ∂ J f(x) =∂x j1 ∂x j2... ∂x j ,kwhere the multi–index J = (j 1 , ..., j k ) is an unordered k−tuple of integers,with entries 1 ≤ j k ≤ p indicating which derivatives are being taken. Moregenerally, if f : X → U is a smooth function from X ≃ R p to U ≃ R q , sou = f(x) = f(f 1 (x), ..., f q (x)), there are q · p k numbersu α J = ∂ J f α ∂ k f α (x)(x) =∂x j1 ∂x j2... ∂x j ,kneeded to represent all the different kth order derivatives of the componentsof f at a point x. Thus pr (n) f : X → U (n) is a function from X to thespace U (n) , and for each x ∈ X, pr (n) f(x) is a vector whose q · p (n) entriesrepresent the values of f and al its derivatives up to order n at the point x.For example, in the case p = 2, q = 1 we have X ≃ R 2 with coordinates(x 1 , x 2 ) = (x, y), and U ≃ R with the single coordinate u = f(x, y). The


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 251second prolongation u (2) = pr (2) f(x, y) is given by [Olver (1986)]((u; u x , u y ; u xx , u xy , u yy ) = f; ∂f∂x , ∂f∂y ; ∂2 f∂x 2 , ∂ 2 )f∂x∂y , ∂2 f∂y 2 , (3.66)all evaluated at (x, y).The nth prolongation pr (n) f(x) is also known as the n−jet of f. In otherwords, the nth prolongation pr (n) f(x) represents the Taylor polynomial ofdegree n for f at the point x, since the derivatives of order ≤ n determinethe Taylor polynomial and vice versa.3.9.2.2 Prolongations of <strong>Diff</strong>erential EquationsA system S of nth order DEs in p independent and q dependent variablesis given as a system of equations [Olver (1986)]∆ r (x, u (n) ) = 0, (r = 1, ..., l), (3.67)involving x = (x 1 , ..., x p ), u = (u 1 , ..., u q ) and the derivatives ofu with respect to x up to order n. The functions ∆(x, u (n) ) =(∆ 1 (x, u (n) ), ..., ∆ l (x, u (n) )) are assumed to be smooth in their arguments,so ∆ : X ×U (n) → R l represents a smooth map from the jet space X ×U (n)to some lD Euclidean space (see section 4.14.12.5 below). The DEs themselvestell where the given map ∆ vanishes on the jet space X × U (n) , andthus determine a submanifold{}S ∆ = (x, u (n) ) : ∆(x, u (n) ) = 0 ⊂ X × U (n) (3.68)of the total the jet space X × U (n) .We can identify the system of DEs (3.67) with its corresponding submanifoldS ∆ (3.68). From this point of view, a smooth solution of the givensystem of DEs is a smooth function u = f(x) such that [Olver (1986)]∆ r (x, pr (n) f(x)) = 0,(r = 1, ..., l),whenever x lies in the domain of f. This is just a restatement of the factthat the derivatives ∂ J f α (x) of f must satisfy the algebraic constraintsimposed by the system of DEs. This condition is equivalent to the statementthat the graph of the prolongation pr (n) f(x) must lie entirely within thesubmanifold S ∆ determined by the system:Γ (n)f≡{(x, pr (n) f(x))}⊂ S ∆ ={}∆(x, u (n) ) = 0 .


252 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWe can thus take an nth order system of DEs to be a submanifold S ∆ in then−jet space X ×U (n) and a solution to be a function u = f(x) such that thegraph of the nth prolongation pr (n) f(x) is contained in the submanifoldS ∆ .For example, consider the case of Laplace equation in the planeu xx + u yy = 0 (remember, u x ≡ ∂ x u).Here p = 2 since there are two independent variables x and y, and q = 1since there is one dependent variable u. Also n = 2 since the equation issecond–order, so S ∆ ⊂ X × U (2) is given by (3.66). A solution u = f(x, y)must satisfy∂ 2 f∂x 2 + ∂2 f∂y 2 = 0for all (x, y). This is the same as requiring that the graph of the secondprolongation pr (2) f lie in S ∆ .3.9.2.3 Prolongations of Group ActionsLet G be a local group of transformations acting on an open subsetM ⊂ X × U of the space of independent and dependent variables. There isan induced local action of G on the n−jet space M (n) , called the nth prolongationpr (n) G of the action of G on M. This prolongation is defined so thatit transforms the derivatives of functions u = f(x) into the correspondingderivatives of the transformed function ũ = ˜f(˜x) [Olver (1986)].More precisely, suppose (x 0 , u (n)0 ) is a given point in M (n) . Choose anysmooth function u = f(x) defined in a neighborhood of x 0 , whose graphΓ f lies in M, and has the given derivatives at x 0 :u (n)0 = pr (n) f(x 0 ), i.e., u α J0 = ∂ J f α (x 0 ).. We then determine the action of the prolonged group oftransformations pr (n) g on the point (x 0 , u (n)0 ) by evaluating the derivativesof the transformed function g · f at ˜x 0 ; explicitly [Olver (1986)]If g is an element of G sufficiently near the identity, the transformed functiong·f as given by (3.65) is defined in a neighborhood of the correspondingpoint (˜x 0 , ũ 0 ) = g · (x 0 , u 0 ), with u 0 = f(x 0 ) being the zeroth order componentsof u (n)0pr (n) g · (x 0 , u (n)0 ) = (˜x 0, ũ (n)0 ),


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 253whereũ (n)0 ≡ pr (n) (g · f)(˜x 0 ).For example, let p = q = 1, so X × U ≃ R 2 , and consider the action ofthe rotation group SO(2). To calculate its first prolongation pr (1) SO(2),first note that X × U (1) ≃ R 3 , with coordinates (x, u, u x ). given a functionu = f(x), the first prolongation is [Olver (1986)]pr (1) f(x) = (f(x), f ′ (x)).Now, given a point (x 0 , u 0 , u 0 x) ∈ X × U (1) , and a rotation in SO(2) characterizedby the angle θ as given above, the corresponding transformedpointpr (1) θ · (x 0 , u 0 , u 0 x) = (˜x 0 , ũ 0 , ũ 0 x)(provided it exists). As for the first–order derivative, we findũ 0 x = sin θ + u x cos θcos θ − u x sin θ .Now, applying the group transformations given above, and dropping the0−indices, we find that the prolonged action pr (1) SO(2) on X × U (1) isgiven by(pr (1) θ · (x, u, u x ) = x cos θ − u sin θ, x sin θ + u cos θ, sin θ + u )x cos θ,cos θ − u x sin θwhich is defined for |θ| < | arccot u x |. Note that even though SO(2) isa linear, globally defined group of transformations, its first prolongationpr (1) SO(2) is both nonlinear and only locally defined. This fact demonstratesthe complexity of the operation of prolonging a group of transformations.In general, for any Lie group G, the first prolongation pr (1) G acts onthe original variables (x, u) exactly the same way that G itself does; onlythe action on the derivative u x gives an new information. Therefore, pr (0) Gagrees with G itself, acting on M (0) = M.3.9.2.4 Prolongations of Vector FieldsProlongation of the infinitesimal generators of the group action turn out tobe the infinitesimal generators of the prolonged group action [Olver (1986)].Let M ⊂ X × U be open and suppose v is a vector–field on M, with


254 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncorresponding local one–parameter group exp(εv). The nth prolongationof v, denoted pr (n) v, will be a vector–field on the n−jet space M (n) , andis defined to be the infinitesimal generator of the corresponding prolongedon–parameter group pr (n) [exp(εv)]. In other words,pr (n) v| (x,u (n) ) =d dε∣ pr (n) [exp(εv)](x, u (n) ) (3.69)ε=0for any (x, u (n) ) ∈ M (n) .For a vector–field v on M, given byv = ξ i (x, u) ∂∂x i + φα (x, u) ∂∂u α ,the nth prolongation pr (n) v is given by [Olver (1986)](i = 1, ..., p, α = 1, ..., q),pr (n) v = ξ i (x, u) ∂∂x i + φα J (x, u (n) ) ∂∂u α ,Jwith φ α 0 = φ α , and J a multiindex defined above.For example, in the case of SO(2) group, the corresponding infinitesimalgenerator iswithv = −u ∂∂x + x ∂∂u ,exp(εv)(x, u) = (x cos ε − u sin ε, x sin ε + u cos ε) ,being the rotation through angle ε. The first prolongation takes the form(pr (1) [exp(εv)](x, u, u x ) = x cos ε − u sin ε, x sin ε + u cos ε, sin ε + u )x cos ε.cos ε − u x sin εAccording to (3.69), the first prolongation of v is get by differentiating theseexpressions with respect to ε and setting ε = 0, which givespr (1) v = −u ∂∂x + x ∂∂u + (1 + u2 x) ∂∂u x.3.9.2.5 General Prolongation FormulaLetv = ξ i (x, u) ∂∂x i + φα (x, u) ∂ , (i = 1, ..., p, α = 1, ..., q), (3.70)∂uα


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 255be a vector–field defined on an open subset M ⊂ X × U. The nth prolongationof v is the vector–field [Olver (1986)]pr (n) v = v + φ α J (x, u (n) ) ∂∂u α , (3.71)Jdefined on the corresponding jet space M (n) ⊂ X × U (n) . The coefficientfunctions φ α J are given by the following formula:φ α (J = D J φ α − ξ i u α )i + ξ i u α J,i , (3.72)where u α i = ∂u α /∂x i , and u α J,i = ∂uα J /∂xi . D J is the total derivative withrespect to the multiindex J, i.e.,D J = D j1 D j2 ...D jk ,while the total derivative with respect to the ordinary index, D i , is definedas follows. Let P (x, u (n) ) be a smooth function of x, u and derivatives of uup to order n, defined on an open subset M (n) ⊂ X ×U (n) . the total derivativeof P with respect to x i is the unique smooth function D i P (x, u (n) )defined on M (n+1) and depending on derivatives of u up to order n + 1,with the recursive property that if u = f(x) is any smooth function thenD i P (x, pr (n+1) f(x)) = ∂ x i{P (x, pr (n) f(x))}.For example, in the case of SO(2) group, with the infinitesimal generatorv = −u ∂∂x + x ∂∂u ,the first prolongation is (as calculated above)whereAlso,pr (1) v = −u ∂∂x + x ∂∂u + ∂φx ,∂u xφ x = D x (φ − ξu x ) + ξu xx = 1 + u 2 x.φ xx = D x φ x − u xx D x ξ = 3u x u xx ,thus the infinitesimal generator of the second prolongation pr (2) SO(2) actingon X × U (2) ispr (2) v = −u ∂∂x + x ∂∂u + (1 + u2 x) ∂∂+ 3u x u xx .∂u x ∂u xx


256 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet v and w be two smooth vector–fields on M ⊂ X ×U. Then their nthprolongations, pr (n) v and pr (n) w respectively, have the linearity propertypr (n) (c 1 v + c 2 w) = c 1 pr (n) v + c 2 pr (n) w,(c 1 , c 2 − constant),and the Lie bracket propertypr (n) [v, w] = [pr (n) v, pr (n) w].3.9.3 Generalized Lie SymmetriesConsider a vector–field (3.70) defined on an open subset M ⊂ X × U. Providedthe coefficient functions ξ i and φ α depend only on x and u, v willgenerate a (local) one–parameter group of transformations exp(εv) actingpointwise on the underlying space M. A significant generalization of the notionof symmetry group is get by relaxing this geometrical assumption, andallowing the coefficient functions ξ i and φ α to also depend on derivativesof u [Olver (1986)].A generalized vector–field is a (formal) expressionv = ξ i ∂[u]∂x i + ∂φα [u] , (i = 1, ..., p, α = 1, ..., q), (3.73)∂uα in which ξ i and φ α are smooth functions. For example,∂v = xu x∂x + u ∂xx∂uis a generalized vector in the case p = q = 1.According to the general prolongation formula (3.71), we can define theprolonged generalized vector–fieldpr (n) v = v + φ α J [u]whose coefficients are as before determined by the formula (3.72). Thus, inour previous example [Olver (1986)],∂∂u α Jpr (n) ∂v = xu x∂x + u ∂xx∂u + [u ∂xxx − (xu xx + u x )u x ] .∂u xGiven a generalized vector–field v, its infinite prolongation (includingall the derivatives) is the formal expressionpr v = ξ i ∂∂x i + φα J∂∂u α J,.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 257Now, a generalized vector–field v is a generalized infinitesimal symmetry ofa system S of differential equations∆ r [u] = ∆ r (x, u (n) ) = 0,(r = 1, ..., l),iffpr v[∆ r ] = 0for every smooth solution m u = f(x) [Olver (1986)].For example, consider the heat equation∆[u] = u t − u xx = 0.The generalized vector–field v = u x∂∂uThushas prolongation∂pr v = u x∂u + u ∂ ∂ ∂xx + u xt + u xxx + ...∂u x ∂u t ∂u xxpr v(∆) = u xt − u xxx = D x (u t − u xx ) = D x ∆,and hence v is a generalized symmetry of the heat equation.3.9.3.1 Noether SymmetriesHere we present some results about Noether symmetries, in particular forthe first–order Lagrangians L(q, ˙q) (see [Batlle et. al. (1989); Pons et. al.(2000)]). We start with a Noether–Lagrangian symmetry,δL = ˙ F ,and we will investigate the conversion of this symmetry to the Hamiltonianformalism. Definingwe can writeG = (∂L/∂ ˙q i ) δq i − F,δ i L δq i + Ġ = 0, (3.74)where δ i L is the Euler–Lagrangian functional derivative of L,δ i L = α i − W ik ¨q k ,


258 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionwhereW ik ≡∂2 L∂ ˙q i ∂ ˙q k and α i ≡ − ∂2 L∂ ˙q i ∂q k˙qk + ∂L∂q i .We consider the general case where the mass matrix, or Hessian (W ij ),may be a singular matrix. In this case there exists a kernel for the pull–backFL ∗ of the Legendre map, i.e., fibre–derivative FL, from the velocity phase–space manifold T M (tangent bundle of the biodynamical manifold M) tothe momentum phase–space manifold T ∗ M (cotangent bundle of M). Thiskernel is spanned by the vector–fieldsΓ µ = γ i ∂µ∂ ˙q i ,where γ i µ are a basis for the null vectors of W ij . The Lagrangian time–evolution differential operator can therefore be expressed as:X = ∂ t + ˙q k ∂∂q k + ak (q, ˙q) ∂∂ ˙q k + λµ Γ µ ≡ X o + λ µ Γ µ ,where a k are functions which are determined by the formalism, and λ µ arearbitrary functions. It is not necessary to use the Hamiltonian techniqueto find the Γ µ , but it does facilitate the calculation:( ) ∂φµγ i µ = FL ∗ , (3.75)∂p iwhere the φ µ are the Hamiltonian primary first class constraints.Notice that the highest derivative in (3.74), ¨q i , appears linearly. BecauseδL is a symmetry, (3.74) is identically satisfied, and therefore the coefficientof ¨q i vanishes:We contract with a null vector γ i µ to find thatW ik δq k − ∂G = 0. (3.76)∂ ˙qiΓ µ G = 0.It follows that G is projectable to a function G H in T ∗ Q; that is, it is thepull–back of a function (not necessarily unique) in T ∗ Q:G = FL ∗ (G H ).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 259This important property is valid for any conserved quantity associatedwith a Noether symmetry. Observe that G H is determined up to the additionof linear combinations of the primary constraints. Substitution of thisresult in (3.76) gives[ ( )]W ik δq k − FL ∗ ∂GH= 0,∂p kand so the brackets enclose a null vector of W ik :( )δq i − FL ∗ ∂GH= r µ γ i∂pµ, (3.77)ifor some r µ (t, q, ˙q).We shall investigate the projectability of variations generated by diffeomorphismsin the following section. Assume that an infinitesimal transformationδq i is projectable:Γ µ δq i = 0.If δq i is projectable, so must be r µ , so that r µ = FL ∗ (r µ H). Then, using(3.75) and (3.77), we see that( ∂(GHδq i = FL ∗ + r µ H φ )µ).∂p iWe now redefine G H to absorb the piece r µ H φ µ, and from now on we willhave( )δq i = FL ∗ ∂GH.∂p iDefineˆp i = ∂L∂ ˙q i ;after eliminating (3.76) times ¨q i from (3.74), we get( ∂L∂q i − ∂ ˆp )˙qk i∂q k FL ∗ ( ∂G H) + ˙q i ∂∂p i ∂q i FL∗ (G H ) + FL ∗ ∂ t G H = 0,which simplifies to∂L∂q i FL∗ ( ∂G H∂p i) + ˙q i FL ∗ ( ∂G H∂q i ) + FL∗ ∂ t G H = 0. (3.78)


260 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow let us invoke two identities [Batlle et. al. (1989)] that are at the coreof the connection between the Lagrangian and the Hamiltonian equationsof motion. They areand˙q i = FL ∗ ( ∂H∂p i) + v µ (q, ˙q)FL ∗ ( ∂φ µ∂p i),∂L∂q i = −FL∗ ( ∂H∂q i ) − vµ (q, ˙q)FL ∗ ( ∂φ µ∂q i );where H is any canonical Hamiltonian, so that FL ∗ (H) = ˙q i (∂L/∂ ˙q i )−L =Ê, the Lagrangian energy, and the functions v µ are determined so as torender the first relation an identity. Notice the important relationΓ µ v ν = δ ν µ,which stems from applying Γ µ to the first identity and taking into accountthatΓ µ ◦ FL ∗ = 0.Substitution of these two identities into (3.78) induces (where { , } denotesthe Poisson bracket)FL ∗ {G H , H} + v µ FL ∗ {G H , φ µ } + FL ∗ ∂ t G H = 0.This result can be split through the action of Γ µ intoandor equivalently,andFL ∗ {G H , H} + FL ∗ ∂ t G H = 0,FL ∗ {G H , φ µ } = 0;{G H , H} + ∂ t G H = pc,{G H , φ µ } = pc,where pc stands for any linear combination of primary constraints. Inthis way, we have arrived at a neat characterization for a generator G H ofNoether transformations in the canonical formalism.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2613.9.4 Application: Biophysical PDEsIn this subsection we consider two most important equations for biophysics:(1) The heat equation, which has been analyzed in muscular mechanicssince the early works of A.V. Hill ([Hill (1938)]); and(2) The Korteveg–de Vries equation, the basic equation for solitary modelsof muscular excitation–contraction dynamics.SupposeS : ∆ r (x, u (n) ) = 0,(r = 1, ..., l),is a system of DEs of maximal rank defined over M ⊂ X × U. If G is a localgroup of transformations acting on M, andpr (n) v[∆ r (x, u (n) )] = 0, whenever ∆(x, u (n) ) = 0, (3.79)(with r = 1, ..., l) for every infinitesimal generator v of G, then G is asymmetry group of the system S [Olver (1986)].3.9.4.1 The Heat EquationRecall that the (1 + 1)D heat equation (with the thermal diffusivity normalizedto unity)u t = u xx (3.80)has two independent variables x and t, and one dependent variable u, sop = 2 and q = 1. Equation (3.80) has the second–order, n = 2, and can beidentified with the linear submanifold M (2) ⊂ X × U (2) determined by thevanishing of ∆(x, t, u (2) ) = u t − u xx .Letv = ξ(x, t, u) ∂∂x + τ(x, t, u) ∂ ∂+ φ(x, t, u)∂t ∂ube a vector–field on X × U. According to (3.79) we need to now the secondprolongationpr (2) v = v + φ x ∂+ φ t ∂+ φ xx ∂+ φ xt ∂+ φ tt∂u x ∂u t ∂u xx ∂u xt∂∂u ttof v. Applying pr (2) v to (3.80) we find the infinitesimal criterion (3.79) tobeφ t = φ xx ,


262 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhich must be satisfied whenever u t = u xx .3.9.4.2 The Korteveg–De Vries EquationRecall that the Korteveg–de Vries equationu t + u xxx + uu x = 0 (3.81)arises in physical systems in which both nonlinear and dispersive effects arerelevant. A vector–fieldv = ξ(x, t, u) ∂∂x + τ(x, t, u) ∂ ∂+ φ(x, t, u)∂t ∂ugenerates a one–parameter symmetry group iffwhenever u satisfies (3.81), etc.φ t + φ xxx + uφ x + u x φ = 0,3.9.5 Lie–Invariant <strong>Geom</strong>etric Objects3.9.5.1 Robot KinematicsIt is well known (see [Blackmore and Leu (1992); Prykarpatsky (1996)]),that motion planning, numerically controlled machining and robotics arejust a few of many areas of manufacturing automation in which the analysisand representation of swept volumes plays a crucial role. The swept volumemodelling is also an important part of task-oriented robot motion planning.A typical motion planning problem consists in a collection of objects movingaround obstacles from an initial to a final configuration. This may includein particular, solving the collision detection problem.When a solid object undergoes a rigid motion, the totality of pointsthrough which it passed constitutes a region in space called the swept volume.To describe the geometrical structure of the swept volume we pose thisproblem as one of geometric study of some manifold swept by surface pointsusing powerful tools from both modern differential geometry and nonlineardynamical systems theory [Ricca (1993); Langer and Perline (1994);Prykarpatsky (1996); Groesen and Jager (1994)] on manifolds. For somespecial cases of the Euclidean motion in the space R 3 one can constructa very rich hydrodynamic system [Blackmore and Leu (1992)] modelling asweep flow, which appears to be a completely integrable Hamiltonian systemhaving a special Lax type representation. To describe in detail these and


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 263other properties of swept volume dynamical systems, we develop Cartan’stheory of Lie–invariant geometric objects generated by closed ideals in theGrassmann’s algebra, following [Blackmore et. al. (1998)].Let a Lie group G act on an analytical manifold Y in the transitive way,that is the action G×Y → ρ Y generates some nonlinear exact representationof the Lie group G on the manifold Y . In the frame of the Cartan’s theory,the representation G × Y → ρ Y can be described by means of a system ofdifferential 1–forms (see section 5.8 below)¯β j = dy j + ξ j i ¯ωi (a, da) (3.82)in the Grassmann algebra Λ(Y ×G) on the product Y ×G, where ¯ω i (a, da) ∈Ta ∗ (G), i = 1, ..., r = dim G is a basis of left–invariant Cartan’s forms of theLie group G at a point a ∈ G, y = {y j : j = 1, ..., n = dim Y } ∈ Y andξ j i : Y × G → R are some smooth real valued functions.The following Cartan Theorem (see [Blackmore et. al. (1998)]) is basicin describing a geometric object invariant with respect to the mentionedabove group action G × Y → ρ Y : The system of differential forms (3.82)is a system of an invariant geometric object iff the following conditions arefulfilled:(1) The coefficients ξ j i ∈ Ck (Y ; R) for all i = 1, ..., r, j = 1, ..., n, are someanalytical functions on Y ; and(2) The differential system (3.82) is completely integrable within theFrobenius–Cartan criterion.The Cartan’s Theorem actually says that the differential system (3.82)can be written down as¯β j = dy j + ξ j i (y)¯ωi (a, da), (3.83)where 1–forms {¯ω i (a, da) : i = 1, ..., r} satisfy the standard Maurer–Cartanequations¯Ω j = d¯ω j + 1 2 cj ik ¯ωi ∧ ¯ω k = 0, (3.84)for all j = 1, ..., r on G, coefficients c j ik∈ R, i, j, k = 1, ..., r, being thecorresponding structure constants of the Lie algebra G of the Lie group G.


264 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.9.5.2 Maurer–Cartan 1–FormsLet be given a Lie group G with the Lie algebra G ≃ T e (G), whose basisis a set {A i ∈ G : i = 1, ..., r}, where r = dim G ≡ dim G. Let also aset U 0 ⊂ {a i ∈ R : i = 1, ..., r} be some open neighborhood of the zeropoint in R r . The exponential mapping exp : U 0 → G 0 , where by definition[Blackmore et. al. (1998)]R r ⊃ U 0 ∋ (a 1 , . . . , a r ) :exp ✲ exp(a i A i)= a ∈ G0 ⊂ G, (3.85)is an analytical mapping of the whole U 0 on some open neighborhood G 0of the unity element e ∈ G. From (3.85) it is easy to find that T e (G) =T e (G 0 ) ≃ G, where e = exp(0) ∈ G. Define now the following left–invariantG−valued differential 1–form on G 0 ⊂ G:¯ω(a, da) = a −1 da = ¯ω j (a, da)A j , (3.86)where A j ∈ G, ¯ω j (a, da) ∈ T ∗ a (G), a ∈ G 0 , j = 1, ..., r. To build effectivelythe unknown forms {¯ω j (a, da) : j = 1, ..., r}, let us consider the followinganalytical one–parameter 1–form ¯ω t (a, da) = ¯ω(a t ; da t ) on G 0 , wherea t = exp ( ta i A i), t ∈ [0, 1], and differentiate this form with respect to theparameter t ∈ [0, 1]. We will get [Blackmore et. al. (1998)]d¯ω t /dt = −a j A j a −1tda t +a −1t a t da j A j +a −1t da t a j A j = −a j [A j , ¯ω t ]+A j da j .(3.87)Having used the Lie identity [A j , A k ] = c i jk A i, j, k = 1, ..., r, and the righthand side of (3.86) in formwe ultimately get that¯ω j (a, da) = ¯ω j k (a)dak , (3.88)ddt (t¯ωj i (ta)) = Aj k t¯ωk i (ta) + δ j i , (3.89)where the matrix A k i , i, k = 1, ..., r, is defined as follows:A k i = c k ija j . (3.90)Thus, the matrix W j i (t) = t¯ωj i (ta), i, j = 1, ..., r, satisfies the following from(3.89) differential equation [Chevalley (1955)]dW/dt = AW + E, W | t=0= 0, (3.91)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 265where E = ‖δ j i ‖ is the unity matrix. The solution of (3.91) is representableasW (t) =∞∑n=1t nn! An−1 (3.92)for all t ∈ [0, 1]. Whence, recalling the above definition of the matrix W (t),we get easily that¯ω j k (a) = W j k (t) ∣∣∣t=1=∞∑(n!) −1 A n−1 . (3.93)Therefore, the following Theorem solves the problem of finding in aneffective algebraic way corresponding to a Lie algebra G the left–invariant 1–form ¯ω(a, da) ∈ Ta ∗ (G)⊗G at any a ∈ G : Let’s be given a Lie algebra G withthe structure constants c k ij ∈ R, i, j, k = 1, ..., r = dim G, related to somebasis {A j ∈ G : j = 1, ..., r}. Then the adjoint to G left–invariant Maurer–Cartan 1–form ¯ω(a, da) is built as follows [Blackmore et. al. (1998)]:n=1¯ω(a, da) = A j ¯ω j k (a)dak , (3.94)where the matrix W = ‖ ¯w j k(a)‖, j, k = 1, ..., r, is given exactly asW =∞∑(n!) −1 A n−1 , A j k = cj ki ai . (3.95)n=1Below we shall try to use the experience gained above in solving ananalogous problem of the theory of connections over a principal fibre bundleP (M; G) as well as over associated with it a fibre bundle P (M; Y, G).3.9.5.3 General Structure of Integrable One–FormsGiven 2−forms generating a closed ideal I(α) in the Grassmann algebraΛ(M), we will denote as above by I(α, β) an augmented ideal in Λ(M; Y ),where the manifold Y will be called in further the representation space ofsome adjoint Lie group G action: G × Y → ρ Y . We can find, therefore, thedetermining relationships for the set of 1–forms {β} and 2–forms {α}{α} = {α j ∈ Λ 2 (M) : j = 1, ..., m α },{β} = {β j ∈ Λ 1 (M × Y ) : j = 1, ..., n = dim Y },(3.96)


266 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsatisfying such equations [Blackmore et. al. (1998)]:dα i = a i k (α) ∧ αk ,dβ j = f j k αk + ω j s ∧ β s ,(3.97)where a i k (α) ∈ Λ1 (M), f j k ∈ Λ0 (M × Y ) and ω j s ∈ Λ 1 (M × Y ) for alli, k = 1, ..., m α , j, s = 1, ..., n. Since the identity d 2 β j ≡ 0 takes place forall j = 1, ..., n, from (3.97) we deduce the following relationship:(dω j k + ωj s ∧ ω s k)∧ β k +(df j s + ω j k f k s + f j l al s(α)As a result of (3.98) we get [Blackmore et. al. (1998)])∧ α s ≡ 0. (3.98)dω j k + ωj s ∧ ω s k∈ I(α, β),df j s + ω j k f k s + f j l al s(α) ∈ I(α, β)(3.99)for all j, k = 1, ..., n, s = 1, ..., m α . The second inclusion in (3.99) gives apossibility to define the 1–forms θ j s = f j l al s(α) satisfying the inclusiondθ j s + ω j k ∧ θk s ∈ I(α, β) ⊕ f j l cl s(α), (3.100)which we obtained having used the identities d 2 α j ≡ 0, j = 1, ..., m α , inthe form c j s(α) ∧ α s ≡ 0,c j s(α) = da j s(α) + a j l (α) ∧ al s(α), (3.101)following from (3.97). Let us suppose further that as s = s 0 the 2–formsc j s 0(α) ≡ 0 for all j = 1, ..., m α . Then as s = s 0 , we can define a set of1–forms θ j = θ j s 0∈ Λ 1 (M × Y ), j = 1, ..., n, satisfying the exact inclusions:dθ j + ω j k ∧ θk = Θ j ∈ I(α, β) (3.102)together with a set of inclusions for 1–forms ω j k ∈ Λ1 (M × Y )dω j k + ωj s ∧ ω s k = Ω j k∈ I(α, β) (3.103)As it follows from the general theory [Sulanke and Wintgen (1972)] of connectionson the fibred frame space P (M; GL(n)) over a base manifold M,we can interpret the equations (3.103) as the equations defining the curvature2–forms Ω j k ∈ Λ2 (P ), as well as interpret the equations (3.102) asthose, defining the torsion 2–forms Θ j ∈ Λ 2 (P ). Since I(α) = 0 = I(α, β)upon the integral submanifold ¯M ⊂ M, the reduced fibred frame spaceP ( ¯M; GL(n)) will have the flat curvature and be torsion free, being as a


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 267result, completely trivialized on ¯M ⊂ M. Consequently, we can formulatethe following Theorem.Let the condition above on the ideals I(α) and I(α, β) be fulfilled. Thenthe set of 1–forms {β} generates the integrable augmented ideal I(α, β) ⊂Λ(M ×Y ) iff there exists some curvature 1–form ω ∈ Λ 1 (P )⊗Gl(n) and torsion1–form θ ∈ Λ 1 (P )⊗R n on the adjoint fibred frame space P (M; GL(n)),satisfying the inclusions [Blackmore et. al. (1998)]dω + ω ∧ ω ∈ I(α, β) ⊗ Gl(n),dθ + ω ∧ θ ∈ I(α, β) ⊗ R n .(3.104)Upon the reduced fibred frame space P ( ¯M; GL(n)) the corresponding curvatureand torsion are vanishing, where ¯M ⊂ M is the integral submanifoldof the ideal I(α) ⊂ Λ(M).3.9.5.4 Lax Integrable Dynamical SystemsConsider some set {β} defining a Cartan’s Lie group G invariant object ona manifold M × Y :β j = dy j + ξ j k (y)bk (z), (3.105)where i = 1, ..., n = dim Y, r = dim G. The set (3.105) defines on themanifold Y a set {ξ} of vector–fields, compiling a representation ρ : G → {ξ}of a given Lie algebra G, that is vector–fields ξ s = ξ j s(y) ∂∂y∈ {ξ}, s =j1, ..., r, enjoy the following Lie algebra G relationships[ξ s , ξ l ] = c k slξ k (3.106)for all s, l, k = 1, ..., r. We can now compute the differentials dβ j ∈ Λ 2 (M ×Y ), j = 1, ..., n, using (3.105) and (3.106) as follows [Blackmore et. al.(1998)]:)dβ j = ∂ξj k(β (y) l − ξ l ∂y s(y)b s (z) ∧ b k (z) + ξ j l k (y)dbk (z) (3.107)which is equal to∂ξ j k (y)(∂y l β l ∧ b k (z) + ξ j ldb l (z) + 1 )2 cl ksdb k (z) ∧ db s (z) ,where {α} ⊂ Λ 2 (M) is some a priori given integrable system of 2–formson M, vanishing upon the integral submanifold ¯M ⊂ M. It is obvious that


268 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioninclusions (3.107) take place iff the following conditions are fulfilled: for allj = 1, ..., rdb j (z) + 1 2 cj ks dbk (z) ∧ db s (z) ∈ I(α). (3.108)The inclusions (3.108) mean in particular, that upon the integral submanifold¯M ⊂ M of the ideal I(α) ⊂ Λ(M) the equalitiesµ ∗ ¯ω j ≡ s ∗ b j (3.109)are true, where ¯ω j ∈ Te ∗ (G), j = 1, ..., r, are the left–invariant Maurer–Cartan forms on the invariance Lie group G. Thus, due to inclusions (3.108)all conditions of Cartan’s Theorem are enjoyed, giving rise to a possibilityto get the set of forms b j (z) ∈ Λ 1 (M) in an explicit form. To do this, letus define a G−valued curvature 1–form ω ∈ Λ 1 (P (M; G)) ⊗ G as follows[Blackmore et. al. (1998)](ω = Ad a −1 Aj b j) + ¯ω (3.110)where ¯ω ∈ G is the standard Maurer–Cartan 1–form on G. This 1–formsatisfies followed by (3.108) the canonical structure inclusion for Γ = A j b j ∈Λ 1 (M) ⊗ G:dΓ + Γ ∧ Γ ∈ I(α) ⊗ G, (3.111)serving as a main relationships determining the form (3.110).3.9.5.5 Application: Burgers Dynamical SystemConsider the Burgers dynamical system on a functional manifold M ⊂C k (R; R):u t = uu x + u xx , (3.112)where u ∈ M and t ∈ R is an evolution (time) parameter. The flow of(3.112) on M can be recast into a set of 2–forms {α} ⊂ Λ 2 (J(R 2 ; R))upon the adjoint jet–manifold J(R 2 ; R) (see section 5.8 below) as follows[Blackmore et. al. (1998)]:{α} = { du (0) ∧ dt − u (1) dx ∧ dt = α 1 , du (0) ∧ dx + u (0) du (0) ∧ dt(+du (1) ∧ dt = α 2 : x, t; u (0) , u (1)) τ∈ M 4 ⊂ J 1 (R 2 ; R)},(3.113)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 269where M 4 is a 4–D submanifold in J 1 (R 2 ; R)) with coordinates (x, t, u (0) =u, u (1) = u x ). The set of 2–forms (3.113) generates the closed ideal I(α),sincedα 1 = dx ∧ α 2 − u (0) dx ∧ α 1 , dα 2 = 0, (3.114)the integral submanifold ¯M = {x, t ∈ R} ⊂ M 4 being defined by thecondition I(α) = 0. We now look for a reduced ‘curvature’ 1–form Γ ∈Λ 1 (M 4 ) ⊗ g, belonging to some (not yet determined) Lie algebra g. This1–form can be represented using (3.113), as follows:Γ = b (x) (u (0) , u (1) )dx + b (t) (u (0) , u (1) )dt, (3.115)where elements b (x) , b (t) ∈ g satisfy [Blackmore et. al. (1998)]∂b (x)∂b= g∂u (0) 2 , (x)∂b= 0, (t)= g∂u (1) ∂u (0) 1 + g 2 u (0) ,∂b (t)= g∂u (1) 2 , [b (x) , b (t) ] = −u (1) g 1 .The set (3.116) has the following unique solution(3.116)b (x) = A 0 +A 1 u (0) ,b (t) = u (1) A 1 + u(0)22 A 1+[A 1 , A 0 ]u (0) +A 2 , (3.117)where A j ∈ g, j = 0, 2, are some constant elements on M of a Lie algebrag under search, satisfying the next Lie structure equations:[A 0 , A 2 ] = 0,[A 0 , [A 1 , A 0 ]] + [A 1 , A 2 ] = 0,[A 1 , [A 1 , A 0 ]] + 1 2 [A 0, A 1 ] = 0.(3.118)From (3.116) one can see that the curvature2–form Ω ∈ span R {A 1 , [A 0 , A 1 ] : A j ∈ g, j = 0, 1}. Therefore, reducingvia the Ambrose–Singer Theorem the associated principal fibered framespace P (M; G = GL(n)) to the principal fibre bundle P (M; G(h)), whereG(h) ⊂ G is the corresponding holonomy Lie group of the connection Γ onP , we need to satisfy the following conditions for the set g(h) ⊂ g to be aLie subalgebra in g : ∇ m x ∇ n t Ω ∈ g(h) for all m, n ∈ Z + .Let us try now to close the above procedure requiring that [Blackmoreet. al. (1998)]g(h) = g(h) 0 = span R {∇ m x ∇ n xΩ ∈ g : m + n = 0} (3.119)


270 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis means thatg(h) 0 = span R {A 1 , A 3 = [A 0 , A 1 ]}. (3.120)To satisfy the set of relations (3.118) we need to use expansions over thebasis (3.120) of the external elements A 0 , A 2 ∈ g(h):A 0 = q 01 A 1 + q 13 A 3 , A 2 = q 21 A 1 + q 23 A 3 . (3.121)Substituting expansions (3.121) into (3.118), we get that q 01 = q 23 =λ, q 21 = −λ 2 /2 and q 03 = −2 for some arbitrary real parameter λ ∈ R,that is g(h) = span R {A 1 , A 3 }, where[A 1 , A 3 ] = A 3 /2; A 0 = λA 1 −2A 3 , A 2 = −λ 2 A 1 /2+λA 3 . (3.122)As a result of (3.122) we can state that the holonomy Lie algebra g(h)is a real 2D one, assuming the following (2 × 2)−matrix representation[Blackmore et. al. (1998)]:( )( )1/4 00 1A 1 =, A 3 = ,0 −1/40 0( )( λ/4 −2−λ 2 )/8 λA 0 =, A 2 =0 −λ/40 λ 2 ./8(3.123)Thereby from (3.115), (3.117) and (3.123) we get the reduced curvature1–form Γ ∈ Λ 1 (M) ⊗ g,Γ = (A 0 + uA 1 )dx + ((u x + u 2 /2)A 1 − uA 3 + A 2 )dt, (3.124)generating parallel transport of vectors from the representation space Y ofthe holonomy Lie algebra g(h):dy + Γy = 0, (3.125)upon the integral submanifold ¯M ⊂ M 4 of the ideal I(α), generated bythe set of 2–forms (3.113). The result (3.125) means also that the Burgersdynamical system (3.112) is endowed with the standard Lax type representation,having the spectral parameter λ ∈ R necessary for its integrabilityin quadratures.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2713.10 Riemannian Manifolds and Their Applications3.10.1 Local Riemannian <strong>Geom</strong>etryAn important class of problems in Riemannian geometry is to understandthe interaction between the curvature and topology on a smooth manifold(see [Cao and Chow (1999)]). A prime example of this interaction is theGauss–Bonnet formula on a closed surface M 2 , which says∫K dA = 2π χ(M), (3.126)Mwhere dA is the area element of a metric g on M, K is the Gaussiancurvature of g, and χ(M) is the Euler characteristic of M.To study the geometry of a smooth manifold we need an additionalstructure: the Riemannian metric tensor. The metric is an inner producton each of the tangent spaces and tells us how to measure angles and distancesinfinitesimally. In local coordinates (x 1 , x 2 , · · · , x n ), the metric g isgiven by g ij (x) dx i ⊗dx j , where (g ij (x)) is a positive definite symmetric matrixat each point x. For a smooth manifold one can differentiate functions.A Riemannian metric defines a natural way of differentiating vector–fields:covariant differentiation. In Euclidean space, one can change the order ofdifferentiation. On a Riemannian manifold the commutator of twice covariantdifferentiating vector–fields is in general nonzero and is called theRiemann curvature tensor, which is a 4−tensor–field on the manifold.For surfaces, the Riemann curvature tensor is equivalent to the Gaussiancurvature K, a scalar function. In dimensions 3 or more, the Riemann curvaturetensor is inherently a tensor–field. In local coordinates, it is denotedby R ijkl , which is anti-symmetric in i and k and in j and l, and symmetricin the pairs {ij} and {kl}. Thus, it can be considered as a bilinear form on2−forms which is called the curvature operator. We now describe heuristicallythe various curvatures associated to the Riemann curvature tensor.Given a point x ∈ M n and 2−plane Π in the tangent space of M at x, wecan define a surface S in M to be the union of all geodesics passing throughx and tangent to Π. In a neighborhood of x, S is a smooth 2D submanifoldof M. We define the sectional curvature K(Π) of the 2−plane to be theGauss curvature of S at x:K(Π) = K S (x).Thus the sectional curvature K of a Riemannian manifold associates to each


272 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2−plane in a tangent space a real number. Given a line L in a tangent space,we can average the sectional curvatures of all planes through L to get theRicci tensor Rc(L). Likewise, given a point x ∈ M, we can average the Riccicurvatures of all lines in the tangent space of x to get the scalar curvatureR(x). In local coordinates, the Ricci tensor is given by R ik = g jl R ijkl andthe scalar curvature is given by R = g ik R ik , where (g ij ) = (g ij ) −1 is theinverse of the metric tensor (g ij ).3.10.1.1 Riemannian Metric on MIn this section we mainly follow [Petersen (1999); Petersen (1998)].Riemann in 1854 observed that around each point m ∈ M one can picka special coordinate system (x 1 , . . . , x n ) such that there is a symmetric(0, 2)−tensor–field g ij (m) called the metric tensor defined asg ij (m) = g(∂ x i, ∂ x j ) = δ ij , ∂ x kg ij (m) = 0.Thus the metric, at the specified point m ∈ M, in the coordinates(x 1 , . . . , x n ) looks like the Euclidean metric on R n . We emphasize thatthese conditions only hold at the specified point m ∈ M. When passing todifferent points it is necessary to pick different coordinates. If a curve γpasses through m, say, γ(0) = m, then the acceleration at 0 is defined byfirstly, writing the curve out in our special coordinatesγ(t) = (γ 1 (t), . . . , γ n (t)),secondly, defining the tangent, velocity vector–field, as˙γ = ˙γ i (t) · ∂ x i,and finally, the acceleration vector–field as¨γ(0) = ¨γ i (0) · ∂ x i.Here, the background idea is that we have a connection.Recall that a connection on a smooth manifold M tells us how to paralleltransport a vector at a point x ∈ M to a vector at a point x ′ ∈ M alonga curve γ ∈ M. Roughly, to parallel transport vectors along curves, it isenough if we can define parallel transport under an infinitesimal displacement:given a vector X at x, we would like to define its parallel transportedversion ˜X after an infinitesimal displacement by ɛv, where v is a tangentvector to M at x.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 273More precisely, a vector–field X along a parameterized curve α : I → Min M is tangent to M along α if X(t) ∈ M α(t) for all for t ∈ I ⊂ R. However,the derivative Ẋ of such a vector–field is, in general, not tangent to M.We can, nevertheless, get a vector–field tangent to M by projecting Ẋ(t)orthogonally onto M α(t) for each t ∈ I. This process of differentiating andthen projecting onto the tangent space to M defines an operation with thesame properties as differentiation, except that now differentiation of vector–fields tangent to M induces vector–fields tangent to M. This operation iscalled covariant differentiation.Let γ : I → M be a parameterized curve in M, and let X be a smoothvector–field tangent to M along α. The absolute covariant derivative ofX is the vector–field ˙¯X tangent to M along α, defined by ˙¯X = Ẋ(t) −[Ẋ(t) · N(α(t))] N(α(t)), where N is an orientation on M. Note that ˙¯X isindependent of the choice of N since replacing N by -N has no effect onthe above formula.Lie bracket (3.7.2) defines a symmetric affine connection ∇ on any manifoldM:[X, Y ] = ∇ X Y − ∇ Y X.In case of a Riemannian manifold M, the connection ∇ is also compatiblewith the Riemannian metrics g on M and is called the Levi–Civitaconnection on T M.For a function f ∈ C k (M, R) and a vector a vector–field X ∈ X k (M)we always have the Lie derivative (3.7)L X f = ∇ X f = df(X).But there is no natural definition for ∇ X Y, where Y ∈ X k (M), unlessone also has a Riemannian metric. Given the tangent field ˙γ, the accelerationcan then be computed by using a Leibniz rule on the r.h.s, if we canmake sense of the derivative of ∂ x i in the direction of ˙γ. This is exactlywhat the covariant derivative ∇ X Y does. If Y ∈ T m M then we can writeY = a i ∂ x i, and therefore∇ X Y = L X a i ∂ x i. (3.127)Since there are several ways of choosing these coordinates, one must checkthat the definition does not depend on the choice. Note that for two vector–fields we define (∇ Y X)(m) = ∇ Y (m) X. In the end we get a connection∇ : X k (M) × X k (M) → X k (M),


274 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhich satisfies (for all f ∈ C k (M, R) and X, Y, Z ∈ X k (M)):(1) Y → ∇ Y X is tensorial, i.e., linear and ∇ fY X = f∇ Y X.(2) X → ∇ Y X is linear.(3) ∇ X (fY ) = (∇ X f)Y (m) + f(m)∇ X Y .(4) ∇ X Y − ∇ Y X = [X, Y ].(5) L X g(Z, Y ) = g(∇ X Z, Y ) + g(Z, ∇ X Y ).A semicolon is commonly used to denote covariant differentiation withrespect to a natural basis vector. If X = ∂ x i, then the components of ∇ X Yin (3.127) are denotedY k; i = ∂ x iY k + Γ k ij Y j , (3.128)where Γ k ij are Christoffel symbols defined in (3.129) below. Similar relationshold for higher–order tensor–fields (with as many terms with Christoffelsymbols as is the tensor valence).Therefore, no matter which coordinates we use, we can now define theacceleration of a curve in the following way:γ(t) = (γ 1 (t), . . . , γ n (t)),˙γ(t) = ˙γ i (t)∂ x i,¨γ(t) = ¨γ i (t)∂ x i + ˙γ i (t)∇ ˙γ(t) ∂ x i.We call γ a geodesic if γ(t) = 0. This is a second–order nonlinear ODE ina fixed coordinate system (x 1 , . . . , x n ) at the specified point m ∈ M. Thuswe see that given any tangent vector X ∈ T m M, there is a unique geodesicγ X (t) with ˙γ X (0) = X. If the manifold M is closed, the geodesic must existfor all time, but in case the manifold M is open this might not be so. Tosee this, take as M any open subset of Euclidean space with the inducedmetric.Given an arbitrary vector–field Y (t) along γ, i.e., Y (t) ∈ T γ(t) M for allt, we can also define the derivative Ẏ ≡ dYdtin the direction of ˙γ by writingY (t) = a i (t)∂ x i,Ẏ (t) = ȧ i (t)∂ x i + a i (t)∇ ˙γ(t) ∂ x i.Here the derivative of the tangent field ˙γ is the acceleration γ. The field Yis said to be parallel iff Ẏ = 0. The equation for a field to be parallel is afirst–order linear ODE, so we see that for any X ∈ T γ(t0)M there is a uniqueparallel field Y (t) defined on the entire domain of γ with the property that


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 275Y (t 0 ) = X. Given two such parallel fields Y, Z ∈ X k (M), we have thatġ(Y, Z) = D ˙γ g(Y, Z) = g(Ẏ , Z) + g(Y, Ż) = 0.Thus X and Y are both of constant length and form constant angles along γ.Hence, ‘parallel translation’ along a curve defines an orthogonal transformationbetween the tangent spaces to the manifold along the curve. However,in contrast to Euclidean space, this parallel translation will depend on thechoice of curve.An infinitesimal distance between the two nearby local points m and non M is defined by an arc–elementds 2 = g ij dx i dx j ,and realized by the curves x i (s) of shortest distance, called geodesics, addressedby the Hilbert 4th problem. In local coordinates (x 1 (s), ..., x n (s))at a point m ∈ M, the geodesic defining equation is a second–order ODE,ẍ i + Γ i jk ẋ j ẋ k = 0,where the overdot denotes the derivative with respect to the affine parameters, ẋ i (s) = d ds xi (s) is the tangent vector to the base geodesic, while theChristoffel symbols Γ i jk = Γi jk(m) of the affine Levi–Civita connection ∇ atthe point m ∈ M are defined, in a holonomic coordinate basis e i asΓ k ij = g kl Γ ijl , with g ij = (g ij ) −1 and (3.129)Γ ijk = 1 2 (∂ x kg ij + ∂ x j g ki − ∂ x ig jk ).Note that the Christoffel symbols (3.129) do not transform as tensors onthe tangent bundle. They are the components of an object on the secondtangent bundle, a spray. However, they do transform as tensors on the jetspace (see section 5.3 below).In nonholonomic coordinates, (3.129) takes the extended formΓ i kl = 1 2 gim (∂ x lg mk + ∂ x k∂g ml − ∂ x m∂g kl + c mkl + c mlk − c klm ) ,where c klm = g mp c p klare the commutation coefficients of the basis, i.e.,[e k , e l ] = c m kl e m.The torsion tensor–field T of the connection ∇ is the function T :X k (M) × X k (M) → X k (M) given byT (X, Y ) = ∇ X Y − ∇ Y X − [X, Y ].


276 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFrom the skew symmetry ([X, Y ] = −[Y, X]) of the Lie bracket, follows theskew symmetry (T (X, Y ) = −T (Y, X)) of the torsion tensor. The mappingT is said to be f−bilinear since it is linear in both arguments and alsosatisfies T (fX, Y ) = fT (X, Y ) for smooth functions f. Since [∂ x i, ∂ x j ] = 0for all 1 ≤ i, j ≤ n, it follows thatT (∂ x i, ∂ x j ) = (Γ k ij − Γ k ji)∂ x k.Consequently, torsion T is a (1, 2) tensor–field, locally given byT = T k i j dx i ⊗ ∂ x k ⊗ dx j ,where the torsion components T k i jare given byT k i j = Γ k ij − Γ k ji.Therefore, the torsion tensor gives a measure of the nonsymmetry of theconnection coefficients. Hence, T = 0 if and only if these coefficients aresymmetric in their subscripts. A connection ∇ with T = 0 is said to betorsion free or symmetric.The connection also enables us to define many other classical conceptsfrom calculus in the setting of Riemannian manifolds. Suppose we have afunction f ∈ C k (M, R). If the manifold is not equipped with a Riemannianmetric, then we have the differential of f defined by df(X) = L X f, whichis a 1−form. The dual concept, the gradient of f, is supposed to be avector–field. But we need a metric g to define it. Namely, ∇f is defined bythe relationshipg(∇f, X) = df(X).Having defined the gradient of a function on a Riemannian manifold, wecan then use the connection to define the Hessian as the linear map∇ 2 f : T M → T M,∇ 2 f(X) = ∇ X ∇f.The corresponding bilinear map is then defined as∇ 2 f(X, Y ) = g(∇ 2 f(X), Y ).One can check that this is a symmetric bilinear form. The Laplacian of f,∆f, is now defined as the trace of the Hessian∆f = Tr(∇ 2 f(X)) = Tr(∇ X ∇f),


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 277which is a linear map. It is also called the Laplace–Beltrami operator, sinceBeltrami first considered this operator on Riemannian manifolds.Riemannian metric has the following mechanical interpretation. Let Mbe a closed Riemannian manifold with the mechanical metric g = g ij v i v j ≡〈v, v〉, with v i = ẋ i . Consider the Lagrangian functionL : T M → R, (x, v) ↦→ 1 〈v, v〉 − U(x) (3.130)2where U(x) is a smooth function on M called the potential. On a fixedlevel of energy E, bigger than the maximum of U, the Lagrangian flowgenerated by (3.130) is conjugate to the geodesic flow with metric ḡ =2(e − U(x))〈v, v〉. Moreover, the reduced action of the Lagrangian is thedistance for g = 〈v, v〉 [Arnold (1989); Abraham et al. (1988)]. Both ofthese statements are known as the Maupertius action principle.3.10.1.2 Geodesics on MFor a C k , k ≥ 2 curve γ : I → M, we define its length on I as∫ ∫√L (γ, I) = | ˙γ| dt = g ( ˙γ, ˙γ)dt.IThis length is independent of our parametrization of the curve γ. Thus thecurve γ can be reparameterized, in such a way that it has unit velocity.The distance between two points m 1 and m 2 on M, d (m 1 , m 2 ) , can now bedefined as the infimum of the lengths of all curves from m 1 to m 2 , i.e.,IL (γ, I) → min .This means that the distance measures the shortest way one can travel fromm 1 to m 2 .If we take a variation V (s, t) : (−ε, ε) × [0, l] → M of a smooth curveγ (t) = V (0, t) parameterized by arc–length L and of length l, then thefirst derivative of the arc–length functionL(s) =dL(0)ds∫ l0| ˙V | dt, is given by∫ l≡ ˙L(0) = g ( ˙γ, X)| l 0 − g (γ, X) dt, (3.131)where X (t) = ∂V∂s(0, t) is the so–called variation vector–field. Equation(3.131) is called the first variation formula. Given any vector–field X alongγ, one can produce a variation whose variational field is X. If the variation0


278 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfixes the endpoints, X (a) = X (b) = 0, then the second term in the formuladrops out, and we note that the length of γ can always be decreased as longas the acceleration of γ is not everywhere zero. Thus the Euler–Lagrangianequations for the arc–length functional are the equations for a curve to bea geodesic.Recall that in local coordinates x i ∈ U, where U is an open subset in theRiemannian manifold M, the geodesics are defined by the geodesic equationẍ i + Γ i jkẋ j ẋ k = 0, (3.132)where overdot means derivative upon the line parameter s, while Γ i jk areChristoffel symbols of the affine Levi–Civita connection ∇ on M. From(3.132) it follows that the linear connection homotopy,¯Γ i jk = sΓ i jk + (1 − s)Γ i jk, (0 ≤ s ≤ 1),determines the same geodesics as the original Γ i jk .3.10.1.3 Riemannian Curvature on MThe Riemann curvature tensor is a rather ominous tensor of type (1, 3);i.e., it has three vector variables and its value is a vector as well. It isdefined through the Lie bracket (3.7.2) asR (X, Y ) Z = ( ∇ [X,Y ] − [∇ X , ∇ Y ] ) Z = ∇ [X,Y ] Z − ∇ X ∇ Y Z + ∇ Y ∇ X Z.This turns out to be a vector valued (1, 3)−tensor–field in the three variablesX, Y, Z ∈ X k (M). We can then create a (0, 4)−tensor,R (X, Y, Z, W ) = g ( ∇ [X,Y ] Z − ∇ X ∇ Y Z + ∇ Y ∇ X Z, W ) .Clearly this tensor is skew–symmetric in X and Y , and also in Z andW ∈ X k (M). This was already known to Riemann, but there are somefurther, more subtle properties that were discovered a little later by Bianchi.The Bianchi symmetry condition readsR(X, Y, Z, W ) = R(Z, W, X, Y ).Thus the Riemann curvature tensor is a symmetric curvature operatorR : Λ 2 T M → Λ 2 T M.The Ricci tensor is the (1, 1)− or (0, 2)−tensor defined byRic(X) = R(∂ x i, X)∂ x i, Ric(X, Y ) = g(R(∂ x i, X)∂ x i, Y ),


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 279for any orthonormal basis (∂ x i). In other words, the Ricci curvature is atrace of the curvature tensor. Similarly one can define the scalar curvatureas the tracescal(m) = Tr (Ric) = Ric(∂ x i, ∂ x i).When the Riemannian manifold has dimension 2, all of these curvaturesare essentially the same. Since dim Λ 2 T M = 1 and is spanned by X ∧ Ywhere X, Y ∈ X k (M) form an orthonormal basis for T m M, we see that thecurvature tensor depends only on the scalar valueK(m) = R(X, Y, X, Y ),which also turns out to be the Gaussian curvature. The Ricci tensor is ahomothetyRic(X) = K(m)X,Ric(Y ) = K(m)Y,and the scalar curvature is twice the Gauss curvature. In dimension 3 thereare also some redundancies as dim T M = dim Λ 2 T M = 3. In particular,the Ricci tensor and the curvature tensor contain the same amount of information.The sectional curvature is a kind of generalization of the Gauss curvaturewhose importance Riemann was already aware of. Given a 2−planeπ ⊂ T m M spanned by an orthonormal basis X, Y ∈ X k (M) it is defined assec(π) = R(X, Y, X, Y ).The remarkable observation by Riemann was that the curvature operator isa homothety, i.e., looks like R = kI on Λ 2 T m M iff all sectional curvaturesof planes in T m M are equal to k. This result is not completely trivial, asthe sectional curvature is not the entire quadratic form associated to thesymmetric operator R. In fact, it is not true that sec ≥ 0 implies that thecurvature operator is nonnegative in the sense that all its eigenvalues arenonnegative. What Riemann did was to show that our special coordinates(x 1 , . . . , x n ) at m can be chosen to be normal at m, i.e., satisfy the conditionx i = δ i jx j , (δ i jx j = g ij )on a neighborhood of m. One can show that such coordinates are actuallyexponential coordinates together with a choice of an orthonormal basis for


280 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionT m M so as to identify T m M with R n . In these coordinates one can thenexpand the metric as follows:g ij = δ ij − 1 3 R ikjlx k x l + O ( r 3) .Now the equations x i = g ij x j evidently give conditions on the curvaturesR ijkl at m.If Γ i jk(m) = 0, the manifold M is flat at the point m. This means thatthe (1, 3) curvature tensor, defined locally at m ∈ M asR l ijk = ∂ x j Γ l ik − ∂ x kΓ l ij + Γ l rjΓ r ik − Γ l rkΓ r ij,also vanishes at that point, i.e., Rijk l (m) = 0.Now, the rate of change of a vector–field A k on the manifold M alongthe curve x i (s) is properly defined by the absolute covariant derivativeDds Ak = ẋ i ∇ i A k = ẋ i ( ∂ x iA k + Γ k ij A j) = Ȧk + Γ k ij ẋ i A j .By applying this result to itself, we can get an expression for the secondcovariant derivative of the vector–field A k along the curve x i (s):D 2ds 2 Ak = d (Ȧk + Γ k ij ẋ i A j) + Γ k ij ẋ ids(Ȧj + Γ j mn ẋ m A n ).In the local coordinates (x 1 (s), ..., x n (s)) at a point m ∈ M, if δx i =δx i (s) denotes the geodesic deviation, i.e., the infinitesimal vector describingperpendicular separation between the two neighboring geodesics, passingthrough two neighboring points m, n ∈ M, then the Jacobi equation ofgeodesic deviation on the manifold M holds:D 2 δx ids 2 + R i jkl ẋ j δx k ẋ l = 0. (3.133)This equation describes the relative acceleration between two infinitesimallyclose facial geodesics, which is proportional to the facial curvature (measuredby the Riemann tensor Rjkl i at a point m ∈ M), and to the geodesicdeviation δx i . Solutions of equation (3.133) are called Jacobi fields.In particular, if the manifold M is a 2D–surface in R 3 , the Riemanncurvature tensor simplifies intoR i jmn = 1 2 R gik (g km g jn − g kn g jm ),


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 281where R denotes the scalar Gaussian curvature. Consequently the equationof geodesic deviation (3.133) also simplifies intoD 2ds 2 δxi + R 2 δxi − R 2 ẋi (g jk ẋ j δx k ) = 0. (3.134)This simplifies even more if we work in a locally Cartesian coordinatesystem; in this case the covariant derivative D2Dsreduces to an ordinary2derivative d2dsand the metric tensor g 2ij reduces to identity matrix I ij , soour 2D equation of geodesic deviation (3.134) reduces into a simple second–order ODE in just two coordinates x i (i = 1, 2)ẍ i + R 2 δxi − R 2 ẋi (I jk ẋ j δx k ) = 0.3.10.2 Global Riemannian <strong>Geom</strong>etry3.10.2.1 The Second Variation FormulaCartan also establishes another important property of manifolds with nonpositivecurvature. First he observes that all spaces of constant zero curvaturehave torsion–free fundamental groups. This is because any isometryof finite order on Euclidean space must have a fixed point (the center ofmass of any orbit is necessarily a fixed point). Then he notices that onecan geometrically describe the L ∞ center of mass of finitely many points{m 1 , . . . , m k } in Euclidean space as the unique minimum for the strictlyconvex function1{x → max (d (m i , x)) 2} .i=1,··· ,k 2In other words, the center of mass is the center of the ball of smallest radiuscontaining {m 1 , . . . , m k } . Now Cartan’s observation from above was thatthe exponential map is expanding and globally distance nondecreasing as amap:(T m M, Euclidean metric) → (T m M, with pull–back metric) .Thus distance functions are convex in nonpositive curvature as well as inEuclidean space. Hence the above argument can in fact be used to concludethat any Riemannian manifold of nonpositive curvature must also havetorsion free fundamental group.Now, let us set up the second variation formula and explain how itis used. We have already seen the first variation formula and how it can


282 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbe used to characterize geodesics. Now suppose that we have a unit speedgeodesic γ (t) parameterized on [0, l] and consider a variation V (s, t) , whereV (0, t) = γ (t). Synge then shows that (¨L ≡ d2 L¨L(0) =∫ l0ds 2 ){g(Ẋ, Ẋ) − (g(Ẋ, ˙γ))2 − g(R(X, ˙γ)X, ˙γ)}dt + g( ˙γ, A)| l 0 ,where X (t) = ∂V∂s(0, t) is the variational vector–field, Ẋ = ∇ ˙γ X, andA (t) = ∇ ∂V X. In the special case where the variation fixes the endpoints,∂si.e., s → V (s, a) and s → V (s, b) are constant, the term with A in it fallsout. We can also assume that the ) variation is perpendicular to the geodesicand then drop the term g(Ẋ, ˙γ . Thus, we arrive at the following simpleform:¨L(0) =∫ l0{g(Ẋ, Ẋ) − g (R (X, ˙γ) X, ˙γ)}dt =∫ l0{|Ẋ|2 − sec( ˙γ, X) |X| 2 }dt.Therefore, if the sectional curvature is nonpositive, we immediately observethat any geodesic locally minimizes length (that is, among close–by curves),even if it does not minimize globally (for instance γ could be a closedgeodesic). On the other hand, in positive curvature we can see that if ageodesic is too long, then it cannot minimize even locally. The motivationfor this result comes from the unit sphere, where we can consider geodesicsof length > π. Globally, we know that it would be shorter to go in theopposite direction. However, if we consider a variation of γ where thevariational field looks like X = sin ( )t · πl E and E is a unit length parallelfield along γ which is also perpendicular to γ, then we get∫ l{ ∣∣∣ } ¨L(0) = Ẋ∣∣ 2 − sec ( ˙γ, X) |X| 2 dt==0∫ l0∫ l0{ (π ) ( 2· cos2t · π )ll( (π ) ( 2· cos2t · πll( − sec ( ˙γ, X) sin 2 t · π ) } dtl) ( − sin 2 t · π ) ) dt = − 1 (l 2 − π 2) ,l 2lwhich is negative if the length l of the geodesic is greater than π. Therefore,the variation gives a family of curves that are both close to and shorter thanγ. In the general case, we can then observe that if sec ≥ 1, then for thesame type of variation we get¨L(0) ≤ − 1 (l 2 − π 2) .2l


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 283Thus we can conclude that, if the space is complete, then the diametermust be ≤ π because in this case any two points are joined by a segment,which cannot minimize if it has length > π. With some minor modificationsone can now conclude that any complete Riemannian manifold (M, g) withsec ≥ k 2 > 0 must satisfy diam(M, g) ≤ π · k −1 . In particular, M must becompact. Since the universal covering of M satisfies the same curvaturehypothesis, the conclusion must also hold for this space; hence M musthave compact universal covering space and finite fundamental group.In odd dimensions all spaces of constant positive curvature must beorientable, as orientation reversing orthogonal transformation on odd–dimensional spheres have fixed points. This can now be generalized tomanifolds of varying positive curvature. Synge did it in the following way:Suppose M is not simply–connected (or not orientable), and use this tofind a shortest closed geodesic in a free homotopy class of curves (that reversesorientation). Now consider parallel translation around this geodesic.As the tangent field to the geodesic is itself a parallel field, we see thatparallel translation preserves the orthogonal complement to the geodesic.This complement is now odd dimensional (even dimensional), and by assumptionparallel translation preserves (reverses) the orientation; thus itmust have a fixed point. In other words, there must exist a closed parallelfield X perpendicular to the closed geodesic γ. We can now use the abovesecond variation formula¨L(0) =∫ l∫ l{|Ẋ|2 − |X| 2 sec ( ˙γ, X)}dt + g ( ˙γ, A)| l 0 = − |X| 2 sec ( ˙γ, X) dt.00Here the boundary term drops out because the variation closes up at theendpoints, and Ẋ = 0 since we used a parallel field. In case the sectionalcurvature is always positive we then see that the above quantity is negative.But this means that the closed geodesic has nearby closed curves which areshorter. However, this is in contradiction with the fact that the geodesicwas constructed as a length minimizing curve in a free homotopy class.In 1941 Myers generalized the diameter bound to the situation whereone only has a lower bound for the Ricci curvature. The idea is thatRic( ˙γ, ˙γ) = ∑ n−1i=1 sec (E i, ˙γ) for any set of vector–fields E i along γ suchthat ˙γ, E 1 , . . ., E n−1 forms an orthonormal frame. Now assume that thefields are parallel and consider the n − 1 variations coming from the variationalvector–fields sin ( )t · πl Ei . Adding up the contributions from the


284 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionvariational formula applied to these fields then inducesn−1∑i=1n−1∑¨L(0) ==i=1∫ l0∫ l{ (π0 l{(n − 1)) ( 2· cos2t · π )l( πl) ( 2· cos2t · π )l( − sec ( ˙γ, E i ) sin 2 t · π ) } dtl(− Ric ( ˙γ, ˙γ) sin 2 t · π ) } dt.lTherefore, if Ric( ˙γ, ˙γ) ≥ (n − 1) k 2 (this is the Ricci curvature of Sk n ), thenn−1∑i=1¨L(0) ≤ (n − 1)∫ l0= − (n − 1) 1 2l{ (πl) ( 2· cos2t · π )l(l 2 k 2 − π 2) ,( − k 2 sin 2 t · π ) } dtlwhich is negative when l > π · k −1 (the diameter of Sk n ). Thus at least oneof the contributions d2 L ids(0) must be negative as well, implying that the2geodesic cannot be a segment in this situation.3.10.2.2 Gauss–Bonnet FormulaIn 1926 Hopf proved that in fact there is a Gauss–Bonnet formula forall even–dimensional hypersurfaces H 2n ⊂ R 2n+1 . The idea is that thedeterminant of the differential of the Gauss map G : H 2n → S 2n is theGaussian curvature of the hypersurface. Moreover, this is an intrinsicallycomputable quantity. If we integrate this over the hypersurface, we get,1Vol S∫H2n det (DG) = deg (G) ,where deg (G) is the Brouwer degree of the Gauss map. Note that thiscan also be done for odd–dimensional surfaces, in particular curves, butin this case the degree of the Gauss map will depend on the embeddingor immersion of the hypersurface. Instead one gets the so–called windingnumber. Hopf then showed, as Dyck had earlier done for surfaces, thatdeg (G) is always half the Euler characteristic of H, thus yielding2Vol S∫H2n det (DG) = χ (H) . (3.135)Since the l.h.s of this formula is in fact intrinsic, it is natural to conjecturethat such a formula should hold for all manifolds.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2853.10.2.3 Ricci Flow on MRicci flow, or the parabolic Einstein equation, was introduced by R. Hamiltonin 1982 [Hamilton (1982)] in the form∂ t g ij = −2R ij . (3.136)Now, because of the minus sign in the front of the Ricci tensor R ij inthis equation, the solution metric g ij to the Ricci flow shrinks in positiveRicci curvature direction while it expands in the negative Ricci curvaturedirection. For example, on the 2−sphere S 2 , any metric of positive Gaussiancurvature will shrink to a point in finite time. Since the Ricci flow (3.136)does not preserve volume in general, one often considers the normalizedRicci flow defined by∂ t g ij = −2R ij + 2 n rg ij, (3.137)where r = ∫ RdV / ∫ dV is the average scalar curvature. Under this normalizedflow, which is equivalent to the (unnormalized) Ricci flow (3.136)by reparameterizing in time t and scaling the metric in space by a functionof t, the volume of the solution metric is constant in time. Also thatEinstein metrics (i.e., R ij = cg ij ) are fixed points of (3.137).Hamilton [Hamilton (1982)] showed that on a closed Riemannian3−manifold M 3 with initial metric of positive Ricci curvature, the solutiong(t) to the normalized Ricci flow (3.137) exists for all time and themetrics g(t) converge exponentially fast, as time t tends to the infinity, toa constant positive sectional curvature metric g ∞ on M 3 .Since the Ricci flow lies in the realm of parabolic partial differentialequations, where the prototype is the heat equation, here is a brief reviewof the heat equation [Cao and Chow (1999)].Let (M n , g) be a Riemannian manifold. Given a C 2 function u : M → R,its Laplacian is defined in local coordinates { x i} to be∆u = Tr ( ∇ 2 u ) = g ij ∇ i ∇ j u,where ∇ i = ∇ ∂x iis its associated covariant derivative (Levi–Civita connection).We say that a C 2 function u : M n × [0, T ) → R, where T ∈ (0, ∞],is a solution to the heat equation if∂ t u = ∆u.


286 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOne of the most important properties satisfied by the heat equation is themaximum principle, which says that for any smooth solution to the heatequation, whatever pointwise bounds hold at t = 0 also hold for t > 0. Letu : M n × [0, T ) → R be a C 2 solution to the heat equation on a completeRiemannian manifold. If C 1 ≤ u (x, 0) ≤ C 2 for all x ∈ M, for someconstants C 1 , C 2 ∈ R, then C 1 ≤ u (x, t) ≤ C 2 for all x ∈ M and t ∈ [0, T )[Cao and Chow (1999)].Now, given a smooth manifold M, a one–parameter family of metricsg (t) , where t ∈ [0, T ) for some T > 0, is a solution to the Ricci flow if(3.136) is valid at all x ∈ M and t ∈ [0, T ). The minus sign in the equation(3.136) makes the Ricci flow a forward heat equation [Cao and Chow (1999)](with the normalization factor 2).In local geodesic coordinates {x i }, we have [Cao and Chow (1999)]g ij (x) = δ ij − 1 (3 R ipjqx p x q + O |x| 3) , therefore, ∆g ij (0) = − 1 3 R ij,where ∆ is the standard Euclidean Laplacian. Hence the Ricci flow is likethe heat equation for a Riemannian metric∂ t g ij = 6∆g ij .The practical study of the Ricci flow is made possible by the followingshort–time existence result: Given any smooth compact Riemannian manifold(M, g o ), there exists a unique smooth solution g(t) to the Ricci flowdefined on some time interval t ∈ [0, ɛ) such that g(0) = g o [Cao and Chow(1999)].Now, given that short–time existence holds for any smooth initial metric,one of the main problems concerning the Ricci flow is to determineunder what conditions the solution to the normalized equation exists forall time and converges to a constant curvature metric. Results in this directionhave been established under various curvature assumptions, mostof them being some sort of positive curvature. Since the Ricci flow (3.136)does not preserve volume in general, one often considers, as we mentionedin the Introduction, the normalized Ricci flow (3.137). Under this flow, thevolume of the solution g(t) is independent of time.To study the long–time existence of the normalized Ricci flow, it isimportant to know what kind of curvature conditions are preserved underthe equation. In general, the Ricci flow tends to preserve some kind ofpositivity of curvatures. For example, positive scalar curvature is preservedin all dimensions. This follows from applying the maximum principle to the


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 287evolution equation for scalar curvature R, which is∂ t R = ∆R + 2 |R ij | 2 .In dimension 3, positive Ricci curvature is preserved under the Ricci flow.This is a special feature of dimension 3 and is related to the fact that theRiemann curvature tensor may be recovered algebraically from the Riccitensor and the metric in dimension 3. Positivity of sectional curvatureis not preserved in general. However, the stronger condition of positivecurvature operator is preserved under the Ricci flow.3.10.2.4 Structure Equations on MLet {X a } m a=1, {Y i } n i=1 be local orthonormal framings on M, N respectivelyand {e i } n i=1 be the induced framing on E defined by e i = Y i ◦ φ, then thereexist smooth local coframings {ω a } m a=1, {η i } n i=1 and {φ∗ η i } n i=1 on T M, T Nand E respectively such that (locally)g =m∑ω 2 a and h =a=1n∑η 2 i .The corresponding first structure equations are [Mustafa (1999)]:i=1dω a = ω b ∧ ω ba , ω ab = −ω ba ,dη i = η j ∧ η ji , η ij = −η ji ,d(φ ∗ η i ) = φ ∗ η j ∧ φ ∗ η ji , φ ∗ η ij = −φ ∗ η ji ,where the unique 1–forms ω ab , η ij , φ ∗ η ijforms. The second structure equations areare the respective connectiondω ab = ω ac ∧ ω cb + Ω M ab, dη ij = η ik ∧ η kj + Ω N ij ,d(φ ∗ η ij ) = φ ∗ η ik ∧ φ ∗ η kj + φ ∗ Ω N ij ,where the curvature 2–forms are given byΩ M ab = − 1 2 RM abcdω c ∧ ω d and Ω N ij = − 1 2 RN ijklη k ∧ η l .The pull back map φ ∗ and the push forward map φ ∗ can be written as[Mustafa (1999)]φ ∗ η i = f ia ω a


288 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor unique functions f ia on U ⊂ M, so thatφ ∗ = e i ⊗ φ ∗ η i = f ia e i ⊗ ω a .Note that φ ∗ is a section of the vector bundle φ −1 T N ⊗ T ∗ M.The covariant differential operators are represented as∇ M X a = ω ab ⊗ X b , ∇ N Y i = η ij ⊗ Y j , ∇ ∗ ω a = −ω ca ⊗ ω c ,where ∇ ∗ is the dual connection on the cotangent bundle T ∗ M.Furthermore, the induced connection ∇ φ on E is∇ φ e i = ( η ij (Y k ) ◦ φ ) e j ⊗ f ka ω a .The components of the Ricci tensor and scalar curvature are definedrespectively byR M ab = R M acbc and R M = R M aa.Given a function f : M → , there exist unique functions f cb = f bc such thatdf c − f b ω cb = f cb ω b , (3.138)where f c = df(X c ) for a local orthonormal frame {X c } m c=1. To prove thiswe take the exterior derivative of df = ∑ mc=1 f cω c and using structureequations, we have0 = [df c ∧ ω c + f bc ω b ∧ ω bc ] = [(df c − f b ω cb ) ∧ ω c ] .Hence by Cartan’s lemma (cf. [Willmore (1993)]), there exist unique functionsf cb = f bc such thatdf c − f b ω cb = f cb ω b .The Laplacian of a function f on M is given by∆f = − Tr(∇df),that is, negative of the usual Laplacian on functions.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2893.10.3 Application: Autonomous Lagrangian Dynamics3.10.3.1 Basis of Lagrangian DynamicsRecall that Riemannian metric g = on the configuration manifold Mis a positive–definite quadratic form g : T M → R, given in local coordinatesq i ∈ U (U open in M) asg ij ↦→ g ij (q, m) dq i dq j , where (3.139)∂x r ∂x sg ij (q, m) = m µ δ rs∂q i ∂q j (3.140)is the covariant material metric tensor defining a relation between internaland external coordinates and including n segmental masses m µ . The quantitiesx r are external coordinates (r, s = 1, . . . , 6n) and i, j = 1, . . . , N ≡6n − h, where h denotes the number of holonomic constraints.The Lagrangian of the system is a quadratic form L : T M → R dependenton velocity v and such that L(v) = 1 2< v, v >. It is locally givenbyL(v) = 1 2 g ij(q, m) v i v j .On the velocity phase–space manifold T M exist:(1) a unique 1−form θ L , defined in local coordinates q i , v i = ˙q i ∈ U v (U vopen in T M) by θ L = L v idq i , where L v i ≡ ∂L/∂v i ; and(2) a unique nondegenerate Lagrangian symplectic 2−form ω L , which isclosed (dω L = 0) and exact (ω L = dθ L = dL v i ∧ dq i ).T M is an orientable manifold, admitting the standard volume given byΩ ωL=N(N+1)(−1) 2N!ω N L ,in local coordinates q i , v i = ˙q i ∈ U v (U v open in T M) it is given byΩ L = dq 1 ∧ · · · ∧ dq N ∧ dv 1 ∧ · · · ∧ dv N .On the velocity phase–space manifold T M we can also define the actionA : T M → R in local coordinates q i , v i = ˙q i ∈ U v (U v open in T M) givenby A = v i L v i, so E = v i L v i − L. The Lagrangian vector–field X L onT M is determined by the condition i XL ω L = dE. Classically, it is given


290 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionby the second–order Lagrangian equationsd ∂Ldt ∂v i= ∂L∂q i . (3.141)For a Lagrangian vector–field X L on M, there is a base integral curveγ 0 (t) = (q i (t), v i (t)) iff γ 0 (t) is a geodesic. This is given by the contravariantvelocity equation˙q i = v i , ˙v i + Γ i jk v j v k = 0. (3.142)Here Γ i jkdenote the Christoffel symbols of the Levi–Civita connection ∇in an open chart U on M, defined on the Riemannian metric g = by(see section 3.10.1.1 above)Γ i jk = g il Γ jkl , Γ ijk = 1 2 (∂ x ig jk + ∂ x j g ki + ∂ x kg ij ). (3.143)The l.h.s ˙¯v i = ˙v i + Γ i jk vj v k in the second part of (3.142) representsthe Bianchi covariant derivative of the velocity with respect to t. Paralleltransport on M is defined by ˙¯v i = 0. When this applies, X L is called thegeodesic spray and its flow the geodesic flow.For the dynamics in the gravitational potential field V : M → R, theLagrangian L : T M → R has an extended formL(v, q) = 1 2 g ijv i v j − V (q),A Lagrangian vector–field X L is still defined by the second–order Lagrangianequations (3.141, 3.142).A general form of the forced, non–conservative Lagrangian equations isgiven asd ∂Ldt ∂v i− ∂L∂q i = F i (t, q i , v i )).Here the F i (t, q i , v i ) represent any kind of covariant forces as a functionsof time, coordinates and momenta. In covariant form we have˙q i = v i , g ij ( ˙v i + Γ i jk v j v k ) = F j (t, q i , v i )).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 2913.10.3.2 Lagrange–Poincaré DynamicsEuler–Poincaré EquationsLet G be a Lie group and let L : T G → R be a left–invariant Lagrangian.Let l : g → R be its restriction to the identity. For a curve g(t) ∈ G, letξ(t) = g(t) −1 · ġ(t); that is, ξ(t) = T g(t) L g(t) −1ġ(t). Then the following areequivalent [Marsden and Ratiu (1999)]:(1) g(t) satisfies the Euler–Lagrangian equations for L on G;(2) The variational principle holds,∫δ L(g(t), ġ(t)) dt = 0for variations with fixed endpoints;(3) The Euler–Poincaré equations hold:d ∂ldt ∂ξ = δlAd∗ ξδξ ;(4) The variational principle holds on g,∫δ l(ξ(t)) dt = 0,using variations of the form δξ = ˙η + [ξ, η], where η vanishes at theendpoints.Lagrange–Poincaré EquationsHere we follow [Marsden and Ratiu (1999)] and drop Euler–Lagrangianequations and variational principles from a general velocity phase–spaceT M to the quotient T M/G by an action of a Lie group G on M. If Lis a G−invariant Lagrangian on T M, it induces a reduced Lagrangian lon T M/G. We introduce a connection A on the principal bundle M →S = M/G, assuming that this quotient is nonsingular. This connectionallows one to split the variables into a horizontal and vertical part. Letinternal variables x α be coordinates for shape–space S = M/G, let η a becoordinates for the Lie algebra g relative to a chosen basis, let l be theLagrangian regarded as a function of the variables x α , ẋ α , η a and let Cdbabe the structure constants of the Lie algebra g of G.If one writes the Euler–Lagrangian equations on T M in a local principalbundle trivialization, with coordinates x α on the base and η a in the fibre,


292 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthen one gets the following system of Hamel equations:d ∂ldt ∂ẋ α = ∂l∂x α , and ddt∂l∂η b = ∂l∂η a Ca dbη a .However, this representation of the equations does not make global intrinsicsense. The introduction of a connection overcomes this, and onecan intrinsically and globally split the original variational principle relativeto horizontal and vertical variations. One gets from one form to the otherby means of the velocity shift given by replacing η a by the vertical partrelative to the affine connectionξ a = A a αẋ α + η a .Here A a α are the local coordinates of the connection A. This change of coordinatesis motivated from the mechanical point of view, since the variablesξ a have the interpretation of the locked angular velocity. The resultingLagrange–Poincaré equations have the following form:d ∂ldt ∂ẋ α − ∂l∂x α = ∂l (∂ξ a Bαβẋ a β + Bαdξ a d) ,d ∂ldt ∂ξ b = ∂l (∂ξ a Bbαẋ a α + Cdbξ a d) .In these equations, Bαβ a are the coordinates of the curvature B of A,B a dα = C a dbA b α, and B a bα = −B a αb.The variables ξ a may be regarded as the rigid part of the variables on theoriginal configuration space, while x α are the internal variables.3.10.4 Core Application: Search for Quantum Gravity3.10.4.1 What Is Quantum Gravity?The landscape of fundamental physics has changed substantially during thelast few decades. Not long ago, our understanding of the weak and stronginteractions was very confused, while general relativity was almost totallydisconnected from the rest of physics and was empirically supported bylittle more than its three classical tests. Then two things have happened.The SU(3) × SU(2) × U(1) Standard Model has found a dramatic empiricalsuccess, showing that quantum field theory (QFT) is capable of describingall accessible fundamental physics, or at least all non–gravitational physics.At the same time, general relativity (GR) has undergone an extraordinary


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 293‘renaissance’, finding widespread application in astrophysics and cosmology,as well as novel vast experimental support – so that today GR is basicphysics needed for describing a variety of physical systems we have accessto, including advanced technological systems [Ashby (1997)].These two parallel developments have moved fundamental physics to aposition in which it has rarely been in the course of its history: We havetoday a group of fundamental laws, the Standard Model and GR, which–even if it cannot be regarded as a satisfactory global picture of Nature–is perhaps the best confirmed set of fundamental theories after Newton’suniversal gravitation and Maxwell’s electromagnetism. More importantly,there aren’t today experimental facts that openly challenge or escape thisset of fundamental laws. In this unprecedented state of affairs, a largenumber of theoretical physicists from different backgrounds have begun toaddress the piece of the puzzle which is clearly missing: combining thetwo halves of the picture and understanding the quantum properties ofthe gravitational field. Equivalently, understanding the quantum propertiesof space–time. Interest and researches in quantum gravity have thusincreased sharply in recent years. And the problem of understanding whatis a quantum space–time is today at the core of fundamental physics.Today we have some well developed and reasonably well defined tentativetheories of quantum gravity. String theory and loop quantum gravityare the two major examples. Within these theories definite physical resultshave been obtained, such as the explicit computation of the ‘quanta of geometry’and the derivation of the black hole entropy formula. Furthermore,a number of fresh new ideas, like noncommutative geometry, have enteredquantum gravity. For an overview of the problem of quantum gravity, see[Isham (1997)].3.10.4.2 Main Approaches to Quantum GravityString theoryString theory is by far the research direction which is presently mostinvestigated. String theory presently exists at two levels. First, there is awell developed set of techniques that define the string perturbation expansionover a given metric background. Second, the understanding ofthe non–perturbative aspects of the theory has much increased in recentyears [Polchinski (1995)] and in the string community there is a widespreadfaith, supported by numerous indications, in the existence of a yet-to-befoundfull non–perturbative theory, capable of generating the perturbation


294 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionexpansion. There are attempts of constructing this non–perturbative theory,generically denoted M theory. The currently popular one is Matrix–theory, of which it is far too early to judge the effectiveness [Matacz (2002);Ishibashi et. al. (1997)].The claim that string theory solves QG is based on two facts. First, thestring perturbation expansion includes the graviton. More precisely, one ofthe string modes is a massless spin two, and helicity ±2, particle. Such aparticle necessarily couples to the energy–momentum tensor of the rest ofthe fields [Weinberg (1964); Weinberg (1980)] and gives general relativityto a first approximation. Second, the perturbation expansion is consistentif the background geometry over which the theory is defined satisfies a certainconsistency condition; this condition turns out to be a high energymodification of the Einstein’s equations. The hope is that such a consistencycondition for the perturbation expansion will emerge as a full–fledgeddynamical equation from the yet–to–be–found non–perturbative theory.From the point of view of the problem of quantum gravity, the relevantphysical results from string theory are two [Rovelli (1997)]:Black hole entropy. The most remarkable physical results for quantumgravity is the derivation of the Bekenstein–Hawking formula for theentropy of a black hole as a function of the horizon area. This beautifulresult has been obtained by [Strominger and Vafa (1996)], and has thenbeen extended in various directions. The result indicates that there issome unexpected internal consistency between string theory and QFTon curved space.Microstructure of space–time. There are indications that in string theorythe space–time continuum is meaningless below the Planck length.An old set of results on very high energy scattering amplitudes indicatesthat there is no way of probing the space–time geometry at very shortdistances. What happens is that in order to probe smaller distance oneneeds higher energy, but at high energy the string ‘opens up from beinga particle to being a true string’ which is spread over space–time, andthere is no way of focusing a string’s collision within a small space–timeregion.More recently, in the non–perturbative formulation of the Matrix–theory[Matacz (2002)], the space–time coordinates of the string x i are replaced bymatrices (X i ) n m. This can perhaps be viewed as a new interpretation of thespace–time structure. The continuous space–time manifold emerges onlyin the long distance region, where these matrices are diagonal and com-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 295mute; while the space–time appears to have a noncommutative discretizedstructure in the short distance regime. This features are still poorly understood,but they have intriguing resonances with noncommutative geometry[Connes et. al. (1997)] and loop quantum gravity [Rovelli (1998)].A key difficulty in string theory is the lack of a complete non–perturbative formulation. During the last year, there has been excitementfor some tentative non–perturbative formulations [Matacz (2002)]; but itis far too early to understand if these attempts will be successful. Manypreviously highly acclaimed ideas have been rapidly forgotten.A distinct and even more serious difficulty of string theory is the lackof a background independent formulation of the theory. In the words of EdWitten:‘Finding the right framework for an intrinsic, background independentformulation of string theory is one of the main problems instring theory, and so far has remained out of reach... This problemis fundamental because it is here that one really has to address thequestion of what kind of geometrical object the string represents.’Most of string theory is conceived in terms of a theory describing excitationsover this or that background, possibly with connections betweendifferent backgrounds. This is also true for (most) non–perturbative formulationssuch as Matrix theory. For instance, the (bosonic part of the)Lagrangian of Matrix–theory isL ∼ 1 (2 Tr Ẋ 2 + 1 )2 [Xi , X j ] 2 . (3.144)The indices that label the matrices X i are raised and lowered with aMinkowski metric, and the theory is Lorentz invariant. In other words,the Lagrangian is reallyL ∼ 1 (g2 Tr 00 g ij Ẋ i Ẋ j + 1 )2 gik g jl [X i , X j ][X k , X l ] , (3.145)where g is the flat metric of the background. This shows that there is anon–dynamical metric, and an implicit flat background in the action of thetheory.However, the world is not formed by a fixed background over whichthings happen. The background itself is dynamical. In particular, forinstance, the theory should contain quantum states that are quantum superpositionsof different backgrounds – and presumably these states play


296 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionan essential role in the deep quantum gravitational regime, namely in situationssuch as the big bang or the final phase of black hole evaporation. Theabsence of a fixed background in nature (or active diffeomorphism invariance)is the key general lessons we have learned from gravitational theories[Rovelli (1997)].There has been a burst of recent activity in an outgrowth of stringtheory denoted string cosmology by [Veneziano (1991)]. The aim of stringcosmology is to extract physical consequences from string theory by applyingit to the big bang. The idea is to start from a Minkowski flat universe;show that this is unstable and therefore will run away from the flat (false–vacuum) state. The evolution then leads to a cosmological model thatstarts off in an inflationary phase. This scenario is described using mini–superspace technology, in the context of the low energy theory that emergeas limit of string theory. Thus, first one freezes all the massive modes ofthe string, then one freezes all massless modes except the zero modes (thespatially constant ones), obtaining a finite dimensional theory, which canbe quantized non–perturbatively.Loop quantum gravityThe second most popular approach to quantum gravity, and the most popularamong relativists, is loop quantum gravity [Rovelli (1998)]. Loop quantumgravity is presently the best developed alternative to string theory.Like strings, it is not far from a complete and consistent theory and ityields a corpus of definite physical predictions, testable in principle, onquantum space–time.Loop quantum gravity, however, attacks the problem from the oppositedirection than string theory. It is a non-perturbative and background independenttheory to start with. In other words, it is deeply rooted intothe conceptual revolution generated by general relativity. In fact, successesand problems of loop quantum gravity are complementary to successes andproblems of strings. Loop quantum gravity is successful in providing aconsistent mathematical and physical picture of non perturbative quantumspace–time; but the connection to the low energy dynamics is not yetcompletely clear.The general idea on which loop quantum gravity is based is the following.The core of quantum mechanics is not identified with the structureof (conventional) QFT, because conventional QFT presupposes a backgroundmetric space–time, and is therefore immediately in conflict with GR.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 297Rather, it is identified with the general structure common to all quantumsystems. The core of GR is identified with the absence of a fixed observablebackground space–time structure, namely with active diffeomorphisminvariance. Loop quantum gravity is thus a quantum theory in the conventionalsense: a Hilbert space and a set of quantum (field) operators,with the requirement that its classical limit is GR with its conventionalmatter couplings. But it is not a QFT over a metric manifold. Rather,it is a ‘quantum field theory on a differentiable manifold’, respecting themanifold’s invariances and where only coordinate independent quantitiesare physical.Technically, loop quantum gravity is based on two inputs [Rovelli (1998);Rovelli (1997)]:• The formulation of classical GR based on the Ashtekar connection[Ashtekar (1986); Ashtekar (1987); Ashtekar (1991)]. The version ofthe connection now most popular is not the original complex one, butan evolution of the same, in which the connection is real.• The choice of the holonomies of this connection, denoted loop variables,as basic variables for the quantum gravitational field [Rovelliand Smolin (1988)].This second choice determines the peculiar kind of quantum theory beingbuilt. Physically, it corresponds to the assumption that excitationswith support on a loop are normalizable states. This is the key technicalassumption on which everything relies.It is important to notice that this assumption fails in conventional 4DYang–Mills theory, because loop-like excitations on a metric manifold aretoo singular: the field needs to be smeared in more dimensions [Rovelli(1997)]. Equivalently, the linear closure of the loop states is a ‘far too big’non-separable state space. This fact is the major source of some particlephysicists’s suspicion at loop quantum gravity. What makes GR differentfrom 4D Yang–Mills theory, however, is non–perturbative diffeomorphisminvariance. The gauge invariant states, in fact, are not localized at all– they are, pictorially speaking, smeared by the (gauge) diffeomorphismgroup all over the coordinates manifold. More precisely, factoring awaythe diffeomorphism group takes us down from the state space of the loopexcitations, which is ‘too big’, to a separable physical state space of theright size. Thus, the consistency of the loop construction relies heavilyon diffeomorphism invariance. In other words, the diff–invariant invariantloop states (more precisely, the diff–invariant spin network states) are not


298 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionphysical excitations of a field on space–time. They are excitations of space–time itself.Loop quantum gravity was briefly described by [Rovelli (1997)] as follows:Definition of theory. The mathematical structure of the theory has beenput on a very solid basis. Early difficulties have been overcome. Inparticular, there were three major problems in the theory: the lack ofa well defined scalar product, the overcompleteness of the loop basis,and the difficulty of treating the reality conditions.• The problem of the lack of a scalar product on the Hilbert spacehas been solved with the definition of a diffeomorphism invariantmeasure on a space of connections [Ashtekar and Lewandowski(1995)]. Later, it has also became clear that the same scalar productcan be defined in a purely algebraic manner [DePietri andRovelli (1996)]. The state space of the theory is therefore a genuineHilbert space H.• The overcompleteness of the loop basis has been solved by the introductionof the spin network states [Rovelli and Smolin (1995)].A spin network is a graph carrying labels (related to SU(2) representationsand called ‘colors’) on its links and its nodes.Each spin network defines a spin network state, and the spin networkstates form a (genuine, non-overcomplete) orthonormal basisin H.• The difficulties with the reality conditions have been circumventedby the use of the real formulation [Barbero (1994);Barbero (1995a); Barbero (1995b); Thiemann (1996)].The kinematics of loop quantum gravity is now defined with alevel of rigor characteristic of mathematical physics [Ashtekar andIsham (1992); Ashtekar et. al. (1995)] and the theory can bedefined using various alternative techniques [DePietri and Rovelli(1996); DePietri (1997)].Hamiltonian constraint. A rigorous definition version of the Hamiltonianconstraint equation has been constructed. This is anomaly free,in the sense that the constraints algebra closes (but see later on). TheHamiltonian has the crucial properties of acting on nodes only, whichimplies that its action is naturally discrete and combinatorial [Rovelliand Smolin (1988); Rovelli and Smolin (1994)]. This fact is at the roots


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 299of the existence of exact solutions [Rovelli and Smolin (1988)], and ofthe possible finiteness of the theoryMatter. The old hope that QFT divergences could be cured by QG hasrecently received an interesting corroboration. The matter part of theHamiltonian constraint is well–defined without need of renormalization.Thus, a main possible stumbling block is over: infinities did not appearin a place where they could very well have appeared [Rovelli (1997)].Black hole entropy. The first important physical result in loop quantumgravity is a computation of black hole entropy [Krasnov (1997); Rovelli(1996a); Rovelli (1996b)].Quanta of geometry. A very exciting development in quantum gravity inthe last years has been by the computations of the quanta of geometry.That is, the computation of the discrete eigenvalues of area and volume.In quantum gravity, any quantity that depends on the metric becomesan operator. In particular, so do the area A of a given (physically defined)surface, or the volume V of a given (physically defined) spatial region.In loop quantum gravity, these operators can be written explicitly. Theyare mathematically well defined self–adjoint operators in the Hilbert spaceH. We know from quantum mechanics that certain physical quantities arequantized, and that we can compute their discrete values by computing theeigenvalues of the corresponding operator. Therefore, if we can computethe eigenvalues of the area and volume operators, we have a physical predictionon the possible quantized values that these quantities can take, atthe Planck scale. These eigenvalues have been computed in loop quantumgravity. Here is for instance the main sequence of the spectrum of the areaA ⃗ j = 8πγ G ∑ √ji (j i + 1). (3.146)i⃗j = (j 1 , . . . , j n ) is an n−tuplet of half–integers, labeling the eigenvalues,G and are the Newton and Planck constants, and γ is a dimensionlessfree parameter, denoted the so–called Immirzi parameter [Immirzi (1997)],not determined by the theory. A similar result holds for the volume. Thespectrum (3.146) has been rederived and completed using various differenttechniques [DePietri and Rovelli (1996)]. These spectra represent solidresults of loop quantum gravity. Under certain additional assumptions onthe behavior of area and volume operators in the presence of matter, theseresults can be interpreted as a corpus of detailed quantitative predictionson hypothetical Planck scale observations.


300 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionBesides its direct relevance, the quantization of the area and thee volumeis of interest because it provides a physical picture of quantum space–time.The states of the spin network basis are eigenstates of some area and volumeoperators. We can say that a spin network carries quanta of area along itslinks, and quanta of volume at its nodes. The magnitude of these quantais determined by the coloring. For instance, the half–integers j 1 . . . j n in(3.146) are the coloring of the spin network’s links that cross the givensurface. Thus, a quantum space–time can be decomposed in a basis of statesthat can be visualized as made by quanta of volume (the intersections)separated by quanta of area (the links). More precisely, we can view aspin network as sitting on the dual of a cellular decomposition of physicalspace. The nodes of the spin network sit in the center of the 3–cells, andtheir coloring determines the (quantized) 3–cell’s volume. The links of thespin network cut the faces of the cellular decomposition, and their color ⃗jdetermine the (quantized) areas of these faces via equation (3.146).3.10.4.3 Traditional Approaches to Quantum GravityDiscrete ApproachesDiscrete quantum gravity is the program of regularizing classical GR interms of some lattice theory, quantize this lattice theory, and then studyan appropriate continuum limit, as one may do in QCD. There are threemain ways of discretizing GR.Regge CalculusRegge introduced the idea of triangulating space–time by means of a simplicialcomplex and using the lengths l i of the links of the complex asgravitational variables [Regge (1961)]. The theory can then be quantizedby integrating over the lengths l i of the links. For a recent review andextensive references see [Williams and Tuckey (1992)]. More recent workhas focused in problems such as the geometry of Regge superspace [Hartleet. al. (1997)] and choice of the integration measure.Dynamical TriangulationsAlternatively, one can keep the length of the links fixed, and capturethe geometry by means of the way in which the simplices are glued together,namely by the triangulation. The Einstein–Hilbert action of Eu-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 301clidean gravity is approximated by a simple function of the total numberof simplices and links, and the theory can be quantized summing overdistinct triangulations (for a detailed introduction, see [Ambjørn et. al.(1998)]). There are two coupling constants in the theory, roughly correspondingto the Newton and cosmological constants. These define atwo dimensional space of theories. The theory has a nontrivial continuumlimit if in this parameter space there is a critical point correspondingto a second order phase transition. The theory has phase transitionand a critical point. The transition separates a phase with crumpledspace–times from a phase with ‘elongated’ spaces which are effectively2D, with characteristic of a branched polymer [Bakker and Smit (1995);Ambjørn et. al. (2001a)]. This polymer structure is surprisingly the sameas the one that emerges from loop quantum gravity at short scale. Nearthe transition, the model appears to produce ‘classical’ S 4 space–times, andthere is evidence for scaling, suggesting a continuum behavior.State Sum ModelsA third road for discretizing GR was opened by a celebrated paper by [Ponzanoand Regge (1968)]. They started from a Regge discretization of 3D GRand introduced a second discretization, by posing the so–called Ponzano–Regge ansatz that the lengths l assigned to the links are discretized as well,in half–integers in Planck unitsl = G j, j = 0,1, 1, . . . (3.147)2(Planck length is G in 3D.) The half integers j associated to the links aredenoted ‘coloring’ of the triangulation. Coloring can be viewed as the assignmentof a SU(2) irreducible representation to each link of the Regge triangulation.The elementary cells of the triangulation are tetrahedra, whichhave six links, colored with six SU(2) representations. SU(2) representationtheory naturally assigns a number to a sextuplet of representations:the Wigner 6−j symbol. Rather magically, the product over all tetrahedraof these 6 − j symbols converges to (the real part of the exponent of) theEinstein–Hilbert action. Thus, Ponzano and Regge were led to propose aquantization of 3D GR based on the partition functionZ ∼ ∑ ∏6 − j(color of the tetrahedron), (3.148)coloringtetrahedra


302 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere we have neglected some coefficients for simplicity. They also providedarguments indicating that this sum is independent from the triangulationof the manifold.The formula (3.148) is simple and elegant, and the idea has recently hadmany surprising and interesting developments. 3D GR was quantized as atopological field theory by Ed Witten in [Witten (1988c)] and using loopquantum gravity in [Ashtekar et. al. (1989)]. The Ponzano–Regge quantizationbased on equation (3.148) was shown to be essentially equivalent tothe TQFT quantization in [Ooguri (1992a)], and to the loop quantum gravityin [Rovelli (1993)] (for an extensive discussion of 3D quantum gravity,see [Carlip and Nelson (1995)]).It turns out that the Ponzano–Regge ansatz (3.147) can be derived fromloop quantum gravity [Rovelli (1993)]. Indeed, (3.147) is the 2D version ofthe 3D formula (3.146), which gives the quantization of the area. Therefore,a key result of quantum gravity of the last years, namely the quantizationof the geometry, derived in the loop formalism from a full fledgednon–perturbative quantization of GR, was anticipated as an ansatz by theintuition of Ponzano and Regge.Hawking’s Euclidean Quantum GravityHawking’s Euclidean quantum gravity is the approach based on his formalsum over Euclidean geometries (i.e., an Euclidean path integral, see chapter6 below)∫Z ∼ N D[g] e − R d 4 x √ gR[g] . (3.149)As far as we understand, Hawking and his close collaborators do not anymoreview this approach as an attempt to directly define a fundamentaltheory. The integral is badly ill defined, and does not lead to any knownviable perturbation expansion. However, the main ideas of this approachare still alive in several ways.First, Hawking’s picture of quantum gravity as a sum–over–space–times,continues to provide a powerful intuitive reference point for most of the researchrelated to quantum gravity. Indeed, many approaches can be seenas attempts to replace the ill–defined and non–renormalizable formal integral(3.149) with a well defined expression. The dynamical triangulationapproach (see above) and the spin foam approach (see below) are examplesof attempts to realize Hawking’s intuition. Influence of Euclidean quantumgravity can also be found in the Atiyah axioms for TQFT.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 303Second, this approach can be used as an approximate method for describingcertain regimes of non–perturbative quantum space–time physics,even if the fundamental dynamics is given by a more complete theory. Inthis spirit, Hawking and collaborators have continued the investigation ofphenomena such as, for instance, pair creation of black holes in a backgroundde Sitter space–time.Effective Perturbative Quantum GravityIf we expand classical GR around, say, the Minkowski metric,g µν (x) = η µν + h µν (x),and construct a conventional QFT for the field h µν (x), we get, as it is wellknow, a non renormalizable theory. A small but intriguing group of papershas recently appeared, based on the proposal of treating this perturbativetheory seriously, as a respectable low energy effective theory by its own.This cannot solve the deep problem of understanding the world in generalrelativistic quantum terms. But it can still be used for studying quantumproperties of space–time in some regimes. This view has been advocatedin a convincing way by John Donoghue, who has developed effective fieldtheory methods for extracting physics from non renormalizable quantumGR [Donoghue (1996)].QFT in Curved Space–TimeQuantum field theory in curved space–time is by now a reasonablyestablished theory (see, e.g., [Wald (1994); Birrel and Davies (1982);Fulling (1989)], predicting physical phenomena of remarkable interest suchas particle creation, vacuum polarization effects and Hawking’s black-holeradiance [Hawking (1975)]. To be sure, there is no direct nor indirect experimentalobservation of any of these phenomena, but the theory is quitecredible as an approximate theory, and many theorists in different fieldswould probably agree that these predicted phenomena are likely to be real.The most natural and general formulation of the theory is within thealgebraic approach [Haag (1992)], in which the primary objects are the localobservables and the states of interest may all be treated on equal footing(as positive linear functionals on the algebra of local observables), even ifthey do not belong to the same Hilbert space.The great merit of QFT on curved space–time is that it has providedus with some very important lessons. The key lesson is that in general


304 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionone loses the notion of a single preferred quantum state that could beregarded as the ‘vacuum’; and that the concept of ‘particle’ becomes vagueand/or observer-dependent in a gravitational context. In a gravitationalcontext, vacuum and particle are necessarily ill defined or approximateconcepts. It is perhaps regrettable that this important lesson has not beenyet absorbed by many scientists working in fundamental theoretical physics[Rovelli (1997)].3.10.4.4 New Approaches to Quantum GravityNoncommutative <strong>Geom</strong>etryNoncommutative geometry is a research program in mathematics andphysics which has recently received wide attention and raised much excitement.The program is based on the idea that space–time may have anoncommutative structure at the Planck scale. A main driving force of thisprogram is the radical, volcanic and extraordinary sequence of ideas of A.Connes [Connes (1994)]. Connes observes that what we know about thestructure of space–time derives from our knowledge of the fundamental interactions:special relativity derives from a careful analysis of Maxwell theory;Newtonian space–time and general relativity, derived both from a carefulanalysis of the gravitational interaction. Recently, we have learned todescribe weak and strong interactions in terms of the SU(3)×SU(2)×U(1)Standard Model. Connes suggests that the Standard Model might hide informationon the minute structure of space–time as well. By making thehypothesis that the Standard Model symmetries reflect the symmetry of anoncommutative microstructure of space–time, Connes and Lott are ableto construct an exceptionally simple and beautiful version of the StandardModel itself, with the impressive result that the Higgs field appears automatically,as the components of the Yang–Mills connection in the internal‘noncommutative’ direction [Connes and Lott (1990)]. The theory admitsa natural extension in which the space–time metric, or the gravitationalfield, is dynamical, leading to GR [Chamseddine and Connes (1996)].The key idea behind a non-commutative space–time is to use algebrainstead of geometry in order to describe spaces. Consider a topological(Hausdorf) space M. Consider all continuous functions f on M. Theseform an algebra A, because they can be multiplied and summed, and thealgebra is commutative. According to a celebrated result, due to Gel’fand,knowledge of the algebra A is equivalent to knowledge of the space M,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 305i.e., M can be reconstructed from A. In particular, the points x of themanifold can be obtained as the 1D irreducible representations x of A,which are all of the form x(f) = f(x). Thus, we can use the algebraof the functions, instead of using the space. In a sense, notices Connes,the algebra is more physical, because we never deal with space–time: wedeal with fields, or coordinates, over space–time. But one can captureRiemannian geometry as well, algebraically. Consider the Hilbert space Hformed by all the spinor fields on a given Riemannian (spin) manifold. LetD be the (curved) Dirac operator, acting on H. We can view A as analgebra of (multiplicative) operators on H. Now, from the triple (H, A, D),which Connes calls ‘spectral triple’, one can reconstruct the Riemannianmanifold. In particular, it is not difficult to see that the distance betweentwo points x and y can be obtained from these data byd(x, y) = sup {f∈A,||D,f||


306 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNull–Surface FormulationA second new set of ideas comes from [Frittelli et. al. (1995)]. These authorshave discovered that the (conformal) information about the geometryis captured by suitable families of null hypersurfaces in space–time, andhave been able to reformulate GR as a theory of self–interacting familiesof surfaces. A remarkable aspect of the theory is that physical informationabout the space–time interior is transferred to null infinity, along nullgeodesics. Thus, the space–time interior is described in terms of how wewould (literally) ‘see it’ from outside. This description is diffeomorphisminvariant, and addresses directly the relational localization characteristic ofGR: the space–time location of a region is determined dynamically by thegravitational field and is captured by when and where we see the space–time region from infinity. This idea may lead to interesting and physicallyrelevant diffeomorphism invariant observables in quantum gravity. A discussionof the quantum gravitational fuzziness of the space–time pointsdetermined by this perspective can be found in [Frittelli et. al. (1997)].Spin Foam ModelsFrom the mathematical point of view, the problem of quantum gravity isto understand what is QFT on a differentiable manifold without metric. Aclass of well understood QFT’s on manifolds exists. These are the topologicalquantum field theories (TQFT). Topological field theories are particularlysimple field theories. They have as many fields as gauges and thereforeno local degree of freedom, but only a finite number of global degrees offreedom. An example is GR in 3D, say on a torus (the theory is equivalentto a Chern–Simons theory). In 3D, the Einstein equations require thatthe geometry is flat, so there are no gravitational waves. Nevertheless, acareful analysis reveals that the radii of the torus are dynamical variables,governed by the theory. Witten has noticed that theories of this kind giverise to interesting quantum models [Witten (1988a)], and [Atiyah (1989)]has provided a beautiful axiomatic definition of a TQFT. Concrete examplesof TQFT have been constructed using Hamiltonian, combinatorial andpath integral methods. The relevance of TQFT for quantum gravity hasbeen suggested by many and the recent developments have confirmed thesesuggestions.Recall that TQFT is a diffeomorphism invariant QFT. Sometimes, theexpression TQFT is used to indicate all diffeomorphism invariant QFT’s.This has lead to a widespread, but incorrect belief that any diffeomorphism


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 307invariant QFT has a finite number of degrees of freedom, unless the invarianceis somehow broken, for instance dynamically. This belief is wrong.The problem of quantum gravity is precisely to define a diffeomorphisminvariant QFT having an infinite number degrees of freedom and ‘local’excitations. Locality in a gravity theory, however, is different from localityin conventional field theory. This point is often source of confusion. Hereis Rovelli’s clarification [Rovelli (1997)]:• In a conventional field theory on a metric space, the degrees of freedomare local in the sense that they can be localized on the metric manifold(an electromagnetic wave is here or there in Minkowski space).• In a diffeomorphism invariant field theory such as general relativity, thedegrees of freedom are still local (gravitational waves exist), but theyare not localized with respect to the manifold. They are neverthelesslocalized with respect to each other (a gravity wave is three metersapart from another gravity wave, or from a black hole).• In a topological field theory, the degrees of freedom are not localized atall: they are global, and in finite number (the radius of a torus is notin a particular position on the torus).The first TQFT directly related to quantum gravity was defined by[Turaev and Viro (1992)]. The Turaev–Viro model is a mathematically rigorousversion of the 3D Ponzano-Regge quantum gravity model describedabove. In the Turaev–Viro theory, the sum (3.148) is made finite by replacingSU(2) with quantum SU(2) q (with a suitable q). Since SU(2) qhas a finite number if irreducible representations, this trick, suggested by[Ooguri (1992a); Ooguri (1992b)], makes the sum finite. The extension ofthis model to four dimensions has been actively searched for a while andhas finally been constructed by [Crane and Yetter (1993)], again followingOoguri’s ideas. The Crane–Yetter (CY) model is the first example of 4DTQFT. It is defined on a simplicial decomposition of the manifold. Thevariables are spins (‘colors’) attached to faces and tetrahedra of the simplicialcomplex. Each 4–simplex contains 10 faces and 5 tetrahedra, andtherefore there are 15 spins associated to it. The action is defined in termsof the quantum Wigner 15 − j symbols, in the same manner in which thePonzano–Regge action is constructed in terms of products of 6−j symbols.Z ∼ ∑ ∏15 − j(color of the 4 − simplex), (3.151)coloring4−simplices(where we have disregarded various factors for simplicity). Crane and Yet-


308 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionter introduced their model independently from loop quantum gravity. However,recall that loop quantum gravity suggests that in 4 dimensions thenaturally discrete geometrical quantities are area and volume, and that itis natural to extend the Ponzano–Regge model to 4D by assigning colorsto faces and tetrahedra.The CY model is not a quantization of 4D GR, nor could it be, being aTQFT in strict sense. Rather, it can be formally derived as a quantizationof SU(2) BF theory. BF theory is a topological field theory with two fields,a connection A, with curvature F , and a 2–form B [Horowitz (1989)], withaction∫S[A, B] = B ∧ F. (3.152)However, there is a strict relation between GR and BF. If we add to SO(3, 1)BF theory the constraint that the 2–form B is the product of two tetrad1–formsB = E ∧ E, (3.153)we get precisely GR. This observation has lead many to suggest that aquantum theory of gravity could be constructed by a suitable modificationof quantum BF theory [Baez (1996c)]. This suggestion has become veryplausible, with the following construction of the spin foam models.The key step in development of the spin foam models was taken by [Barbieri(1997)], studying the ‘quantum geometry’ of the simplices that play arole in loop quantum gravity. Barbieri discovered a simple relation betweenthe quantum operators representing the areas of the faces of the tetrahedra.This relation turns out to be the quantum version of the constraint (3.153),which turns BF theory into GR. [Barret and Crane (1997)] added the Barbierirelation to (the SO(3, 1) version of) the CY model. This is equivalentto replacing the the 15-j Wigner symbol, with a different function A BC ofthe colors of the 4–simplex. This replacement defines a ‘modified TQFT’,which has a chance of having general relativity as its classical limit.The Barret–Crane model is not a TQFT in strict sense. In particular,it is not independent from the triangulation. Thus, a continuum theory hasto be formally defined by some suitable sum over triangulationsZ ∼ ∑ ∑ ∏A BC (color of the 4 − simplex). (3.154)triang coloring 4−simplicesThis essential aspect of the construction, however, is not yet understood.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 309The Barret Crane model can virtually be obtained also from loop quantumgravity. This is an unexpected convergence of two very different linesof research. Loop quantum gravity is formulated canonically in the frozentime formalism. While the frozen time formalism is in principle complete,in practice it is cumbersome, and anti-intuitive. Our intuition is four dimensional,not three dimensional. An old problem in loop quantum gravityhas been to derive a space–time version of the theory. A space–time formulationof quantum mechanics is provided by the sum over histories. A sumover histories can be derived from the Hamiltonian formalism, as Feynmandid originally. Loop quantum gravity provides a mathematically well definedHamiltonian formalism, and one can therefore follow Feynman stepsand construct a sum over histories quantum gravity starting from the loopformalism. This has been done in [Reisenberger and Rovelli (1997)]. Thesum over histories turns out to have the form of a sum over surfaces.More precisely, the transition amplitude between two spin networkstates turns out to be given by a sum of terms, where each term can be representedby a (2D) branched ‘colored’ surface in space–time. A branchedcolored surface is formed by elementary surface elements carrying a label,that meet on edges, also carrying a labelled; edges, in turn meet in vertices(or branching points, see Figure 3.11). The contribution of one such sur-Fig. 3.11A branched surface with two vertices.


310 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfaces to the sum over histories is the product of one term per each branchingpoint of the surface. The branching points represent the ‘vertices’ of thistheory, in the sense of Feynman. The contribution of each vertex can becomputed algebraically from the ‘colors’ (half integers) of the adjacent surfaceelements and edges. Thus, space–time loop quantum gravity is definedby the partition functionZ ∼ ∑surfaces∑colorings∏verticesA loop (color of the vertex) (3.155)The vertex A loop is determined by a matrix elements of the Hamiltonianconstraint. The fact that one obtains a sum over surfaces is not too surprising,since the time evolution of a loop is a surface. Indeed, the time evolutionof a spin network (with colors on links and nodes) is a surface (withcolors on surface elements and edges) and the Hamiltonian constraint generatesbranching points in the same manner in which conventional Hamiltoniansgenerate the vertices of the Feynman diagrams.Now, (3.155) has the same structure of the Barret–Crane model (3.151).To see this, simply notice that we can view each branched colored surface aslocated on the lattice dual to a triangulation. Then each vertex correspondto a 4-simplex; the coloring of the two models matches exactly (elementarysurfaces → faces, edges → tetrahedra); and summing over surfacescorresponds to summing over triangulations. The main difference is thedifferent weight at the vertices. The Barret–Crane vertex A BC can be readas a covariant definition a Hamiltonian constraint in loop quantum gravity.Thus, the space–time formulation of loop quantum GR is a simple modificationof a TQFT. This approach provides a 4D pictorial intuition of quantumspace–time, analogous to the Feynman graphs description of quantumfield dynamics. John Baez has introduced the term ‘spin foam’ for thebranched colored surfaces of the model, in honor of John Wheeler’s intuitionson the quantum microstructure of space–time. Spin foams are aprecise mathematical implementation of Wheeler’s ‘space–time foam’ suggestions.3.10.4.5 Black Hole EntropyA focal point of the research in quantum gravity in the last years hasbeen the discussion of black hole (BH) entropy. This problem has beendiscussed from a large variety of perspectives and within many differentresearch programs.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 311Let us very briefly recall the origin of the problem. In classical GR,future event horizons behave in a manner that has a peculiar thermodynamicalflavor. This remark, together with a detailed physical analysis ofthe behavior of hot matter in the vicinity of horizons, prompted Bekensteinto suggest that there is entropy associated to every horizon. The suggestionwas first consider ridicule, because it implies that a black hole is hotand radiates. But then Steven Hawking, in a celebrated work [Hawking(1975)], showed that QFT in curved space–time predicts that a black holeemits thermal radiation, precisely at the temperature predicted by Bekenstein,and Bekenstein courageous suggestion was fully vindicated. Sincethen, the entropy of a BH has been indirectly computed in a surprisingvariety of manners, to the point that BH entropy and BH radiance arenow considered almost an established fact by the community, although, ofcourse, they were never observed nor, presumably, they are going to beobserved soon. This confidence, perhaps a bit surprising to outsiders, isrelated to the fact thermodynamics is powerful in indicating general propertiesof systems, even if we do not control its microphysics. Many hopethat the Bekenstein–Hawking radiation could play for quantum gravity arole analogous to the role played by the black body radiation for quantummechanics. Thus, indirect arguments indicate that a Schwarzschild BH hasan entropyS = 1 4AG(3.156)The remaining challenge is to derive this formula from first principles [Rovelli(1997)].Later in the book we will continue our exposition of various approachesto quantum gravity.3.10.5 Basics of Morse and (Co)Bordism Theories3.10.5.1 Morse Theory on Smooth ManifoldsAt the same time the variational formulae were discovered, a related technique,called Morse theory, was introduced into Riemannian geometry. Thistheory was developed by Morse, first for functions on manifolds in 1925,and then in 1934, for the loop space. The latter theory, as we shall see, setsup a very nice connection between the first and second variation formulaefrom the previous section and the topology of M. It is this relationship thatwe shall explore at a general level here. In section 5 we shall then see how


312 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthis theory was applied in various specific settings.If we have a proper function f : M → R, then its Hessian (as a quadraticform) is in fact well defined at its critical points without specifying anunderlying Riemannian metric. The nullity of f at a critical point is definedas the dimension of the kernel of ∇ 2 f, while the index is the number ofnegative eigenvalues counted with multiplicity. A function is said to bea Morse function if the nullity at any of its critical points is zero. Notethat this guarantees in particular that all critical points are isolated. Thefirst fundamental Theorem of Morse theory is that one can determine thetopological structure of a manifold from a Morse function. More specifically,if one can order the critical points x 1 , . . . , x k so that f (x 1 ) < · · · < f (x k )and the index of x i is denoted λ i , then M has the structure of a CW complexwith a cell of dimension λ i for each i. Note that in case M is closed then x 1must be a minimum and so λ 1 = 0, while x k is a maximum and λ k = n. Theclassical example of Milnor of this Theorem in action is a torus in 3–spaceand f the height function.We are now left with the problem of trying to find appropriate Morsefunctions. While there are always plenty of such functions, there does notseem to be a natural way of finding one. However, there are natural choicesfor Morse functions on the loop space to a Riemannian manifold. Thisis, somewhat inconveniently, infinite–dimensional. Still, one can developMorse theory as above for suitable functions, and moreover the loop spaceof a manifold determines the topology of the underlying manifold.If m, p ∈ M, then we denote by Ω mp the space of all C k paths from mto p. The first observation about this space is thatπ i+1 (M) = π i (Ω mp ) .To see this, just fix a path from m to q and then join this path to everycurve in Ω mp . In this way Ω mp is identified with Ω m , the space of loops fixedat m. For this space the above relationship between the homotopy groupsis almost self-evident.On the space Ω mp we have two naturally defined functions, the arc–length and energy functionals:∫L (γ, I) = | ˙γ| dt, and E (γ, I) = 1 ∫| ˙γ| 2 dt.2IWhile the energy functional is easier to work with, it is the arc–lengthfunctional that we are really interested in. In order to make things workout nicely for the arc–length functional, it is convenient to parameterizeI


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 313all curves on [0, 1] and proportionally to arc–length. We shall think ofΩ mp as an infinite–dimensional manifold. For each curve γ ∈ Ω mp thenatural choice for the tangent space consists of the vector–fields along γwhich vanish at the endpoints of γ. This is because these vector–fields areexactly the variational fields for curves through γ in Ω mp , i.e., fixed endpointvariations of γ. An inner product on the tangent space is then naturallydefined by(X, Y ) =∫ 10g (X, Y ) dt.Now the first variation formula for arc–length tells us that the gradient forL at γ is -∇ ˙γ ˙γ. Actually this cannot be quite right, as -∇ ˙γ ˙γ does not vanishat the endpoints. The real gradient is gotten in the same way we find thegradient for a function on a surface in space, namely, by projecting it downinto the correct tangent space. In any case we note that the critical pointsfor L are exactly the geodesics from m to p. The second variation formulatells us that the Hessian of L at these critical points is given by∇ 2 L (X) = Ẍ + R (X, ˙γ) ˙γ,at least for vector–fields X which are perpendicular to γ. Again we ignorethe fact that we have the same trouble with endpoint conditions as above.We now need to impose the Morse condition that this Hessian is not allowedto have any kernel. The vector–fields J for which ¨J + R (J, ˙γ) ˙γ = 0 arecalled Jacobi fields. Thus we have to Figure out whether there are anyJacobi fields which vanish at the endpoints of γ. The first observation isthat Jacobi fields must always come from geodesic variations. The Jacobifields which vanish at m can therefore be found using the exponential mapexp m . If the Jacobi field also has to vanish at p, then p must be a criticalvalue for exp m . Now Sard’s Theorem asserts that the set of critical valueshas measure zero. For given m ∈ M it will therefore be true that thearc–length functional on Ω mp is a Morse function for almost all p ∈ M.Note that it may not be possible to choose p = m, the simplest examplebeing the standard sphere. We are now left with trying to decide what theindex should be. This is the dimension of the largest subspace on whichthe Hessian is negative definite. It turns out that this index can also becomputed using Jacobi fields and is in fact always finite. Thus one cancalculate the topology of Ω mp , and hence M, by finding all the geodesicsfrom m to p and then computing their index.


314 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn geometrical situations it is often unrealistic to suppose that one cancalculate the index precisely, but as we shall see it is often possible to givenlower bounds for the index. As an example, note that if M is not simply–connected, then Ω mp is not connected. Each curve of minimal length inthe path components is a geodesic from m to p which is a local minimumfor the arc–length functional. Such geodesics evidently have index zero. Inparticular, if one can show that all geodesics, except for the minimal onesfrom m to p, have index > 0, then the manifold must be simply–connected.We will apply Morse theory in biodynamics/robotic in section (3.13.5.2)below.3.10.5.2 (Co)Bordism Theory on Smooth Manifolds(Co)bordism appeared as a revival of Poincaré’s unsuccessful 1895 attemptsto define homology using only manifolds. Smooth manifolds (withoutboundary) are again considered as ‘negligible’ when they are boundariesof smooth manifolds–with–boundary. But there is a big difference, whichkeeps definition of ‘addition’ of manifolds from running into the difficultiesencountered by Poincaré; it is now the disjoint union. The (unoriented)(co)bordism relation between two compact smooth manifolds M 1 , M 2 ofsame dimension n means that their disjoint union ∂W = M 1 ∪ M 2 is theboundary ∂W of an (n + 1)D smooth manifold–with–boundary W . Thisis an equivalence relation, and the classes for that relation of nD manifoldsform a commutative group N n in which every element has order 2. Thedirect sum N • = ⊕ n≥0 N n is a ring for the multiplication of classes deducedfrom the Cartesian product of manifolds.More precisely, a manifold M is said to be a (co)bordism from A to Bif exists a diffeomorphism from a disjoint sum, ϕ ∈ diff(A ∗ ∪ B, ∂M). Two(co)bordisms M(ϕ) and M ′ (ϕ ′ ) are equivalent if there is a Φ ∈ diff(M, M ′ )such that ϕ ′ = Φ ◦ ϕ. The equivalence class of (co)bordisms is denoted byM(A, B) ∈ Cob(A, B) [Stong (1968)].Composition c Cob of (co)bordisms comes from gluing of manifolds [Baezand Dolan (1995)]. Let ϕ ′ ∈ diff(C ∗ ∪ D, ∂N). One can glue (co)bordismM with N by identifying B with C ∗ , (ϕ ′ ) −1 ◦ ϕ ∈ diff(B, C ∗ ). We get theglued (co)bordism(M ◦ N)(A, D) and a semigroup operation,c(A, B, D) : Cob(A, B) × Cob(B, D) −→ Cob(A, D).A surgery is an operation of cutting a manifold M and gluing to cylin-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 315ders. A surgery gives new (co)bordism: from M(A, B) into N(A, B). Thedisjoint sum of M(A, B) with N(C, D) is a (co)bordism (M ∪ N)(A ∪C, B ∪ D). We got a 2–graph of (co)bordism Cob with Cob 0 = Man d ,Cob 1 = Man d+1 , whose 2–cells from Cob 2 are surgery operations.There is an n−category of (co)bordisms BO [Leinster (2003)] with:• 0−cells: 0−manifolds, where ‘manifold’ means ‘compact, smooth, orientedmanifold’. A typical 0−cell is • • • • .• 1−cells: 1−manifolds with corners, i.e., (co)bordisms between0−manifolds, such as(this being a 1−cell from the4−point manifold to the 2−point 0−manifold).• 2−cells: 2−manifolds with corners, such as• 3−cells, 4−cells,... are defined similarly;• Composition is gluing of manifolds.The (co)bordisms theme was taken a step further by [Baez and Dolan(1995)], when when they started a programme to understand the subtlerelations between certain TMFT models for manifolds of different dimensions,frequently referred to as the dimensional ladder. This programme isbased on higher–dimensional algebra, a generalization of the theory of categoriesand functors to n−categories and n−functors. In this framework atopological quantum field theory (TMFT) becomes an n−functor from then−category BO of n−cobordisms to the n−category of n−Hilbert spaces.


316 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.11 Finsler Manifolds and Their ApplicationsRecall that Finsler geometry is such a generalization of Riemannian geometry,that is closely related to multivariable calculus of variations.3.11.1 Definition of a Finsler ManifoldLet M be a real, smooth, connected, finite–dimensional manifold. The pair(M, F ) is called a Finsler manifold iff there exists a fundamental functionF : T M → R, not necessary reversible (i.e., F (x, −y) need not be equal toF (x, y)), that satisfies the following set of axioms (see, e.g., [Udriste andNeagu (1999)]):F1 F (x, y) > 0 for all x ∈ M, y ≠ 0.F2 F (x, λy) = |λ|F (x, y) for all λ ∈ R, (x, y) ∈ T M.F3 the fundamental metric tensor g ij on M, given byg ij (x, y) = 1 ∂ 2 F 22 ∂y i ∂y j ,is positive definite.F4 F is smooth (C ∞ ) at every point (x, y) ∈ T M with y ≠ 0 and continuous(C 0 ) at every (x, 0) ∈ T M. Then, the absolute Finsler energyfunction is given byF 2 (x, y) = g ij (x, y)y i y j .Let c = c(t) : [a, b] → M be a∣smooth regular curve on∣M. For anytwo vector–fields X(t) = X i ∂(t) ∣c(t)∂xand Y (t) = Y i ∂(t) ∣c(t) i ∂xalong theicurve c = c(t), we introduce the scalar (inner) product [Chern (1996)]g(X, Y )(c) = g ij (c, ċ)X i Y jalong the curve c.In particular, if X = Y then we have ‖X‖ = √ g(X, X). The vector–fields X and Y are orthogonal along the curve c, denoted by X⊥Y , iffg(X, Y ) = 0.Let CΓ(N) = (L i jk , N j i, Ci jk) be the Cartan canonical N−linear metricconnection determined by the metric tensor g ij (x, y). The coefficients of


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 317this connection are expressed by [Udriste and Neagu (1999)]L i jk = 1 2 gim (δgmkδx j+ δg jmδx kNj i = 1 ∂ (Γi2 ∂y j kl y k y l) = 1 2where− δg )jkδx m , Cjk i = 1 (∂gmk2 gim ∂y j + ∂g jm∂y k − ∂g )jk∂y m ,∂Γ i 00∂y j ,δδx i = ∂∂x i + N j ∂i∂y j .Γi jk = 1 2 gim (∂gmk∂x j+ ∂g jm∂x k− ∂g )jk∂x m ,Let X∣be a vector–field along the curve c expressed locally by X(t) =X i ∂(t) ∣c(t)∂x. Using the Cartan N−linear connection, we define the covariantderivative ∇Xdti of X(t) along the curve c(t), by [Udriste and Neagu(1999)]Sincewe have∇Xdt= {Ẋi + X m [L i mk(c, ċ)ċ k + C i mk(c, ċ) δ δt (ċk )]} ∂∂x i ∣∣∣∣c(t).δδt (ċk ) = ¨c k + Nl k (c, ċ)ċ l ,∇X=dt{Ẋi + X m [Γ i mk(c, ċ)ċ k + Cmk(c, i ċ)¨c k ]} ∂ ∣∣∣∣c(t)∂x i ,(3.157)whereΓ i mk(c, ċ) = L i mk(c, ċ) + C i ml(c, ċ)N l k(c, ċ).In particular, c is a geodesic iff∇ċdt = 0.Since CΓ(N) is a metric connection, we have( ) (d∇Xdt [g(X, Y )] = g dt , Y + g X, ∇Y ).dt3.11.2 Energy Functional, Variations and ExtremaLet x 0 , x 1 ∈ M be two points not necessarily distinct. We introduce theΩ−set on M, asΩ = {c : [0, 1] → M | c is piecewise C ∞ regular curve, c(0) = x 0 , c(1) = x 1 }.For every p ∈ R − {0}, we can define the p−energy functional on M


318 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction[Udriste and Neagu (1999)]E p : Ω → R + , asE p (c) =∫ 10[g ij (c, ċ)ċ i ċ j ] p/2 dt =∫ 10[g(ċ, ċ)] p/2 dt =In particular, for p = 1 we get the length functionalL(c) =∫ 1and for p = 2 we get the energy functionalE(c) =0∫ 10‖ċ‖dt,‖ċ‖ 2 dt.∫ 10‖ċ‖ p dt.Also, for any naturally parametrized curve (i.e., ‖ċ‖ = const) we haveE p (c) = (L(c)) p = (E(c)) p/2 .Note that the p−energy of a curve is dependent of parametrization if p ≠ 1.For every curve c ∈ Ω, we define the tangent space T c Ω asT c Ω = {X : [0, 1] → T M | X is continuous, piecewise C ∞ , X(t) ∈ T c(t) M,for all t ∈ [0, 1], X(0) = X(1) = 0}.Let (c s ) s∈(−ɛ,ɛ) ⊂ Ω be a one–parameter variation of the curve c ∈ Ω. WedefineUsing the equalityX(t) = dc sds (0, t) ∈ T cΩ.( ) ( ∇ċs∇g∂s , ċ s = g∂t( ) ) ∂cs, ċ s ,∂swe can prove the following Theorem: The first variation of the p−energy is1pdE p (c s )(0) = − ∑ dst−∫ 10g(X, ∆ t (‖ċ‖ p−2 ċ))(‖ċ‖ p−4 g X, ‖ċ‖ 2 ∇ċ( ) ) ∇ċdt + (p − 2)g dt , ċ ċ dt,where ∆ t (‖ċ‖ p−2 ċ) = (‖ċ‖ p−2 ċ) t + − (‖ċ‖ p−2 ċ) t − represents the jump of‖ċ‖ p−2 ċ at the discontinuity point t ∈ (0, 1) [Udriste and Neagu (1999)].The curve c is a critical point of E p iff c is a geodesic.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 319In particular, for p = 1 the curve c is a reparametrized geodesic.Now, let c ∈ Ω be a critical point for E p (i.e., the curve c is a geodesic).Let (c s1s 2) s1,s 2∈(−ɛ,ɛ) ⊂ Ω be a two–parameter variation of c. Using thenotations:X(t) = ∂c s 1s 2∂s 1(0, 0, t) ∈ T c Ω, Y (t) = ∂c s 1s 2∂s 2(0, 0, t) ∈ T c Ω,‖ċ‖ = v = constant, and I p (X, Y ) = ∂2 E p (c s1s 2)∂s 1 ∂s 2(0, 0),we get the following Theorem: The second variation of the p−energy is[Udriste and Neagu (1999)]1pv p−4 I p(X, Y ) = − ∑ ( ( ∇Xg Y, v 2 ∆ tdtt∫ 1( [ ]∇− g Y, v 2 ∇X+ R 2 (X, ċ)ċ + (p − 2)g0 dt dt))=( ∇Xdt− ( )∇Xdt(where ∆ ∇X t dtt +t −discontinuity point t ∈ (0, 1); also, if Rijk lof the Finsler curvature tensor, then) ( ( ∇X+ (p − 2)g ∆ tdt([ ∇ ∇X+ R 2 (X, ċ)ċdt dt) ) ), ċ ċrepresents the jump of ∇Xdtat the(c, ċ) represents the components]) ), ċ ċ dt,R 2 (X, ċ)ċ = Rijk(c, l ċ)ċ i ċ j X k ∂∂x l = Rl jk(c, ċ)ċ j X k ∂∂x l .In particular, we haveRjk i = δN ji iδx k −δN kδx j , and Ri hjk = δLi hjδx k −δLi hkδx j +Ls hjL i sk−L s hkL i sj+ChsR i jk.sMoreover, using the Ricci identities for the deflection tensors, we also haveR i jk = R i mjky m = R i 0jk.I p (X, Y ) = 0 (for all Y ∈ T c Ω) iff X is a Jacobi field, i.e.,∇ ∇X+ R 2 (X, ċ)ċ = 0.dt dtIn these conditions we have the following definition: A point c(b) (0 ≤a < b < 1) of a geodesic c ∈ Ω is called a conjugate point of a point c(a)along the curve c(t), if there exists a non–zero Jacobi field which vanishesat t ∈ {a, b}.


320 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow, integrating by parts and using the property of metric connection,we find∫11( ∇Xpv p−4 I p(X, Y ) = v[g20 dt , ∇Y )]− R 2 (X, ċ, Y, ċ)dt(+ (p − 2)g ċ, ∇X ) (g ċ, ∇Y )dt,dt dtwhere R 2 (X, ċ, Y, ċ) = g(R 2 (Y, ċ)ċ, X) = R 0i0j (c(t), ċ(t))X i Y j .Let R ijk = g jm Rik m . In any Finsler space the following identity is satisfied,R ijk + R jki + R kij = 0,get by the Bianchi identities. As R 0i0j = R i0j = R j0i = R 0j0i we getR 2 (X, ċ, Y, ċ) = R 2 (Y, ċ, X, ċ).The quadratic form associated to the Hessian of the p−energy is givenbyI p (X) = I p (X, X) =Let∫ 10v 2 [ ∥∥∥∥∇Xdt∥2− R 2 (X, ċ, X, ċ)T ⊥ c Ω = {X ∈ T c Ω | g(X, ċ) = 0},]and+(p−2)[ (g ċ, ∇X )] 2dt.dtT ′ cΩ = {X ∈ T c Ω | X = fċ, where f : [0, 1] → R is continuous,piecewise C ∞ , f(0) = f(1) = 0}.Let c be a geodesic and p ∈ R − {0, 1}. Then I p (T cΩ) ′ ≥ 0 for p ∈(−∞, 0) ∪ (1, ∞), and I p (T cΩ) ′ ≤ 0 for p ∈ (0, 1). Moreover, in both cases:I p (X) = 0 iff X = 0. To prove it, let X = fċ ∈ T cΩ. ′ Then we have [Udristeand Neagu (1999)]1v p−4 I p(X) = p= p∫ 10∫ 10{v 2 [ g(f ′ ċ, f ′ ċ) − R 2 (fċ, ċ, fċ, ċ) ] + (p − 2) [g(ċ, f ′ ċ)] 2} dt[v 4 (f ′ ) 2 + (p − 2)v 4 (f ′ ) 2] ∫ 1dt = p(p − 1)v 4 (f ′ ) 2 dt.Moreover, if I p (X) = 0, then f ′ = 0, which means that f is constant.The conditions f(0) = f(1) = 0 imply that f = 0.0


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 321As I p (T ′ cΩ) is positive definite for p ∈ (−∞, 0) ∪ (1, ∞) and negativedefinite for p ∈ (0, 1), it is sufficient to study the behavior of I p restrictedto T ⊥ c Ω. Since X⊥ċ and the curve c is a geodesic it followsHence, for all X ∈ T ⊥ c Ω, we have1pv p−2 I p(X) =∫ 10(g ċ, ∇X )= 0.dt[ ∥∥∥∥∇Xdt∥2− R 2 (X, ċ, X, ċ)]dt = I(X).3.11.3 Application: Finsler–Lagrangian Field TheoryIn this subsection we present generalized Finsler–Lagrangian field theory.The geometrical background of this theory relies on the notion of generalizedLagrangian space, GL n = (M, g ij (x k , y k )), which is a real nD manifold Mwith local coordinates {x i }, (i = 1, ..., n) and a symmetric fundamentalmetric tensor–field g ij = g ij (x k , y k ) of rank n and constant signature on T[Miron et. al. (1988); Miron and Anastasiei (1994)].From physical point of view, the fundamental metric tensor representsa unified gravitational field on T M, which consists of one external(x)−gravitational field spanned by points {x i }, and the one internal(y)−gravitational field spanned by directions {y i } and equipped with somemicroscopic character of the space–time structure.The field theory developed on a generalized Lagrangian space GL n relieson a fixed a priori nonlinear connection Γ = (Nj i (x, y)) on the tangent bundleT M. This plays the role of mapping operator of the internal (y)−fieldonto the external (x)−field, and prescribes the interaction between (x)−and (y)−fields. From geometrical point of view, the nonlinear connectionallows the construction of the adapted bases [Miron et. al. (1988);Miron and Anastasiei (1994)]{ δδx i =∂∂x i − N j i}∂∂y j , ∂∂y i ⊂ X (T M),{dx i , δy i = dy i + N i jdx j } ⊂ X ∗ (T M).As to the spatial structure, the most important thing is to determinethe Cartan canonical connection CΓ = (L i jk , Ci jk ) with respect to g ij, which


322 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncomes from the metric conditionsg ij|k = δg ijδx k −Lm ikg mj −L m jkg mi = 0,g ij;k = ∂g ij∂y k −Cm ikg mj −C m jkg mi = 0,where “ |k ” and “ ;k ” are the local h− and v− covariant derivatives of CΓ.The importance of the Cartan connection comes from its main role playedin the generalized Finsler–Lagrangian theory of physical fields.Regarding the unified field g ij (x, y) of GL n , the authors of [Miron et. al.(1988); Miron and Anastasiei (1994)] constructed a Sasakian metric on T M,G = g ij dx i ⊗ dx j + g ij δy i ⊗ δy j .In this context, the Einstein equations for the gravitational potentialsg ij (x, y) of a generalized Lagrangian space GL n , (n > 2), are postulatedas being the Einstein equations attached to CΓ and G,R ij − 1 2 Rg ij = KT H ij ,S ij − 1 2 Sg ij = KT V ij ,′ P ij = KT 1 ij,′′ P ij = −KT 2 ij,where R ij = Rijm m , S ij = Sijm m , ′ P ij = Pijm m , ′′ P ij = Pimj m are the Riccitensors of CΓ, R = g ij R ij and S = g ij S ij are the scalar curvatures, Tij H,Tij V , T ij 1 , T ij 2 are the components of the energy–momentum tensor T, andK is the Einstein constant (equal to 0 for vacuum). Moreover, the energy–momentum tensors TijH and T ij V satisfy the conservation laws [Miron et. al.(1988); Miron and Anastasiei (1994)]KT H j|m m = −1 hm(Pjs Rhm s + 2R s2mjPs m ), KT V j|m m = 0.The generalized Lagrangian theory of electromagnetism relies on thecanonical Liouville vector–field C = y i ∂∂yand the Cartan connection CΓiof the generalized Lagrangian space GL n . In this context, we can introducethe electromagnetic two—form on T M [Miron and Anastasiei (1994)]F = F ij δy i ∧ dx j + f ij δy i ∧ δy j ,whereF ij = 1 2 [(g imy m ) |j − (g jm y m ) |i ], f ij = 1 2 [(g imy m ) ;j − (g jm y m ) ;i ].Using the Bianchi identities attached to the Cartan connection CΓ, theyconclude that the electromagnetic components F ij and f ij are governed by


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 323the following Maxwell–type equationsF ij|k + F jk|i + F ki|j = −[C imr y m + (g im y m ) |r ]R r jk,F ij;k + F jk;i + F ki;j = −(f ij|k + f jk|i + f ki|j ), f ij;k + f jk;i + f ki;j = 0.3.11.4 Riemann–Finsler Approach to Information<strong>Geom</strong>etry3.11.4.1 Model Specification and Parameter EstimationModel as a Parametric Family of Probability DistributionsFrom a statistical standpoint, observed data is a random sample froman unknown population. Ideally, the goal of modelling is to deduce thepopulation that generated the observed data. Formally, a model is definedas a parametric family of probability distributions (see [Myung and Pitt(2003)]).Let us use f(y|w) to denote the probability distribution function thatgives the probability of observing data, y = (y 1 , . . . , y m ), given the model’sparameter vector, w = (w 1 , . . . , w k ). Under the assumption that individualobservations, y i ’s, are independent of one another, f(y|w) can be rewrittenas a product of individual probability distribution functions,f (y = (y 1 , ..., y m ) |w) = f (y 1 |w) f (y 2 |w) ... f (y m |w) . (3.158)Parameter EstimationOnce a model is specified with its parameters and data have been collected,the model’s ability to fit the data can be assessed. Model fit ismeasured by finding parameter values of the model that give the ‘best’ fitto the data in some defined sense – a procedure called parameter estimationin statistics.There are two generally accepted methods of parameter estimation:least–squares estimation (LSE) and maximum likelihood estimation (MLE).In LSE, we seek the parameter values that minimize the sum of squares error(SSE) between observed data and a model’s predictions:SSE(w) =m∑(y i − y i,prd (w)) 2 ,i=1where y i,prd (w) denotes the model’s prediction for observation y i . In MLE,we seek the parameter values that are most likely to have produced the data.


324 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis is obtained by maximizing the log–likelihood of the observed data:m∑loglik (w) = ln f(y i |w).i=1By maximizing either the likelihood or the log–likelihood, the same solutionis obtained because the two are monotonically related to each other.In practice, the log–likelihood is preferred for computational ease. The parametersthat minimize the sum of squares error or the log–likelihood arecalled the LSE or MLE estimates, respectively.For normally distributed data with constant variance, LSE and MLEare equivalent in the sense that both methods yield the same parameterestimates. For non–normal data such as proportions and response times,however, LSE estimates tend to differ from MLE estimates. Although LSEis often the ‘de facto’ method of estimation in cognitive psychology, MLEis a preferred method of estimation in statistics, especially for non–normaldata. In particular, MLE is well–suited for statistical inference in hypothesistesting and model selection. Finding LSE or MLE estimates generallyrequires use of a numerical optimization procedure.3.11.4.2 Model Evaluation and TestingQualitative CriteriaA model satisfies the explanatory adequacy criterion if its assumptionsare plausible and consistent with established findings, and importantly, thetheoretical account is reasonable for the cognitive process of interest. Inother words, the model must be able to do more than redescribe observeddata. The model must also be interpretable in the sense that the modelmakes sense and is understandable. Importantly, the components of themodel, especially, its parameters, must be linked to psychological processesand constructs. Finally, the model is said to be faithful to the extent thatthe model’s ability to capture the underlying mental process originatesfrom the theoretical principles embodied in the model, rather than fromthe choices made in its computational instantiation.3.11.4.3 Quantitative CriteriaFalsifiabilityThis is a necessary condition for testing a model or theory, refers towhether there exist potential observations that a model cannot describe [Pop-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 325per (1959)]. If so, then the model is said to be falsifiable. An unfalsifiablemodel is one that can describe all possible data patterns in a given experimentalsituation. There is no point in testing an unfalsifiable model.A heuristic rule for determining a model’s falsifiability is already familiarto us: The model is falsifiable if the number of its free parameters is less thanthe number of data observations. This counting rule, however, turns out tobe imperfect, in particular, for nonlinear models. To remedy limitations ofthe counting rule, [Bamber and Santen (1985)] provided a formal rule forassessing a model’s falsifiability, which yielded the counting rule as a specialcase. The rule states that a model is falsifiable if the rank of its Jacobianmatrix is less than the number of data observations for all values of theparameters. Recall that the Jacobian matrix is defined in terms of partialderivatives as: J ij (w) = ∂E (y j ) /∂w i (i = 1, ..., k; j = 1, ..., m) where E(x)stands for the expectation of a random variable x.Goodness of FitA model should also give a good description of the observed data. Goodnessof fit refers to the model’s ability to fit the particular set of observeddata. Common examples of goodness of fit measures are the minimizedsum of squares error (SSE), the mean squared error (MSE), the root meansquared error (RMSE), the percent variance accounted for (PVAF), andthe maximum likelihood (ML).The first four of these, defined below, are related to one another in away that one can be written in terms of another:MSE = SSE (wLSE) ∗ /m,√RMSE = SSE (wLSE ∗ ) /m,P V AF = 100 (1 − SSE (w ∗ LSE) /SST ) ,ML = f (y|w ∗ MLE) .Here wLSE ∗ is the parameter that minimizes SSE(w), that is, an LSE estimate,and SST stands for the sum of squares total defined as SST =∑i (y i − y mean ) 2 . ML is the probability distribution function maximizedwith respect to the model’s parameters, evaluated at wMLE ∗ , which is obtainedthrough MLE.


326 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionComplexityNot only should a model describe the data in hand well, but it shouldalso do so in the least complex (i.e., simplest) way. Intuitively, complexityhas to do with a model’s inherent flexibility that enables it to fit a widerange of data patterns. There seem to be at least two dimensions of modelcomplexity, the number of parameters and the model’s functional form. Thelatter refers to the way the parameters are combined in the model equation.The more parameters a model has, the more complex it is. Importantly also,two models with the same number of parameters but different functionalforms can differ significantly in their complexity. For example, it seemsunlikely that two one–parameter models, y = x + w and y = e wx areequally complex. The latter is probably much better at fitting data thanthe former.It turns out that one can devise a quantitative measure of model complexitythat takes into account both dimensions of complexity and at thesame time is theoretically justified as well as intuitive. One example is thegeometric complexity (GC) of a model [Pitt et. al. (2002)] defined as:GC = k 2 ln n ∫2π + ln dw √ det I(w), (3.159)orGC = parametric complexity + functional complexity,where k is the number of parameters, n is the sample size, I(w) is theFisher information matrix (or, covariance matrix) defined asI ij (w) = −E [ ∂ 2 ln f(y|w)/∂w i ∂w j], i, j = 1, ..., k, (3.160)orI ij = −Expect.Value(Hessian(loglik (w))).Functional form effects of complexity are reflected in the second termof GC through I(w). How do we interpret geometric complexity? Themeaning of geometric complexity is related to the number of ‘different’(i.e., distinguishable) probability distributions that a model can accountfor. The more distinguishable distributions that the model can describe byfinely tuning its parameter values, the more complex it is ([Myung et. al.(2000a)]). For example, when geometric complexity is calculated for thefollowing two-parameter psychophysical models, Stevens’ law (y = w 1 x w2 )and Fechner’s logarithmic law (y = w 1 ln (x + w 2 )), the former turns outto be more complex than the latter [Pitt et. al. (2002)].


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 327GeneralizabilityThe fourth quantitative criterion for model evaluation is generalizability.This criterion is defined as a model’s ability to fit not only the observed dataat hand, but also new, as yet unseen data samples from the same probabilitydistribution. In other words, model evaluation should not be focused solelyon how well a model fits observed data, but how well it fits data generatedby the cognitive process underlying the data. This goal will be achievedbest when generalizability is considered.To summarize, these four quantitative criteria work together to assistin model evaluation and guide (even constrain) model development andselection. The model must be sufficiently complex, but not too complex, tocapture the regularity in the data. Both a good fit to the data and goodgeneralizability will ensure an appropriate degree of complexity, so that themodel captures the regularity in the data. In addition, because of its broadfocus, generalizability will constrain the power of the model, thus makingit falsifiable. Although all four criteria are inter-related, generalizabilitymay be the most important. By making it the guiding principle in modelevaluation and selection, one cannot go wrong.3.11.4.4 Selection Among <strong>Diff</strong>erent ModelsSince a model’s generalizability is not directly observable, it must be estimatedusing observed data. The measure developed for this purpose tradesoff a model’s fit to the data with its complexity, the aim being to select themodel that is complex enough to capture the regularity in the data, but notoverly complex to capture the ever-present random variation. Looked at inthis way, generalizability embodies the principle of Occam’s razor (or principleof parsimony, i.e., the requirement of maximal simplicity of cognitivemodels).Model Selection MethodsNow, we describe specific measures of generalizability. Four representativegeneralizability criteria are introduced. They are the Akaike InformationCriterion (AIC), the Bayesian Information Criterion (BIC), crossvalidation(CV), and minimum description length (MDL) (see a special Journalof Mathematical Psychology issue on model selection, in particular [Myunget. al. (2000b)]). In all four methods, the maximized log–likelihood is usedas a goodness–of–fit measure, but they differ in how model complexity isconceptualized and measured.


328 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAIC and BICAIC and BIC for a given model are defined as follows:AIC = −2 ln f(y|w ∗ ) + 2k,BIC = −2 ln f(y|w ∗ ) + k ln n,where w ∗ is a MLE estimate, k is the number of parameters and n is thesample size. For normally distributed errors with constant variance, thefirst term of both criteria, -2 ln f(y|w ∗ ), is reduced to (n·ln(SSE(w ∗ )) + c o )where c o is a constant that does not depend upon the model. In eachcriterion, the first term represents a lack of fit measure, the second termrepresents a complexity measure, and together they represent a lack ofgeneralizability measure. A lower value of the criterion means better generalizability.Therefore, the model that minimizes a given criterion shouldbe chosen.Complexity in AIC and BIC is a function of only the number of parameters.Functional form, another important dimension of model complexity,is not considered. For this reason, these methods are not recommended forcomparing models with the same number of parameters but different functionalforms. The other two selection methods, CV and MDL, describednext, are sensitive to functional form as well as the number of parameters.Cross–ValidationIn CV, a model’s generalizability is estimated without defining an explicitmeasure of complexity. Instead, models with more complexity thannecessary to capture the regularity in the data are penalized through aresampling procedure, which is performed as follows: The observed datasample is divided into two sub–samples, calibration and validation. Thecalibration sample is then used to find the best–fitting values of a model’sparameters by MLE or LSE. These values, denoted by wcal ∗ , are then fixedand fitted, without any further tuning of the parameters, to the validationsample, denoted by y val . The resulting fit to y val by wcal ∗ is calledas the model’s CV index and is taken as the model’s generalizability estimate.If desired, this single–division–based CV index may be replaced bythe average CV index calculated from multiple divisions of calibration andvalidation samples. The latter is a more accurate estimate of the model’sgeneralizability, though it is also more computationally demanding.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 329The main attraction of cross–validation is its ease of use. All that isneeded is a simple resampling routine that can easily be programmed on anydesktop computer. The second attraction is that unlike AIC and BIC, CVis sensitive to the functional form dimension of model complexity, thoughhow it works is unclear because of the implicit nature of the method. Forthese reasons, the method can be used in all modelling situations, includingthe case of comparing among models that differ in functional form but havethe same number of parameters.Minimum Description LengthMDL is a selection method that has its origin in algorithmic coding theoryin computer science. According to MDL, both models and data areviewed as codes that can be compressed. The basic idea of this approach isthat regularities in data necessarily imply the existence of statistical redundancyand that the redundancy can be used to compress the data [Grunwald(1999); Grunwald (2000); Grunwald et. al. (2005)]. Put another way,the amount of regularity in data is directly related to the data descriptionlength. The shorter the description of the data by the model, the betterthe approximation of the underlying regularity, and thus, the higher themodel’s generalizability is. Formally, MDL is defined as:MDL = − ln f(y|w ∗ ) + k 2 ln n ∫2π + ln dw √ det I(w), (3.161)orMDL = lack–of–fit measure + param. complexity + functional complexity,where the first term is the same lack of fit measure as in AIC and BIC;the second and third terms together represent the geometric complexitymeasure (3.159). In coding theory, MDL is interpreted as the length in bitsof the shortest possible code that describes the data unambiguously withthe help of a model. The model with the minimum value of MDL encodesthe most regularity in the data, and therefore should be preferred.The second term in (3.161), which captures the effects of model complexitydue to the number of parameter (k), is a logarithmic function ofsample size n. In contrast, the third term, which captures functional formeffects, is not sensitive to sample size. This means that as sample size increases,the relative contribution of the effects due to functional form tothose due to the number of parameters will be gradually reduced. Therefore,functional form effects can be ignored for sufficiently large n, in which


330 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncase the MDL value becomes approximately equal to one half of the BICvalue.Probably the most desirable property of MDL over other selection methodsis that its complexity measure takes into account the effects of bothdimensions of model complexity, the number of parameters and functionalform. The MDL complexity measure, unlike CV, shows explicitly how bothfactors contribute to model complexity. In short, MDL is a sharper andmore accurate method than these three competitors. The price that is paidfor MDL’s superior performance is its computational cost. MDL can belaborious to calculate. First, the Fisher information matrix (3.160) mustbe obtained by calculating the second derivatives (i.e., Hessian matrix)of the log–likelihood function, ln f(y|w). This calculation can be non–trivial, though not impossible. Second, the square–root of the determinantof the Fisher information matrix must be integrated over the parameterspace. This generally requires use of a numerical integration method suchas Markov Chain Monte Carlo (see e.g., [Gilks et. al. (1996)]).3.11.4.5 Riemannian <strong>Geom</strong>etry of MinimumDescription LengthFrom a geometric perspective, a parametric model family of probability distributions(3.158) forms a Riemannian manifold embedded in the space ofall probability distributions (see [Amari (1985); Amari and Nagaoka (2000);McCullagh (1987)]). Every distribution is a point in this space, and thecollection of points created by varying the parameters of the model inducesa manifold in which ‘similar’ distributions are mapped to ‘nearby’ points.The infinitesimal distance between points separated by the infinitesimal parameterdifferences dw i is given byds 2 = g ij (w) dw i dw j ,where g ij (w) is the Riemannian metric tensor. The Fisher information,I ij (w), defined by (3.160), is the natural metric on a manifold of distributionsin the context of statistical inference [Amari (1985)]. We argue thatthe MDL measure of model fitness has an attractive interpretation in sucha geometric context.The first term in MDL equation (3.161) estimates the accuracy of themodel since the likelihood f(y|w ∗ ) measures the ability of the model to fitthe observed data. The second and third terms are supposed to penalizemodel complexity; we will show that they have interesting geometric inter-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 331pretations. Given the metric I ij (w) = g ij (w) on the space of parameters,the infinitesimal volume element on the parameter manifold isdV = dw √ det I(w) =k∏dw l√ det I(w).The Riemannian volume of the parameter manifold is obtained by integratingdV over the space of parameters:∫ ∫V M = dV = dw √ det I(w).In other words, the third term (functional complexity) in MDL penalizesmodels that occupy a large volume in the space of distributions.In fact, the volume measure V M is related to the number of ‘distinguishable’probability distributions indexed by the model. Because of theway the model family is embedded in the space of distributions, two differentparameter values can index very similar distributions. If complexityis related to volumes occupied by model manifolds, the measure of volumeshould count only different, or distinguishable, distributions, and not theartificial coordinate volume. It is shown in [Myung et. al. (2000a)] that thevolume V M achieves this goal.l=1Selecting Among Qualitative ModelsApplication of any of the preceding selection methods requires that themodels are quantitative models, each defined as a parametric family of probabilitydistributions.Pseudo–probabilistic MDL ApproachThe ‘pseudo–probabilistic’ approach [Grunwald (1999)] for selectingamong qualitative models derives a selection criterion that is similar tothe MDL criterion for quantitative models, but it is a formulation that iscloser to the original spirit of the MDL principle, which states:‘Given a data set D and a model M, the description length of the data,DL M (D), is given by the sum of (a) the description length of the data whenencoded with help of the model, DL(D|M), and (b) the description lengthof the model itself, DL(M) : DL M (D) = DL(D|M)+DL(M). Among a setof competing models, the best model is the one that minimizes DLM(D).’The above MDL principle is broad enough to include the MDL criterionfor quantitative models as a specific instantiation. The first, lack–of–fit


332 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionterm of the quantitative criterion (-ln f (y|w ∗ )) can be seen as DL(D|M),whereas the second and third terms ( k 2 ln n 2π +ln ∫ dw √ det I (w)) representgeometric complexity as DL(M). Likewise, a computable criterion that implementsthe above principle can be obtained with the pseudo–probabilisticapproach. It is derived from the Kraft–Inequality Theorem in coding theory([Li and Vitanyi (1997)]). The Theorem proves that one can alwaysassociate arbitrary models with their ‘equivalent’ probability distributionsin a procedure called entropification [Grunwald (1999)].MDL Criterion for Qualitative ModelsEntropification proceeds as follows. We first ‘construct’ a parametricfamily of probability distributions for a given qualitative model in the followingform:()m∑p (y = (y 1 , ..., y m ) |w) = exp −w Err (y i,obs − y i,prd (w)) /Z(w).i=1In this equation, Err(x) is an error function that measures the model’sprediction performance such as Err(x) = |x| or x 2 , w is a scalar parameter,and Z(w) is the normalizing factor defined asZ(w) = ∑ y 1... ∑ y mp (y = (y 1 , ..., y m ) |w) .The above formulation requires that each observation y i be represented bya discrete variable that takes on a finite number of possible values representingthe model’s qualitative (e.g., ordinal) predictions.Once a suitable error function, Err(x), is chosen, the above probabilitydistribution function is then used to fit observed data, and the best–fittingparameter w ∗ is sought by MLE. The description length of the data encodedwith the help of the model is then obtained by taking the minus logarithmof the maximum likelihood (ML),DL (D|M) = − ln p (y|w ∗ ) .The second term, DL(M), the description length of the model itself, isobtained by counting the number of different data patterns the model canaccount for and then taking the logarithm of the resulting number. Puttingthese together, the desired MDL criterion for a qualitative model is given


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 333byMDL qual = w ∗m∑i=1Err (y i,obs − y i,prd (w ∗ )) + ln Z (w ∗ ) + ln N,where N is the number of all possible data patterns or data sets that themodel predicts.3.11.4.6 Finsler Approach to Information <strong>Geom</strong>etryRecall that information geometry has emerged from investigating the geometricalstructure of a family of probability distributions, and has beenapplied successfully to various areas including statistical inference, controltheory and multi–terminal information theory (see [Amari (1985);Amari and Nagaoka (2000)]). In this subsection we give a brief review on amore general approach to information geometry, based on Finsler geometry(see subsection 3.11 above).A parameter–space of probability distributions, defined byM = {x : p = p(r, x) is a probability distribution on R}represents a smooth manifold, called the probability manifold [Amari(1985); Amari and Nagaoka (2000)]. On a probability manifold M we candefine a probability divergence D = D(x, y), as∫D(x, y) =Mp(r, y)p(r, x)f(p(r, x) ) dr,where f(·) is a convex function such that f(1) = 0, f ′′ (1) = 1,which satisfies the following conditions [Shen (2005)]D(x, y) > 0 if x ≠ y,D(x, y) = 0 if x = y,D(x, y) ≠ D(y, x)in general.On the other hand, if d = d(x, y) is a probability distance on a probabilitymanifold M, satisfying the following standard conditions:d(x, x) = 0,d(x, y) > 0, if x ≠ y,d(x, y) ≤ d(x, r) + d(r, y)(triangle inequality),


334 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthen for any function ψ = ψ(h) with ψ(0) = 0, ψ(h) > 0 for h > 0, theprobability divergence on M is defined asD(x, y) := ψ(d(x, y)). (3.162)Recall that a Finsler metric L = L(x, y) is a function of tangent vectorsy at a point x ∈ M, with the following properties:L(x, ty) = t 2 L(x, y) for t > 0, (3.163)g ij (x, y) := 1 ∂ 2 L2 ∂y i (x, y) > 0,∂yj F x (y) := √ L(x, y), F x (u + v) ≤ F x (u) + F x (v).This means that there is an inner product g y at a pint x ∈ M, such thatg y (u, v) = g ij (x, y)u i v j ,so that our Finsler metric L(x, y) ∈ M, given by (3.163), becomesL(x, y) = g y (u, v) = g ij (x)y i y j .Therefore, in a special case when g ij (x, y) = g ij (x) are independent of y, theFinsler metric L(x, y) becomes a standard Riemannian metric g ij (x)y i y j .In this way, all the material from the previous subsection can be generalizedto Finsler geometry.Now, D(x, y) ∈ M, given by (3.162), is called the regular divergence, if2D(x, x + y) = L(x, y) + 1 2 L x k(x, y)yk + 1 3 H(x, y) + o(|y|3 ),where H = H(x, y) ∈ M is homogenous function of degree 3 in y, i.e.,H(x, ty) = t 3 H(x, y) for t > 0.A pair {L, H} ∈ M is called a Finsler information structure [Shen (2005)].In a particular case when L(x, y) = g ij (x)y i y j is a Riemannian metric,and H(x, y) = H ijk (x)y i y j y k is a polynomial, then we have affine informationstructure {L, H} ∈ M, which is described by a family of affine connections,called α−connections by [Amari (1985); Amari and Nagaoka (2000)].However, in general, the induced information structure {L, H} ∈ M is notaffine, i.e., L(x, y) is not Riemannian and H(x, y) is not polynomial.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 3353.12 Symplectic Manifolds and Their Applications3.12.1 Symplectic AlgebraSymplectic algebra works in the category of symplectic vector spaces V i andlinear symplectic mappings t ∈ L(V i , V j ) [Puta (1993)].Let V be a nD real vector space and L 2 (V, R) the space of all bilinearmaps from V × V to R. We say that a bilinear map ω ∈ L 2 (V, R) isnondegenerate, i.e., if ω(v 1 , v 2 ) = 0 for all v 2 ∈ V implies v 1 = 0.If {e 1 , ..., e n } is a basis of V and {e 1 , ..., e n } is the dual basis, ω ij =ω(e i , e j ) is the matrix of ω. A bilinear map ω ∈ L 2 (V, R) is nondegenerateiff its matrix ω ij is nonsingular. The transpose ω t of ω is defined byω t (e i , e j ) = ω(e j , e i ). ω is symmetric if ω t = ω, and skew–symmetric ifω t = −ω.Let A 2 (V ) denote the space of skew–symmetric bilinear maps on V .An element ω ∈ A 2 (V ) is called a 2−form on V . If ω ∈ A 2 (V ) is nondegenerate(then)in the basis {e 1 , ..., e n } its matrix ω(e i , e j ) has the form0 InJ = .−I n 0A symplectic form on a real vector space V of dimension 2n is a nondegenerate2−form ω ∈ A 2 (V ). The pair (V, ω) is called a symplecticvector space. If (V 1 , ω 1 ) and (V 2 , ω 2 ) are symplectic vector spaces, a linearmap t ∈ L(V 1 , V 2 ) is a symplectomorphism (i.e., a symplectic mapping) ifft ∗ ω 2 = ω 1 . If (V, ω) is a symplectic vector space, we have an orientationΩ ω on V given byΩ ω =n(n−1)(−1) 2n!ω n .Let (V, ω) be a 2nD symplectic vector space and t ∈ L(V, V ) a symplectomorphism.Then t is volume preserving, i.e., t ∗ (Ω ω ) = Ω ω , anddet Ωω (t) = 1.The set of all symplectomorphisms t : V → V of a 2nD symplecticvector space (V, ω) forms a group under composition, called the symplecticgroup, denoted by Sp(V, ω).In( matrix ) notation, there is a basis of V in which the matrix of ω is0 InJ = , such that J−I n 0−1 = J t = −J, and J 2 = −I. For t ∈ L(V, V )with matrix T = [Tj i ] relative to this basis, the condition t ∈ Sp(V, ω), i.e.,


336 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiont ∗ ω = ω, becomesT t JT = J.In general, by definition a matrix A ∈ M 2n×2n (R) is symplectic iff A t JA =J.Let (V, ω) be a symplectic vector space, t ∈ Sp(V, ω) and λ ∈ C aneigenvalue of t. Then λ −1 , ¯λ and ¯λ −1 are eigenvalues of t.3.12.2 Symplectic <strong>Geom</strong>etrySymplectic geometry is a globalization of symplectic algebra [Puta (1993)];it works in the category Symplec of symplectic manifolds M and symplecticdiffeomorphisms f. The phase–space of a conservative dynamicalsystem is a symplectic manifold, and its time evolution is a one–parameterfamily of symplectic diffeomorphisms.A symplectic form or a symplectic structure on a smooth (i.e., C k )manifold M is a nondegenerate closed 2−form ω on M, i.e., for each x ∈M ω(x) is nondegenerate, and dω = 0. A symplectic manifold is a pair(M, ω) where M is a smooth 2nD manifold and ω is a symplectic formon it. If (M 1 , ω 1 ) and (M 2 , ω 2 ) are symplectic manifolds then a smoothmap f : M 1 → M 2 is called symplectic map or canonical transformation iff ∗ ω 2 = ω 1 .For example, any symplectic vector space (V, ω) is also a symplecticmanifold; the requirement dω = 0 is automatically satisfied since ω is aconstant map. Also, any orientable, compact surface Σ is a symplecticmanifold; any non–vanishing 2−form (volume element) ω on Σ is a symplecticform on Σ.If (M, ω) is a symplectic manifold then it is orientable with the standardvolume formΩ ω =n(n−1)(−1) 2n!If f : M → M is a symplectic map, then f is volume preserving, det Ωω (f) =1 and f is a local diffeomorphism.In general, if (M, ω) is a 2nD compact symplectic manifold then ω n is avolume element on M, so the de Rham cohomology class [ω n ] ∈ H 2n (M, R)is nonzero. Since [ω n ] = [ω] n , [ω] ∈ H 2 (M, R) and all of its powers throughthe nth must be nonzero as well. The existence of such an element ofω n ,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 337H 2 (M, R) is a necessary condition for the compact manifold to admit asymplectic structure.However, if M is a 2nD compact manifold without boundary, then theredoes not exist any exact symplectic structure, ω = dθ on M, as its totalvolume is zero (by Stokes’ Theorem),∫MΩ ω =n(n−1)(−1) 2n!∫Mω n =n(n−1)(−1) 2n!∫Md(θ ∧ ω n−1 ) = 0.For example, spheres S 2n do not admit a symplectic structure for n ≥2, since the second de Rham group vanishes, i.e., H 2 (S 2n , R) = 0. Thisargument applies to any compact manifold without boundary and havingH 2 (M, R) = 0.In mechanics, the phase–space is the cotangent bundle T ∗ M of a configurationspace M. There is a natural symplectic structure on T ∗ M thatis usually defined as follows. Let M be a smooth nD manifold and picklocal coordinates {dq 1 , ..., dq n }. Then {dq 1 , ..., dq n } defines a basis of thetangent space T ∗ q M, and by writing θ ∈ T ∗ q M as θ = p i dq i we get localcoordinates {q 1 , ..., q n , p 1 , ..., p n } on T ∗ M. Define the canonical symplecticform ω on T ∗ M byω = dp i ∧ dq i .This 2−form ω is obviously independent of the choice of coordinates{q 1 , ..., q n } and independent of the base point {q 1 , ..., q n , p 1 , ..., p n } ∈ Tq ∗ M;therefore, it is locally constant, and so dω = 0.The canonical 1−form θ on T ∗ M is the unique 1−form with the propertythat, for any 1−form β which is a section of T ∗ M we have β ∗ θ = θ.Let f : M → M be a diffeomorphism. Then T ∗ f preserves the canonical1−form θ on T ∗ M, i.e., (T ∗ f) ∗ θ = θ. Thus T ∗ f is symplectic diffeomorphism.If (M, ω) is a 2nD symplectic manifold then about each point x ∈ Mthere are local coordinates {q 1 , ..., q n , p 1 , ..., p n } such that ω = dp i ∧ dq i .These coordinates are called canonical or symplectic. By the DarbouxTheorem, ω is constant in this local chart, i.e., dω = 0.


338 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.12.3 Application: Autonomous Hamiltonian Mechanics3.12.3.1 Basics of Hamiltonian MechanicsIn this section we present classical Hamiltonian dynamics. Let (M, ω) be asymplectic manifold and H ∈ C ∞ (M, R) a smooth real valued function onM. The vector–field X H determined by the conditioni XH ω + dH = 0,is called Hamiltonian vector–field with Hamiltonian energy function H. Atriple (M, ω, H) is called a Hamiltonian mechanical system [Marsden andRatiu (1999); Puta (1993)].Nondegeneracy of ω guarantees that X H exists, but only in the nD case.Let {q 1 , ..., q n , p 1 , ..., p n } be canonical coordinates on M, i.e., ω = dp i ∧dq i . Then in these coordinates we have( ∂HX H =∂p i∂∂q i − ∂H∂q i)∂.∂p iAs a consequence, ( (q i (t)), (p i (t)) ) is an integral curve of X H (for i =1, ..., n) iff Hamiltonian equations hold,˙q i = ∂ pi H, ṗ i = −∂ q iH. (3.164)Let (M, ω, H) be a Hamiltonian mechanical system and let γ(t) be anintegral curve of X H . Then H (γ(t)) is constant in t. Moreover, if φ t is theflow of X H , then H ◦ φ t = H for each t.Let (M, ω, H) be a Hamiltonian mechanical system and φ t be the flow ofX H . Then, by the Liouville Theorem, for each t, φ ∗ t ω = ω, ( d dt φ∗ t ω = 0, soφ ∗ t ω is constant in t), that is, φ t is symplectic, and it preserves the volumeΩ ω .A convenient criterion for symplectomorphisms is that they preserve theform of Hamiltonian equations. More precisely, let (M, ω) be a symplecticmanifold and f : M → M a diffeomorphism. Then f is symplectic iff forall H ∈ C ∞ (M, R) we have f ∗ (X H ) = X H◦f .A vector–field X ∈ X (M) on a symplectic manifold (M, ω) is calledlocally Hamiltonian iff L X ω = 0, where L denotes the Lie derivative. Fromthe equality L [X,Y ] ω = L X L Y ω−L Y L X ω, it follows that the locally Hamiltonianvector–fields on M form a Lie subalgebra of X (M).Let (M, ω) be a symplectic manifold and f, g ∈ C ∞ (M, R). The Poisson


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 339bracket of f and g is the function{f, g} ω = −ω(X f , X g ) = −L Xf g = L Xg f.Also, for f 0 ∈ C ∞ (M, R), the map g ↦−→ {f 0 , g} ω is a derivation.connection between the Lie bracket and the Poisson bracket isThe[X f , X g ] = −X {f,g}ω ⇐⇒ dω = 0.The real vector space C ∞ (M, R) together with the Poisson bracket onit forms an infinite–dimensional Lie algebra called the algebra of classicalobservables.In canonical coordinates {q 1 , ..., q n , p 1 , ..., p n } on (M, ω) the Poissonbracket of two functions f, g ∈ C ∞ (M, R) is given byFrom this definition follows:{f, g} ω = ∂f∂q i ∂g∂p i− ∂f∂p i∂g∂q i .{q i , q j } ω = 0, {p i , p j } ω = 0, {q i , p j } ω = δ i j.Let (M, ω) be a symplectic manifold and f : M → M a diffeomorphism.Then f is symplectic iff it preserves the Poisson bracket.Let (M, ω, H) be a Hamiltonian mechanical system and φ t the flow ofX H . Then for each function f ∈ C ∞ (M, R) we have the equations of motionin the Poisson bracket notation:ddt (f ◦ φ t) = {f ◦ φ t , H} ω= {f, H} ω◦ φ t .Also, f is called a constant of motion, or a first integral, if it satisfies thefollowing condition{f, H} ω= 0.If f and g are constants of motion then their Poisson bracket is also aconstant of motion.A Hamiltonian mechanical system (M, ω, H) is said to be integrableif there exists n = 1 2 dim(M) linearly–independent functions K 1 =H, K 2 , ..., K n such that for each i, j = 1, 2, ..., n:{K i , H} ω= 0, {K i , K j } ω= 0.


340 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionReal 1−DOF Hamiltonian DynamicsA vector–field X(t) on the momentum phase–space manifold M can begiven by a system of canonical equations of motion˙q = f(q, p, t, µ), ṗ = g(q, p, t, µ), (3.165)where t is time, µ is a parameter, q ∈ S 1 , p ∈ R×S 1 are coordinates andmomenta, respectively, while f and g are smooth functions on the phase–space R×S 1 .If time t does not explicitly appear in the functions f and g, the vector–field X is called autonomous. In this case equation (3.165) simplifies as˙q = f(q, p, µ), ṗ = g(q, p, µ). (3.166)By a solution curve of the vector–field X we mean a map x = (q, p),from some interval I ⊂ R into the phase–space manifold M, such thatt ↦→ x(t). The map x(t) = (q(t), p(t)) geometrically represents a curve inM, and equations (3.165) or (3.166) give the tangent vector at each pointof the curve.To specify an initial condition on the vector–field X, byx(t, t 0 , x 0 ) = (q(t, t 0 , q 0 ), p(t, t 0 , p 0 )),geometrically means to distinguish a solution curve by a particular pointx(t 0 ) = x 0 in the phase–space manifold M. Similarly, it may be usefulto explicitly display the parametric dependence of solution curves,as x(t, t 0 , x 0 , µ) = (q(t, t 0 , q 0 , µ q ), p(t, t 0 , p 0 , µ p )), where µ q , µ p denoteq−depen-dent and p−dependent parameters, respectively.The solution curve x(t, t 0 , x 0 ) of the vector–field X, may be also referredas the phase trajectory through the point x 0 at t = t 0 . Its graph over t isrefereed to as an integral curve; more precisely, graphx(t, t 0 , x 0 ) ≡ {(x, t) ∈ M × R : x = x(t, t 0 , x 0 ), t ∈ I ⊂ R}.Let x 0 = (q 0 , p 0 ) be a point on M. By the orbit through x 0 , denotedO(x 0 ), we mean the set of points in M that lie on a trajectory passingthrough x 0 ; more precisely, for x 0 ∈ U, U open in M, the orbit through x 0is given by O(x 0 ) = {x ∈ R×S 1 : x = x(t, t 0 , x 0 ), t ∈ I ⊂ R}.Consider a general autonomous vector–field X on the phase–space manifoldM, given by equation ẋ = f(x), x = (q, p) ∈ M. An equilibrium solution,singularity, or fixed point of X is a point ¯x ∈ M such that f(¯x) = 0,i.e., a solution which does not change in time.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 341Any solution ¯x(t) of an autonomous vector–field X on M is stable ifsolutions starting ‘close’ to ¯x(t) at a given time remain close to ¯x(t) for alllater times. It is asymptotically stable if nearby solutions actually convergeto ¯x(t) as t −→ ∞. In order to determine the stability of ¯x(t) we mustunderstand the nature of solutions near ¯x(t), which is done by linearizationof the vector–field X. The solution of the linearized vector–field Yis asymptotically stable if all eigenvalues have negative real parts. In thatcase the fixed point x = ¯x of associated nonlinear vector–field X is alsoasymptotically stable. A fixed point ¯x is called hyperbolic point if none ofthe eigenvalues of Y have zero real part; in that case the orbit structurenear ¯x is essentially the same for X and Y .In the case of autonomous vector–fields on M we have also an importantproperty of Hamiltonian flow. If x(t) = (q(t), p(t)) is a solution of ẋ =f(x), x ∈ M, then so is x(t + τ) for any τ ∈ R. Also, for any x 0 ∈ M thereexists only one solution of an autonomous vector–field passing through thispoint. The autonomous vector–fieldẋ = f(x)has the following properties (compare with the section (6.289) above):(1) x(t, x 0 ) is C ∞ ;(2) x(0, x 0 ) = x 0 ; and(3) x(t + s, x 0 ) = x(t, x(s, x 0 )).These properties show that the solutions of an autonomous vector–fieldform a one–parameter group of diffeomorphisms of the phase–space manifoldM. This is refereed to as a phase–flow and denoted by φ t (x) or φ(t, x).Consider a flow φ(t, x) generated by vector–field ẋ = f(x). A pointx 0 = (q 0 , p 0 ) on M is called an ω−limit point of x, = (q, p) ∈ M, denotedω(x), if there exists a sequence {t i }, t i ↦→ ∞, such that φ(t i , x) ↦→ x 0 .Similarly, α−limit points are defined by taking a sequence {t i }, t i ↦→ −∞.The set of all ω−limit points of a flow is called the ω−limit set. Theα−limit set is similarly defined.A point x 0 = (q 0 , p 0 ) on M is called nonwandering if for any openneighborhood U ⊂ M of x 0 , there exists some t ≠ 0 such that φ(t, U)∩U ≠0. The set of all nonwandering points of a flow is called the nonwanderingset of that particular map or flow.A closed invariant subset A ⊂ M is called an attracting set if there issome open neighborhood U ⊂ M of A such that φ(t, x) ∈ U and φ(t, x) ↦→∞ for any x ∈ U and t ≥ 0. The domain or basin of attraction of A is given


342 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionby ∪ t≤0 φ(t, U). In practice, a way of locating attracting sets is to first finda trapping region, i.e., a closed, connected subset V ⊂ M such that for anyt ≥ 0 φ(t, V ) ⊂ V . Then ∩ t 0 φ(t, V ) = A is an attracting set.As a first example of one–DOF dynamical systems, let us consider avector–field x = (q, p) ∈ R×R of a simple harmonic oscillator, given byequations˙q = p, ṗ = −q. (3.167)Here, the solution passing through the point (q, p) = (1, 0) at t = 0 isgiven by (q(t), p(t)) = (cos t, − sin t); the integral curve passing through(q, p) = (1, 0) at t = 0 is given by {(q, p, t) ∈ R×R×R : (q(t), p(t)) =(cos t, − sin t)}, for all t ∈ R; the orbit passing through (q, p) = (1, 0) isgiven by the circle q 2 + p 2 = 1.A one–DOF dynamical system is called Hamiltonian system if thereexists a first integral or a function of the dependent variables (q, p) whoselevel curves give the orbits of the vector–field X = X H , i.e., a total–energyHamiltonian function H = H(q, p) : U → R, (U open set on the phase–space manifold M), such that the vector–field X H is given by Hamiltoniancanonical equations (3.164). In (3.164), the first, ˙q−equation, is calledthe velocity equation and serves as a definition of the momentum, whilethe second, ṗ−equation is called the force equation, and represents theNewtonian second law of motion.The simple harmonic oscillator (3.167) is a Hamiltonian system with aHamiltonian function H = p22 + q22. It has a fixed point – center (havingpurely imaginary eigenvalues) at (q, p) = (0, 0) and is surrounded by aone–parameter family of periodic orbits given by the Hamiltonian H.A nice example of one–DOF dynamical system with a Hamiltonianstructure is a damped Duffing oscillator (see, e.g., [Wiggins (1990)]). Thisis a plane Hamiltonian vector–field x = (q, p) ∈ R 2 , given by Hamiltonianequations˙q = p ≡ f(q, p), ṗ = q − q 3 − δp ≡ g(q, p, δ), δ ≥ 0. (3.168)For the special parameter value δ = 0, we have an undamped Duffing oscillatorwith a first integral represented by Hamiltonian function H = p2q 22 + q44, wherep222 −corresponds to the kinetic energy (with a mass scaledto unity), and − q22 + q44≡ V (x) corresponds to the potential energy of theoscillator.In general, if the first integral, i.e., a Hamiltonian function H, is definedby H = p22 + V (x), then the momentum is given by p = ±√ 2 √ H − V (x).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 343All one–DOF Hamiltonian systems are integrable and all the solutions lie onlevel curves of the Hamiltonian function, which are topologically equivalentwith the circle S 1 . This is actually a general characteristic of all n−DOFintegrable Hamiltonian systems: their bounded motions lie on nD invarianttori T n = S 1 × · · · × S 1 , or homoclinic orbits. The homoclinic orbitis sometimes called a separatrix because it is the boundary between twodistinctly different types of motion.For example, in case of a damped Duffing oscillator (3.168) with δ ≠ 0,we have∂ q f + ∂ p g = −δ,and according to the Bendixon criterion for δ > 0 it has no closed orbits.The vector–field X given by equations (3.168) has three fixed pointsgiven by (q, p) = (0, 0), (±1, 0). The eigenvalues λ√ 1,2 of the associatedlinearized vector–field are given by λ 1,2 = −δ/2 ± 1 2δ 2 + 4, for the fixed√point (0, 0), and by λ 1,2 = −δ/2 ± 1 2δ 2 − 8, for the fixed point (±1, 0).Hence, for δ > 0, (0, 0) is unstable and (±1, 0) are asymptotically stable;for δ = 0, (±1, 0) are stable in the linear approximation (see, e.g., [Wiggins(1990)]).Another example of one–DOF Hamiltonian systems is a simple pendulum(again, all physical constants are scaled to unity), given by Hamiltonianfunction H = p22− cos q. This is the first integral of the cylindricalHamiltonian vector–field (q, p) ∈ S 1 × R, defined by canonical equations˙q = p, ṗ = − sin q.This vector–field has fixed points at (0, 0), which is a center (i.e., the eigenvaluesare purely imaginary), and at (±π, 0), which are saddles, but sincethe phase–space manifold is the cylinder, these are really the same point.The basis of human arm and leg dynamics represents the coupling oftwo uniaxial, SO(2)−joints. The study of two DOF Hamiltonian dynamicswe shall start with the most simple case of two linearly coupled linearundamped oscillators with parameters scaled to unity. Under general conditionswe can perform a change of variables to canonical coordinates (the‘normal modes’) (q i , p i ), i = 1, 2, so that the vector–field X H is given by˙q 1 = p 1 , ˙q 2 = p 2 , ṗ 1 = −ω 2 1q 1 , ṗ 2 = −ω 2 2q 2 .This system is integrable, since we have two independent functions of


344 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(q i , p i ), i.e., HamiltoniansH 1 = p2 12 + ω2 1(q 1 ) 2, H 2 = p2 222 + ω2 2(q 2 ) 2.2The level curves of these functions are compact sets (topological circles);therefore, the orbits in the 4D phase–space R 4 actually lie on the two–torusT 2 . By making the appropriate change of variables, it can be shown (see,e.g., [Wiggins (1990)]) that the whole dynamics of the two linearly coupledlinear undamped oscillators is actually contained in the equations˙ θ 1 = ω 1 ,˙ θ 2 = ω 2 , (θ 1 , θ 2 ) ∈ S 1 × S 2 ≡ T 2 . (3.169)The flow on the two–torus T 2 , generated by (3.169), is simple to calculateand is given byθ 1 (t) = ω 1 t + θ 10 , θ 1 (t) = ω 1 t + θ 10 , (mod 2π),and θ 1 and θ 2 are called the longitude and latitude. However, orbits underthis flow will depend on how ω 1 and ω 2 are related. If ω 1 and ω 2 arecommensurate (i.e., the equation mω 1 +nω 2 = 0, (n, m) ∈ Z has solutions),then every phase curve of (3.169) is closed. However, if ω 1 and ω 2 areincommensurate i.e., upper equation has no solutions), then every phasecurve of (3.169) is everywhere dense on T 2 .Somewhat deeper understanding of Hamiltonian dynamics is related tothe method of action–angle variables. The easiest way to introduce this ideais to consider again a simple harmonic oscillator (3.167). If we transformequations (3.167) into polar coordinates using q = r sin θ, p = r cos θ,then the equations of the vector–field become ṙ = 0, ˙θ = 1, having theobvious solution r = const, θ = t+θ 0 . For this example polar coordinateswork nicely because the system (3.167) is linear and, therefore, all of theperiodic orbits have the same period.For the general, nonlinear one–DOF Hamiltonian system (3.164) wewill seek a coordinate transformation that has the same effect. Namely, wewill seek a coordinate transformation (q, p) ↦→ (θ(q, p), I(q, p)) with inversetransformation (θ, I) ↦→ (q(I, θ), p(I, θ)) such that the vector–field (3.164)in the action–angle (θ, I) coordinates satisfies the following conditions: (i)I ˙ = 0; (ii) θ changes linearly in time on the closed orbits with ˙θ = Ω(I).We might even think of I and θ heuristically as ’nonlinear polar coordinates’.In such a coordinate system Hamiltonian function takes the formH = H(I), and also, Ω(I) = ∂ I H, i.e., specifying I specifies a periodicorbit.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 345The action variable I(q, p) geometrically represents an area enclosed byany∫closed curve, which is constant in time. It is defined as an integral I =12πp dq, where H denotes the periodic orbit defined by H(q, p) = H =Hconst. If the period of each periodic orbit defined by H(q, p) = H = constis denoted by T (H), the angle variable θ(q, p) is defined byθ(q, p) =2π t(q, p),T (H)where t = t(q, p) represents the time taken for the solution starting from(q 0 , p 0 ) to reach (q, p).For the system with Hamiltonian H = p22+ V (x) and momentum p =± √ 2 √ H − V (x) the action is given by I = √ ∫2 qmax√H − V (q) dq, andthe angle is given by θ(q, p) =2πT (H)∫ qmaxq minπ q mindq√2√H−V (q).Closely related to the action–angle variables is the perturbation theory(see [Nayfeh (1973)]). To explain the main idea of this theory, let us consideran ɛ−perturbed vector–field periodic in t which can be in component formgiven as (with (q, p) ∈ R 2 )˙q = f 1 (q, p) + ɛg 1 (q, p, t, ɛ), ṗ = f 2 (q, p) + ɛg 2 (q, p, t, ɛ). (3.170)Setting ɛ = 0 we get the unperturbed Hamiltonian system with a smoothscalar–valued function H(q, p) for which holdsf 1 (q, p) =∂H(q, p)∂H(q, p), f 2 (q, p) = − ,∂p∂qso, the perturbed system (3.170) gets the symmetric canonical form˙q =∂H(q, p)∂p∂H(q, p)+ ɛg 1 (q, p, t, ɛ), ṗ = − + ɛg 2 (q, p, t, ɛ).∂qThe perturbation (g 1 , g 2 ) need not be Hamiltonian, although in the casewhere perturbation is Hamiltonian versus the case where it is not, thedynamics are very different.Now, if we transform the coordinates of the perturbed vector–field usingthe action–angle transformation for the unperturbed Hamiltonian vector–field, we get˙ I = ɛ( ∂I∂q g 1 + ∂I˙θ = Ω(I) + ɛ)∂p g 2 ≡ ɛF (I, θ, t, ɛ), (3.171)( ∂θ∂q g 1 + ∂θ )∂p g 2 ≡ Ω(I) + ɛG(I, θ, t, ɛ), where


346 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionF (I, θ, t, ɛ) = ∂I∂q (q(I, θ), p(I, θ)) g 1((q(I, θ), p(I, θ), t, ɛ)+ ∂I∂p (q(I, θ), p(I, θ)) g 2((q(I, θ), p(I, θ), t, ɛ),G(I, θ, t, ɛ) = ∂θ∂q (q(I, θ), p(I, θ)) g 1((q(I, θ), p(I, θ), t, ɛ)+ ∂θ∂p (q(I, θ), p(I, θ)) g 2((q(I, θ), p(I, θ), t, ɛ).andHere, F and G are 2π periodic in θ and T = 2π/ω periodic in t.Finally, we shall explain in brief the most important idea in the dynamicalsystems theory, the idea of Poincaré maps. The idea of reducingthe study of continuous time systems (flows) to the study of an associateddiscrete time system (map) is due to Poincaré who first utilized it in theend of the last Century in his studies of the three body problem in celestialmechanics. Nowadays virtually any discrete time system that is associatedwith an ordinary differential equation is refereed to as a Poincaré map[Wiggins (1990)]. This technique offers several advantages in the study ofdynamical systems, including dimensional reduction, global dynamics andconceptual clarity. However, construction of a Poincaré map requires someknowledge of the phase–space of a dynamical system. One of the techniqueswhich can be used for construction of Poincaré maps is the perturbationmethod.To construct the Poincaré map for the system (3.171), we have to rewriteit as an autonomous system˙ I = ɛF (I, θ, φ, ɛ), ˙θ = Ω(I) + ɛG(I, θ, φ, ɛ), ˙φ = ω, (3.172)(where (I, θ, φ) ∈ R + ×S 1 ×S 1 . We construct a global cross–section Σ tothis vector–field defined as Σ φ 0 = {(I, θ, φ)|φ = φ 0 }. If we denote the(I, θ) components of solutions of (3.172) by (I ɛ (t), θ ɛ (t)) and the (I, θ)components of solutions of (3.172) for ɛ = 0 by (I 0 , Ω(I 0 )t + θ 0 ), thenthe perturbed Poincaré map is given byP ɛ : Σ φ 0 → Σφ 0, (Iɛ (0), θ ɛ (0)) ↦→ (I ɛ (T ), θ ɛ (T )),and the mth iterate of the Poincaré map is given byP m ɛ : Σ φ 0 → Σφ 0, (Iɛ (0), θ ɛ (0)) ↦→ (I ɛ (mT ), θ ɛ (mT )).Now we can approximate the solutions to the perturbed problem as


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 347linear, constant–coefficient approximationI ɛ (t) = I 0 + ɛI 1 (t) + O(ɛ 2 ), θ ɛ (t) = θ 0 + Ω(I 0 )t + ɛθ 1 (t) + O(ɛ 2 ),where we have chosen I ɛ (0) = I 0 , θ ɛ (0) = θ 0 .As a last example of one–DOF Hamiltonian dynamics we shall analyzea damped, forced Duffing oscillator, given by canonical equations [Wiggins(1990)]˙q = p, ṗ = q − q 3 − δp + γ cos ωt, δ, γ, ω ≥ 0, (q, p) ∈ R 2 .(3.173)where δ, γ, and ω are real parameters physically meaning dissipation, amplitudeof forcing and frequency, respectively.The perturbed system (3.173) is given by˙q = p, ṗ = q − q 3 + ɛ(γ cos ωt − δp), (3.174)where ɛ−perturbation is assumed small. Then the unperturbed system reads˙q = p, ṗ = q − q 3 .It is conservative with Hamiltonian functionH(q, p) = p22 − q22 + q44 . (3.175)In the unperturbed phase–space all orbits are given by the level sets ofthe Hamiltonian (3.175). There are three equilibrium points at the followingcoordinates: (q, p) = (±1, 0) – centers, and (q, p) = (0, 0) – saddle. Thesaddle point is connected to itself by two homoclinic orbits given byq 0 +(t) = ( √ 2(cosh t) −1 , − √ 2(cosh t) −1 tanh t),q 0 −(t) = −q 0 +(t).There are two families of periodic orbits q±(t), k where k represents the ellipticmodulus related to the Hamiltonian by H(q±(t)) k ≡ H(k) =k2 −1(2−k 2 )2 ,inside the corresponding homoclinic orbits q 0 ±(t), with the period T (k) =2K(k) √ 2 − k 2 (K(k) is the complete elliptic integral of the first kind.Also, there exists a family of periodic orbits outside the homoclinicorbits with the period T (k) = 4K(k) √ k 2 − 1.The perturbed system (3.174) can be rewritten as a third–order autonomoussystem˙q = p, ṗ = q − q 3 + ɛ(γ cos φ − δp), ˙φ = ω,


348 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere (q, p, φ) ∈ R 2 ×S 1 , S 1 is the circle of length 2π/ω and φ(t) = ωt + φ 0 .We form the global cross–section to the flowΣ φ 0= {(q, p, φ)|φ = φ0 ∈ [0, 2π/ω]}and the associated Poincaré map is given byP : Σ φ 0 → Σφ 0, (q(0), p(0)) ↦→ (q(2π/ω), p(2π/ω)).A detailed analysis of the perturbed Poincaré map for the damped,forced Duffing oscillator is related to the Melnikov function (see [Wiggins(1990)]).Complex 1–DOF Hamiltonian DynamicsRecall that setting z = q + ip, z ∈ C, i = √ −1, Hamiltonianequations ˙q = ∂H/∂p, ṗ = −∂H/∂q may be written in complex notationas [Abraham and Marsden (1978); Marsden and Ratiu (1999);Wiggins (1990)]ż = −2i ∂H∂¯z . (3.176)Let U be an open set in the complex phase–space manifold M C (i.e.,manifold M modelled on C). A C 0 function γ : [a, b] → U ⊂ M C , t ↦→γ(t) represents a solution curve γ(t) = q(t) + ip(t) of a complex Hamiltoniansystem (3.176). For example, the curve γ(θ) = cos θ + i sin θ, 0 ≤θ ≤ 2π is the unit circle. γ(t) is a parameterized curve. We call γ(a) thebeginning point, and γ(b) the end point of the curve. By a point on thecurve we mean a point w such that w = γ(t) for some t ∈ [a, b].The derivative ˙γ(t) is defined in the usual way, namely˙γ(t) = ˙q(t) + iṗ(t),so that the usual rules for the derivative of a sum, product, quotient, andchain rule are valid. The speed is defined as usual to be | ˙γ(t)|. Also, iff : U → M C represents a holomorphic, or analytic function, then thecomposite f ◦ γ is differentiable (as a function of the real variable t) and(f ◦ γ) ′ (t) = f ′ (γ(t)) ˙γ(t).Recall that a path represents a sequence of C 1 −curves,γ = {γ 1 , γ 2 , . . . , γ n },


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 349such that the end point of γ j , (j = 1, . . . , n) is equal to the beginning pointof γ j+1 . If γ j is defined on the interval [a j , b j ], this means thatγ j (b j ) = γ j+1 (a j+1 ).We call γ 1 (a 1 ) the beginning point of γ j , and γ n (b n ) the end point of γ j .The path is said to lie in an open set U ⊂ M C if each curve γ j lies in U,i.e., for each t, the point γ j (t) lies in U.An open set U is connected if given two points α and β in U, thereexists a path γ = γ 1 , γ 2 , . . . , γ n in U such that α is the beginning pointof γ 1 and β is the end point of γ n ; in other words, if there is a path γ inU which joins α to β. If U is a connected open set and f a holomorphicfunction on U such that f ′ = 0, then f is a constant. If g is a function onU such that f ′ = g, then f is called a primitive of g on U. Primitives canbe either find out by integration or written down directly.Let f be a C 0 −function on an open set U, and suppose that γ is a curvein U, meaning that all values γ(t) lie in U for a ≤ t ≤ b. The integral of falong γ is defined as∫γ∫f =γf(z) =∫ baf(γ(t)) ˙γ(t) dt.For example, let f(z) = 1/z, and γ(θ) = e iθ . Then ˙γ(θ) = ie iθ . Wewant to find the value of the integral of f over the circle, ∫ dz/z, so 0 ≤ θ ≤γ2π. By definition, this integral is equal to ∫ 2πie iθ /e iθ dθ = i ∫ 2πdθ = 2πi.0 0∫The length L(γ) is defined to be the integral of the speed, L(γ) =b| ˙γ(t)| dt.aIf γ = γ 1 , γ 2 , . . . , γ n is a path, then the integral of a C 0 −function f onan open set U is defined as ∫ γ f = ∑ n∫i=1f, i.e., the sum of the integralsγ iof f over each curve γ i (i = 1, . . . , n of the path γ. The length of a path isdefined as L(γ) = ∑ ni=1 L(γ i).Let f be continuous on an open set U ⊂ M C , and suppose that f hasa primitive g, that is, g is holomorphic and g ′ = f. Let α, β be two pointsin U, and let γ be a path in U joining α to β. Then ∫ f = g(β) − g(α);γthis integral is independent of the path and depends only on the beginningand end point of the path.A closed path is a path whose beginning point is equal to its end point.If f is a C 0 −function on an open set U ⊂ M C admitting a holomorphicprimitive g, and γ is any closed path in U, then ∫ γ f = 0.Let γ, η be two paths defined over the same interval [a, b] in an open set


350 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionU ⊂ M C . Recall (see Introduction) that γ is homotopic to η if there existsa C 0 −function ψ : [a, b] × [c, d] → U defined on a rectangle [a, b] × [c, d] ⊂U, such that ψ(t, c) = γ(t) and ψ(t, d) = η(t) for all t ∈ [a, b]. Foreach number s ∈ [c, d] we may view the function |psi s (t) = ψ(t, s) as acontinuous curve defined on [a, b], and we may view the family of continuouscurves ψ s as a deformation of the path γ to the path η. It is said that thehomotopy ψ leaves the end points fixed if we have ψ(a, s) = γ(a) andψ(b, s) = γ(b) for all values of s ∈ [c, d]. Similarly, when we speak of ahomotopy of closed paths, we assume that each path ψ s is a closed path.Let γ, η be paths in an open set U ⊂ M C having the same beginning andend points. Assume that they are homotopic in U. Let f be holomorphicon U. Then ∫ γ f = ∫ f. The same holds for closed homotopic paths in U.ηIn particular, if γ is homotopic to a point in U, then ∫ f = 0. Also, it isγsaid that an open set U ⊂ M C is simply–connected if it is connected and ifevery closed path in U is homotopic to a point.In the previous example we found that12πI∫γ1dz = 1,zif γ is a circle around the origin, oriented counterclockwise. Now we definefor any closed path γ its winding number with respect to a point α to beW (γ, α) = 1 ∫12πi z − α dz,provided the path does not pass through α. If γ is a closed path, thenW (γ, α) is an integer.A closed path γ ∈ U ⊂ M C is homologous to 0 in U if∫1dz = 0,z − αγfor every point α not in U, or in other words, W (γ, α) = 0 for every suchpoint.Similarly, let γ, η be closed paths in an open set U ⊂ M C . We say thatthey are homologous in U, and write γ ∼ η, if W (γ, α) = W (η, α) for everypoint α in the complement of U. We say that γ is homologous to 0 in U,and write γ ∼ 0, if W (γ, α) = 0 for every point α in the complement of U.If γ and η are closed paths in U and are homotopic, then they arehomologous. If γ and η are closed paths in U and are close together, thenthey are homologous.γ


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 351Let γ 1 , . . . , γ n be curves in an open set U ⊂ M C , and let m 1 , . . . , m n beintegers. A formal sum γ = m 1 γ 1 + · · · + m n γ n = ∑ ni=1 m iγ i is called achain in U. The chain is called closed if it is a finite sum of closed paths.If γ is the chain as above, then ∫ γ f = ∑ i m ∫i f. If γ and η are closedγ ichains in U, then W (γ + η, α) = W (γ, α) + W (η, α). We say that γ andη are homologous in U, and write γ ∼ η, if W (γ, α) = W (η, α) for everypoint α in the complement of U. We say that γ is homologous to 0 in U,and write γ ∼ 0, if W (γ, α) = 0 for every point α in the complement of U.Recall that the Cauchy Theorem states that if γ is a closed chain in anopen set U ⊂ M C , and γ is homologous to 0 in U, then ∫ f = 0. If γ andγη are closed chains in U, and γ ∼ η in U, then ∫ γ f = ∫ η f.∫It follows from Cauchy’s Theorem that if γ and η are homologous, thenγ f = ∫ f for all holomorphic functions f on U [Abraham and Marsdenη(1978); Wiggins (1990)].3.12.3.2 Library of Basic Hamiltonian SystemsIn this subsection, we present some basic Hamiltonian systems used byhuman–like biodynamics (for more details, see [Puta (1993)]).1D Harmonic OscillatorIn this case we have {p, q} as canonical coordinates on R 2M = T ∗ R ≃ R 2 ,ω = dp ∧ dq,H = 1 2(p 2 + y ) , X H = p ∂ ∂q − q ∂ ∂p ,and Hamiltonian equations read˙q = p,ṗ = −q.For each f, g ∈ C ∞ (R 2 , R) the Poisson bracket is given byComplex Plane{f, g} ω = ∂f ∂g∂q ∂p − ∂f ∂g∂p ∂q .Let T ∗ R ≃ R 2 have the canonical symplectic structure ω = dp ∧ dq.


352 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWriting z = q + ip, we haveω = 1 dz ∧ d¯z,2i X{f, g} ω = i 2( ∂f ∂g∂z ∂¯z − ∂f∂¯z( ∂HH = i∂z∂g∂z),∂∂z − ∂H∂¯zso, the Hamiltonian equations, ˙q = ∂ p H, ṗ = −∂ q H, become2D Harmonic Oscillatorż = −2i ∂H∂¯z .)∂,∂¯zIn this case we have {q 1 , y, p 1 , p 2 } as canonical coordinates on R 4M = T ∗ R 2 ≃ R 4 , ω = dp 1 ∧ dq 1 + dp 2 ∧ dq 2 ,H = 1 [p22 1 + p 2 2 + (q 1 ) 2 + (y) 2] .The functions f = p i p j + q i q j and g = p i q j + p j q i , (for i, j = 1, 2), areconstants of motion.nD Harmonic OscillatorIn this case we have (i = 1, ..., n)M = T ∗ R n ≃ R 2n , ω = dp i ∧ dq i ,H = 1 n∑ [p22 i + (q i ) 2] .i=1The system is integrable in an open set of T ∗ R n with:K 1 = H, K 2 = p 2 2 + (y) 2 , ..., K n = p 2 n + (q n ) 2 .Toda MoleculeConsider three mass–points on the line with coordinates q i , (i = 1, 2, 3),and satisfying the ODEs:¨q i = −∂ q iU, where U = e q1 −q 2 + e q2 −q 3 − e q3 −q 1 .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 353This is a Hamiltonian system with {q i , p i } as canonical coordinates on R 6 ,M = T ∗ R 3 ≃ R 6 , ω = dp i ∧ dq i ,H = 1 (p22 1 + p 2 2 + p 2 )3 + U.The Toda molecule (3.12.3.2) is an integrable Hamiltonian system in anopen set of T ∗ R 3 with:K 1 = H, K 2 = p 1 + p 2 + p 3 ,K 3 = 1 9 (p 1 + p 2 + p 3 ) (p 2 + p 3 − 2p 1 ) (p 3 + p 1 − 2p 2 ) − (p 1 + p 2 − 2p 3 ) e q1 −q 2− (p 2 + p 3 − 2p 1 ) e q2 −q 3 − (p 3 + p 1 − 2p 2 ) e q3 −q 1 .3–Point Vortex ProblemThe motion of three–point vortices for an ideal incompressible fluid inthe plane is given by the equations:˙q j = − 12πṗ j = 12π∑Γ i (p j − p i ) /rij,2i≠j∑ (Γ i q i − q j) /rij,2i≠jr 2 ij = ( q i − q j) 2+ (pj − p i ) 2 ,where i, j = 1, 2, 3, and Γ i are three nonzero constants. This mechanicalsystem is Hamiltonian if we take:M = T ∗ R 3 ≃ R 6 , ω = dp i ∧ dq i , (i = 1, ..., 3),H = − 1 3∑Γ i Γ i ln (r ij ) .4πi,j=1Moreover, it is integrable in an open set of T ∗ R 3 with:K 1 = H, K 2 =3∑i=1( 3∑) 2K 3 = Γ i q i + K2.2i=1Γ i[ (qi ) 2+ p2i],The Newton’s Second Law as a Hamiltonian System


354 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn the case of conservative forces, Newton’s law of motion can be writtenon R 3n asm i¨q i = −∂ q iU,(i = 1, 2, ..., 3n).Its symplectic formulation reads:M = T ∗ R 3 ≃ R 6 , ω = dp i ∧ dq i ,H =3n∑i=1p 2 i2m i+ U.The Hamiltonian vector–field X H is( )piX H = ∂m q i − ∂ q iU ∂ pi ,igiving the Hamiltonian equations˙q i = p im i,ṗ i = −∂ q iU.Rigid Body Fixed in a PointThe configuration space of a rigid body fixed in a point is SO(3), thegroup of proper orthogonal transformations of R 3 to itself, while the correspondingphase–space is its cotangent bundle, T ∗ SO(3). The motion of arigid body is a geodesic with respect to a left–invariant Riemannian metric(the inertia tensor) on SO(3). The momentum map J : P → R 3 for the leftSO(3)−action is right translation to the identity. We identify so(3) ∗ withso(3) via the Killing form and identify R 3 with so(3) via the map v ↦→ ˆv,where ˆv(w) = v × w (× being the standard cross product). Points in so(3) ∗are regarded as the left reduction of T ∗ SO(3) by G = SO(3) and are theangular momenta as seen from a body–fixed frame.A Segment of a Human–Like BodyA rigid body with a fixed point is a basic model of a single segment of thehuman (or robot) body. This is a left–invariant Hamiltonian mechanicalsystem on the phase–space T ∗ SO(3). The differentiable structure on SO(3)is defined using the traditional Euler angles {ϕ, ψ, θ}. More precisely, a local


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 355chart is(ϕ, ψ, θ) ∈ R 3 ↦−→ A ∈ SO(3), 0 < ϕ, ψ < 2π; 0 < θ < π, where⎡⎤cos ψ cos ϕ − cos θ sin ϕ sin ψ cos ψ cos ϕ + cos θ cos ϕ sin ψ sin θ sin ψA = ⎣ − sin ψ cos ϕ − cos θ sin ϕ sin ψ − sin ψ sin ϕ + cos θ cos ϕ cos ψ sin θ cos ψ ⎦sin θ sin ϕ − sin θ cos ϕ cos θThe corresponding conjugate momenta are denoted by p ϕ , p ψ , p θ , so{ϕ, ψ, θ, p ϕ , p ψ , p θ } is the phase–space T ∗ SO(3). Thus, we haveM = T ∗ SO(3), ω = dp ϕ ∧ dϕ + dp ψ ∧ dψ + dp θ ∧ dθ, H = 1 2 K,K = [(p ϕ − p ψ cos θ) sin ψ + p θ sin θ cos ψ] 2I 1 sin 2 θ+ [(p ϕ − p ψ cos θ) cos ψ − p θ sin θ sin ψ] 2I 2 sin 2 + p2 ψ,θI 3where I 1 , I 2 , I 3 are the moments of inertia, diagonalizing the inertia tensorof the body.The Hamiltonian equations are˙ϕ = ∂H∂p ϕ,ṗ ϕ = − ∂H∂ϕ ,˙ψ =∂H∂p ψ,ṗ ψ = − ∂H∂ψ ,˙θ =∂H∂p θ,ṗ θ = − ∂H∂θ .For each f, g ∈ C ∞ (T ∗ SO(3), R) the Poisson bracket is given by{f, g} ω = ∂f ∂g− ∂f ∂g∂ϕ ∂p ϕ ∂p ϕ ∂ϕ + ∂f ∂g− ∂f ∂g∂ψ ∂p ψ ∂p ψ ∂ψ+ ∂f ∂g− ∂f ∂g∂θ ∂p θ ∂p θ ∂θ .The Heavy Top – ContinuedRecall (see (3.8.4.2) above) that the heavy top is by definition a rigidbody moving about a fixed point in a 3D space [Puta (1993)]. The rigidityof the top means that the distances between points of the body are fixed


356 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionas the body moves. In this case we haveM = T ∗ SO(3),ω = dp ϕ ∧ dϕ + dp ψ ∧ dψ + dp θ ∧ dθ,H = 1 K + mgl cos θ,2K = [(p ϕ − p ψ cos θ) sin ψ + p θ sin θ cos ψ] 2I 1 sin 2 θ+ [(p ϕ − p ψ cos θ) cos ψ − p θ sin θ sin ψ] 2I 2 sin 2 + p2 ψ,θI 3where I 1 , I 2 , I 3 are the moments of inertia, m is the total mass, g is thegravitational acceleration and l is the length of the vector determining thecenter of mass at t = 0.The Hamiltonian equations are˙ϕ = ∂H∂p ϕ,ṗ ϕ = − ∂H∂ϕ ,˙ψ =∂H∂p ψ,ṗ ψ = − ∂H∂ψ ,˙θ =∂H∂p θ,ṗ θ = − ∂H∂θ .For each f, g ∈ C ∞ (T ∗ SO(3), R) the Poisson bracket is given by{f, g} ω = ∂f ∂g− ∂f ∂g∂ϕ ∂p ϕ ∂p ϕ ∂ϕ + ∂f ∂g− ∂f ∂g∂ψ ∂p ψ ∂p ψ ∂ψ+ ∂f ∂g− ∂f ∂g∂θ ∂p θ ∂p θ ∂θ .The Hamiltonian H is invariant under rotations about the z−axis,i.e., ϕ is a cyclic variable, so p ϕ is a constant of motion. The momentummap for this S 1 −-action is J(ϕ, ψ, θ, p ϕ , p ψ , p θ ) = p ϕ . The reducedphase–space J −1 (p ϕ )/S 1 can be identified with T ∗ S 2 and it is parameterizedby {ψ, θ, p ψ , p θ }. The equations of motion for ψ, θ are just Hamiltonianequations for H with p ϕ held constant.Two Coupled PendulaThe configuration space of the system of two coupled pendula in theplane is T 2 = {(θ 1 , θ 2 )}, where the θs are the two pendulum angles, the


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 357phase–space is T ∗ T 2 with its canonical symplectic structure and the HamiltonianH is given by [Puta (1993)]H = 1 2 (p2 ϕ + p 2 ψ) + V ( √ 2ψ),whereϕ = θ 1 + θ 2√2, ψ = θ 1 − θ 2√2.The circle group S 1 acts on a torus T 2 by θ ·(θ 1 +θ 2 ) = (θ +θ 1 , θ +θ 2 ) andhence the induced momentum map for the lifted action to T ∗ T 2 is given byJ(ϕ, ψ, p ϕ , p ψ ) = p ϕ . Therefore, the reduced phase–space J −1 (p ϕ )/S 1 issymplectically diffeomorphic to T ∗ S 1 with its canonical symplectic structureω µ = dp ψ ∧ dψ. The reduced Hamiltonian H µ is H µ = 1 2 p2 ψ +V ( √ 2ψ), and Hamiltonian equations for H µ are˙ψ = p ψ ,ṗ ψ = − √ 2 ˙V ( √ 2ψ).The Plane 2–Body ProblemThe plane two body problem can be formulated as the triple (M, ω, H)where [Puta (1993)]M = T ∗ ( (0, ∞) × S 1) ,H = (p 2 r + p 2 θ)/r 2 − 1/r.ω = dp r ∧ dr + dp θ ∧ dθ,The Lie group G = SO(2) ≃ S 1 acts on the configuration space M =(0, ∞) × S 1 by rotations, i.e., if R ϕ ∈ SO(2) thenφ : (R ϕ , (r, θ)) ↦→ (r, θ + ϕ, p r , p θ ).The corresponding momentum map isThe 3–Body ProblemJ(r, θ, p r , p θ ) = p θ .There is a vast literature on the restricted three–body problem (see[Meyer and Hall (1992)]). Among other things, there are investigations ofthe equilibriums points and their stability, investigations of the existence,stability and bifurcation of periodic orbits, and investigations of collisionsand ejection orbits. The restricted problem is said to be a limit of the fullthree–body problem as one of the masses tends to zero, and so to each result


358 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor the restricted problem there should be a corresponding result for thefull three–body problem.The restricted three–body problem is a Hamiltonian system of differentialequations which describes the motion of an infinitesimal particle (thesatellite) moving under the gravitational influence of two particles of finitemass (the primaries) which are moving on a circular orbit of the Keplerproblem [Meyer (2005)].Since the motion of the primaries is given, the restricted problem hastwo DOF for the planar problem and three DOF for the spatial problem.However, the full problem has six DOF in the planar case and nine DOFin the spatial case. Thus, at first the restricted problem seems too smallto reflect the full complexity of the full problem; but when the symmetriesof the full problem are taken into account the dimension gap narrowsconsiderably.The Hamiltonian of the full problem is invariant under Euclidean motions,i.e., translations and rotations, which begets the integrals of linearand angular momentum. Translations and rotations induce ignorable coordinates.Holding the integrals fixed and dropping the ignorable coordinatesreduces the full problem from six to three DOF in the planar case andfrom nine to four DOF in the spatial case. Thus the full problem on thereduced space is only one DOF larger than the restricted problem in eitherthe planar or the spatial case [Meyer (2005)].The full 3–body problem in 3D space has 9 DOF. By placing the centerof mass at the origin and setting linear momentum equal to zero the problemreduces one with six DOF. This can be done using Jacobi coordinates. TheHamiltonian of the full 3–body problem in rotating (about the z−-axis)Jacobi coordinates (u 0 , u 1 , u 2 , v 0 , v 1 , v 2 ) isH = ‖ v 0 ‖ 22M 0− u T 0 Jv 0 + ‖ v 1 ‖ 22M 1− u T 1 Jv 1 − m 0m 1‖ u 1 ‖+ ‖ v 2 ‖ 22M 2− u T 2 Jv 2 −m 1 m 2‖ u 2 − α 0 u 1 ‖ − m 2 m 0‖ u 2 + α 1 u 1 ‖where u i , v i ∈ R 3 ,M 0 = m 0 + m 1 + m 2 , M 1 = m 0 m 1 /(m 0 + m 1 ),M 2 = m 2 (m 0 + m 1 )/(m 0 + m 1 + m 2 ),α 0 = m 0 /(m 0 + m 1 ), α 1 = m 1 /(m 0 + m 1 ),


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 359( ) 0 1 0and J = −1 0 0 .0 0 0In these coordinates u 0 is the center of mass, v 0 is total linear momentum,and total angular momentum is: A = u 0 × v 0 + u 1 × v 1 + u 2 × v 2 . See[Meyer and Hall (1992)] for further details.n−DOF Hamiltonian DynamicsClassically, n−DOF Hamiltonian dynamics combines the ideas of differentialequations and variational principles (see [Abraham and Marsden(1978); Arnold (1989); Marsden and Ratiu (1999); Wiggins (1990)]). AsHamiltonian first realized, many of the systems of mechanics and opticscan be put into the special form (compare (3.164))˙q i = ∂H∂p i(q i , p i , t),ṗ i = − ∂H∂q i (qi , p i , t),(i = 1, . . . , n),or an associated variational form (summing upon the repeated index is usedin the following text)∫δ (p i dq i − H) dt = 0.Here the state of the system is given as a point (q 1 , . . . , q n , p 1 , . . . , p n ) inphase–space, the q’s are the configuration coordinates, the p’s are the momenta,t is time, and H = H(q i , p i , t) is a total–energy function calledHamiltonian. The variables (q i , p i ) are called canonical coordinates.If H = H(q i , p i ) does not depend explicitly on time, the system is saidto be autonomous. In this case, it is easy to verify that H is conserved.The search for other conserved quantities led to a new notion of solvingHamiltonian systems. Instead of finding formulae for the coordinates as afunction of time, one searches for constants of the motion (integrals). Ifone can find n integrals I i (q i , p i ) which are in involution:[I i , I j ] = ∂I i∂q k ∂I j∂p k− ∂I i ∂I j= 0, (i ≠ j),∂p k ∂qk and independent (the vectors ∇I i are independent ‘almost everywhere’),then associated variables φ i can be derived which evolve linearly in time:˙φ i = ∂H∂I i(I i ).Such a system is integrable in the sense of Liouville [Arnold (1989)]. Ifthe sets I = const are bounded, then they are nD tori T n in phase–space.


360 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionChoosing irreducible cycles, γ i , on the tori, one can define a preferred set ofintegrals J i = ∫ γ ip i dq i , called action variables, for which the correspondingφ i are angle variables mod 1 on T n . The quantities ω i (J) = ∂H∂J i(J i ) arecalled the frequencies on T n .Another feature of Hamiltonian systems noticed by Liouville is thepreservation of phase–space volume ∫ (dq) n (dp) n . A more general resultis that Poincaré’s integral ∫ p i dq i is conserved around any loop followingthe flow [Arnold (1989)]. This is the property that really distinguishesHamiltonian differential equations from general ones.The major problem with the notion of integrability is that most systemsare not integrable. This was first appreciated when Poincaré proved thatthe circular restricted three–body problem has no integral analytic in themass ratio. The perturbation expansions which gave excellent predictionsof motion of the planets do not converge. The basic reason is that amongthe invariant tori of integrable systems is a dense subset on which thefrequencies ω i are commensurate, i.e, m i ω i = 0 for some non–zero integervector m i . However, most systems have no commensurate tori, becausethey can be destroyed by arbitrarily small perturbation.Poincaré went on to examine what really does happen. The key techniquehe used was geometrical analysis: instead of manipulating formulaefor canonical transformations as Jacobi and others did, he pictured the orbitsin phase–space. An important step in this qualitative ODE theory wasthe idea of surface of section. If Σ is a codimension–one surface (i.e., ofdimension one less than that of the phase–space) transverse to a flow, thenthe sequence {x j } of successive intersections of an orbit with Σ gives a lotof information about that orbit. For example, if {x j } is periodic then it correspondsto a periodic orbit. If {x j } is confined to a subset of codimensionm on Σ then so is the orbit of the flow, etc.. The flow induces a mappingof Σ to itself; the map takes a point in Σ to the point at which it firstreturns to Σ (assuming there is on). Since the surface of section has onedimension less than the phase–space it is easier to picture the dynamics ofthe return map than the flow. In fact, for Hamiltonian systems one can doeven better; since H is conserved, Σ decomposes into a one–parameter familyof codimension two surfaces parameterized by the value of the energy, areduction of two dimensions.This led Poincaré to the ideas of stable and unstable manifolds for hyperbolicperiodic orbits, which are extensions of the stable and unstableeigenspaces for associated linear systems, and their intersections, known as


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 361hetero– and homo–clinic points, whose orbits converge to one periodic orbitin the past and to another (or the same) in the future. He showed that havingintersected once, the invariant manifolds must intersect infinitely often.Moreover the existence of one heteroclinic orbit implies the existence of aninfinity of others.The distance between the stable and unstable manifolds can be quantifiedby Melnikov’s integral. This leads to a technique for proving thenon–existence of integrals for a slightly perturbed, integrable Hamiltonian.For integrable systems, nearby orbits separate linearly in time. However,dynamical systems can have exponentially separating orbits. Let δx be atangent vector at the phase–space point x and δx t be the evolved vectorfollowing the orbit of x. Then, recall that the average rate of exponentiationof δx t is the Lyapunov exponent λ (see, e.g., [Chen and Dong (1998)])λ(x, δx) =lim 1/t ln |δx t|.t−→∞If λ is nonzero, then the predictions one can make will be valid for a timeonly logarithmic in the precision. Therefore, although deterministic in principle,a system need not be predictable in practice.A concrete example of the complexity of behavior of typical Hamiltoniansystems is provided by the ‘horseshoe’, a type of invariant set found nearhomoclinic orbits. Its points can be labelled by doubly infinite sequences of0’s and 1’s corresponding to which half of a horseshoe shaped set the orbitis in at successive times. For every sequence, no matter how complicated,there is an orbit which has that symbol sequence. This implies, e.g., that asimple pendulum in a sufficiently strongly modulated time–periodic gravitationalfield has an initial condition such that the pendulum will turn overonce each period when there is 1 in the sequence and not if there is a 0 forany sequence of 0’s and 1’s.3.12.3.3 Hamilton–Poisson MechanicsNow, instead of using symplectic structures arising in Hamiltonian mechanics,we propose the more general Poisson manifold (g ∗ , {F, G}). Here g ∗ isa chosen Lie algebra with a (±) Lie–Poisson bracket {F, G} ± (µ)) and carriesan abstract Poisson evolution equation F ˙ = {F, H}. This approachis well–defined in both the finite– and the infinite–dimensional case. Itis equivalent to the strong symplectic approach when this exists and offersa viable formulation for Poisson manifolds which are not symplectic(for technical details, see see [Weinstein (1990); Abraham et al. (1988);


362 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionMarsden and Ratiu (1999); Puta (1993); <strong>Ivancevic</strong> and Pearce (2001a)]).Let E 1 and E 2 be Banach spaces. A continuous bilinear functional: E 1 × E 2 −→ R is nondegenerate if < x, y > = 0 implies x = 0 andy = 0 for all x ∈ E 1 and y ∈ E 2 . We say E 1 and E 2 are in duality if thereis a nondegenerate bilinear functional : E 1 × E 2 −→ R. This functionalis also referred to as an L 2 −pairing of E 1 with E 2 .Recall that a Lie algebra consists of a vector space g (usually a Banachspace) carrying a bilinear skew–symmetric operation [, ] : g × g → g, calledthe commutator or Lie bracket. This represents a pairing [ξ, η] = ξη − ηξof elements ξ, η ∈ g and satisfies Jacobi identity[[ξ, η], µ] + [[η, µ], ξ] + [[µ, ξ], η] = 0.Let g be a (finite– or infinite–dimensional) Lie algebra and g ∗ its dualLie algebra, that is, the vector space L 2 paired with g via the inner product: g ∗ × g → R. If g is nD, this pairing reduces to the usual action(interior product) of forms on vectors. The standard way of describing anynD Lie algebra g is to give its n 3 Lie structural constants γ k ij , defined by[ξ i , ξ j ] = γ k ij ξ k, in some basis ξ i , (i = 1, . . . , n)For any two smooth functions F : g ∗ → R, we define the Fréchet derivativeD on the space L(g ∗ , R) of all linear diffeomorphisms from g ∗ to R asa map DF : g ∗ → L(g ∗ , R); µ ↦→ DF (µ). Further, we define the functionalderivative δF /δµ ∈ g byDF (µ) · δµ = < δµ, δFδµ >with arbitrary ‘variations’ δµ ∈ g ∗ .For any two smooth functions F, G : g ∗ → R, we define the (±) Lie–Poisson bracket by[ δF{F, G} ± (µ) = ± < µ,δµ , δG ]> . (3.1)δµHere µ ∈ g ∗ , [ξ, µ] is the Lie bracket in g and δF /δµ, δG/δµ ∈ g are thefunctional derivatives of F and G.The (±) Lie–Poisson bracket (3.1) is clearly a bilinear and skew–symmetric operation. It also satisfies the Jacobi identity{{F, G}, H} ± (µ) + {{G, H}, F } ± (µ) + {{H, F }, G} ± (µ) = 0


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 363thus confirming that g ∗ is a Lie algebra, as well as Leibniz’ rule{F G, H} ± (µ) = F {G, H} ± (µ) + G{F, H} ± (µ). (3.177)If g is a nD phase–space manifold with structure constants γ k ij , the (±)Lie–Poisson bracket (3.177) becomes{F, G} ± (µ) = ±µ k γ k δF δGij . (3.178)δµ i δµ jThe (±) Lie–Poisson bracket represents a Lie–algebra generalizationof the classical nD Poisson bracket [F, G] = ω(X f , X g ) on the symplecticphase–space manifold (P, ω) for any two real–valued smooth functions F, G :P −→ R.As in the classical case, any two smooth functions F, G : g ∗ −→ R are ininvolution if {F, G} ± (µ) = 0.The Lie–Poisson Theorem states that a Lie algebra g ∗ with a ± Lie–Poisson bracket {F, G} ± (µ) represents a Poisson manifold (g ∗ , {F, G} ± (µ)).Given a smooth Hamiltonian function H : g ∗ → R on the Poissonmanifold (g ∗ , {F, G} ± (µ)), the time evolution of any smooth function F :g ∗ → R is given by the abstract Poisson evolution equation˙ F = {F, H}. (3.179)3.12.3.4 Completely Integrable Hamiltonian SystemsIn order to integrate a system of 2n ODEs, we must know 2n first integrals.It turns out that if we are given a canonical system of ODEs, it is oftensufficient to know only n first integrals [Arnold (1989)].Liouville Theorem on Completely Integrable SystemsRecall that a function F is a first integral of a system Ξ with Hamiltonianfunction H iff H and F are in involution on the system’s phase–spaceP (which is the cotangent bundle of the system’s configuration manifoldT ∗ M), i.e., iff the Poisson bracket of H and F is identically equal to zeroon P , {H, F } ≡ 0.Liouville proved that if, in a system Ξ with n DOF (i.e., with a 2nDphase–space P = T ∗ M), n independent first integrals in involution areknown, then the system is integrable by quadratures.Here is the exact formulation of the Liouville Theorem [Arnold (1989)]:Suppose that we are given n functions in involution on a symplectic 2nD


364 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmanifold:F 1 , ..., F n ; {F i , F j } ≡ 0, (i, j = 1, ..., n).Consider a level set of the functions F i :M f = {x : F i (x) = f i },(i = 1, ..., n).Assume that the n functions F i are independent on M f (i.e., the n 1−formsdF i are linearly independent at each point of M f ). Then1. M f is a smooth manifold, invariant under the phase–flow with Hamiltonianfunction H = F 1 .2. If the manifold M f is compact and connected, then it is diffeomorphicto the n−torusT n = {(ϕ 1 , ..., ϕ n ) mod 2π}.3. The phase–flow with Hamiltonian function H determines a conditionallyperiodic motion on M f , i.e., in angular coordinates ϕ i = (ϕ 1 , ..., ϕ n )we have˙ϕ i = ω i , ω i = ω i (f i ), (i = 1, ..., n).4. The canonical equations with Hamiltonian function H can be integratedby quadratures.For the proof of this Theorem see [Arnold (1989)].As an example with 3 DOF, we consider a heavy symmetrical Lagrangiantop fixed at a point on its axis. Three first integrals are immediatelyobvious: H, M z and M 3 . It is easy to verify that the integralsM z and M 3 are in involution. Furthermore, the manifold H = h in thephase–space is compact. Therefore, we can say without any calculationsthat the motion of the top is conditionally periodic: the phase trajectoriesfill up the 3D torus T 3 , given by: H = c 1 , M z = c 2 , M 3 = c 3 . The correspondingthree frequencies are called frequencies of fundamental rotation,precession, and nutation.Other examples arise from the following observation: if a canonical systemcan be integrated by the method of Hamiltonian–Jacobi, then it has nfirst integrals in involution. The method consists of a canonical transformation(p i , q i ) → (P i , Q i ) such that the Q i are first integrals, while thefunctions Q i and Q i are in involution.The Liouville Theorem, as formulated above, covers all the problems ofdynamics which have been integrated to the present day [Arnold (1989)].


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 365Action–Angle VariablesUnder the hypothesis of the Liouville Theorem, we can find symplecticcoordinates (I i , ϕ i ) such that the first integrals F i depend only on I i and ϕ i(for i = 1, ..., n) are angular coordinates on the n−torus T n ≃ M f = {x :F i (x) = f i }, which is invariant with respect to the phase–flow. We chooseangular coordinates ϕ i on M f so that the phase–flow with Hamiltonianfunction H = F 1 takes an especially simple form [Arnold (1989)]:˙ϕ i = ω i (f i ), ϕ i (t) = ϕ i (0) + ω i t.Now we look at a neighborhood of the n−manifold M f = T n in the system’s2nD phase–space P .In the coordinates (F i , ϕ i ) the phase–flow with Hamiltonian functionH = F 1 can be written in the form of the simple system of 2n ODEs˙ F i = 0, ˙ϕ i = ω i (F i ), (i = 1, ..., n), (3.180)which is easily integrated: F i (t) = F i (0), ϕ i (t) = ϕ i (0) + ω i (F i (0)) t.Thus, in order to integrate explicitly the original canonical system ofODEs, it is sufficient to find the variables ϕ i in explicit form. It turnsout that this can be done using only quadratures. A construction of thevariables ϕ i is given below [Arnold (1989)].In general, the variables (F i , ϕ i ) are not symplectic coordinates. However,there are functions of F i , which we denote by I i = I i (F i ), (i = 1, ..., n),such that the variables (I i , ϕ i ) are symplectic coordinates: the original symplecticstructure dp i ∧dq i is expressed in them as dI i ∧dϕ i . The variables I ihave physical dimensions of action and are called action variables; togetherwith the angle variables ϕ i they form the action–angle system of canonicalcoordinates in a neighborhood of M f = T n .The quantities I i are first integrals of the system with Hamiltonianfunction H = F 1 , since they are functions of the first integrals F i . Inturn, the variables F i can be expressed in terms of I i and, in particular,H = F 1 = H(I i ). In action–angle variables, the ODEs of our flow (3.180)have the form˙ I i = 0, ˙ϕ i = ω i (I i ), (i = 1, ..., n).A system with one DOF in the phase plane (p, q) is given by the Hamiltonianfunction H(p, q). In order to construct the action–angle variables,we look for a canonical transformation (p, q) → (I, ϕ) satisfying the two


366 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionconditions:I = I(h),∮M hdϕ = 2π. (3.181)The action variable in the system with one DOF given by the Hamiltonianfunction H(p, q) is the quantityI(h) = 12π Π(h) = 1 ∮pdq,2π M hwhich is the area bounded by the phase curve H = h. Arnold statesthe following Theorem: Set S(I, q) = ∫ qq 0pdq| H=h(I) is a generating function.Then a canonical transformation (p, q) → (I, ϕ) satisfying conditions(3.181) is given by( )∂S(I, q) ∂S(I, q) ∂S(I, q)p = , ϕ = , H , q = h(I).∂q∂I∂qWe turn now to systems with n DOF given in R 2n = {(p i , q i ), i =1, ..., n} by a Hamiltonian function H(p i , q i ) and having n first integrals ininvolution F 1 = H, F 2 ..., F n . Let γ 1 , ..., γ n be a basis for the 1D cycles onthe torus M f = T n (the increase of the coordinate ϕ i on the cycle γ j isequal to 2π if i = j and 0 if i ≠ j). We setI i (f i ) = 12π∮M hp i dq i , (i = 1, ..., n). (3.182)The n quantities I i (f i ) given by formula (3.182) are called the actionvariables [Arnold (1989)].We assume now that, for the given values f i of the n integrals F i , then quantities I i are independent, det(∂I i /∂f i )| fi ≠ 0. Then in a neighborhoodof the torus M f = T n we can take the variables I i , ϕ i as symplecticcoordinates, i.e., the transformation (p i , q i ) → (I i , ϕ i ) is canonical, i.e.,dp i ∧ dq i = dI i ∧ dϕ i ,(i = 1, ..., n).Now, let m be a point on M f , in a neighborhood of which the n variablesq i are coordinates of M f , such that the submanifold M f ⊂ R 2n is given byn equations of the form p i = p i (I i , q i ), q i (m) = q0. i In a simply–connectedneighborhood of the point q0 i a single–valued function is defined,S(I i , q i ) =∫ qq 0p i (I i , q i ) dq i ,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 367and we can use it as the generating function of a canonical transformation(p i , q i ) → (I i , ϕ i ):p i = ∂S∂q i ,ϕi = ∂S∂I i.A Universal Model for Completely Integrable SystemsA Hamiltonian system on a 2nD symplectic manifold M is said to becompletely integrable if it has n first integrals in involution, which are functionallyindependent on some open dense submanifold of M. This definitionof a completely integrable system is usually found, with some minor variants,in any modern text on symplectic mechanics [Arnold (1989); Abrahamand Marsden (1978); Libermann and Marle (1987); Marmo et. al. (1995);Thirring (1979)].Starting with this definition, one uses the so–called Liouville–ArnoldTheorem to introduce action–angle variables and write the Hamiltoniansystem in the form˙ I k = 0,˙φk = ∂H∂I k= ν k (I),where k ∈ {1, . . . , n}. The corresponding flow is given byI k (t) = I k (0), φ k (t) = φ k (0) + ν k t. (3.183)The main interest in completely integrable systems relies on the fact thatthey can be integrated by quadratures [Arnold (1989)].However, it is clear that even if ν k dI k is not an exact (or even aclosed) 1–form, as long as ˙ν k = 0, the system can always be integrated byquadratures.If we consider the Abelian Lie group R n , we can construct a Hamiltonianaction of R n on T ∗ R n induced by the group addition: R n × T ∗ R n →T ∗ R n . This can be generalized to the Hamiltonian action [Alekseevskyet. al. (1997)]R n × T ∗ (R k × T n−k ) → T ∗ (R k × T n−k ),of R n , where T m stands for the mD torus, and reduces to R n × T ∗ T n orT n × T ∗ T n , when k = 0.By using the standard symplectic structure on T ∗ R n , we find the momentummap µ : T ∗ R n −→ (R n ) s , (q, p) ↦→ p, induced by the naturalaction of R n on itself via translations, which is a Poisson map if (R n ) s is


368 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwith the (trivial) natural Poisson structure of the dual of a Lie algebra.It is now clear that any function on (R n ) s , when pulled back to T ∗ R n orT ∗ T n , induces a Hamiltonian system which is completely integrable (inthe Liouville sense). Because the level sets of this function carry on theaction of R n , the completely integrable system induces a 1D subgroup ofthe action of R n on the given level set. However, the specific subgroup willdepend on the particular level set, i.e., the ‘frequencies’ are first integrals.The property of being integrable by quadratures is captured by the factthat it is a subgroup of the R n −action on each level set.It is now clear, how we can preserve this property, while giving up therequirement that our system is Hamiltonian. We can indeed consider any1–form η on (R n ) s and pull it back to T ∗ R n or T ∗ T n , then associatedvector–field Γ η = Λ 0 (µ s (η)), where Λ 0 is the canonical Poisson structurein the cotangent bundle, is no more Hamiltonian, but it is still integrableby quadratures. In action–angle variables, if η = ν k dI k is the 1–form on(R n ) s , the associated equations of motion on T ∗ T n will be [Alekseevskyet. al. (1997)]˙ I k = 0, ˙φk = ν k ,with ˙ν k = 0, therefore the flow will be as in (3.183), even though ∂ I j ν k ≠∂ I kν j .We can now generalize this construction to any Lie group G. We considerthe Hamiltonian action G × T ∗ G → T ∗ G, of G on the cotangentbundle, induced by the right action of G on itself. The associated momentummap µ : T ∗ G ≃ G s × G −→ G s . It is a Poisson map with respect tothe natural Poisson structure on G s (see, e.g., [Alekseevsky et. al. (1994);Libermann and Marle (1987)]).Now, we consider any differential 1–form η on G s which is annihilatedby the natural Poisson structure Λ G ∗ on G s associated with the Liebracket. Such form we call a Casimir form. We define the vector–fieldΓ η = Λ 0 (µ s (η)). Then, the corresponding dynamical system can be writtenas [Alekseevsky et. al. (1997)]g −1 ġ = η(g, p) = η(p), ṗ = 0,since ω 0 = d(< p, g −1 dg >) (see [Alekseevsky et. al. (1994)]). Here weinterpret the covector η(p) on G s as a vector of G. Again, our system canbe integrated by quadratures, because on each level set, get by fixing p’sin G s , our dynamical system coincides with a one–parameter group of the


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 369action of G on that particular level set.We give a familiar example: the rigid rotator and its generalizations[Alekseevsky et. al. (1997)]. In the case of G = SO(3) the (right) momentummapµ : T ∗ SO(3) −→ so(3) ∗is a Poisson map onto so(3) ∗ with the linear Poisson structureΛ so(3) ∗ = ε ijk p i ∂ pj ⊗ ∂ pk .Casimir 1–forms for Λ so(3) ∗ read η = F dH 0 , where H 0 = ∑ p 2 i /2 is the‘free Hamiltonian’ and F = F (p) is an arbitrary function. Clearly, F dH 0 isnot a closed form in general, but (p i ) are first integrals for the dynamicalsystem Γ η = Λ 0 (µ s (η)). It is easy to see thatΓ η = F (p)Γ 0 = F (p)p i ̂Xi ,where ̂X i are left–invariant vector–fields on SO(3), corresponding to thebasis (X i ) of so(3) identified with (dp i ). Here we used the identificationT ∗ SO(3) ≃ SO(3)×so(3) ∗ given by the momentum map µ. In other words,the dynamics is given byṗ i = 0,g −1 ġ = F (p)p i X i ∈ so(3),and it is completely integrable, since it reduces to left–invariant dynamicson SO(3) for every value of p. We recognize the usual isotropic rigid rotator,when F (p) = 1.We can generalize our construction once more, replacing the cotangentbundle T ∗ G by its deformation, namely a group double D(G, Λ G ) associatedwith a Lie–Poisson structure Λ G on G (see e.g., [Lu (1990)]). This double,denoted simply by D, carry on a natural Poisson tensor–field Λ + D whichis non–degenerate on the open–dense subset D + = G · G s ∩ G s · G of D(here G s ⊂ D is the dual group of G with respect to Λ G ). We refer to Das being complete if D + = D. Identifying D with G × G s if D is complete(or D + with an open submanifold of G × G s in general case; we assumecompleteness for simplicity) via the group product, we can write Λ + Din‘coordinates’ (g, u) ∈ G × G s in the form [Alekseevsky et. al. (1997)]Λ + D (g, u) = Λ G(g) + Λ G ∗(u) − X l i(g) ∧ Y ri (u), (3.184)where Xil and Y ir are, respectively, the left– and right–invariant vector–fields on G and G s relative to dual bases X i and Y i in the Lie algebras G


370 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand G s , and where Λ G and Λ G ∗are the corresponding Lie–Poisson tensorson G and G s (see [Lu (1990)]). It is clear now that the projections µ G ∗and µ G of (D, Λ + D ) onto (G, Λ G) and (G s , Λ G ∗), respectively, are Poissonmaps. Note that we get the cotangent bundle (D, Λ + D ) = (T ∗ G, Λ 0 ) if weput Λ G = 0.The group G acts on (D, Λ + D) by left translations which, in generalare not canonical transformations. However, this is a Poisson action withrespect to the inner Poisson structure Λ G on G, which is sufficient to developthe momentum map reduction theory (see [Lu (1991)]). For our purposes,let us take a Casimir 1–form η for Λ G ∗, i.e., Λ G ∗(η) = 0. By means of themomentum mapµ G ∗ : D −→ G s , we define the vector–field on D [Alekseevsky et. al. (1997)]:Γ η = Λ + D (µs G ∗(η)).In ‘coordinates’ (g, u), due to the fact that η is a Casimir, we getΓ η (g, u) =< Y ri , η > (u)X l i(g),so that Γ η is associated with the Legendre mapL η : D ≃ G × G s −→ T G ≃ G × G,L η (g, u) =< Y ri , η > (u)X i ,which can be viewed also as a map L η : G s −→ G. Thus we get the followingTheorem [Alekseevsky et. al. (1997)]: The dynamics Γ η on the group doubleD(G, Λ G ), associated with a 1–form η which is a Casimir for the Lie–Poissonstructure Λ G ∗ on the dual group, is given by the system of equations˙u = 0,g −1 ġ =< Y ri , η > (u)X i ∈ G,and is therefore completely integrable by quadratures.We have seen that if we concentrate on the possibility of integrating oursystem by quadratures, then we can do without the requirement that thesystem is Hamiltonian.By considering again the equations of motion in action–angle variables,we classically have˙ I k = 0,˙φk = ν k (I).Clearly, if we have˙ I k = F k (I), ˙φk = A j k (I)φ j, (3.185)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 371and we are able to integrate the first equation by quadratures, we againhave the possibility to integrate by quadratures the system (3.185), if onlythe matrices (A j k(I(t))) commute [Alekseevsky et. al. (1997)]:(∫ t)φ(t) = exp A(I(s))ds φ 0 .0Because φ k are discontinues functions on the torus, we have to be morecareful here. However, we show how this idea works for double groups. Inthe case when the 1–form η on G s is not a Casimir 1–form for the Lie–Poisson structure Λ G ∗, we get, in view of (3.184),Γ η (g, u) =< Y ri , η > (u)X l i(g) + Λ G ∗(η)(u).Now, the momenta evolve according to the dynamics Λ G ∗(η) on G s (whichcan be interpreted, as we will see later, as being associated with an interactionof the system with an external field) and ‘control’ the evolutionof the field of velocities on G (being left–invariant for a fixed time) by a‘variation of constants’. Let us summarize our observations in the followingTheorem [Alekseevsky et. al. (1997)]: The vector–field Γ η on the doublegroup D(G, Λ G ), associated with a 1–form η on G s , defines the followingdynamics˙u = Λ G ∗(η)(u),g −1 ġ =< Y ri , α > (u)X i ∈ G, (3.186)and is therefore completely integrable, if only we are able to integrate theequation (3.186) and < Yi r,η > (u(t))X i lie in a commutative subalgebraof G for all t.Finally, we can weaken the assumptions of the previous Theorem. It issufficient to assume [Alekseevsky et. al. (1997)] thatg −1 ġ(t) = exp(tX)A(t) exp(−tX),for some A(t), X ∈ G, such that X + A(t) lie in a commutative subalgebraof G for all t (e.g., A(t) = const), to assure that (3.186) is integrable byquadratures. Indeed, in the new variablethe equation (3.186) readsg 1 (t) = exp(−tX)g(t) exp(tX),ġ 1 (t) = g 1 (t)(X + A(t)) − Xg 1 (t),


372 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand, since the right– and the left–multiplications commute, we find that[Alekseevsky et. al. (1997)](g(t) = g 0 exp tX +∫ t0)A(s)ds exp(−tX).This procedure is similar to what is known as the Dirac interaction picturein the quantum evolution.3.12.3.5 Momentum Map and Symplectic ReductionLet (M, ω) be a connected symplectic manifold and φ : G × M → M asymplectic action of the Lie group G on M, that is, for each g ∈ G the mapφ g : M → M is a symplectic diffeomorphism. If for each ξ ∈ g there existsa globally defined function Ĵ(ξ) : M → R such that ξ M = XĴ(ξ) , then themap J : M → g ∗ , given byJ : x ∈ M ↦→ J(x) ∈ g ∗ ,J(x)(ξ) = Ĵ(ξ)(x)is called the momentum map for φ [Marsden and Ratiu (1999); Puta (1993)].Since φ is symplectic, φ exp(tξ) is a one–parameter family of canonicaltransformations, i.e., φ ∗ exp(tξ)ω = ω, hence ξ M is locally Hamiltonian andnot generally Hamiltonian. That is why not every symplectic action has amomentum map. φ : G × M → M is Hamiltonian iff Ĵ : g → Ck (M, R) isa Lie algebra homomorphism.Let H : M → R be G − −invariant, that is H ( φ g (x) ) = H(x) forall x ∈ M and g ∈ G. Then Ĵ(ξ) is a constant of motion for dynamicsgenerated by H.Let φ be a symplectic action of G on (M, ω) with the momentum mapJ. Suppose H : M → R is G − −invariant under this action. Then theNoether’s Theorem states that J is a constant of motion of H, i.e., J ◦ φ t =J, where φ t is the flow of X H .A Hamiltonian action is a symplectic action with an Ad ∗ –equivariantmomentum map J, i.e.,J ( φ g (x) ) = Ad ∗ g−1 (J(x)) ,for all x ∈ M and g ∈ G.Let φ be a symplectic action of a Lie group G on (M, ω). Assume thatthe symplectic form ω on M is exact, i.e., ω = dθ, and that the action φof G on M leaves the one form θ ∈ M invariant. Then J : M → g ∗ given


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 373by (J(x)) (ξ) = ( i ξM θ ) (x) is an Ad ∗ −-equivariant momentum map of theaction.In particular, in the case of the cotangent bundle (M = T ∗ M, ω = dθ)of a mechanical configuration manifold M, we can lift up an action φ of aLie group G on M to get an action of G on T ∗ M. To perform this lift, letG act on M by transformations φ g : M → M and define the lifted actionto the cotangent bundle by (φ g ∗ : T ∗ M → T ∗ M by pushing forward oneforms, (φ g ) ∗ (α) · v = α ( T φ −1g v ) ,where α ∈ Tq ∗ M and v ∈ T φg (q)M. Thelifted action (φ g ) ∗ preserves the canonical one form θ on T ∗ M and themomentum map for (φ g ) ∗ is given byJ : T ∗ M → g ∗ , J (α q ) (ξ) = α q (ξ M (q)) .For example, let M = R n , G = R n and let G act on R n by translations:φ : (t, q) ∈ R n × R n ↦→ t + q ∈ R n .Then g = R n and for each ξ ∈ g we have ξ R n(q) = ξ.In case of the group of rotations in R 3 , M = R 3 , G = SO(3) and letG act on R 3 by φ(A, q) = A · q. Then g ≃ R 3 and for each ξ ∈ g we haveξ R 3(q) = ξ × q.Let G act transitively on (M, ω) by a Hamiltonian action. Then J(M) ={Ad ∗ g(J(x)) |g ∈ G} is a coadjoint orbit.−1Now, let (M, ω) be a symplectic manifold, G a Lie group and φ : G ×M → M a Hamiltonian action of G on M with Ad ∗ –equivariant momentummap J : M → g ∗ . Let µ ∈ g ∗ be a regular value of J; then J −1 (µ) isa submanifold of M such that dim ( J −1 (µ) ) = dim (M) − dim (G). LetG µ = {g ∈ G|Ad ∗ gµ = µ} be the isotropy subgroup of µ for the coadjointaction. By Ad ∗ −-equivariance, if x ∈ J −1 (µ) then φ g (x) = J −1 (µ) for allg ∈ G, i.e., J −1 (µ) is invariant under the induced G µ −-action and we canform the quotient space M µ = J −1 (µ)/G µ , called the reduced phase–spaceat µ ∈ g ∗ .Let (M, ω) be a symplectic 2nD manifold and let f 1 , ..., f k be k functionsin involution, i.e., {f i , f j } ω = 0, i = 1, ..., k. Because the flow of X fi andX fj commute, we can use them to define a symplectic action of G = R k onM. Here µ ∈ R k is in the range space of f 1 ×...×f k and J = f 1 ×...×f k is themomentum map of this action. Assume that {df 1 , ..., df k } are independentat each point, so µ is a regular value for J. Since G is Abelian, G µ = G sowe get a symplectic manifold J −1 (µ)/G of dimension 2n − 2k. If k = n wehave integrable systems.


374 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(For example, let G = SO(3) and (M, ω) = R 6 , ∑ 3i=1 dp i ∧ dq), i andthe action of G on R 6 is given by φ : (R, (q, p)) ↦→ (R q , R p ). Then themomentum map is the well known angular momentum and for each µ ∈g ∗ ≃ R 3 µ ≠ 0, G µ ≃ S 1 and the reduced phase–space (M µ , ω µ ) is (T ∗ R,ω = dp i ∧ dq i ), so that dim (M µ ) = dim (M) − dim (G) − dim (G µ ). Thisreduction is in celestial mechanics called by Jacobi ’the elimination of thenodes’.The equations of motion: f ˙ = {f, H} ω on M reduce to the equations ofmotion: f ˙ µ = {f µ , H µ } ωµ on M µ (see [Marsden and Ratiu (1999)]).3.12.4 Multisymplectic <strong>Geom</strong>etryMultisymplectic geometry constitutes the general framework for a geometrical,covariant formulation of classical field theory. Here, covariant formulationmeans that space–like and time–like directions on a given space–time be treated on equal footing. With this principle, one can constructa covariant form of the Legendre transformation which associates to everyfield variable as many conjugated momenta, the multimomenta, asthere are space–time dimensions. These, together with the field variables,those of nD space–time, and an extra variable, the energy variable,span the multiphase–space [Kijowski and Szczyrba (1976)]. For arecent exposition on the differential geometry of this construction, see [Gotay(1991a)]. Multiphase–space, together with a closed, nondegeneratedifferential (n + 1)−form, the multisymplectic form, is an example of amultisymplectic manifold. Among the achievements of the multisymplecticapproach is a geometrical formulation of the relation of infinitesimalsymmetries and covariantly conserved quantities (Noether’s Theorem), see[León et. al. (2004)] for a recent review, and [Gotay and Marsden (1992);Forger and Römer (2004)] for a clarification of the improvement techniques(‘Belinfante–Rosenfeld formula’) of the energy–momentum tensor in classicalfield theory. Multisymplectic geometry also gives convenient sets ofvariational integrators for the numerical study of partial differential equations[Marsden et. al. (1998)].Since in multisymplectic geometry, the symplectic two–form of classicalmechanics is replaced by a closed differential form of higher tensordegree, multivector–fields and differential forms have their natural appearance.(See [Paufler and Römer (2002)] for an interpretation of multivector–fields as describing solutions to field equations infinitesimally.) Multivector–


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 375fields form a graded Lie algebra with the Schouten bracket (see [Kosmann(2004)] for a review and unified viewpoint). Using the multisymplectic(n + 1)−form, one can construct a new bracket for the differential forms,the Poisson forms [Forger et. al. (2004)], generalizing a well–known formulafor the Poisson brackets related to a symplectic two–form. A remarkablefact is that in order to establish a Jacobi identity, the multisymplectic formhas to have a potential, a condition that is not seen in symplectic geometry.Further, the admissible differential forms, the Poisson forms, are subject tothe rather strong restrictions on their dependence on the multimomentumvariables [Gotay (1991b)]. In particular, (n − 1)−forms in this contextcan be shown to arise exactly from the covariantly conserved currents ofNoether symmetries, which allows their pairing with space–like hypersurfacesto yield conserved charges in a geometrical way.The Hamiltonian, infinite dimensional formulation of classical field theoryrequires the choice of a space–like hypersurface (‘Cauchy surface’),which manifestly breaks the general covariance of the theory at hand.For (n − 1)−forms, the above new bracket reduces to the Peierls–deWittbracket after integration over the space–like hypersurface [Gotay and Nester(1980)]. With the choice of a hypersurface, a constraint analysis a‘la Dirac[Henneaux and Teitelboim (1992); Gotay et. al. (2004)] can be performed[Landsman (1995)]. Again, the necessary breaking of general covarianceraises the need for an alternative formulation of all this [Marsden and Weinstein(1974)]; first attempts have been made to carry out a Marsden–Weinstein reduction [Munteanu et. al. (2004)] for multisymplectic manifoldswith symmetries. However, not very much is known about how toquantize a multisymplectic geometry, see [Bashkirov and Sardanashvily(2004)] for an approach using a path integral.So far, everything was valid for field theories of first–order, i.e., wherethe Lagrangian depends on the fields and their first derivatives. Higherorder theories can be reduced to first–order ones for the price of introducingauxiliary fields. A direct treatment would involve higher order jet bundles[Saunders (1989)]. A definition of the covariant Legendre transform andthe multiphase–space has been given for this case [Gotay (1991a)].3.13 Application: Biodynamics–RoboticsRecall from [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)] that modern unified geometricalbasis for both human biodynamics and humanoid robotics repre-


376 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsents the constrained SE(3)−group, i.e., the so–called special Euclideangroup of rigid–body motions in 3D space (see, e.g., [Murray et al. (1994);Park and Chung (2005)]). In other words, during human movement, in eachmovable human joint there is an action of a constrained SE(3)−group.Therefore, constrained SE(3)−group represents general kinematics ofhuman–like joints. The corresponding nonlinear dynamics problem (resolvedmainly for aircraft and spacecraft dynamics) is called the dynamicson SE(3)−group, while the associated nonlinear control problem (resolvedmainly for general helicopter control) is called the control on SE(3)−group.Recall that the Euclidean SE(3)−group is defined as a semidirect (noncommutative)product of 3D rotations and 3D translations, SE(3) :=SO(3) ✄ R 3 [Murray et al. (1994); Park and Chung (2005); <strong>Ivancevic</strong> and<strong>Ivancevic</strong> (2006)]). Its most important subgroups are the following:SubgroupSO(3), group of rotations in 3D(a spherical joint)SE(2), special Euclidean group in 2D(all planar motions)SO(2), group of rotations in 2Dsubgroup of SE(2) − group(a revolute joint)R 3 , group of translations in 3D(all spatial displacements)DefinitionSet of all proper orthogonal3 × 3 − rotational matricesSet of all 3 × 3 − matrices:⎡⎤cos θ sin θ r x⎣ − sin θ cos θ r y⎦0 0 1Set of all proper orthogonal2 × 2 − rotational matricesincluded in SE(2) − groupEuclidean 3D vector spaceIn the next subsection we give detailed analysis of these subgroups, aswell as the total SE(3)−group.3.13.1 Muscle–Driven Hamiltonian BiodynamicsWe will develop our Hamiltonian geometry on the configuration biodynamicalmanifold M in three steps, following the standard symplectic geometryprescription (see subsection 3.12 above):Step A Find a symplectic momentum phase–space (P, ω).Recall that a symplectic structure on a smooth manifold M is a nondegenerateclosed 2−form ω on M, i.e., for each x ∈ M, ω(x) is nondegenerate,and dω = 0.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 377Let Tx ∗ M be a cotangent space to M at m. The cotangent bundle T ∗ Mrepresents a union ∪ m∈M Tx ∗ M, together with the standard topology onT ∗ M and a natural smooth manifold structure, the dimension of which istwice the dimension of M. A 1−form θ on M represents a section θ : M →T ∗ M of the cotangent bundle T ∗ M.P = T ∗ M is our momentum phase–space. On P there is a nondegeneratesymplectic 2−form ω is defined in local joint coordinates q i , p i ∈ U,U open in P , as ω = dq i ∧ dp i (’∧’ denotes the wedge or exterior product).In that case the coordinates q i , p i ∈ U are called canonical. In a usualprocedure the canonical 1−form θ is first defined as θ = p i dq i , and thenthe canonical 2–form ω is defined as ω = −dθ.A symplectic phase–space manifold is a pair (P, ω).Step B Find a Hamiltonian vector–field X H on (P, ω).Let (P, ω) be a symplectic manifold. A vector–field X : P → T P iscalled Hamiltonian if there is a smooth function F : P −→ R such thati X ω = dF (i X ω denotes the interior product or contraction of the vector–field X and the 2–form ω). X is locally Hamiltonian if i X ω is closed.Let the smooth real–valued Hamiltonian function H : P → R, representingthe total biodynamical energy H(q, p) = T (p) + V (q) (T and Vdenote kinetic and potential energy of the system, respectively), be givenin local canonical coordinates q i , p i ∈ U, U open in P . The Hamiltonianvector–field X H , condition by i XH ω = dH, is actually defined via symplecticmatrix J, in a local chart U, asX H = J∇H = ( ∂ pi H, −∂ q iH ) , J =( )0 I,−I 0where I denotes the n × n identity matrix and ∇ is the gradient operator.Step C Find a Hamiltonian phase–flow φ t of X H .Let (P, ω) be a symplectic phase–space manifold and X H = J∇H aHamiltonian vector–field corresponding to a smooth real–valued Hamiltonianfunction H : P → R, on it. If a unique one–parameter group ofdiffeomorphisms φ t : P → P exists so that d dt | t=0 φ t x = J∇H(x), it iscalled the Hamiltonian phase–flow.A smooth curve t ↦→ ( q i (t), p i (t) ) on (P, ω) represents an integral curveof the Hamiltonian vector–field X H = J∇H, if in the local canonical coordinatesq i , p i ∈ U, U open in P , Hamiltonian canonical equations (3.164)hold.An integral curve is said to be maximal if it is not a restriction of anintegral curve defined on a larger interval of R. It follows from the standard


378 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionTheorem on the existence and uniqueness of the solution of a system ofODEs with smooth r.h.s, that if the manifold (P, ω) is Hausdorff, then forany point x = (q i , p i ) ∈ U, U open in P , there exists a maximal integralcurve of X H = J∇H, passing for t = 0, through point x. In case X H iscomplete, i.e., X H is C p and (P, ω) is compact, the maximal integral curveof X H is the Hamiltonian phase–flow φ t : U → U.The phase–flow φ t is symplectic if ω is constant along φ t , i.e., φ ∗ t ω = ω(φ ∗ t ω denotes the pull–back of ω by φ t ),iff L XH ω = 0(L XH ω denotes the Lie derivative of ω upon X H ).Symplectic phase–flow φ t consists of canonical transformations on(P, ω), i.e., diffeomorphisms in canonical coordinates q i , p i ∈ U, U openon all (P, ω) which leave ω invariant. In this case the Liouville Theoremis valid: φ t preserves the phase volume on (P, ω). Also, the system’s totalenergy H is conserved along φ t , i.e., H ◦ φ t = φ t .Recall that the Riemannian metrics g = on the configuration manifoldM is a positive–definite quadratic form g : T M → R, in local coordinatesq i ∈ U, U open in M, given by (3.139–3.140) above. Given themetrics g ij , the system’s Hamiltonian function represents a momentump−-dependent quadratic form H : T ∗ M → R – the system’s kinetic energyH(p) = T (p) = 1 2 < p, p >, in local canonical coordinates qi , p i ∈ U p ,U p open in T ∗ M, given byH(p) = 1 2 gij (q, m) p i p j , (3.187)where g ij (q, m) = g −1ij (q, m) denotes the inverse (contravariant) materialmetric tensorg ij (q, m) =n∑χ=1m χ δ rs∂q i∂x r ∂q j∂x s .T ∗ M is an orientable manifold, admitting the standard volume formΩ ωH=N(N+1)(−1) 2N!ω N H.For Hamiltonian vector–field, X H on M, there is a base integral curveγ 0 (t) = ( q i (t), p i (t) ) iff γ 0 (t) is a geodesic, given by the one–form forceequation˙¯p i ≡ ṗ i + Γ i jk g jl g km p l p m = 0, with ˙q k = g ki p i , (3.188)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 379where Γ i jkdenote Christoffel symbols of an affine Levi–Civita connectionon M, defined upon the Riemannian metric g = by (3.143).The l.h.s ˙¯p i of the covariant momentum equation (3.188) represents theintrinsic or Bianchi covariant derivative of the momentum with respectto time t. Basic relation ˙¯p i = 0 defines the parallel transport on T N ,the simplest form of human–motion dynamics. In that case Hamiltonianvector–field X H is called the geodesic spray and its phase–flow is called thegeodesic flow.For Earthly dynamics in the gravitational potential field V : M → R,the Hamiltonian H : T ∗ M → R (3.187) extends into potential formH(p, q) = 1 2 gij p i p j + V (q),with Hamiltonian vector–field X H = J∇H still defined by canonical equations(3.164).A general form of a driven, non–conservative Hamiltonian equationsreads:˙q i = ∂ pi H, ṗ i = F i − ∂ q iH, (3.189)where F i = F i (t, q, p) represent any kind of joint–driving covariant torques,including active neuro–muscular–like controls, as functions of time, anglesand momenta, as well as passive dissipative and elastic joint torques. Inthe covariant momentum formulation (3.188), the non–conservative Hamiltonianequations (3.189) become˙¯p i ≡ ṗ i + Γ i jk g jl g km p l p m = F i , with ˙q k = g ki p i .3.13.2 Hamiltonian–Poisson Biodynamical SystemsRecall from subsection 3.12.3.3 above that Hamiltonian–Poisson mechanicsis a generalized form of classical Hamiltonian mechanics. Let (P, {}) be aPoisson manifold and H ∈ C ∞ (P, R) a smooth real valued function on P .The vector–field X H defined byX H (F ) = {F, H},is the Hamiltonian vector–field with energy function H. The triple(P, {}, H) we call the Hamiltonian–Poisson biodynamical system (HPBS)[Marsden and Ratiu (1999); Puta (1993); <strong>Ivancevic</strong> and Pearce (2001a)].


380 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe map F ↦→ {F, H} is a derivation on the space C ∞ (P, R), hence it definesa vector–field on P . The map F ∈ C ∞ (P, R) ↦→ X F ∈ X (P ) is a Liealgebra anti–homomorphism, i.e., [X F , X g ] = −X {F,g} .Let (P, {}, H) be a HPBS and φ t the flow of X H . Then for all F ∈C ∞ (P, R) we have the conservation of energy:H ◦ φ t = H,and the equations of motion in Poisson bracket form,ddt (F ◦ φ t) = {F, H} ◦ φ t = {F ◦ φ t , H},that is, the above Poisson evolution equation (3.179) holds. Now, the functionF is constant along the integral curves of the Hamiltonian vector–fieldX H iff{F, H} = 0.φ t preserves the Poisson structure.Next we present two main examples of HPBS.‘Ball–and–Socket’ Joint Dynamics in Euler Vector FormThe dynamics of human body–segments, classically modelled via Lagrangianformalism (see [Hatze (1977b); <strong>Ivancevic</strong> (1991); <strong>Ivancevic</strong> et al.(1995); <strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]), may be also prescribed by Euler’sequations of rigid body dynamics. The equations of motion for a free rigidbody, described by an observer fixed on the moving body, are usually givenby Euler’s vector equationṗ = p × w. (3.190)Here p, w ∈ R 3 , p i = I i w i and I i (i = 1, 2, 3) are the principal momentsof inertia, the coordinate system in the segment is chosen so that the axesare principal axes, w is the angular velocity of the body and p is the correspondingangular momentum.The kinetic energy of the segment is the Hamiltonian function H : R 3 →R given by [<strong>Ivancevic</strong> and Pearce (2001a)]H(p) = 1 2 p · wand is a conserved quantity for (3.190).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 381The vector space R 3 is a Lie algebra with respect to the bracket operationgiven by the usual cross product. The space R 3 is paired with itselfvia the usual dot product. So if F : R 3 → R, then δF /δp = ∇F (p) andthe (–) Lie–Poisson bracket {F, G} − (p) is given via (3.178) by the tripleproduct{F, G} − (p) = −p · (∇F (p) × ∇G(p)).Euler’s vector equation (3.190) represents a generalized Hamiltoniansystem in R 3 relative to the Hamiltonian function H(p) and the (–) Lie–Poisson bracket {F, G} − (p). Thus the Poisson manifold (R 3 , {F, G} − (p)) isdefined and the abstract Poisson equation is equivalent to Euler’s equation(3.190) for a body segment and associated joint.Solitary Model of Muscular ContractionRecall that the so–called sliding filament theory of muscular contractionwas developed in 1950s by Nobel Laureate A. Huxley [Huxley andNiedergerke (1954); Huxley (1957)]. At a deeper level, the basis of themolecular model of muscular contraction is represented by oscillationsof Amid I peptide groups with associated dipole electric momentum insidea spiral structure of myosin filament molecules (see [Davydov (1981);Davydov (1991)]).There is a simultaneous resonant interaction and strain interaction generatinga collective interaction directed along the axis of the spiral. Theresonance excitation jumping from one peptide group to another can berepresented as an exciton, the local molecule strain caused by the staticeffect of excitation as a phonon and the resultant collective interaction asa soliton.The simplest model of Davydov’s solitary particle–waves is given by thenonlinear Schrödinger equation [<strong>Ivancevic</strong> and Pearce (2001a)]i∂ t ψ = −∂ x 2ψ + 2χ|ψ| 2 ψ, (3.191)for -∞ < x < +∞. Here ψ(x, t) is a smooth complex–valued wave functionwith initial condition ψ(x, t)| t=0 = ψ(x) and χ is a nonlinear parameter. Inthe linear limit (χ = 0) (3.191) becomes the ordinary Schrödinger equationfor the wave function of the free 1D particle with mass m = 1/2.We may define the infinite–dimensional phase–space manifold P ={(ψ, ¯ψ) ∈ S(R, C)}, where S(R, C) is the Schwartz space of rapidly–decreasing complex–valued functions defined on R). We define also the


382 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionalgebra χ(P) of observables on P consisting of real–analytic functionalderivatives δF /δψ, δF /δ ¯ψ ∈ S(R, C).The Hamiltonian function H : P −→ R is given by∫ (+∞∣∣∣∣ )2∂ψH(ψ) =∂x ∣ + χ|ψ| 4 dx−∞and is equal to the total energy of the soliton. It is a conserved quantityfor (4.3) (see [Seiler (1995)]).The Poisson bracket on χ(P) represents a direct generalization of theclassical nD Poisson bracket{F, G} + (ψ) = i∫ +∞−∞( δFδψδG δF−δ ¯ψ δ ¯ψ)δGdx. (3.192)δψIt manifestly exhibits skew–symmetry and satisfies Jacobi identity. Thefunctionals are given by δF /δψ = −i{F, ¯ψ} and δF /δ ¯ψ = i{F, ψ}. Thereforethe algebra of observables χ(P) represents the Lie algebra and thePoisson bracket is the (+) Lie–Poisson bracket {F, G} + (ψ).The nonlinear Schrödinger equation (3.191) for the solitary particle–wave is a Hamiltonian system on the Lie algebra χ(P) relative to the (+)Lie–Poisson bracket {F, G} + (ψ) and Hamiltonian function H(ψ). Thereforethe Poisson manifold (χ(P), {F, G} + (ψ)) is defined and the abstractPoisson evolution equation (3.179), which holds for any smooth functionF : χ(P) →R, is equivalent to equation (3.191).A more subtle model of soliton dynamics is provided by the Korteveg–deVries equation [<strong>Ivancevic</strong> and Pearce (2001a)]f t − 6ff x + f xxx = 0, (f x = ∂ x f), (3.193)where x ∈ R and f is a real–valued smooth function defined on R (comparewith (3.81) above). This equation is related to the ordinary Schrödingerequation by the inverse scattering method [Seiler (1995); <strong>Ivancevic</strong> andPearce (2001a)].We may define the infinite–dimensional phase–space manifold V = {f ∈S(R)}, where S(R) is the Schwartz space of rapidly–decreasing real–valuedfunctions R). We define further χ(V) to be the algebra of observablesconsisting of functional derivatives δF /δf ∈ S(R).The Hamiltonian H : V → R is given byH(f) =∫ +∞−∞(f 3 + 1 2 f 2 x) dx


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 383and gives the total energy of the soliton. It is a conserved quantity for(3.193) (see [Seiler (1995)]).As a real–valued analogue to (3.192), the (+) Lie–Poisson bracket onχ(V) is given via (3.177) by{F, G} + (f) =∫ +∞−∞δFδfd δGdx δf dx.Again it possesses skew–symmetry and satisfies Jacobi identity. The functionalsare given by δF /δf = {F, f}.The Korteveg–de Vries equation (KdV1), describing the behavior of themolecular solitary particle–wave, is a Hamiltonian system on the Lie algebraχ(V) relative to the (+) Lie–Poisson bracket {F, G} + (f) and the Hamiltonianfunction H(f). Therefore, the Poisson manifold (χ(V), {F, G} + (f))is defined and the abstract Poisson evolution equation (3.179), which holdsfor any smooth function F : χ(V) →R, is equivalent to (3.193).3.13.3 Lie–Poisson Neurodynamics ClassifierA Lie–Poisson neuro–classifier is a tensor–field–system {µ} = (q, p, ω) ona Poisson manifold (g ∗ , {F, H(µ)} ± (µ)). Like a GBAM neuro–classifier, itconsists of continual activation (q, p)−dynamics and self–organized learningω–dynamics. In this case, both dynamics are defined by “neural activationform” of the abstract Lie–Poisson evolution equation˙ F = {S(F ), H(µ)}, (3.194)where S(·) = tanh(·) denotes the sigmoid activation function. A Hamiltonianfunction H(µ), representing the total network energy, is given in theformH(µ) = 1 2 ω ijδ ij + 1 2 ωij δ ij ,(i, j = 1, ..., n),where δ ij and δ ij are Kronecker tensors, while ω ij = ω ij (q i ) and ω ij =ω ij (p i ) correspond to the contravariant and covariant components of thefunctional–coupling synaptic tensor ω = ω(q, p), defined respectively byω ij = ε q i q j , ω ij = τ p i p j ,with random coefficients ε and τ.Activation (q, p)−-dynamics are given by˙q i = I i + {S(q i ), H(µ)},ṗ i = J i + {S(p i ), H(µ)},


384 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere I i and J i represent the two input features.Two types of self–organized learning ω−-dynamics are presented andcompared:Lie–Poisson learning dynamics, in which synaptic update law is given byinhibitory–covariant and excitatory–contravariant form of equation (3.194):˙ω ij = {S(ω ij ), H(µ)},˙ω ij = {S(ω ij ), H(µ)},respectively.<strong>Diff</strong>erential Hebbian learning (see [Kosko (1992)] for details), in bothinhibitory–covariant and excitatory–contravariant learning form:˙ω ij = −ω ij + Φ ij (q i , p i ), ˙ω ij = −ω ij + Φ ij (q i , p i ),with innovations defined in both variance–forms as:Φ ij = S i (q i )S j (p j ) + Ṡi(q i )Ṡj(p j ), Φ ij = S i (q i )S j (p j ) + Ṡi (q i )Ṡj (p j ).3.13.4 Biodynamical Functors3.13.4.1 The Covariant Force FunctorRecall (see subsection 2.1.4.3 above) that int the realm of biodynamicsthe central concept is the covariant force law, F i = mg ij a j [<strong>Ivancevic</strong> and<strong>Ivancevic</strong> (2006)]. In categorical language, it represents the covariant forcefunctor F ∗ defined by commutative diagram:T T ∗ M✻F i = ṗ iF ∗✲ T T M✻a i = ˙¯v iT ∗ M = {x i , p i } T M = {x i , v i }❅■p❅i❅ v i = ẋ i❅ ✒M = {x i }which states that the force 1–form F i = ṗ i , defined on the mixed tangent–cotangent bundle T T ∗ M, causes the acceleration vector–field a i = ˙¯v i , definedon the second tangent bundle T T M of the configuration manifoldM.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 385The corresponding contravariant acceleration functor is defined as itsinverse map F ∗ : T T M −→ T T ∗ M.In the following subsections we present several Lie functors, as they areused in modern biodynamical research, all being different formulations ofthe covariant force law, F i = mg ij a j , and giving different Lie representationsof the fundamental covariant force functor F ∗ : T T ∗ M −→ T T M.3.13.4.2 Lie–Lagrangian Biodynamical FunctorNow we develop the Lie–Lagrangian biodynamical functor using a modern,nonlinear formulation of the classical robotics structure (see [<strong>Ivancevic</strong>(2005b); <strong>Ivancevic</strong> (2005c)]):Kinematics → Dynamics → ControlLie groups → Exterior Lagrangian → Lie derivativeThe conservative part of generalized Lagrangian formalism, as used inbiodynamics, is derived from Lagrangian conservative energy function. Itdescribes the motion of the conservative skeleton, which is free of controland dissipation. According to the Liouville Theorem, this conservativedynamics is structurally unstable due to the phase–space spreading effect,caused by the growth of entropy (see [<strong>Ivancevic</strong> (1991); <strong>Ivancevic</strong> et al.(1995); <strong>Ivancevic</strong> and Snoswell (2001); <strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]. Thedissipative part is derived from nonlinear dissipative function, and describesquadratic joint dampings, which prevent entropy growth. Its driving partrepresents equivalent muscular torques F i acting in all DOF (or just inactive joints, as used in the affine input control), in the form of force–timeand force–velocity signals.Joint KinematicsRecall that human joints represented by internal coordinates x i (i =1, . . . , n), constitute an nD smooth biodynamical configuration manifoldM (see Figure 3.6). Now we are going to perform some categorical transformationson the biodynamical configuration manifold M. If we applythe functor Lie to the category • [SO(k) i ] of rotational Lie groups SO(k) iand their homomorphisms we get the category • [so(k) i ] of correspondingtangent Lie algebras so(k) i and their homomorphisms. If we further applythe isomorphic functor Dual to the category • [so(k) i ] we get the dualcategory ∗ •[so(k) ∗ i ] of cotangent, or, canonical Lie algebras so(k)∗ i and their


386 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionhomomorphisms. To go directly from • [SO(k) i ] to ∗ •[so(k) ∗ i ] we use thecanonical functor Can [<strong>Ivancevic</strong> and Snoswell (2001); <strong>Ivancevic</strong> (2002);<strong>Ivancevic</strong> and Beagley (2005); <strong>Ivancevic</strong> (2005)]. Therefore we have a commutativetriangle• [SO(k) i ]❅❅❅❅❅❅❘CanLGALie ✠∗•[so(k) i ] ✲∼ • [so(k) ∗ i ]=DualBoth the tangent algebras so(k) i and the cotangent algebras so(k) ∗ i containinfinitesimal group generators, angular velocities ẋ i = ẋ φ i in the firstcase and canonical angular momenta p i = p φi in the second. As Lie groupgenerators, angular velocities and angular momenta satisfy the respectivecommutation relations [ẋ φ i , ẋψ i] = ɛφψθẋ θi and [p φi , p ψi ] = ɛ θ φψ p θ i, wherethe structure constants ɛ φψθand ɛ θ φψconstitute totally antisymmetric third–order tensors.In this way, the functor Dual G : Lie ∼ = Can establishes a geometricalduality between kinematics of angular velocities ẋ i (involved in Lagrangianformalism on the tangent bundle of M) and that of angular momenta p i(involved in Hamiltonian formalism on the cotangent bundle of M). Thisis analyzed below. In other words, we have two functors Lie and Canfrom a category of Lie groups (of which • [SO(k) i ] is a subcategory) into acategory of their Lie algebras (of which • [so(k) i ] and ∗ •[so(k) ∗ i ] are subcategories),and a natural equivalence (functor isomorphism) between themdefined by the functor Dual G . (As angular momenta p i are in a bijectivecorrespondence with angular velocities ẋ i , every component of the functorDual G is invertible.)Applying the functor Lie to the biodynamical configuration manifoldM (Figure 3.6), we get the product–tree of the same anthropomorphicstructure, but having tangent Lie algebras so(k) i as vertices, instead of thegroups SO(k) i . Again, applying the functor Can to M, we get the product–tree of the same anthropomorphic structure, but this time having cotangentLie algebras so(k) ∗ i as vertices.The functor Lie defines the second–order Lagrangian formalism on thetangent bundle T M (i.e., the velocity phase–space manifold) while the func-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 387tor Can defines the first–order canonical Hamiltonian formalism on thecotangent bundle T ∗ M (i.e., the momentum phase–space manifold). Asthese two formalisms are related by the isomorphic functor Dual, they areequivalent. In this section we shall follow the Lagrangian functor Lie, usingthe powerful formalism of exterior differential systems and integral variationalprinciples [Griffiths (1983); Choquet-Bruhat and DeWitt-Morete(1982)]. For the parallel, Hamiltonian treatment along the functor Can,more suitable for chaos theory and stochastic generalizations, see [<strong>Ivancevic</strong>and Snoswell (2001); <strong>Ivancevic</strong> (2002)].Exterior Lagrangian DynamicsLet Ω p (M) = ∑ ω I dx I denote the space of differential p−forms on M.That is, if multi–index I ⊂ {1, . . . , n} is a subset of p elements then wehave a p−form dx I = dx i 1 ∧ dx i 2 ∧ · · · ∧ dx i p on M. We define the exteriorderivative on M as dω = ∑ ∂ω I∂x pdx p ∧ dx I (compare with (5.8) above).Now, from exterior differential systems point of view (see subsection3.6.2 above as well as [Griffiths (1983)]), human–like motion representsan n DOF neuro–musculo–skeletal system Ξ, evolving in time on its nDconfiguration manifold M, (with local coordinates x i , i = 1, ..., n) as wellas on its tangent bundle T M (with local coordinates (x i ; ẋ i )).For the system Ξ we will consider a well–posed variational problem(I, ω; ϕ), on an associated (2n+1)−-D jet space X = J 1 (R, M) ∼ = R×T M,with local canonical variables (t; x i ; ẋ i ).Here, (I, ω) is called a Pfaffian exterior differential system on X (see[Griffiths (1983)]), given locally as{ θ i = dx i − ẋ i ω = 0, (3.195)ω ≡ dt ≠ 0with the structure equationsdθ i = −dẋ i ∧ ω.Integral manifolds N ∈ J 1 (R, M) of the Pfaffian system (I, ω) are locallyone–jets t → (t, x(t), ẋ(t)) of curves x = x(t) : R → M.ϕ is a 1−formϕ = L ω, (3.196)where L = L(t, x, ẋ) is the system’s Lagrangian function defined on X, havingboth coordinate and velocity partial derivatives, respectively denoted


388 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionbyL x i ≡ ∂ x iL, and Lẋi ≡ ∂ẋiL.A variational problem (I, ω; ϕ) is said to be strongly non–degenerate,or well–posed [Griffiths (1983)], if the determinant of the matrix of mixedvelocity partials of the Lagrangian is positive definite, i.e.,det ‖Lẋiẋ j ‖ > 0.The extended Pfaffian system⎧⎨⎩θ i = 0dLẋi − L x i ω = 0ω ≠ 0.generates classical Euler–Lagrangian equationsddt L ẋ i = L xi, (3.197)describing the control–free, dissipation–free, conservative skeleton dynamics.If an integral manifold N satisfies the Euler–Lagrangian equations(3.197) of a well–posed variational problem on X thenddt(∫N tϕfor any admissible variation N t ∈ N that satisfies the endpoint conditionsω = θ i = 0.Theorem: Under the above conditions, both the Lagrangian dynamicswith initial conditions{ddt L ẋ i = L x ix(t 0 ) = x 0 , ẋ(t 0 ) = ẋ 0)t=0= 0and the Lagrangian dynamics with endpoint conditions{ddt L ẋ i = L x ix(t 0 ) = x 0 , x(t 1 ) = x 1have unique solutions. For the proof, see [Griffiths (1983)].Now, if M is a smooth Riemannian manifold, its metric g =< . > islocally given by a positive definite quadratic formds 2 = g ij (x) dx i dx j , (3.198)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 389where the metric tensor g ij is a C ∞ symmetric matrix g(x) = ‖g ij (x)‖.Kinetic energy of the system Ξ is a function T = T (x, ẋ) on the tangentbundle T M, which induces a positive definite quadratic form in each fibreT x M ⊂ T M. In local coordinates, it is related to the Riemannian metric(3.198) by: T ω 2 = 1 2 ds2 .If potential energy of the system Ξ is a function U = U(x) on M, thenthe autonomous Lagrangian is defined as L(x, ẋ) = T (x, ẋ) − U(x), i.e.,kinetic minus potential energy.The condition of well–posedness is satisfied, asdet ‖Lẋiẋ j ‖ = det ‖g ij(x)‖ > 0.Now, the covariant Euler–Lagrangian equations (3.197) expand asd (gij (x(t)) ẋ j (t) ) = 1 dt2(∂x ig jk (x(t)) ẋ j (t) ẋ k (t) ) − F i (x(t)), (3.199)where F i (x(t)) = ∂U(x(t))∂ẋdenote the gradient force 1–forms.iLetting ∥ g ij (x) ∥ be the inverse matrix to ‖gij (x)‖ and introducing theChristoffel symbolsΓ i jk = g il Γ jkl , Γ jkl = 1 2 (∂ x j g kl + ∂ x kg jl − ∂ x lg jk )the equations (3.199) lead to the classical contravariant form (see [<strong>Ivancevic</strong>(1991); <strong>Ivancevic</strong> and Pearce (2001b); <strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)])ẍ i (t) + Γ i jk(x(t)) ẋ j (t) ẋ k (t) = −F i (x(t)), (3.200)where F i (x(t)) = g ij (x) ∂U(x(t))∂ẋdenote the gradient force vector–fields.jThe above Theorem implies that both the Lagrangian dynamics withinitial conditions{ ẍi (t) + Γ i jk (x(t)) ẋj (t) ẋ k (t) = −F i (x(t))(3.201)x(t 0 ) = x 0 , ẋ(t 0 ) = ẋ 0and the Lagrangian dynamics with endpoint conditions{ ẍi (t) + Γ i jk (x(t)) ẋj (t) ẋ k (t) = −F i (x(t))x(t 0 ) = x 0 , x(t 1 ) = x 1(3.202)have unique solutions. We consider the system (3.201) to be the valid basisof human–like dynamics, and the system (3.202) to be the valid basis of thefinite biodynamics control.


390 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow, recall that any smooth n−manifold M induces an n−categoryΠ n (M), its fundamental n−groupoid. In Π n (M), 0–cells are points in M;1–cells are paths in M (i.e., parameterized smooth maps f : [0, 1] → M);2–cells are smooth homotopies (denoted by ≃) of paths relative to endpoints(i.e., parameterized smooth maps h : [0, 1]×[0, 1] → M); 3–cells are smoothhomotopies of homotopies of paths in M (i.e., parameterized smooth mapsj : [0, 1]×[0, 1]×[0, 1] → M). Categorical composition is defined by pastingpaths and homotopies, which gives the recursive homotopy dynamics (seebelow).On the other hand, to describe the biodynamical realism, we have togeneralize (3.200), so to include any other type of external contravariantforces (including excitation and contraction dynamics of muscular–like actuators,as well as nonlinear dissipative joint forces) to the r.h.s of (3.200).In this way, we get the general form of contravariant Lagrangian dynamicsẍ i (t) + Γ i jk(x(t)) ẋ j (t) ẋ k (t) = F i (t, x(t), ẋ(t)) , (3.203)or, in exterior, covariant formddt L ẋ i − L x i = F i (t, x(t), ẋ(t)) . (3.204)Recursive homotopy dynamics:0 − cell : x 0 • x 0 ∈ M; in the higher cells below: t, s ∈ [0, 1];1 − cell : x 0 •f ✲ • x1 f : x 0 ≃ x 1 ∈ M,f : [0, 1] → M, f : x 0 ↦→ x 1 , x 1 = f(x 0 ), f(0) = x 0 , f(1) = x 1 ;e.g., linear path: f(t) = (1 − t) x 0 + t x 1 ;e.g., Euler–Lagrangian f − dynamics with endpoint conditions (x 0 , x 1 ) :ddt f ẋ i = f x i, with x(0) = x 0, x(1) = x 1 , (i = 1, ..., n);or


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 3912 − cell : x 0 •f∨ h❘ • x 1 h : f ≃ g ∈ M,✒gh : [0, 1] × [0, 1] → M, h : f ↦→ g, g = h(f(x 0 )),h(x 0 , 0) = f(x 0 ), h(x 0 , 1) = g(x 0 ), h(0, t) = x 0 , h(1, t) = x 1e.g., linear homotopy: h(x 0 , t) = (1 − t) f(x 0 ) + t g(x 0 );e.g., homotopy between two Euler–Lagrangian (f, g) − dynamicswith the same endpoint conditions (x 0 , x 1 ) :ddt f ẋ i = f x i, and ddt g ẋ i = g x i with x(0) = x 0, x(1) = x 1 ;3 − cell : x 0 •hfj> gi❘• x 1 j : h ≃ i ∈ M,✒j : [0, 1] × [0, 1] × [0, 1] → M, j : h ↦→ i, i = j(h(f(x 0 )))j(x 0 , t, 0) = h(f(x 0 )), j(x 0 , t, 1) = i(f(x 0 )),j(x 0 , 0, s) = f(x 0 ), j(x 0 , 1, s) = g(x 0 ),j(0, t, s) = x 0 , j(1, t, s) = x 1e.g., linear composite homotopy: j(x 0 , t, s) = (1 − t) h(f(x 0 )) + t i(f(x 0 ));or, homotopy between two homotopies between above two Euler-Lagrangian (f, g) − dynamics with the same endpoint conditions (x 0 , x 1 ).or3.13.4.3 Lie–Hamiltonian Biodynamical FunctorThe three fundamental and interrelated obstacles facing any researcher inthe field of human–like musculo–skeletal dynamics, could be identified as[<strong>Ivancevic</strong> and Snoswell (2001)]:(1) Deterministic chaos,(2) Stochastic forces, and(3) Imprecision of measurement (or estimation) of the system numbers(SN): inputs, parameters and initial conditions.


392 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall that the deterministic chaos is manifested as an irregular andunpredictable time evolution of purely deterministic nonlinear systems. Ifa nonlinear system is started twice, from slightly different initial conditions,its time evolution differs exponentially, while in case of a linear system, thedifference in time evolution is linear.Again, recall that the stochastic dynamics is based on the concept ofMarkov process 7 , which represents the probabilistic analogue to the deterministicdynamics. The property of a Markov chain of prime importancefor human–motion dynamics is the existence of an invariant distribution ofstates: we start with an initial state x 0 whose absolute probability is 1.Ultimately the states should be distributed according to a specified distribution.Recall that Brownian dynamics represents the phase–space trajectoriesof a collection of particles that individually obey Langevin rate equations(see [Gardiner (1985)]) in the field of force (i.e., the particles interact witheach other via some deterministic force). For one free particle the Langevinequation of motion is given bym ˙v = R(t) − βv,where m denotes the mass of the particle and v its velocity. The r.h.srepresents the coupling to a heat bath; the effect of the random force R(t)is to heat the particle. To balance overheating (on the average), the particleis subjected to friction β.Noe, between pure deterministic (in which all DOF of the system inconsideration are explicitly taken into account, leading to classical dynamicalequations like Hamiltonian) and pure stochastic dynamics (Markovprocess), there is so–called hybrid dynamics, particularly the Brownian dynamics,in which some of DOF are represented only through their stochasticinfluence on others.System theory and artificial intelligence have long investigated the topicof uncertainty in measurement, modelling and simulation. Research inartificial intelligence has enriched the spectrum of available techniquesto deal with uncertainty by proposing a theory of possibility, based onthe theory of fuzzy sets (see [Yager (1987); Dubois and Prade (1980);Cox (1992); Cox (1994)]). The field of qualitative reasoning and simulation[Berleant and Kuipers (1992)] is also interested in modelling in-7 Recall that the Markov process is characterized by a lack of memory, i.e., the statisticalproperties of the immediate future are uniquely determined by the present, regardlessof the past (see [Gardiner (1985)]).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 393completely known systems where qualitative values are expressed by intervals.However, qualitative simulation techniques reveal a low predictivepower in presence of complex models. In this section we have combinedqualitative and quantitative methods, in spirit of [Bontempi (1995);<strong>Ivancevic</strong> and Snoswell (2001)].In this section we will deal with the general biodynamics from the pointof view that mathematically and logically approaches a general theory ofsystems, i.e., that makes the unique framework for both linear and nonlinear,discrete and continuous, deterministic and stochastic, crisp and fuzzy,SISO and MIMO–systems, and generalizes the robot dynamics elaboratedin the literature (see [Vukobratovic (1970); Vukobratovic et al. (1970);Vukobratovic and Stepanenko (1972); Vukobratovic and Stepanenko (1973);Vukobratovic (1975); Igarashi and Nogai (1992); Hurmuzlu (1993); Shihet al. (1993); Shih and Klein (1993)]), including all necessary DOF tomatch the physiologically realistic human–like motion. Yet, we wish toavoid all the mentioned fundamental system obstacles. To achieve this goalwe have formulated the general biodynamics functor machine, covering aunion of the three intersected frameworks:(1) Muscle–driven, dissipative, Hamiltonian (nonlinear, both discrete andcontinuous) MIMO–system;(2) Stochastic forces (including dissipative fluctuations and ‘Master’jumps); and(3) Fuzzy system numbers.The Abstract Functor MachineIn this subsection we define the abstract functor machine [<strong>Ivancevic</strong> andSnoswell (2001)] (compare with [Anderson et al. (1976)]) by a two–stepgeneralization of the Kalman’s modular theory of linear MIMO–systems[Kalman et. al. (1969); Kalman (1960)]. The first generalization putsthe Kalman’s theory into the category Vect of vector spaces and linearoperators (see [MacLane (1971)] for technical details about categorical language),thus formulating the unique, categorical formalism valid both forthe discrete– and continuous–time MIMO–systems.We start with the unique, continual–sequential state equationẋ(t + 1) = Ax(t) + Bu(t), y(t) = Cx(t), (3.205)where the nD vector spaces of state X ∋ x, input U ∋ u, and outputY ∋ y have the corresponding linear operators, respectively A : X → X,


394 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionB : U → X, and C : X → Y . The modular system theory comprises thesystem dynamics, given by a pair (X, A), together with a reachability mape : U → X of the pair (B, A), and an observability map m : X → Y ofthe pair (A, C). If the reachability map e is surjection the system dynamics(X, A) is called reachable; if the observability map m is injection the systemdynamics (X, A) is called observable. If the system dynamics (X, A) is bothreachable and observable, a composition r = m ◦ e : U → Y defines thetotal system’s response, which is given by solution of equation (3.205). If theunique solution to the continual–sequential state equation exists, it givesthe answer to the (minimal) realization problem: find the system S thatrealizes the given response r = m ◦ e : U → Y (in the smallest number ofdiscrete states and in the shortest time).In categorical language, the system dynamics in the category Vect is apair (X, A), where X ∈ Ob(Vect) is an object in Vect and A : X → X ∈Mor(Vect) is a Vect–morphism. A decomposable system in Vect is such asextuple S ≡ (X, A, U, B, Y, C) that (X, A) is the system dynamics in Vect,a Vect−-morphism B : U → X is an input map, and a Vect–morphismC : X → Y is an output map. Any object in Vect is characterized bymutually dual 8 notions of its degree (a number of its input morphisms) andits codegree (a number of its output morphisms). Similarly, any decomposablesystem S in Vect has a reachability map given by an epimorphisme = A ◦ B : U → X and its dual observability map given by a monomorphismm = C ◦ A : X → Y ; their composition r = m ◦ e : U → Y inMor(Vect) defines the total system’s response in Vect given by the uniquesolution of the continual–sequential state equation (3.205).The second generalization gives an extension of the continual–sequentialMIMO–system theory: from the linear category Vect – to an arbitrarynonlinear category K. We do this extension (see [<strong>Ivancevic</strong> and Snoswell(2001)]) by formally applying the action of the nonlinear process–functorF : K ⇒ K on the decomposable system S ≡ (X, A, U, B, Y, C) in Vect.Under the action of the process functor F the linear system dynamics (X, A)in Vect transforms into a nonlinear F−-dynamics (F[X], F[A]) in K, creatingthe functor machine in K represented by a nonlinear decomposablesystem F[S] ≡ (F[X], F[A], F[U], F[B], F[Y ], F[C]). The reachability maptransforms into the input process F[e] = F[A] ◦ F[B] : F[U] → F[X], whileits dual, observability map transforms into the output process F[m] =8 Recall that in categorical language duality means reversing the (arrows of) morphisms;the knowledge of one of the two mutually dual terms automatically implies theknowledge of the other.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 395F[C] ◦ F[A] : F[X] → F[Y ]. In this way the total response of the linearsystem r = m ◦ e : U → Y in Mor(Vect) transforms into the nonlinearsystem behavior F[r] = F[m] ◦ F[e] : F[U] → F[Y ] in Mor(K). Obviously,F[r], if exists, is given by a nonlinear F−-transform of the linear stateequation (3.205).The purpose of this section is to formulate a nonlinear F–transformfor the linear state equation (3.205) for biodynamics, i.e., the biodynamicsfunctor machine. In subsequent sections we give a three–step developmentof a fuzzy–stochastic–Hamiltonian formulation for the biodynamics functormachine F[S], with a corresponding nonlinear system behavior F[r].Muscle–Driven, Dissipative, Hamiltonian BiodynamicsIn this subsection we choose the functor Can, as the first–order Hamiltonianformalism is more suitable for both stochastic and fuzzy generalizationsto follow. Recall that the general deterministic Hamiltonian biodynamics,representing the canonical functor Can : S • [SO(n) i ] ⇒ S• ∗ [so(n) ∗ i ], is givenby dissipative, driven δ−Hamiltonian equations,˙q i = ∂H∂p i+ ∂R∂p i, (3.206)ṗ i = F i − ∂H∂q i + ∂R∂q i , (3.207)q i (0) = q i 0, p i (0) = p 0 i , (3.208)including contravariant equation (3.206) – the velocity vector–field, andcovariant equation (3.207) – the force 1−form, together with initial jointangles and momenta (3.208). Here (i = 1, . . . , N), and R = R(q, p) denotesthe Raileigh nonlinear (biquadratic) dissipation function, and F i =F i (t, q, p) are covariant driving torques of equivalent muscular actuators, resemblingmuscular excitation and contraction dynamics in rotational form.The velocity vector–field (3.206) and the force 1−form (3.207) togetherdefine the generalized Hamiltonian vector–field X H , which geometricallyrepresents the section of the momentum phase–space manifold T ∗ M, whichis itself the cotangent bundle of the biodynamical configuration manifoldM; the Hamiltonian (total energy) function H = H(q, p) is its generatingfunction.As a Lie group, the configuration manifold M is Hausdorff [Abrahamet al. (1988); Marsden and Ratiu (1999); Postnikov (1986)]. Therefore, forx = (q i , p i ) ∈ U p , U p open in T ∗ M, there exists a unique one–parameter


396 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongroup of diffeomorphisms φ δt: T ∗ M → T ∗ M, the generalized deterministicδ−Hamiltonian phase–flowφ δt: G 1 × T ∗ M → T ∗ M : (p(0), q(0)) ↦→ (p(t), q(t)), (3.209)(φ δt◦ φ δs= φ δt+s, φ δ0= identity),given by (3.206–3.208) such thatddt | t=0 φ δtx = J∇H(x).The δ−Hamiltonian system (3.206–3.208), with its δ−Hamiltonianphase–flow φ δt(3.209), i.e., the canonical functor Can, represents our first,continual–deterministic model for the biodynamics functor machine F[S]with the nonlinear system behavior F[r]. In the two subsequent sectionswe generalize this model to include discrete stochastic forces and fuzzy SN.Stochastic–Lie–Hamiltonian Biodynamical FunctorIn terms of the Markov stochastic process, we can interpret the deterministicδ−Hamiltonian biodynamical system (3.206–3.208) as deterministicdrift corresponding to the Liouville equation. Thus, we cannaturally (in the sense of Langevin) add the covariant vector σ i (t) ofstochastic forces (diffusion fluctuations and discontinuous–Master jumps)σ i (t) = B ij [q i (t), t] dW j (t) to the canonical force equation. In this way weget stochastic σ−Hamiltonian biodynamical system, a stochastic transformationStoch[Can] of the canonical functor Can,( ∂Hdq i =dp i =∂p i+ ∂R∂p i)dt, (3.210)(F i − ∂H∂q i + ∂R )∂q i dt + σ i (t), (3.211)σ i (t) = B ij [q i (t), t] dW j (t), q i (0) = q i 0, p i (0) = p 0 i .In our low–dimensional example–case of symmetrical 3D load–lifting,the velocity and force σ−Hamiltonian biodynamics equations (3.210–3.211)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 397become⎛ ⎧ ⎡ ⎛⎞ ⎤ ⎪⎨2−1⎫ ⎞dq i ⎜= ⎝p i [J i ] −1 ⎢i∑⎪⎬+ ⎣m i⎝ L j cos q j ⎠ ⎥ ⎦ + ∂R ⎟⎠ dt,⎪ ⎩ ⎪ ⎭ ∂p ij=1⎛dp i = B ij [q i (t), t] dW j (t) + ⎝F i − g−10−i∑j=iL j sin q j p i p j⎡⎣m i( i∑k=110−i∑j=iL j m j sin q j) 3⎤L k cos q k ⎦−1⎞+ ∂R ⎟∂q i ⎠ dt.Recall that Ito quadratic cotangent bundle I ∗ Q N is defined as a WhitneysumI ∗ Q N = T ∗ Q N ⊕ SQ N ,where SQ N corresponds to stochastic tensor bundle, whose elements are2nd–order tensor–fields composed of continual diffusion fluctuations anddiscontinuous jumps at every point of the manifold Q N . On I ∗ Q N is defineda non–degenerate, stochastic 2−form α which is closed, i.e., dα = 0, andexact, i.e., α = dβ, where 1−form β represents a section β : Q N → I ∗ Q Nof the Ito bundle I ∗ Q N .Now, the stochastic Hamiltonian vector–field Ξ H represents a sectionΞ H : Q N → IQ N of the Ito quadratic tangent bundle IQ N , also defined asa Whitney sumIQ N = T M ⊕ SQ N .The quadratic character of Ito stochastic fibre–bundles corresponds tothe second term (trace of the 2nd–order tensor–field) of associate stochasticTaylor expansion (see [Elworthy (1982); Mayer (1981)]).Through stochastic σ−Hamiltonian biodynamical system (3.210–3.211),the deterministic δ−Hamiltonian phase–flow φ δt(3.209), extends intostochastic σ−Hamiltonian phase–flow φ σtφ σt: G 1 × I ∗ M → I ∗ M : (p(0), q(0)) ↦→ (p(t), q(t)), (3.212)(φ σt◦ φ σs= φ σt+s, φ σ0= identity),where I ∗ M denotes Ito quadratic cotangent bundle (see [Elworthy (1982);Mayer (1981)]) of biodynamical configuration manifold M.


398 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionBesides the σ−Hamiltonian phase–flow φ σt(3.212), including N individualrandom–phase trajectories, we can also define (see [Elworthy (1982)])an average or mean 〈σ〉 − Hamiltonian flow 〈φ〉 σt〈φ〉 σt: G 1 × I ∗ M → I ∗ M : (〈p(0)〉 , 〈q(0)〉) ↦→ (〈p(t)〉 , 〈q(t)〉),(〈φ〉 σt◦ 〈φ〉 σs= 〈φ〉 σt+s, 〈φ〉 σ0= identity),which stochastically corresponds to the trajectory of the center ofmass in the human–like dynamics, approximatively lumbo–sacral spinalSO(3)−joint.The necessary conditions for existence of a unique non–anticipating solutionof the σ−Hamiltonian biodynamical system in a fixed time intervalare Lipschitz condition and growth condition (see [Elworthy (1982);Mayer (1981)]). For constructing an approximate solution a simple iterativeCauchy–Euler procedure could be used to calculate (qk+1 i , pk+1 i ) from theknowledge of (qk i , pk i ) on the mesh of time points tk , k = 1, . . . , s, by addingdiscrete δ–Hamiltonian drift–terms A i (qk i )∆tk and A i (p k i )∆tk ,as well as astochastic term B ij (qi k, tk )∆W j k .σ−Hamiltonian biodynamical system (3.210–3.211), with its σ−Hamiltonian phase–flow φ σt(3.212), i.e., the functorStoch[Can], represents our second, continual–discrete stochastic model forthe biodynamics functor machine F[S] with the nonlinear system behaviorF[r]. In the next section we generalize this model once more to includefuzzy SN.Fuzzy–Stochastic–Lie–Hamiltonian FunctorGenerally, a fuzzy differential equation model (FDE–model, for short)is a symbolic description expressing a state of incomplete knowledge of thecontinuous world, and is thus an abstraction of an infinite set of ODEsmodels. Qualitative simulation (see [Berleant and Kuipers (1992)]) predictsthe set of possible behaviors consistent with a FDE model and aninitial state. Specifically, as a FDE we consider an ordinary deterministic(i.e., crisp) differential equation (CDE) in which some of the parameters(i.e., coefficients) or initial conditions are fuzzy numbers, i.e., uncertain andrepresented in a possibilistic form. As a solution of a FDE we consider atime evolution of a fuzzy region of uncertainty in the system’s phase–space,which corresponds to its the possibility distribution.Recall that a fuzzy number is formally defined as a convex, normalizedfuzzy set [Dubois and Prade (1980); Cox (1992); Cox (1994)]. The concept


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 399of fuzzy numbers is an extension of the notion of real numbers: it encodesapproximate quantitative knowledge. It is not probabilistic, but rather apossibilistic distribution. The mathematics of fuzzy numbers is founded onthe extension principle, introduced by Zadeh [Yager (1987)]. This principlegives a general method for extending standard mathematical concepts inorder to deal with fuzzy quantities [Dubois and Prade (1980)].Let Φ : Y 1 × Y 2 × · · · × Y n → Z be a deterministic map such thatz = Φ(y 1 , y 2 , . . . , y n ) for all z ∈ Z, y i ∈ Y i . The extension principle allowsus to induce from n input fuzzy sets ȳ i on Y i an output fuzzy set ¯z on Zthrough Φ given byµ¯z (t) =sup min(µȳ1(s 1 ), . . . , µȳn(s n )),t=Φ(s 1 ,...,s n )or µ¯z (t) = 0 if Φ −1 (t) = ∅,where Φ −1 (t) denotes the inverse image of t and µȳi is the membershipfunction of ȳ i , (i = 1, . . . , n).The extension principle gives a method to calculate the fuzzy valueof a fuzzy map but, in practice, its application is not feasible because ofthe infinite number of computations it would require. The simplest wayof efficiently applying the extension principle is in the form of iterativerepetition of several crisp Hamiltonian simulations (see [Bontempi (1995);<strong>Ivancevic</strong> and Snoswell (2001); Pearce and <strong>Ivancevic</strong> (2003); Pearce and<strong>Ivancevic</strong> (2004)]), within the range of included fuzzy SN.Fuzzification of the crisp deterministic δ−Hamiltonian biodynamicalsystem (3.206–3.208) gives the fuzzified µ−-Hamiltonian biodynamical system,namely δ−-Hamiltonian biodynamical system with fuzzy SN, i.e., thefuzzy transformation Fuzzy[Can] of the canonical functor Can˙q i ∂H(q, p, σ)= + ∂R ,∂p i ∂p i(3.213)ṗ i = ¯F ∂H(q, p, σ)i (q, p, σ) −∂q i + ∂R∂q i , (3.214)q i (0) = ¯q i 0, p i (0) = ¯p 0 i , (i = 1, . . . , N). (3.215)Here σ = σ µ (with µ ≥ 1) denote fuzzy sets of conservative parameters(segment lengths, masses and moments of inertia), dissipative joint dampingsand actuator parameters (amplitudes and frequencies), while the bar(.) ¯ over a variable (.) denotes the corresponding fuzzified variable.


400 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn our example–case of symmetrical 3D load–lifting, the fuzzifiedµ−Hamil-tonian biodynamical system (3.213–3.215) becomes⎧ ⎡ ⎛⎪⎨˙q i = p i [⎪ ¯J i ] −1 ⎢i∑+ ⎣ ¯m i⎝⎩j=1ṗ i = ¯F10−i∑i (t, q i , p i , {σ} µ ) − g−10−i∑j=i¯L j sin q j p i p j⎡j=i⎣ ¯m i( i∑k=1⎞ ⎤ 2−1⎫⎪⎬¯L j cos q j ⎠ ⎥ ⎦ + ∂R ,⎪ ⎭ ∂p i¯L j ¯m j sin q j¯L k cos q k ) 3⎤q i (0) = ¯q i 0, p i (0) = ¯p 0 i , (i = 1, . . . , 9).⎦−1+ ∂R∂q iIn this way, the crisp δ−-Hamiltonian phase–flow φ δt(3.209) extendsinto fuzzy–deterministic µ−-Hamiltonian phase–flow φ µtφ µt: G 1 × T ∗ M → T ∗ M : (¯p 0 i , ¯q i 0) ↦→ (p(t), q(t)),(φ µt◦ φ µs= φ µt+s, φ µ0= identity).Similarly, fuzzification of crisp stochastic σ−Hamiltonian biodynamicalsystem (3.210–3.211) gives fuzzy–stochastic [µσ]−Hamiltonian biodynamicalsystem, namely stochastic σ−Hamiltonian biodynamical system withfuzzy SN, i.e., the fuzzy–stochastic transformation Fuzzy[Stoch[Can]] of thecanonical functor Can( ∂H(q, p, σ)dq i =∂p i+ ∂R∂p i)dp i = B ij [q i (t), t] dW j (t) +dt, (3.216)(∂H(q, p, σ)¯F i (q, p, σ) −∂q i + ∂R )∂q i dt, (3.217)q i (0) = ¯q i 0, p i (0) = ¯p 0 i . (3.218)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 401In our example–case of symmetrical 3D load–lifting, the velocity andforce [µσ]−Hamiltonian biodynamics equations (3.216–3.217) become⎛ ⎧ ⎡ ⎛⎪⎨dq i ⎜= ⎝p i [⎪ ¯J i ] −1 ⎢i∑+ ⎣ ¯m i⎝⎩j=1¯L j cos q j ⎞⎠dp i = B ij [q i (t), t] dW j (t) + ⎝ ¯Fi (t, q i , p i , {σ} µ ) − g−10−i∑j=i¯L j sin q j p i p j⎡⎛⎣ ¯m i( i∑k=1¯L k cos q k ) 3⎤⎤ 2−1⎫ ⎞⎪⎬⎥ ⎦ + ∂R ⎟⎠ dt,⎪ ⎭ ∂p i⎦−110−i∑j=i¯L j ¯m j sin q j⎞+ ∂R ⎟∂q i ⎠ dt.In this way, the crisp stochastic σ−-Hamiltonian phase–flow φ σt(3.212)extends into fuzzy–stochastic [µσ]–Hamiltonian phase–flow φ [µσ]tφ [µσ]t : G 1 × I ∗ M → I ∗ M : (¯p 0 i , ¯q i 0) ↦→ (p(t), q(t)), (3.219)(φ [µσ]t ◦ φ [µσ]s = φ [µσ]t+s , φ [µσ]0 = identity).[µσ]−Hamiltonian biodynamical system (3.216–3.218), with its phase–flow φ [µσ]t (3.219), i.e., the functor Fuzzy[Stoch[Can]], represents our final,continual–discrete and fuzzy–stochastic model for the biodynamics functormachine F[S] with the nonlinear system behavior F[r].3.13.5 Biodynamical Topology3.13.5.1 (Co)Chain Complexes in BiodynamicsIn this section we present the category of (co)chain complexes, as used inmodern biodynamics. The central concept in cohomology theory is thecategory S • (C) of generalized cochain complexes in an Abelian category C[Dieudonne (1988)]. The objects of the category S • (C) are infinite sequencesA • : · · · −→ A n−1 d n−1 ✲ An d n ✲ A n+1 −→ · · ·where, for each n ∈ Z, A n is an object of C and d n a morphism of C, withthe conditionsd n−1 ◦ d n = 0


402 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor every n ∈ Z. When A n = 0 for n < 0, one speaks of cochain complexes.The d n are called coboundary operators.The morphisms of the category S • (C) are sequences f • = (f n ) : A • →B • where, for each n ∈ Z, f n : A n → B n is a morphism of C, and in thediagram· · · −→ A n−1 d n−1 ✲ An d n ✲ A n+1 −→ · · ·f n−1 | ↓ f n | ↓ f n+1 | ↓ (3.220)· · · −→ B n−1 d n−1 ✲ Bn d n ✲ B n+1 −→ · · ·all squares are commutative; one says the f n commute with the coboundaryoperators. One has Im d n+1 ⊂ Ker d n ⊂ A n for every n ∈ Z; the quotientH n (A • ) = Ker d n / Im d n+1 is called the nth cohomology object of A • . From(3.220) it follows that there is a morphismdeduced canonically from f • , andH n (f • ) : H n (A • ) → H n (B • )(A • , f • ) ⇒ (H n (A • ), H n (f • ))is a covariant functor from S • (C) to C.The cohomology exact sequence: if three cochain complexes A • , B • , C •are elements of a short exact sequence of morphisms0 −→ A • −→ B • −→ C • −→ 0then there exists an infinite sequence of canonically defined morphismsd n : H n (C • ) → H n−1 (A • ) such that the sequence· · · −→ H n (A • ) −→ H n (B • ) −→ H n (C • ) −→ H n−1 (A • ) −→ · · ·is exact, that is the image of each homomorphism in the sequence is exactlythe kernel of the next one.The dual to the category S • (C) is the category of S • (C) of generalizedchain complexes. Its objects and morphisms are get by formal inversion ofall arrows and lowering all indices.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 403Biodynamical (Co)HomologiesLet M • denote the Abelian category of cochains, (i.e., p−-forms) on thebiodynamical configuration manifold M (see Figure 3.7). When C = M • ,we have the category S • (M • ) of generalized cochain complexes A • in M • ,and if A ′ = 0 for n < 0 we have a subcategory S • DR (M• ) of the de Rhamdifferential complexes in M •A • DR : 0 → Ω 0 (M)d−→ Ω 1 (M)d−→ Ω 2 (M) · · ·d−→ Ω n (M)d−→ · · · .Here A ′ = Ω n (M) is the vector space over R of all p–forms ω on M (forp = 0 the smooth functions on M) and d n = d : Ω n−1 (M) → Ω n (M) is theexterior differential. A form ω ∈ Ω n (M) such that dω = 0 is a closed formor n–cocycle. A form ω ∈ Ω n (M) such that ω = dθ, where θ ∈ Ω n−1 (M), isan exact form or n–coboundary. Let Z n (M) = Ker d (resp. B n (M) = Im ddenote a real vector space of cocycles (resp. coboundaries) of degree n.Since d n+1 ◦ d n = d 2 = 0, we have B n (M) ⊂ Z n (M). The quotient vectorspaceH n DR(M) = Ker d/ Im d = Z n (M)/B n (M)is the de Rham cohomology group. The elements of HDR n (M) representequivalence sets of cocycles. Two cocycles ω 1 , ω 2 belong to the sameequivalence set, or are cohomologous (written ω 1 ∼ ω 2 ) iff they differ by acoboundary ω 1 − ω 2 = dθ. The de Rham’s cohomology class of any formω ∈ Ω n (M) is [ω] ∈ HDR n (M). The de Rham differential complex (1) canbe considered as a system of second–order DEs d 2 θ = 0, θ ∈ Ω n−1 (M)having a solution represented by Z n (M) = Ker d.Analogously let M • denote the Abelian category of chains on the configurationmanifold M. When C = M • , we have the category S • (M • ) ofgeneralized chain complexes A • in M • , and if A n = 0 for n < 0 we have asubcategory S• C (M • ) of chain complexes in M •A • : 0 ← C 0 (M)∂←− C 1 (M)∂←− C 2 (M) · · ·∂←− C n (M)∂←− · · · .Here A n = C n (M) is the vector space over R of all finite chains C on themanifold M and ∂ n = ∂ : C n+1 (M) → C n (M). A finite chain C suchthat ∂C = 0 is an n−cycle. A finite chain C such that C = ∂B is ann−boundary. Let Z n (M) = Ker ∂ (resp. B n (M) = Im ∂) denote a realvector space of cycles (resp. boundaries) of degree n. Since ∂ n+1 ◦ ∂ n =


404 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction∂ 2 = 0, we have B n (M) ⊂ Z n (M). The quotient vector spaceH C n (M) = Ker ∂/ Im ∂ = Z n (M)/B n (M)is the n−homology group. The elements of H C n (M) are equivalence setsof cycles. Two cycles C 1 , C 2 belong to the same equivalence set, or arehomologous (written C 1 ∼ C 2 ), iff they differ by a boundary C 1 −C 2 = ∂B).The homology class of a finite chain C ∈ C n (M) is [C] ∈ H C n (M).The dimension of the n−cohomology (resp. n−homology) group equalsthe nth Betti number b n (resp. b n ) of the manifold M. Poincaré lemmasays that on an open set U ∈ M diffeomorphic to R N , all closed forms(cycles) of degree p ≥ 1 are exact (boundaries). That is, the Betti numberssatisfy b p = 0 (resp. b = 0), for p = 1, . . . , n.The de Rham Theorem states the following. The map Φ: H n ×H n → Rgiven by ([C], [ω]) → 〈C, ω〉 for C ∈ Z n , ω ∈ Z n is a bilinear nondegeneratemap which establishes the duality of the groups (vector spaces) H n and H nand the equality b n = b n .Configuration Manifold Reduction and its Euler CharacteristicRecall (see subsection (3.8.4.3) above), that for the purpose of high–level control, the rotational biodynamical configuration manifold M (Figure3.6), could be first, reduced to an n−torus, and second, transformed intoan n−cube ‘hyper–joystick’, using the following topological techniques (see[<strong>Ivancevic</strong> and Pearce (2001b); <strong>Ivancevic</strong> (2002); <strong>Ivancevic</strong> (2005)]).Let S 1 denote the constrained unit circle in the complex plane, whichis an Abelian Lie group. Firstly, we propose two reduction homeomorphisms,using the noncommutative semidirect product ‘✄’ of the constrainedSO(2)−groups:SO(3) SO(2) ✄ SO(2) ✄ SO(2), and SO(2) ≈ S 1 .Next, let I n be the unit cube [0, 1] n in R n and ‘∼’ an equivalence relationon R n get by ‘gluing’ together the opposite sides of I n , preserving theirorientation. Therefore, the manifold M can be represented as the quotientspace of R n by the space of the integral lattice points in R n , that is anoriented and constrained nD torus T n :n∏R n /Z n = I n / ∼ ≈ Si 1 ≡ {(q i , i = 1, . . . , N) : mod 2π} = T n . (3.221)i=1Now, using the de Rham Theorem and the homotopy Axiom for the deRham cohomologies, we can calculate the Euler–Poincaré characteristics


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 405for T n as well as for its two bundles, T T n and T ∗ T n , as (see [<strong>Ivancevic</strong>(2002); <strong>Ivancevic</strong> (2005)])χ(T n , T T n ) =n∑(−1) p b p , where b p are the Betti numbers defined asp=1b 0 = 1, b 1 = n, . . . b p =( np), . . . b n−1 = n, b n = 1, (p = 0, ..., n).3.13.5.2 Morse Theory in BiodynamicsMorse <strong>Geom</strong>etry of a Biodynamical ManifoldRecall that on any smooth manifold M there exist many Riemannianmetrics g. Each of these metrics is locally defined in a particular point q ∈ Mas a symmetric (0, 2) tensor–field such that g| q : T q M × T q M → R is a positivelydefined inner product for each point q ∈ M. In an open local chartU ∈ M containing the point q, this metric is given as g| q ↦→ g ij (q) dq i dq j .With each metric g| q there is associated a local geodesic on M.Now, two main global geodesics problems on the biodynamical configurationmanifold M with the Riemannian metrics g, can be formulated asfollows (compare with subsection 3.10.5.1 above):(1) Is there a minimal geodesic γ 0 (t) between two points A and B on M? Inother words, does an arc of geodesic γ 0 (t) with extremities A, B actuallyhave minimum length among all rectifiable curves γ(t) = (q i (t), p i (t))joining A and B?(2) How many geodesic arcs are there joining two points A and B on M?Locally these problems have a complete answer: each point of the biodynamicsmanifold M has an open neighborhood V such that for any twodistinct points A, B of V there is exactly one arc of a geodesic containedin V and joining A and B, and it is the unique minimal geodesic betweenA and B.Recall (see subsection (3.10.5.1) above), that seven decades ago, Morseconsidered the set Ω = Ω(M; A, B) of piecewise smooth paths on a Riemannianmanifold M having fixed extremities A, B, defined as continuousmaps γ : [0, 1] → M such that γ(0) = A, γ(1) = B, and there were a finitenumber of pointst 0 = 0 < t 1 < t 2 < · · · < t m−1 < t m = 1, (3.222)


406 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsuch that in every closed interval [t i , t i+1 ], γ was a C ∞ −function.parametrization was always chosen such that for t j ≤ t ≤ t j+1 ,Thet − t j = t j+1 − t jl j∫ tt j‖ dγdu ‖ du, with ∫ tj+1l j = ‖ dγt jdu ‖ du.(3.223)In other words, t − t j was proportional to the length of the image of [t j , t]by γ. ThenL(γ) =m∑l j ,the length of γ, was a function of γ in Ω. A minimal arc from A to B shouldbe a path γ for which L(γ) is minimum in Ω, and a geodesic arc from A toB should be a path that is a ‘critical point’ for the function L. This at firsthas no meaning, since Ω is not a differential manifold; the whole of Morse’stheory consists in showing that it is possible to substitute for Ω genuinedifferential manifolds to which his results on critical points can be applied([Morse (1934)]).To study the geodesics joining two points A, B it is convenient, insteadof working with the length L(γ), to work with the energy of a path γ :[A, B] → M, defined by ([Dieudonne (1988)])E B A (γ) =∫ BAj=0‖ dγdu ‖2 du. (3.224)With the chosen parametrization (3.223), E(γ) = (B − A)L(γ) 2 , and theextremals of E are again the geodesics, but the computations are easierwith E.Morse theory can be divided into several steps (see [Milnor (1963)]).Step 1 is essentially a presentation of the classical Lagrangian methodthat brings to light the analogy with the critical points of a C ∞ − functionon M. No topology is put on Ω; a variation of a path γ ∈ Ω is a continuousmap α into M, defined in a product ] − ε, ε[ × [0, 1] with the followingproperties:(1) α(0, t) = γ(t);(2) α(u, 0) = A, α(u, 1) = B for -ε < u < ε; and(3) There is a decomposition (3.222) such that α is C ∞ in each set] − ε, ε[ × [t i , t i+1 ].


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 407A variation vector–field t ↦→ W (t) is associated to each variation α,where W (t) is a tangent vector in the tangent space T γ(t) M to M, definedbyW (t) = ∂ u α(0, t). (3.225)It is a continuous map of [0, 1] into the tangent bundle T M, smooth in eachinterval [t i , t i+1 ]. These maps are the substitute for the tangent vectors atthe point γ; they form an infinite–dimensional vector space written T Ω(γ).More generally the interval ] − ε, ε[ can be replaced in the definition ofa variation by a neighborhood of 0 in some R n , defining an n−parametervariation.A critical path γ 0 ∈ Ω for a function F : Ω −→ R is defined by thecondition that for every variation α of γ 0 the functionu ↦→ F (α(u, ·))is derivable for u = 0 and its derivative is 0.Step 2 is a modern presentation of the formulas of Riemannian geometry,giving the first variation and second variation of the energy (3.224) of apath γ 0 ∈ Ω, which form the basis of Jacobi results.First consider an arbitrary path ω 0 ∈ Ω, its velocity ˙ω(t) = dω/dt, andits acceleration in the Riemannian sense¨ω(t) = ∇ t ˙ω(t),where ∇ t denotes the Bianchi covariant derivative. They belong to T ω(t) Mfor each t ∈ [0, 1], are defined and continuous in each interval [t i , t i+1 ] inwhich ω is smooth, and have limits at both extremities. Now let α be avariation of ω and t ↦→ W (t) be the corresponding variation vector–field(3.225). The first variation formula gives the first derivative1 d2 du E(α(u, ·))| u=0 = − ∑ i(W (t i )| ˙ω(t i +) − ˙ω(t i −)) −∫ 10(W (t)|¨ω(t)) dt,where (x|y) denotes the scalar product of two vectors in a tangent space.It follows from this formula that γ 0 ∈ Ω is a critical path for E iff γ is ageodesic.Next, fix such a geodesic γ and consider a two–parameter variation:α : U × [0, 1] → M,


408 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere U is a neighborhood of 0 in R 2 , so thatα(0, 0, t) = γ(t), ∂ u1 α(0, 0, t) = W 1 (t), ∂ u2 α(0, 0, t) = W 2 (t),in which W 1 and W 2 are in T Ω(γ). The second variation formula gives themixed second derivative1E(α(u 1 , u 2 , ·)))| (0,0) = − ∑ 2 ∂u 1 ∂u 2i∂ 2−∫ 10(W 2 (t i )|∇ t W 1 (t i +) − ∇ t W 1 (t i −))(W 2 (t)|∇ 2 t W 1 (t) + R(V (t) ∧ W 1 (t)) · V (t)) dt, (3.226)where Z ↦→ R(X ∧ Y ) · Z is the curvature of the Levi–Civita connection.The l.h.s of (3.226) is thus a bilinear symmetric form(W 1 , W 2 ) ↦→ E ∗∗ (W 1 , W 2 )on the product T Ω(γ) × T Ω(γ). For a one–parameter variation αd 2E ∗∗ (W, W ) = 1 2 du 2 E(α(u, ·))| u=0,from which it follows that if γ is a minimal geodesic in Ω, E ∗∗ (W, W ) ≥ 0in T Ω(γ). As usual, we shall speak of E ∗∗ indifferently as a symmetricbilinear form or as a quadratic form W ↦→ E ∗∗ (W, W ).Formula (3.226) naturally leads to the junction with Jacobi work (see[Dieudonne (1988)]): consider the smooth vector–fields t ↦→ J(t) alongγ ∈ M, satisfying the equation∇ 2 t J(t) + R(V (t) ∧ J(t)) · V (t) = 0 for 0 ≤ t ≤ 1. (3.227)With respect to a frame along γ moving by parallel translation on M thisrelation is equivalent to a system of n linear homogeneous ODEs of order 2with C ∞ −coefficients; the solutions J of (3.227) are called the Jacobi fieldsalong γ and form a vector space of dimension 2n. If for a value a ∈]0, 1] ofthe parameter t there exists a Jacobi field along γ that is not identically 0but vanishes for t = 0 and t = a, then the points A = γ(0) and r = γ(a)are conjugate along γ with a multiplicity equal to the dimension of thevector space of Jacobi fields vanishing for t = 0 and t = a.Jacobi fields on the biodynamical configuration manifold M may also bedefined as variation vector–fields for geodesic variations of the path γ ∈ M:they are C ∞ −mapsα : ] − ε, ε[ × [0, 1] → M,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 409such that for any u ∈ ] − ε, ε[ , t ↦→ α(u, t) is a geodesic and α(0, t) = γ(t).It can be proved that the Jacobi fields along γ ∈ M that vanish at Aand B (hence belong to T Ω(γ)) are exactly the vector–fields J ∈ T Ω(γ)such thatE ∗∗ (J, W ) = 0for every W ∈ T Ω(γ). Although T Ω(γ) is infinite–dimensional, the formE ∗∗ is again called degenerate if the vector space of the Jacobi fields vanishingat A and B is note reduced to 0 and the dimension of that vectorspace is called the nullity of E ∗∗ . Therefore, E ∗∗ is thus degenerate iff Aand B are conjugate along γ and the nullity of E ∗∗ is the multiplicity of B.Step 3 is the beginning of Morse’s contributions (see [Milnor (1963)]).He first considered a fixed geodesic γ : [0, 1] → M with extremities A =γ(0), B = γ(1) and the bilinear symmetric form E ∗∗ : T Ω(γ)×T Ω(γ) → R.By analogy with the finite–dimensional quadratic form, the index of E ∗∗ isdefined as the maximum dimension of a vector subspace of T Ω(γ) in whichE ∗∗ is strictly negative (i.e., nondegenerate and taking values E ∗∗ (W, W )


410 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondone by considering the geodesic arc γ τ : [0, τ] → M, the restriction of γto [0, τ], and its energyE(γ τ ) = τ∫ τ0‖ dγdu ‖2 du.E∗∗ τ is the corresponding quadratic form on T Ω(γ τ ), and λ(τ) is its index;one studies the variation of λ(τ) when tau varies from 0 to 1, and λ(1) isthe index of E ∗∗ .The index Theorem says: the index of E ∗∗ is the sum of the multiplicitiesof the points conjugate to A along B and distinct from B.We have seen that the dimension of T Ω(γ; t 0 , t 1 , · · · , t m ) is finite; itfollows that the index of E ∗∗ is always finite, and therefore the number ofpoints conjugate to A along γ is also finite.Step 4 of Morse theory introduces a topology on the set Ω = Ω(M; A, B).On the biodynamical configuration manifold M the usual topology can bedefined by a distance ρ(A, B), the g.l.b. of the lengths of all piecewisesmooth paths joining A and B. For any pair of paths ω 1 , ω 2 in Ω(M; A, B),consider the function d(ω 1 , ω 2 ) ∈ M√ ∫ 1d(ω 1 , ω 2 ) = sup ρ(ω 1 (t), ω 2 (t)) + (ṡ 1 − ṡ 2 ) 2 dt,0t1where s 1 (t) (resp. s 2 (t)) is the length of the path τ ↦→ ω 1 (τ) (resp. τ ↦→ω 2 (τ)) defined in [0, t]. This distance on Ω such that the function ω ↦→EA B (ω) is continuous for that distance.Morse Homology of a Biodynamical ManifoldMorse Functions and Boundary Operators. Let f : M → Rrepresents a C ∞ −function on the biodynamical configuration manifold M.Recall that z = (q, p) ∈ M is the critical point of f if df(z) ≡ df[(q, p)] = 0.In local coordinates (x 1 , ..., x n ) = (q 1 , ..., q n , p 1 , ..., p n ) in a neighborhoodof z, this means ∂f∂x(z) = 0 for i = 1, ..., n. The Hessian of f at a criticalipoint z defines a symmetric bilinear form ∇df(z) = d 2 f(z) ( on T z M, inlocal coordinates (x 1 , ..., x n ) represented by the matrix ∂ 2 f∂x i ∂x). Indexjand nullity of this matrix are called index and nullity of the critical pointz of f.Now, we assume that all critical points z 1 , ..., z n of f ∈ M are nondegeneratein the sense that the Hessians d 2 f(z i ), i = 1, ..., m, have maximalrank. Let z be such a critical point of f of Morse index s (= number0


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 411of negative eigenvalues of d 2 f(z i ), counted with multiplicity). The eigenvectorscorresponding to these negative eigenvalues then span a subspaceV z ⊂ T z M of dimension s. We choose an orthonormal basis e 1 , ..., e s ofV z w.r.t. the Riemannian metric g on M (induced by the system’s kineticenergy), with dual basis dx 1 , ..., dx s . This basis then defines an orientationof V z which we may also represent by the s−-form dx 1 ∧ ... ∧ dx s . We nowlet z ′ be another critical point of f, of Morse index s − 1. We considerpaths γ(t) of the steepest descent of f from z to z ′ , i.e., integral curves ofthe vector–field -∇f(γ). Thus γ(t) defines the gradient flow of f˙γ(t) = −∇f (γ(t)) ,with{limt→−∞ γ(t) = z,lim t→∞ γ(t) = z ′ . (3.228)A path γ(t) obviously depends on the Riemannian metric g on M as∇f = g ij ∂ x if ∂ x j f.From [Smale (1960); Smale (1967)] it follows that for a generic metric g,the Hessian ∇df(y) has only nondegenerate eigenvalues. Having a metricg induced by the system’s kinetic energy, we let Ṽy ⊂ T y M be the spacespanned by the eigenvectors corresponding to the s − 1 lowest eigenvalues.Since z ′ has Morse index s−1, ∇df(z ′ ) = d 2 f(z ′ ) has precisely s−1 negativeeigenvalues. Therefore, Ṽ z ′ ≡ lim t→∞ Ṽ γ(t) = V z ′, while the unit tangentvector of γ at z ′ ˙γ(t), i.e., lim t→∞ ‖ ˙γ(t)‖, lies in the space of directions correspondingto positive eigenvalues and is thus orthogonal to V z ′. Likewise,the unit tangent vector v z of γ at z, while contained in V z , is orthogonal toṼ z , because it corresponds to the largest one among the s negative eigenvaluesof d 2 f(z). Taking the interior product i(v z ) dx 1 ∧ ... ∧ dx s defines anorientation of Ṽz. Since Ṽy depends smoothly on y, we may transport theorientation of Ṽz to Ṽz ′ along γ. We then define n γ = +1 or -1, dependingon whether this orientation of Ṽz ′ coincides with the chosen orientation ofV z ′ or not, and further define n(z, z ′ ) = ∑ γ n γ, where the sum is takenover all such paths γ of the steepest descent from p to p ′ .Now, let M s be the set of critical points of f of Morse index s, and letHf s be the vector space over R spanned by the elements of M s . We definea boundary operatorδ : H s−1f→ Hf s , by putting, for z ′ ∈ M s−1 ,δ(z ′ ) = ∑n(z ′ , z) z, and extending δ by linearity.n∈M s


412 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis operator satisfies δ 2 = 0 and therefore defines a cohomology theory.Using Conley’s continuation principle, Floer [Floer (1988)] showed thatthe resulting cohomology theories resulting from different choices of f arecanonically isomorphic.In his QFT–based rewriting the Morse topology, Ed Witten [Witten(1982)] considered also the operators:d t = e −tf de tf , their adjoints : d ∗ t = e tf de −tf ,as well as their Laplacian: ∆ t = d t d ∗ t + d ∗ t d t .For t = 0, ∆ 0 is the standard Hodge–de Rham Laplacian, whereas fort → ∞, one has the following expansion∆ t = dd ∗ + d ∗ d + t 2 ‖df‖ 2 + t ∑ k,j∂ 2 h∂x k ∂x j [i ∂ x k, dxj ],where (∂ x k) k=1,...,n is an orthonormal frame at the point under consideration.This becomes very large for t → ∞, except at the critical points of f,i.e., where df = 0. Therefore, the eigenvalues of ∆ t will concentrate nearthe critical points of f for t → ∞, and we get an interpolation between deRham cohomology and Morse cohomology.Morse Homology on M. Now, following [Milinkovic (1999); <strong>Ivancevic</strong>and Pearce (2006)], for any Morse function f on the configuration manifoldM we denote by Crit p (f) the set of its critical points of index p and defineC p (f) as a free Abelian group generated by Crit p (f). Consider the gradientflow generated by (3.228). Denote by M f,g (M) the set of all γ : R → Msatisfying (3.228) such thatThe spaces∫ +∞−∞2dγ∣ dt ∣ dt < ∞.M f,g (x − , x + ) = {γ ∈ M f,g (M) | γ(t) → x ± as t → ±∞}are smooth manifolds of dimension m(x + ) − m(x − ), where m(x) denotesthe Morse index of a critical point x. Note thatM f,g (x, y) ∼ = W u g (x) ∩ W s g (y),


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 413where W s g (y) and W u g (x) are the stable and unstable manifolds of the gradientflow (3.228). For generic g the intersection above is transverse (Morse–Smale condition). The group R acts on M f,g (x, y) by γ ↦→ γ(· + t). WedenotêM f,g (x, y) = M f,g (x, y)/R.The manifolds ̂Mf,g (x, y) can be given a coherent orientation σ(see [Schwarz (1993)]).Now, we can define the boundary operator, as∑∂ : C p (f) → C p−1 (f), ∂x =n(x, y)y,y∈Crit p−1(f)where n(x, y) is the number of points in 0D manifold ̂M f,g (x, y) countedwith the sign with respect to the orientation σ. The proof of ∂ ◦ ∂ = 0 isbased on gluing and cobordism arguments [Schwarz (1993)]. Now Morsehomology groups are defined byHpMorse (f) = Ker(∂)/Im(∂).For generic choices of Morse functions f 1 and f 2 the groups H p (f 1 ) andH p (f 2 ) are isomorphic. Furthermore, they are isomorphic to the singularhomology group of M, i.e.,H Morsep(f) ∼ = H sing (M),for generic f [Milnor (1965)].The construction of isomorphism is given (see [Milinkovic (1999);<strong>Ivancevic</strong> and Pearce (2006)]) asph αβ : H p (f α ) → H p (f β ), (3.229)for generic Morse functions f α , f β . Consider the ‘connecting trajectories’,i.e., the solutions of non–autonomous equation˙γ = −∇f αβt , (3.230)where f αβt is a homotopy connecting f α and f β such that for some R > 0{f αβ f α for t ≤ −Rt ≡f β for t ≥ R .


414 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFor x α ∈ Crit p (f α ) and x β ∈ Crit p (f β ) denoteM f αβ ,g(x α , x β ) = {γ : γ satisfies (3.230) andlim γ =t→−∞ xα , lim γ = x β }.t→∞As before, M f αβ ,g is a smooth finite–dimensional manifold. Now, define(h αβ ) ♯ : C p (f α ) → C p (f β ), by∑(h αβ ) ♯ x α =n(x α , x β )x β , for x α ∈ Crit p (f α ),x β ∈Crit p(f β )where n(x α , x β ) is the algebraic number of points in 0D manifoldM f αβ ,g(x α , x β ) counted with the signs defined by the orientation ofM f αβ ,g. Homomorphisms (h αβ ) ♯ commute with ∂ and thus define the homomorphismsh αβ in homology which, in addition, satisfy h αβ ◦h βγ = h αγ .Now, if we fix a Morse function f : M → R instead of a metric g,we establish the isomorphism (see [Milinkovic (1999); <strong>Ivancevic</strong> and Pearce(2006)])h αβ : H p (g α , f) → H p (g β , f)between the two Morse homology groups defined by means of two genericmetrics g α and g β in a similar way, by considering the ‘connecting trajectories’,˙γ = −∇ gαβ tf. (3.231)Here g αβt is a homotopy connecting g α and g β such that for some R > 0{g αβ g α for t ≤ −R,t ≡g β for t ≥ R,and ∇ g is a gradient defined by metric g.Note that f is decreasing along the trajectories solving autonomousgradient equation (3.228). Therefore, the boundary operator ∂ preservesthe downward filtration given by level sets of f. In other words, if we denoteCrit λ p(f) = Crit p (f) ∩ f −1 ((−∞, λ]),andC λ p (f) = free Abelian group generated by Crit λ p(f),then the boundary operator ∂ restricts to ∂ λ : C λ p (f) → C λ p−1(f). Obviously,∂ λ ◦ ∂ λ = 0, thus we can define the relative Morse homology groupsH λ p (f) = Ker(∂ λ )/Im(∂ λ ).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 415Following the standard algebraic construction, we define (relative)Morse cohomology. We setand defineC p λ (f) = Hom(Cλ p (f), Z),δ λand: C p λ(f) → Cp+1λ(f), 〈δ λ a, x〉 = 〈a, ∂ λ x〉H p λ (f) = Ker(δλ )/Im(δ λ ).Since Crit p (f) is finite, we have H λ p (f) = H p (f) and H p λ (f) = Hp (f).3.13.5.3 Hodge–De Rham Theory in BiodynamicsHodge LaplacianA single biodynamical configuration manifold M can be equipped withmany different Riemannian metrics g in local coordinates (apart from theone generated by its kinetic energy)g = g ij (u 1 , u 2 , ..., u n ) du i du j .Beltrami had shown that it is always possible for such a metric to definean operator (depending on the metric) that generalizes the usual Laplacianon R n and therefore induces the notion of harmonic functions on theRiemannian manifold [Choquet-Bruhat and DeWitt-Morete (1982)].Hodge theory was described by H. Weyl as ‘one of the landmarks inthe history of mathematics in the 20th Century’. Hodge showed that it waspossible to define a notion of harmonic exterior differential form: the metricg on M canonically defines a metric on the tangent bundle T M, hence also,by standard multilinear algebra, a metric on any bundle of tensors on M. Inparticular, let (α, β) ↦→ g p (α, β) be the positive nondegenerate symmetricbilinear form defined on the vector space of p−forms on M. As M isorientable, this defines a duality between p−forms and (n − p)−forms: toeach p−form α is associated a (n−p)−form ∗α, defined by the linear Hodgestar operator ∗ (see subsection 3.6.3.7), characterized by the relationsβ ∧ (∗α) = g p (α, β) v, ∗ ∗ α = (−1) p(n−p) α,for all p−forms α, β, where v is the volume form on the Riemannian manifoldM. If d is the exterior derivative, it has a transposed (adjoint) operator


416 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor that duality, the codifferential δ, defined asδ = −(∗) ◦ d ◦ (∗),which maps p−forms onto (p − 1)−forms, such thatδα = (−1) np+n+1 ∗ d ∗ α.asRecall (from subsection 3.6.2 above) that the Hodge Laplacian, defined∆ = d ◦ δ + δ ◦ d,transforms p−forms into p−forms and generalizes Beltrami’s Laplacian(3.47), which is the special case for p = 0 (up to a sign). This definesharmonic (real or complex valued) p−forms as those for which ∆α = 0, orequivalently, dα = δα = 0.In other words, let dv be the volume element of the chosen metric g.Then for every p−form α we can define a norm functional∫‖α‖ = (α, ∗α) g dv,for which the Euler–Lagrangian equation becomes ∆α = 0.Now, the pth Betti number of M can be defined asXb p = dim Ker ∆ p ,so that the Euler–Poincaré characteristics of M is given byχ(M) =n∑(−1) p b p =p=0n∑(−1) p dim Ker ∆ p . (3.232)Finally, for any (p − 1)−form α, (p + 1)−form β, and harmonic p−formγ (∆γ = 0) on the biodynamical configuration manifold M, the celebratedHodge–de Rham decomposition of a p−form ω [Griffiths (1983b); Voisin(2002)] givesp=0ω = dα + δβ + γ.Now, recall from section 3.14, that a large class of symplectic manifoldsis given by the Kähler manifolds. Let M be a smooth manifold and ga Riemannian metric on M. Let J be a complex structure on M, thatis, J : T M → T M, J 2 = −Id, and J is g−-orthogonal. M is called aKähler manifold if ∇j = 0, where ∇ is the Levi–Civita connection of g


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 417and J is regarded as a (1, 1) tensor–field. Define a 2−form ω on M byω(X, Y ) = g(JX, Y ), for each vector–field X, Y on M. Then (M, ω) is asymplectic manifold.Hodge theory takes place on the cohomology of the compact orientableconfiguration manifold M and reflects the subtle interplay of the followingbasic additional linear structures one can impose on M:• Symplectic structure ω ∈ Γ C ∞(M, Λ 2 TM ∨ ), where ω is nondegenerate,dω = 0.• Riemannian structure g ∈ Γ C ∞(M, S 2 TM ∨ ), where g is positive definite.• Complex structure J ∈ Γ C ∞(M, End(T M )), where J 2 = − id, and J isintegrable.The data (M, ω, g, J) satisfy the Kähler condition if ω, g and J arecompatible in the sense thatω(•, J(•)) = g(•, •),where • is the strong compatibility condition allowing the comparison ofdifferent cohomology theories.Recall that the de Rham cohomology of (M, J) is defined as()KerHDR(M) k =ImdΩ k (M) −→ Ω k+1 (M)() .Ω k−1 d(M) −→ Ω k (M)de Rham cohomology classes are represented by harmonic (natural) differentialforms.Let (M, g) be a compact oriented (real or complex) Riemannian manifold.Let dv be the volume element of g. Then for every k−form α we candefine∫‖α‖ = (α, ᾱ) g dv.MThe Euler–Lagrangian equation for the norm functional turns out to bedα = δα = 0. A k−form α ∈ Ω k (M) is called harmonic if it satisfies one ofthe following equivalent conditions:• α is closed and ‖α‖ ≤ ‖α + dβ‖ for all β ∈ Ω k−1 (M).• dα = δα = 0.• ∆α = 0, where ∆ = dδ + δd is the Hodge Laplacian.


418 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionHodge–Weyl Theorem [Griffiths (1983b); Voisin (2002)] states that everyde Rham cohomology class has a unique harmonic representative.Heat Kernel and Thermodynamics on MBesides pure mechanical consideration of biodynamical system, there isanother biophysical point of view – thermodynamical, compatible with thehuman motion [Hill (1938)]. Namely, the heat equation on the biodynamicalconfiguration manifold M,∂ t a(t) = ∆a(t), with initial condition a(0) = α,has a unique solution for every t ∈ [0, ∞) and every p−form α on M.If we think of α as an initial temperature distribution on M then as theconfiguration manifold cools down, according to the classical heat equation,the temperature should approach a steady state which should be harmonic[Davies (1989)].To prove this, we define a stationary and hence harmonic operatorH(α) = lim t→∞ a(t). Also, a map α → G(α) withG(α) =∫ ∞0a(t) dtis orthogonal to the space of harmonic forms and satisfies∆G(α) =∫ ∞0∆a(t) dt = −∫ ∞0∂ t a(t) dt = α − H(α).Here, the map α → H(α) is called harmonic projection and the map α →G(α) is called Green’s operator.In particular, for each p−form α we get a unique decompositionα = H(α) + ∆G(α).This proves the existence of a harmonic representative in every de Rhamcohomology class, as follows.Let α ∈ Ω p (M) be a closed form. Thenα = H(α) + dd ∗ G(α) + d ∗ dG(α).But the three terms in this sum are orthogonal and so‖d ∗ dG(α)‖ = 〈d ∗ dG(α), α〉 = 〈dG(α), dα〉 = 0,since α is closed. Thus H(α) is cohomologous to α.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 419This thermal reflection on the biodynamics topology complies with thebasic biophysics of human muscles (see [Hill (1938)]).3.13.5.4 Lagrangian–Hamiltonian Duality in BiodynamicsThe present section uncovers the underlying dual geometro–topologicalstructure beneath the general biodynamics. It presents a parallel developmentof Hamiltonian and Lagrangian formulations of biodynamics (see[<strong>Ivancevic</strong> and Snoswell (2001); <strong>Ivancevic</strong> (2002); <strong>Ivancevic</strong> and Pearce(2001b); <strong>Ivancevic</strong> and Pearce (2001b); <strong>Ivancevic</strong> (2005)]), proves bothdifferential–geometrical and algebraic–topo-logical dualities between thesetwo formulations, and finally establishes a unique functorial relation betweenbiodynamics geometry and biodynamics topology.Lagrangian formulation of biodynamics is performed on the tangentbundle T M, while Hamiltonian formulation is performed on the cotangentbundle T ∗ M. Both Riemannian and symplectic geometry are used. Thegeometrical duality (see [Kolar et al. (1993); Choquet-Bruhat and DeWitt-Morete (1982)]) of Lie groups and algebras between these two biodynamicsformulations is proved as an existence of natural equivalence between Lieand canonical functors. The topological duality (see [Dodson and Parker(1997)]) between these two biodynamics formulations is proved as an existenceof natural equivalence between Lagrangian and Hamiltonian functorsin both homology and cohomology categories. In the case of reduced configurationmanifold, the Betti numbers and Euler–Poincaré characteristicare given.<strong>Geom</strong>etrical Duality Theorem for MTheorem. There is a geometrical duality between rotational Lagrangianand Hamiltonian biodynamical formulations on M (as given by Figure 3.6).In categorical terms, there is a unique natural geometrical equivalenceDual G : Lie ∼ = Canin biodynamics (symbols are described in the next subsection).Proof. The proof has two parts: Lie–functorial and geometrical.Lie–Functorial Proof. If we apply the functor Lie on the category • [SO(n) i ](for n = 2, 3 and i = 1, . . . , N) of rotational Lie groups SO(n) i (and theirhomomorphisms) we get the category • [so(n) i ] of corresponding tangentLie algebras so(n) i (and their homomorphisms). If we further apply the


420 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionisomorphic functor Dual to the category • [so(n) i ] we get the dual category∗ •[so(n) ∗ i ] of cotangent, or, canonical Lie algebras so(n)∗ i (and theirhomomorphisms). To go directly from • [SO(n) i ] to ∗ •[so(n) ∗ i ] we use thecanonical functor Can. Therefore, we have a commutative triangle:Lie ✠•[so(n) i]• [SO(n) i ]❅❅❅❅❅❅❘CanLGA∼ =Dual A✲∗•[so(n) ∗ i ]Applying the functor Lie on the biodynamical configuration manifoldM, we get the product–tree of the same anthropomorphic structure, buthaving tangent Lie algebras so(n) i as vertices, instead of the groups SO(n) i .Again, applying the functor Can on M, we get the product–tree of the sameanthropomorphic structure, but this time having cotangent Lie algebrasso(n) ∗ i as vertices. Both the tangent algebras so(n) i and the cotangentalgebras so(n) ∗ i contain infinitesimal group generators: angular velocities˙q i = ˙q φ i – in the first case, and canonical angular momenta pi = p φi – in thesecond case [<strong>Ivancevic</strong> and Snoswell (2001)]. As Lie group generators, boththe angular velocities and the angular momenta satisfy the commutationrelations: [ ˙q φ i , ˙qψ i] = ɛφψθ˙q θi and [p φi , p ψi ] = ɛ θ φψ p θ i, respectively, wherethe structure constants ɛ φψθand ɛ θ φψconstitute the totally antisymmetricthird–order tensors.In this way, the functor Dual G : Lie ∼ = Can establishes the unique geometricalduality between kinematics of angular velocities ˙q i (involved inLagrangian formalism on the tangent bundle of M) and kinematics of angularmomenta p i (involved in Hamiltonian formalism on the cotangentbundle of M), which is analyzed below. In other words, we have two functors,Lie and Can, from the category of Lie groups (of which • [SO(n) i ] isa subcategory) into the category of (their) Lie algebras (of which • [so(n) i ]and ∗ •[so(n) ∗ i ] are subcategories), and a unique natural equivalence betweenthem defined by the functor Dual G . (As angular momenta p i are in a bijectivecorrespondence with angular velocities ˙q i , every component of thefunctor Dual G is invertible.) <strong>Geom</strong>etrical Proof. <strong>Geom</strong>etrical proof is given along the lines of Riemannianand symplectic geometry of mechanical systems, as follows (see 3.13.1


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 421above, as well as [Marsden and Ratiu (1999); <strong>Ivancevic</strong> and Snoswell (2001);<strong>Ivancevic</strong> (2002); <strong>Ivancevic</strong> and Pearce (2001b); <strong>Ivancevic</strong> (2005)]). Recallthat the Riemannian metric g = on the configuration manifold M isa positive–definite quadratic form g : T M → R, given in local coordinatesq i ∈ U (U open in M) asg ij ↦→ g ij (q, m) dq i dq j ,∂x r ∂x sg ij (q, m) = m µ δ rs∂q i ∂q jwhereis the covariant material metric tensor g, defining a relation between internaland external coordinates and including n segmental masses m µ .The quantities x r are external coordinates (r, s = 1, . . . , 6n) and i, j =1, . . . , N ≡ 6n − h, where h denotes the number of holonomic constraints.The Lagrangian of the system is a quadratic form L : T M → R dependenton velocity v and such that L(v) = 1 2< v, v >. It is given byL(v) = 1 2 g ij(q, m) v i v jin local coordinates q i , v i = ˙q i ∈ U v (U v open in T M). The Hamiltonianof the system is a quadratic form H : T ∗ M → R dependent on momentump and such that H(p) = 1 2< p, p >. It is given byH(p) = 1 2 gij (q, m) p i p jin local canonical coordinates q i , p i ∈ U p (U p open in T ∗ M). The inverse(contravariant) metric tensor g −1 , is defined asg ij (q, m) = m µ δ rs∂q i∂x r ∂q j∂x s .For any smooth function L on T M, the fibre derivative, or Legendretransformation, is a diffeomorphism FL : T M → T ∗ M, F(w) · v = , from the momentum phase–space manifold to the velocity phase–space manifold associated with the metric g = . In local coordinatesq i , v i = ˙q i ∈ U v (U v open in T M), FL is given by (q i , v i ) ↦→ (q i , p i ).Recall that on the momentum phase–space manifold T ∗ M exists:(i) A unique canonical 1−form θ H with the property that, for any 1−formβ on the configuration manifold M, we have β ∗ θ H = β. In local canonicalcoordinates q i , p i ∈ U p (U p open in T ∗ M) it is given by θ H = p i dq i .(ii) A unique nondegenerate Hamiltonian symplectic 2−form ω H , which is


422 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionclosed (dω H = 0) and exact (ω H = dθ H = dp i ∧dq i ). Each body segmenthas, in the general SO(3) case, a sub–phase–space manifold T ∗ SO(3) withω (sub)H= dp φ ∧ dφ + dp ψ ∧ dψ + dp θ ∧ dθ.Analogously, on the velocity phase–space manifold T M exists:(i) A unique 1−form θ L , defined by the pull–back θ L = (FL) ∗ θ H of θ Hby FL. In local coordinates q i , v i = ˙q i ∈ U v (U v open in T M) it is givenby θ L = L v idq i , where L v i ≡ ∂L/∂v i .(ii) A unique nondegenerate Lagrangian symplectic 2−form ω L , defined bythe pull–back ω L = (FL) ∗ ω H of ω H by FL, which is closed (dω L = 0)and exact (ω L = dθ L = dL v i ∧ dq i ).Both T ∗ M and T M are orientable manifolds, admitting the standardvolumes given respectively byΩ ωH , =N(N+1)(−1) 2N!ω N H, and Ω ωL =N(N+1)(−1) 2N!ω N L ,in local coordinates q i , p i ∈ U p (U p open in T ∗ M), resp. q i , v i = ˙q i ∈ U v(U v open in T M). They are given byΩ H = dq 1 ∧ · · · ∧ dq N ∧ dp 1 ∧ · · · ∧ dp N , andΩ L = dq 1 ∧ · · · ∧ dq N ∧ dv 1 ∧ · · · ∧ dv N .On the velocity phase–space manifold T M we can also define the actionA : T M → R by A(v) = FL(v) · v and the energy E = A − L. In localcoordinates q i , v i = ˙q i ∈ U v (U v open in T M) we have A = v i L v i, soE = v i L v i − L. The Lagrangian vector–field X L on T M is determinedby the condition i XL ω L = dE. Classically, it is given by the second–orderLagrangian equationsddt L v i = L qi. (3.233)The Hamiltonian vector–field X H is defined on the momentum phase–space manifold T ∗ M by the condition i XH ω = dH. ( The condition ) may be0 Iexpressed equivalently as X H = J∇H, where J = .−I 0In local canonical coordinates q i , p i ∈ U p (U p open in T ∗ M) the vector–field X H is classically given by the first–order Hamiltonian canonical equations˙q i = ∂ pi H,p˙i = − ∂ q iH. (3.234)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 423As a Lie group, the configuration manifold M is Hausdorff. Thereforefor x = (q i , p i ) ∈ U p (U p open in T ∗ M) there exists a uniqueone–parameter group of diffeomorphisms φ t : T ∗ M → T ∗ M such thatddt | t=0 φ t x = J∇H(x). This is termed Hamiltonian phase–flow and representsthe maximal integral curve t ↦→ (q i (t), p i (t)) of the Hamiltonianvector–field X H passing through the point x for t = 0.The flow φ t is symplectic if ω H is constant along it (that is, φ ∗ t ω H = ω H )iff its Lie derivative vanishes (that is, L XH ω H = 0). A symplectic flow consistsof canonical transformations on T ∗ M, that is, local diffeomorphismsthat leave ω H invariant. By Liouville Theorem, a symplectic flow φ t preservesthe phase volume on T ∗ M. Also, the total energy H = E of thesystem is conserved along φ t , that is, H ◦ φ t = φ t .Lagrangian flow can be defined analogously (see [Abraham and Marsden(1978); Marsden and Ratiu (1999)]).For a Lagrangian (resp. a Hamiltonian) vector–field X L (resp. X H )on M, there is a base integral curve γ 0 (t) = (q i (t), v i (t)) (resp. γ 0 (t) =(q i (t), p i (t))) iffγ 0 (t) is a geodesic. This is given by the contravariant velocityequation˙q i = v i , ˙v i + Γ i jk v j v k = 0, (3.235)in the former case, and by the covariant momentum equation˙q k = g ki p i , ṗ i + Γ i jk g jl g km p l p m = 0, (3.236)in the latter. As before, Γ i jkdenote the Christoffel symbols of an affineconnection ∇ in an open chart U on M, defined by the Riemannian metricg = as: Γ i jk = ( ) gil Γ jkl , Γ jkl = 1 2 ∂q j g kl + ∂ q kg jl − ∂ q lg jk .The l.h.s ˙¯v i = ˙v i + Γ i jk vj v k (resp. ˙¯p i = ṗ i + Γ i jk gjl g km p l p m ) inthe second parts of (3.235) and (3.236) represent the Bianchi covariantderivative of the velocity (resp. momentum) with respect to t. Paralleltransport on M is defined by ˙¯v i = 0, (resp. ˙¯p i = 0). When this applies,X L (resp. X H ) is called the geodesic spray and its flow the geodesic flow.For the dynamics in the gravitational potential field V : M → R, theLagrangian L : T M → R (resp. the Hamiltonian H : T ∗ M → R) has anextended formL(v, q) = 1 2 g ijv i v j − V (q),(resp.H(p, q) = 1 2 gij p i p j + V (q)).


424 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionA Lagrangian vector–field X L (resp. Hamiltonian vector–field X H ) isstill defined by the second–order Lagrangian equations (3.233, 3.235) (resp.first–order Hamiltonian equations (3.234, 3.236)).The fibre derivative FL : T M → T ∗ M thus maps Lagrangian equations(3.233, 3.235) into Hamiltonian equations (3.234, 3.236). Clearly thereexists a diffeomorphism FH : T ∗ M → T M, such that FL = (FH) −1 . Inlocal canonical coordinates q i , p i ∈ U p (U p , open in T ∗ M) this is given by(q i , p i ) ↦→ (q i , v i ) and thus maps Hamiltonian equations (3.234, 3.236) intoLagrangian equations (3.233, 3.235).A general form of the forced, non–conservative Hamiltonian equations(resp. Lagrangian equations) is given as(resp.˙ q i =∂H ,∂p id ∂Ldt ∂v ip˙i = − ∂H∂q i + F i (t, q i , p i ),− ∂L∂q i = F i (t, q i , v i )).Here the F i (t, q i , p i ) (resp. F i (t, q i , v i )) represent any kind of covariantforces, including dissipative and elastic joint forces, as well as actuatordrives and control forces, as a function of time, coordinates and momenta.In covariant form we have(resp.˙ q k = g ki p i , ṗ i + Γ i jk g jl g km p l p m = F i (t, q i , p i ),˙ qi = v i , ˙v i + Γ i jk v j v k = g ij F j (t, q i , v i )). This proves the existence of the unique natural geometrical equivalencein the rotational biodynamics.Dual G : Lie ∼ = CanTopological Duality Theorem for MIn this section we want to prove that the general biodynamics can beequivalently described in terms of two topologically dual functors Lag andHam, from <strong>Diff</strong>, the category of smooth manifolds (and their smooth maps)of class C p , into Bund, the category of vector bundles (and vector–bundlemaps) of class C p−1 , with p ≥ 1. Lag is physically represented by thesecond–order Lagrangian formalism on T M ∈ Bund, while Ham is physicallyrepresented by the first–order Hamiltonian formalism on T ∗ M ∈ Bund.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 425Theorem. There is a topological duality between Lagrangian and Hamiltonianformalisms on M (as given by Figure 3.6). In categorical terms, thereis a unique natural topological equivalenceDual T : Lag ∼ = Hamin the general biodynamics.Proof. The proof has two parts: cohomological and homological.Cohomological Proof. If C = H • M (resp. C = L • M) represents theAbelian category of cochains on the momentum phase–space manifoldT ∗ M (resp. the velocity phase–space manifold T M), we have the categoryS • (H • M) (resp. S • (L • M)) of generalized cochain complexes A • in H • M(resp. L • M) and if A ′ = 0 for n < 0 we have a subcategory S • DR (H• M)(resp. S • DR (L• M)) of de Rham differential complexes in S • (H • M) (resp.S • (L • M))A • DR : 0 → Ω 0 (T ∗ M)d−→ Ω 2 (T ∗ M)(resp. A • DR : 0 → Ω 0 (T M)· · ·d−→ Ω 1 (T ∗ dM) −→d−→ · · ·d−→ Ω 1 (T M)d−→ Ω N d(T M) −→ · · · ),d−→ Ω N (T ∗ M)d−→ Ω 2 (T M)d−→ · · ·d−→where A ′ = Ω N (T ∗ M) (resp. A ′ = Ω N (T M)) is the vector space of allN−-forms on T ∗ M (resp. T M) over R.Let Z N (T ∗ M) = Ker(d) (resp. Z N (T ) = Ker(d)) and B N (T ∗ M) =Im(d) (resp. B N (T M) = Im(d)) denote respectively the real vector spacesof cocycles and coboundaries of degree N. Since d N+1 d N = d 2 = 0, itfollows that B N (T ∗ M) ⊂ Z N (T ∗ M) (resp. B N (T M) ⊂ Z N (T M)). Thequotient vector space(resp.H N DR(T ∗ M) = Ker(d)/ Im(d) = Z N (T ∗ M)/B N (T ∗ M)H N DR(T M) = Ker(d)/ Im(d) = Z N (T M)/B N (T M)),we refer to as the de Rham cohomology group (vector space) of T ∗ M (resp.T M). The elements of HDR N (T ∗ M) (resp. HDR N (T M)) are equivalence setsof cocycles. Two cocycles ω 1 and ω 2 are cohomologous, or belong to thesame equivalence set (written ω 1 ∼ ω 2 ) iff they differ by a coboundaryω 1 − ω 2 = dθ . Any form ω H ∈ Ω N (T ∗ M) (resp. ω L ∈ Ω N (T M) has a deRham cohomology class [ω H ] ∈ HDR N (T ∗ M) (resp. [ω L ] ∈ HDR N (T M)).Hamiltonian symplectic form ω H = dp i ∧ dq i on T ∗ M (resp. Lagrangiansymplectic form ω L = dL v i ∧ dq i on T M) is by definition both


426 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona closed 2−form or two–cocycle and an exact 2−form or two–coboundary.Therefore the 2D–de Rham cohomology group of human motion is definedas a quotient vector space(resp.H 2 DR(T ∗ M) = Z 2 (T ∗ M)/B 2 (T ∗ M)H 2 DR(T M) = Z 2 (T M)/B 2 (T M)).As T ∗ M (resp. T M) is a compact Hamiltonian symplectic (resp. Lagrangiansymplectic) manifold of dimension 2N, it follows that ω N H (resp.ω N L ) is a volume element on T ∗ M (resp. T M), and the 2ND de Rham’s cohomologyclass [ ωH] N ∈ H2NDR (T ∗ M) (resp. [ ]ω N L ∈ H2NDR (T M)) is nonzero.Since [ ωH] N = [ωH ] N [ ](resp. ωNL = [ωL ] N ), then [ω H ] ∈ HDR 2 (T ∗ M)(resp. [ω L ] ∈ HDR 2 (T M) ) and all of its powers up to the N−-th must bezero as well. The existence of such an element is a necessary condition forT ∗ M (resp. T M) to admit a Hamiltonian symplectic structure ω H (resp.Lagrangian symplectic structure ω L ).The de Rham complex A • DR on T ∗ M (resp. T M) can be consideredas a system of second–order ODEs d 2 θ H = 0, θ H ∈ Ω N (T ∗ M) (resp.d 2 θ L = 0, θ L ∈ Ω N (T M)) having a solution represented by Z N (T ∗ M)(resp. Z N (T M)). In local coordinates q i , p i ∈ U p (U p open in T ∗ M) (resp.q i , v i ∈ U v (U v open in T M)) we have d 2 θ H = d 2 (p i dq i ) = d(dp i ∧ dq i ) =0, (resp. d 2 θ L = d 2 (L v idq i ) = d(dL v i ∧ dq i ) = 0). Homological Proof. If C = H • M, (resp. C = L • M) represents an Abeliancategory of chains on T ∗ M (resp. T M), we have a category S • (H • M) (resp.S • (L • M)) of generalized chain complexes A • in H • M (resp. L • M), andif A = 0 for n < 0 we have a subcategory S• C (H • M) (resp. S• C (L • M)) ofchain complexes in H • M (resp. L • M)A • : 0 ← C 0 (T ∗ M)· · ·(resp. A • : 0 ← C 0 (T M)∂←− C 1 (T ∗ M)∂←− C n (T ∗ ∂M) ←− · · ·∂←− C 1 (T M)∂· · · ←− C n ∂(T M) ←− · · · ).∂←− C 2 (T ∗ M)∂←− C 2 (T M)∂←− · · ·∂←− · · ·Here A N = C N (T ∗ M) (resp. A N = C N (T M)) is the vector space of allfinite chains C on T ∗ M (resp. T M) over R, and ∂ N = ∂ : C N+1 (T ∗ M) →C N (T ∗ M) (resp. ∂ N = ∂ : C N+1 (T M) → C N (T M)). A finite chain Csuch that ∂C = 0 is an N−cycle. A finite chain C such that C = ∂Bis an N−boundary. Let Z N (T ∗ M) = Ker(∂) (resp. Z N (T M) = Ker(∂))and B N (T ∗ M) = Im(∂) (resp. B N (T M) = Im(∂)) denote respectively


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 427real vector spaces of cycles and boundaries of degree N. Since ∂ N−1 ∂ N =∂ 2 = 0, then B N (T ∗ M) ⊂ Z N (T ∗ M) (resp. B N (T M) ⊂ Z N (T M)). Thequotient vector space(resp.H C N(T ∗ M) = Z N (T ∗ M)/B N (T ∗ M)H C N(T M) = Z N (T M)/B N (T M))represents an ND biodynamics homology group (vector space). The elementsof HN C(T ∗ M) (resp. HN C (T M)) are equivalence sets of cycles. Twocycles C 1 and C 2 are homologous, or belong to the same equivalence set(written C 1 ∼ C 2 ) iff they differ by a boundary C 1 − C 2 = ∂B. The homologyclass of a finite chain C ∈ C N (T ∗ M) (resp. C ∈ C N (T M)) is[C] ∈ HN C(T ∗ M) (resp. [C] ∈ HN C (T M)). Lagrangian Versus Hamiltonian DualityIn this way, we have proved a commutativity of a triangle:Lag ✠TanBund<strong>Diff</strong>ManMFB∼ =Dual THam❅❅❅❅❅ ❅❘✲ CotBundwhich implies the existence of the unique natural topological equivalencein the rotational biodynamics.Dual T : Lag ∼ = HamGlobally Dual Structure of Rotational BiodynamicsTheorem. Global dual structure of the rotational biodynamics is definedby the unique natural equivalenceDyn : Dual G∼ = DualT .


428 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionProof. This unique functorial relation, uncovering the natural equivalencebetween geometrical and topological structures of biodynamics:• [SO(n) i ]❅❅❅❅❅❅❘CanLGALie ✠∗•[so(n) i ] ✲∼ • [so(n) ∗ i ]=Dual GF⊣❄✻GLag ✠TanBund<strong>Diff</strong>Man❅❅❅ HamMFB∼ =Dual T❅❅❅❘✲CotBund– has been established by parallel development of Lagrangian and Hamiltonianbiodynamics formulations, i.e., functors Lag(Lie) and Ham(Can). 3.14 Complex and Kähler Manifolds and TheirApplicationsJust as a smooth manifold has enough structure to define the notion ofdifferentiable functions, a complex manifold is one with enough structureto define the notion of holomorphic (or, analytic) functions f : X → C.Namely, if we demand that the transition functions φ j ◦ φ −1i in the chartsU i on M (see Figure 3.12) satisfy the Cauchy–Riemann equations∂ x u = ∂ y v, ∂ y u = −∂ x v,then the analytic properties of f can be studied using its coordinate representativef ◦ φ −1i with assurance that the conclusions drawn are patch


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 429independent. Introducing local complex coordinates in the charts U i onM, the φ i can be expressed as maps from U i to an open set in C n 2 , withφ j ◦ φ −1i being a holomorphic map from C n 2 to C n 2 . Clearly, n must beeven for this to make sense. In local complex coordinates, we recall thata function h : C n 2 → C n 2 is holomorphic if h(z 1 , ¯z 1 , ..., z n 2 , ¯z n 2 ) is actuallyindependent of all the ¯z j .In a given patch on any even–dimensional manifold, we can always introducelocal complex coordinates by, for instance, forming the combinationsz j = x j + ix n 2 +j , where the x j are local real coordinates on M. The realtest is whether the transition functions from one patch to another — whenexpressed in terms of the local complex coordinates — are holomorphicmaps. If they are, we say that M is a complex manifold of complex dimensiond = n/2. The local complex coordinates with holomorphic transitionfunctions give M with a complex structure (see [Greene (1996)]).Fig. 3.12 The charts for a complex manifold M have complex coordinates (see text forexplanation).Given a smooth manifold with even real dimension n, it can be a difficultquestion to determine whether or not a complex structure exists. On theother hand, if some smooth manifold M does admit a complex structure,we are not able to decide whether it is unique, i.e., there may be numerousinequivalent ways of defining complex coordinates on M.Now, in the same way as a homeomorphism defines an equivalence betweentopological manifolds, and a diffeomorphism defines an equivalencebetween smooth manifolds, a biholomorphism defines an equivalence betweencomplex manifolds. If M and N are complex manifolds, we considerthem to be equivalent if there is a map φ : M → N which in additionto being a diffeomorphism, is also a holomorphic map. That is, when expressedin terms of the complex structures on M and N respectively, φ is


430 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionholomorphic. It is not hard to show that this necessarily implies that φ −1is holomorphic as well and hence φ is known as a biholomorphism. Such amap allows us to identify the complex structures on M and N and hencethey are isomorphic as complex manifolds.These definitions are important because there are pairs of smooth manifoldsM and N which are homeomorphic but not diffeomorphic, as well as,there are complex manifolds M and N which are diffeomorphic but not biholomorphic.This means that if one simply ignored the fact that M and Nadmit local complex coordinates (with holomorphic transition functions),and one only worked in real coordinates, there would be no distinction betweenM and N. The difference between them only arises from the way inwhich complex coordinates have been laid down upon them.Again, recall that a tangent space to a manifold M at a point p isthe closest flat approximation to M at that point. A convenient basis forthe tangent space of M at p consists of the n linearly independent partialderivatives,T p M : {∂ x 1| p , ..., ∂ x n| p }. (3.237)A vector v ∈ T p M can then be expressed as v = v α ∂ x α| p .Also, a convenient basis for the dual, cotangent space T ∗ p M, is the basisof one–forms, which is dual to (3.237) and usually denoted byT ∗ p M : {dx 1 | p , ..., dx n | p }, (3.238)where, by definition, dx i : T p M → R is a linear map with dx i p(∂ x j | p ) = δ i j.Now, if M is a complex manifold of complex dimension d = n/2, there isa notion of the complexified tangent space of M, denoted by T p M C , whichis the same as the real tangent space T p M except that we allow complexcoefficients to be used in the vector space manipulations. This is oftendenoted by writing T p M C = T p M ⊗ C. We can still take our basis tobe as in (3.237) with an arbitrary vector v ∈ T p M C being expressed asv = v α ∂∂x| α p , where the v α can now be complex numbers. In fact, it isconvenient to rearrange the basis vectors in (3.237) to more directly reflectthe underlying complex structure. Specifically, we take the following linearcombinations of basis vectors in (3.237) to be our new basis vectors:T p M C : {(∂ x 1 + i∂ x d+1)| p , ..., (3.239)(∂ x d + i∂ x 2D)| p , (∂ x 1 − i∂ x d+1)| p , ..., (∂ x d − i∂ x 2D)| p }.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 431In terms of complex coordinates we can write the basis (3.239) asT p M C : {∂ z 1| p , ..., ∂ z d| p , ∂¯z 1| p , ..., ∂¯z d| p }.From the point of view of real vector spaces, ∂ x j | p and i∂ x j | p would beconsidered linearly independent and hence T p M C has real dimension 4D.In exact analogy with the real case, we can define the dual to T p M C ,which we denote by T ∗ p M C = T ∗ p M ⊗ C, with the one–forms basisT ∗ p M C : {dz 1 | p , ..., dz d | p , d¯z 1 | p , ..., d¯z d | p }.For certain types of complex manifolds M, it is worthwhile to refine thedefinition of the complexified tangent and cotangent spaces, which pullsapart the holomorphic and anti–holomorphic directions in each of thesetwo vector spaces. That is, we can writeT p M C = T p M (1,0) ⊕ T p M (0,1) ,where T p M (1,0) is the vector space spanned by {∂ z 1| p , ..., ∂ z d| p } andT p M (0,1) is the vector space spanned by {∂¯z 1| p , ..., ∂¯z d| p }. Similarly, wecan writeT ∗ p M C = T ∗ p M (1,0) ⊕ T ∗ p M (0,1) ,where Tp ∗ M (1,0) is the vector space spanned by {dz 1 | p , ..., dz d | p } andTp ∗ M (0,1) is the vector space spanned by {d¯z 1 | p , ..., d¯z d | p }. We call T p M (1,0)the holomorphic tangent space; it has complex dimension d and we callTp ∗ M 1,0 the holomorphic cotangent space. It also has complex dimensiond. Their complements are known as the anti–holomorphic tangent andcotangent spaces respectively [Greene (1996)].Now, a complex vector bundle is a vector bundle π : E → M whose fiberbundle π −1 (x) is a complex vector space. It is not necessarily a complexmanifold, even if its base manifold M is a complex manifold. If a complexvector bundle also has the structure of a complex manifold, and isholomorphic, then it is called a holomorphic vector bundle.3.14.1 Complex Metrics: Hermitian and KählerIf M is a complex manifold, there is a natural extension of the metric g toa mapg : T p M C × T p M C → C,


432 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondefined in the following way. Let r, s, u, v be four vectors in the tangentspace T p M to a complex manifold M. Using them, we can construct, forexample, two vectors w (1) = r + is and w (2) = u + iv which lie in T p M C .Then, we evaluate g on w (1) and w (2) by linearity:g(w (1) , w (2) ) = g(r + is, u + iv) = g(r, u) − g(s, v) + i [g(r, v) + g(s, u)] .We can define components of this extension of the original metric (whichwe have called by the same symbol) with respect to complex coordinatesin the usual way: g ij = g( ∂ ∂∂z, i ∂z), g j i¯j = g( ∂ ∂∂z, i ∂) and so forth. The¯z¯jreality of our original metric g and its symmetry implies that in complexcoordinates we have g ij = g ji , g i¯j = g¯ji and g ij = gī¯j , g i¯j = gīj .Now, recall that a Hermitian metric on a complex vector bundle assignsa Hermitian inner product to every fiber bundle. The basic example is thetrivial bundle π : U × C 2 → U, where U is an open set in R n . Then apositive definite Hermitian matrix H defines a Hermitian metric by〈v, w〉 = v T H ¯w,where ¯w is the complex conjugate of w. By a partition of unity, any complexvector bundle has a Hermitian metric.In local coordinates of a complex manifold M, a metric g is Hermitianif g ij = gī¯j = 0. In this case, only the mixed type components of g arenonzero and hence it can be written asg = g i¯j dz i ⊗ d¯z¯j + gīj d¯zī ⊗ dz j .With a little bit of algebra one can work out the constraint this implies forthe original metric written in real coordinates. Formally, if J is a complexstructure acting on the real tangent space T p M, i.e.J : T p M → T p M with J 2 = −I,then the Hermiticity condition on g is g(Jv (1) , Jv (2) ) = g(v (1) , v (2) ).On a holomorphic vector bundle with a Hermitian metric h, there is aunique connection compatible with h and the complex structure. Namely,it must be ∇ = ∂ + ¯∂.In the special case of a complex manifold, the complexified tangentbundle T M ⊗ C may have a Hermitian metric, in which case its real part isa Riemannian metric and its imaginary part is a nondegenerate alternatingmultilinear form ω. When ω is closed, i.e., in this case a symplectic form,then ω is called the Kähler form. Formally, given a Hermitian metric g on


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 433M, we can build a form in Ω 1,1 (M) — that is, a form of type (1, 1) in thefollowing way:ω = ig i¯j dz i ⊗ d¯z¯j − ig¯ji d¯z¯j ⊗ dz i .By the symmetry of g, we can write this asω = ig i¯j dz i ∧ d¯z j .Now, if ω is closed, that is, if dJ = 0, then ω is called a Kähler form and Mis called a Kähler manifold. At first sight, this Kählerity condition mightnot seem too restrictive. However, it leads to remarkable simplifications inthe resulting differential geometry on M.A Kähler structure on a complex manifold M combines a Riemannianmetric on the underlying real manifold with the complex structure. Sucha structure brings together geometry and complex analysis, and the mainexamples come from algebraic geometry. When M has n complex dimensions,then it has 2n real dimensions. A Kähler structure is related to theunitary group U(n), which embeds in SO(2n) as the orthogonal matricesthat preserve the almost complex structure (multiplication by i). In a coordinatechart, the complex structure of M defines a multiplication by i andthe metric defines orthogonality for tangent vectors. On a Kähler manifold,these two notions (and their derivatives) are related.A Kähler manifold is a complex manifold for which the exterior derivativeof the fundamental form ω associated with the given Hermitian metricvanishes, so dω = 0. In other words, it is a complex manifold with a Kählerstructure. It has a Kähler form, so it is also a symplectic manifold. It hasa Kähler metric, so it is also a Riemannian manifold.The simplest example of a Kähler manifold is a Riemann surface, whichis a complex manifold of dimension 1. In this case, the imaginary part ofany Hermitian metric must be a closed form since all 2−forms are closedon a real 2D manifold.In other words, a Kähler form is a closed two–form ω on a complexmanifold M which is also the negative imaginary part of a Hermitian metrich = g−iw. In this case, M is called a Kähler manifold and g, the real part ofthe Hermitian metric, is called a Kähler metric. The Kähler form combinesthe metric and the complex structure, g(M, Y ) = ω(M, JY ),where ω isthe almost complex structure induced by multiplication by i. Since theKähler form comes from a Hermitian metric, it is preserved by ω, sinceh(M, Y ) = h(JX, JY ). The equation dω = 0 implies that the metric and


434 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe complex structure are related. It gives M a Kähler structure, and hasmany implications.In particular, on C 2 , the Kähler form can be written asω = − i 2(dz1 ∧ dz 1 + dz 2 ∧ dz 2)= dx1 ∧ dy 1 + dx 2 ∧ dy 2 ,where z n = x n + i y n . In general, the Kähler form can be written incoordinatesω = g ij dz i ∧ dz j ,where g ij is a Hermitian metric, the real part of which is the Kähler metric.Locally, a Kähler form can be written as i∂ ¯∂f, where f is a function calleda Kähler potential. The Kähler form is a real (1, 1)−complex form. TheKähler potential is a real–valued function f on a Kähler manifold for whichthe Kähler form ω can be written as ω = i∂ ¯∂f, where,∂ = ∂ zk dz k and ¯∂ = ∂¯zk d¯z k .In local coordinates, the fact that dJ = 0 for a Kähler manifold MimpliesThis implies thatdJ = (∂ + ¯∂)ig i¯j dz i ∧ d¯z¯j = 0.∂ z lg i¯j = ∂ z ig l¯j (3.240)and similary with z and ¯z interchanged. From this we see that locally wecan express g i¯j asg i¯j =∂2 φ∂z i ∂¯z¯j .That is, ω = i∂ ¯∂φ, where φ is a locally defined function in the patch whoselocal coordinates we are using, which is known as the Kähler potential.If ω on M is a Kähler form, the conditions (3.240) imply that thereare numerous cancellations in (3.240). so that the only nonzero Christoffelsymbols (of the standard Levi–Civita connection) in complex coordinatesare those of the form Γ l jkholomorphic. Specifically,and Γ¯l¯j¯k, with all indices holomorphic or anti–Γ l jk = g l¯s ∂ z j g k¯s and Γ¯l¯j¯k = g¯ls ∂¯z¯jg¯ks .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 435The curvature tensor also greatly simplifies. The only non–zero componentsof the Riemann tensor, when written in complex coordinates, havethe form R i¯jk¯l (up to index permutations consistent with symmetries of thecurvature tensor). And we haveas well as the Ricci tensorR i¯jk¯l = g i¯s ∂ z kΓ¯s¯j¯l ,Rīj = R¯kī¯kj = −∂ z j Γ¯kī¯k .Since the Kähler form ω is closed, it represents a cohomology class in thede Rham cohomology. On a compact manifold, it cannot be exact becauseω n /n! ≠ 0 is the volume form determined by the metric. In the special caseof a projective variety, the Kähler form represents an integral cohomologyclass. That is, it integrates to an integer on any 1D submanifold, i.e., analgebraic curve. The Kodaira Embedding Theorem says that if the Kählerform represents an integral cohomology class on a compact manifold, thenit must be a projective variety. There exist Kähler forms which are notprojective algebraic, but it is an open question whether or not any Kählermanifold can be deformed to a projective variety (in the compact case).A Kähler form satisfies Wirtinger’s inequality,|ω(M, Y )| ≤ |M ∧ Y | ,where the r.h.s is the volume of the parallelogram formed by the tangentvectors M and Y . Corresponding inequalities hold for the exterior powersof ω. Equality holds iff M and Y form a complex subspace. Therefore, thereis a calibration form, and the complex submanifolds of a Kähler manifoldare calibrated submanifolds. In particular, the complex submanifolds arelocally volume minimizing in a Kähler manifold. For example, the graphof a holomorphic function is a locally area–minimizing surface in C 2 = R 4 .Kähler identities is a collection of identities which hold on a Kählermanifold, also called the Hodge identities. Let ω be a Kähler form, d = ∂+ ¯∂be the exterior derivative, [A, B] = AB − BA be the commutator of twodifferential operators, and A ∗ denote the formal adjoint of A. The followingoperators also act on differential forms α on a Kähler manifold:L(α) = α ∧ ω, Λ(α) = L ∗ (α) = α⌋ω, d c = −JdJ,where J is the almost complex structure, J = −I, and ⌋ denotes the interior


436 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionproduct. Then we have[L, ¯∂] = [L, ∂] = 0, [Λ, ¯∂ ∗ ] = [Λ, ∂ ∗ ] = 0,[L, ¯∂ ∗ ] = −i∂, [L, ∂ ∗ ] = i ¯∂, [Λ, ¯∂] = −i∂ ∗ , [Λ, ∂] = −i ¯∂.These identities have many implications. For example, the two operators∆ d = dd ∗ + d ∗ d and ∆ ¯∂ = ¯∂ ¯∂ ∗ + ¯∂ ∗ ¯∂(called Laplacians because they are elliptic Laplacian–like operators) satisfy∆ d = 2∆ ¯∂.At this point, assume that M is also a compact manifold. Along withHodge’s Theorem, this equality of Laplacians proves the Hodge decomposition.The operators L and Λ commute with these Laplacians. By Hodge’sTheorem, they act on cohomology, which is represented by harmonic forms.Moreover, definingH = [L, Λ] = ∑ (p + q − n) Π p,q ,where Π p,q is projection onto the (p, q)−Dolbeault cohomology, they satisfy[L, Λ] = H, [H, L] = −2L, [H, Λ] = 2L.In other words, these operators give a group representation of the speciallinear Lie algebra sl 2 (C) on the complex cohomology of a compact Kählermanifold (Lefschetz Theorem).3.14.2 Calabi–Yau ManifoldsA Calabi–Yau manifold is a Kähler manifold of complex dimension n witha covariant constant holomorphic n−form. Equivalently it is a Riemannianmanifold with holonomy contained in SU(n).It is convenient for our purposes to play down the role of the complexstructure in describing such manifolds and to emphasize instead the roleof three closed forms, satisfying certain algebraic identities. We have theKähler 2–form ω and the real and imaginary parts Ω 1 and Ω 2 of the covariantconstant n−form. These satisfy some identities [Hitchin (1997);Atiyah and Hitchin (1988)]:(i) ω is non–degenerate(ii) Ω 1 + iΩ 2 is locally decomposable and non-vanishing(iii) Ω 1 ∧ ω = Ω 2 ∧ ω = 0


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 437(iv) (Ω 1 + iΩ 2 ) ∧ (Ω 1 − iΩ 2 ) = ω n (resp. iω n ) if n is even (resp. odd)(v) dω = 0, dΩ 1 = 0, dΩ 2 = 0.These conditions (together with a positivity condition) we now showserve to characterize CY manifolds. Firstly if Ω c = Ω 1 + iΩ 2 is locallydecomposable as θ 1 ∧ θ 2 ∧ ... ∧ θ n , then take the subbundle Λ of T ∗ M ⊗ Cspanned by θ 1 , . . . , θ n . By (iv) and the fact that ω n ≠ 0, we haveθ 1 ∧ · · · ∧ θ n ∧ ¯θ 1 ∧ · · · ∧ ¯θ n ≠ 0and so T ∗ M = Λ + ¯Λ and we have an almost–complex structure. In thisdescription a 1−form θ is of type (1, 0) iff Ω c ∧ θ = 0. Since from (v)dΩ 1 = dΩ 2 = 0, this means that Ω c ∧ dθ = 0. Writingdθ = ∑ a ij θ i ∧ θ j + ∑ b ij θ i ∧ ¯θ j + ∑ c ij¯θi ∧ ¯θ j (3.241)we see that c ij = 0. Thus the ideal generated by Λ is closed under exteriordifferentiation, and (by the Newlander–Nirenberg Theorem) the structureis integrable.Similarly, applying the decomposition of 2–forms (3.241) to ω, (iii) impliesthat the (0, 2)−component vanishes, and since ω is real, it is of type(1, 1). It is closed by (v), so if the Hermitian form so defined is positivedefinite, then we have a Kähler metric.Since Ω c is closed and of type (n, 0) it is a non–vanishing holomorphicsection s of the canonical bundle. Relative to the trivialization s, the Hermitianconnection has connection form given by ∂ ln(‖s‖ 2 ). But property(iv) implies that it has constant length, so the connection form vanishesand s = Ω c is covariant constant.3.14.3 Special Lagrangian SubmanifoldsA submanifold L of a symplectic manifold X is Lagrangian if ω restricts tozero on L and dim X = 2 dim L. A submanifold of a CY manifold is specialLagrangian if in addition Ω = Ω 1 restricts to zero on L. This conditioninvolves only two out of the three forms, and in many respects what weshall be doing is to treat them both, the 2–form ω and the n−form Ω, onthe same footing.Here we need to emphasize the following remarks:1. We could relax the definition a little since Ω c is a chosen holomorphicn−form: any constant multiple of Ω c would also be covariant constant, sounder some circumstances we may need to say that L is special Lagrangianif, for some non–zero c 1 , c 2 ∈ R, c 1 Ω 1 + c 2 Ω 2 = 0.


438 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction2. On a special Lagrangian submanifold L, the n−form Ω 2 restricts to anon–vanishing form, so in particular L is always oriented.Examples of special Lagrangian submanifolds are difficult to find, andso far consist of three types [Hitchin (1997); Atiyah and Hitchin (1988)]:• Complex Lagrangian submanifolds of hyperkähler manifolds;• Fixed points of a real structure on a CY manifold;• Explicit examples for non–compact CY manifolds;The hyperkähler examples arise easily. In this case we have n = 2k andthree Kähler forms ω 1 , ω 2 , ω 3 corresponding to the three complex structuresI, J, K of the hyperkähler manifold. With respect to the complex structureI the form ω c = (ω 2 + iω 3 ) is a holomorphic symplectic form. If L is acomplex Lagrangian submanifold (i.e. L is a complex submanifold and ω cvanishes on L), then the real and imaginary parts of this, ω 2 and ω 3 , vanishon L. Thus ω = ω 2 vanishes and if k is odd (resp. even), the real (resp.imaginary) part of Ω c = (ω 3 + iω 1 ) k vanishes. Using the complex structureJ instead of I, we see that L is special Lagrangian. For examples here, wecan take any holomorphic curve in a K3 surface S, or its symmetric productin the Hilbert scheme S [m] , which is hyperkähler from [Hitchin (1997);Atiyah and Hitchin (1988)].If X is a CY manifold with a real structure — an antiholomorphicinvolution σ, for which σ ∗ ω = −ω and σ ∗ Ω = −Ω, then the fixed pointset (the set of real points of X) is easily seen to be a special Lagrangiansubmanifold L. All CY metrics on compact manifolds are produced by theexistence Theorem of Yau. In particular T ∗ S n (with the complex structureof an affine quadric) has a complete CY metric for which the zero sectionis special Lagrangian.3.14.4 Dolbeault Cohomology and Hodge NumbersA generalization of the real–valued de Rham cohomology to complex manifoldsis called the Dolbeault cohomology. On complex mD manifolds, wehave local coordinates z i and ¯z i . One can now study (p, q)−forms, whichare forms containing p factors of dz i and q factors of d¯z j :ω = ω i1...i p,j 1...j q(z, ¯z) dz i1 ∧ · · · ∧ dz ip ∧ d¯z j1 ∧ · · · ∧ d¯z jq .Moreover, one can introduce two exterior derivative operators ∂ and ¯∂,


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 439where ∂ is defined by∂ω ≡ ∂ω i 1...i p,j 1...j q∂z k dz k ∧ dz i1 ∧ · · · ∧ dz ip ∧ d¯z j1 ∧ · · · ∧ d¯z jq ,and ¯∂ is defined similarly by differentiating with respect to ¯z k and addinga factor of d¯z k . Again, both of these operators square to zero. We can nowconstruct two cohomologies – one for each of these operators – but as wewill see, in the cases that we are interested in, the information containedin them is the same. Conventionally, one uses the cohomology defined bythe ¯∂−operator.For complex manifolds, the Hodge Theorem also holds: each cohomologyclass H p,q (M) contains a unique harmonic form. Here, a harmonic formω h is a form for which the complex Laplacian∆ ¯∂ = ¯∂ ¯∂ ∗ + ¯∂ ∗ ¯∂has a zero eigenvalue: ∆ ¯∂ω h = 0. In general, this operator does not equalthe ordinary Laplacian, but one can prove that in the case where M is aKähler manifold,∆ = 2∆ ¯∂ = 2∆ ∂ .In other words, on a Kähler manifold the notion of a harmonic form isthe same, independently of which exterior derivative one uses. As a firstconsequence, we find that the vector spaces H p,q∂(M) and Hp,q(M) both¯∂equal the vector space of harmonic (p, q)−forms, so the two cohomologiesare indeed equal. Moreover, every (p, q)−form is a (p + q)−form in thede Rham cohomology, and by the above result we see that a harmonic(p, q)−form can also be viewed as a de Rham harmonic (p + q)−form.Conversely, any de Rham p−form can be written as a sum of Dolbeaultforms:ω p = ω p,0 + ω p−1,1 + . . . + ω 0,p . (3.242)Acting on this with the Laplacian, one sees that for a harmonic p−form,∆ω p = ∆ ¯∂ω p = ∆ ¯∂ω p,0 + ∆ ¯∂ω p−1,1 + . . . + ∆ ¯∂ω 0,p = 0.Since ∆ ¯∂ does not change the degree of a form, ∆ ¯∂ω p1,p 2is also a(p 1 , p 2 )−form. Therefore, the r.h.s. can only vanish if each term vanishesseparately, so all the terms on the r.h.s. of (3.242) must be harmonicforms. Summarizing, we have shown that the vector space of harmonic deRham p−forms is a direct sum of the vector spaces of harmonic Dolbeault


440 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(p 1 , p 2 )−forms with p 1 +p 2 = p. Since the harmonic forms represent the cohomologyclasses in a 1–1 way, we find the important result that for Kählermanifolds,H p (M) = H p,0 (M) ⊕ H p−1,1 (M) ⊕ · · · ⊕ H 0,p (M).That is, the Dolbeault cohomology can be viewed as a refinement of the deRham cohomology. In particular, we haveb p = h p,0 + h p−1,1 + . . . + h 0,p ,where h p,q = dim H p,q (M) are called the Hodge numbers of M.The Hodge numbers of a Kähler manifold give us several topologicalinvariants, but not all of them are independent. In particular, the followingtwo relations hold:h p,q = h q,p , h p,q = h m−p,m−q . (3.243)The first relation immediately follows if we realize that ω ↦→ ω maps∂−harmonic (p, q)−forms to ¯∂−harmonic (q, p)−forms, and hence can beviewed as an invertible map between the two respective cohomologies. Aswe have seen, the ∂−cohomology and the ¯∂−cohomology coincide on aKähler manifold, so the first of the above two equations follows.The second relation can be proved using the map∫(α, ω) ↦→ α ∧ ωfrom H p,q × H m−p,m−q to C. It can be shown that this map is nondegenerate,and hence that H p,q and H m−p,m−q can be viewed as dual vectorspaces. In particular, it follows that these vector spaces have the samedimension, which is the statement in the second line of (3.243).Note that the last argument also holds for de Rham cohomology, inwhich case we find the relation b p = b n−p between the Betti numbers.We also know that H n−p (M) is dual to H n−p (M), so combining thesestatements we find an identification between the vector spaces H p (M)and H n−p (M). Recall that this identification between p−form cohomologyclasses and (n − p)−cycle homology classes represents the Poincaréduality. Intuitively, take a certain (n − p)−cycle Σ representing a homologyclass in H n−p . One can now try to define a ‘delta function’ δ(Σ) whichis localized on this cycle. Locally, Σ can be parameterized by setting pcoordinates equal to zero, so δ(Σ) is a ‘pD delta function’ – that is, it is anobject which is naturally integrated over pD submanifolds: a p−form. ThisM


. ..<strong>Applied</strong> Manifold <strong>Geom</strong>etry 441intuition can be made precise, and one can indeed view the cohomologyclass of the resulting ‘delta–function’ p−form as the Poincar é dual to Σ.Going back to the relations (3.243), we see that the Hodge numbers of aKähler manifold can be nicely written in a so–called Hodge diamond form:h 0,0h 1,0 h 0,1. ..h m,0 · · · h 0,m. ..h m,m−1 h m−1,mh m,mThe integers in this diamond are symmetrical under the reflection in itshorizontal and vertical axes.. ..3.15 Conformal Killing–Riemannian <strong>Geom</strong>etryIn this section we present some basic facts from conformal Killing–Riemannian geometry. In mechanics it is well–known that symmetries ofLagrangian or Hamiltonian result in conservation laws, that are used todeduce constants of motion for the trajectories (geodesics) on the configurationmanifold M. The same constants of motion are get using geometricallanguage, where a Killing vector–field is the standard tool for the descriptionof symmetry [Misner et al. (1973)]. A Killing vector–field ξ i is avector–field on a Riemannian manifold M with metrics g, which in coordinatesx j ∈ M satisfies the Killing equationξ i;j + ξ j;i = ξ (i;j) = 0, or L ξ ig ij = 0, (3.244)where semicolon denotes the covariant derivative on M, the indexed bracketdenotes the tensor symmetry, and L is the Lie derivative.The conformal Killing vector–fields are, by definition, infinitesimal conformalsymmetries i.e., the flow of such vector–fields preserves the conformalclass of the metric. The number of linearly–independent conformal Killingfields measures the degree of conformal symmetry of the manifold. Thisnumber is bounded by 1 2(n + 1)(n + 2), where n is the dimension of themanifold. It is the maximal one if the manifold is conformally flat [Baum(2000)].


442 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow, to properly initialize our conformal geometry, recall that conformaltwistor spinor–fields ϕ were introduced by R. Penrose into physics (see[Penrose (1967); Penrose and Rindler (1984); Penrose and Rindler (1986)])as solutions of the conformally covariant twistor equation∇ S Xϕ + 1 X · Dϕ = 0,nfor each vector–fields X on a Riemannian manifold (M, g), where D is theDirac operator. Each twistor spinor–field ϕ on (M, g) defines a conformalvector–field V ϕ on M byg(V ϕ , X) = i k+1 〈X · ϕ, ϕ〉.Also, each twistor spinor–field ϕ that satisfies the Dirac equation on (M, g),Dϕ = µϕ,is called a Killing spinor–field. Each twistor spinor–field without zeros on(M, g) can be transformed by a conformal change of the metric g into aKilling spinor–field [Baum (2000)].3.15.1 Conformal Killing Vector–Fields and Forms on MThe space of all conformal Killing vector–fields forms the Lie algebra of theisometry group of a Riemannian manifold (M, g) and the number of linearlyindependent Killing vector–fields measures the degree of symmetry ofM. It is known that this number is bounded from above by the dimensionof the isometry group of the standard sphere and, on compact manifolds,equality is attained if and only if the manifold M is isometric to the standardsphere or the real projective space. Slightly more generally one canconsider conformal vector–fields, i.e., vector–fields with a flow preserving agiven conformal class of metrics. There are several geometrical conditionswhich force a conformal vector–field to be Killing [Semmelmann (2002)].A natural generalization of conformal vector–fields are the conformalKilling forms [Yano (1952)], also called twistor forms [Moroianu and Semmelmann(2003)]. These are p−forms α satisfying for any vector–field Xon the manifold M the Killing–Yano equation∇ X α − 1p+1 X ⌋ dα + 1n−p+1 X∗ ∧ d ∗ α = 0, (3.245)where n is the dimension of the manifold (M, g), ∇ denotes the covariantderivative of the Levi–Civita connection on M, X ∗ is 1−form dual to X and


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 443⌋ is the operation dual to the wedge product on M. It is easy to see thata conformal Killing 1−form is dual to a conformal vector–field. Coclosedconformal Killing p−forms are called Killing forms. For p = 1 they aredual to Killing vector–fields.Let α be a Killing p−form and let γ be a geodesic on (M, g), i.e.,∇ ˙γ ˙γ = 0. Then∇ ˙γ ( ˙γ⌋ α) = (∇ ˙γ ˙γ)⌋ α + ˙γ⌋ ∇ ˙γ α = 0,i.e., ˙γ⌋ α is a (p − 1)−form parallel along the geodesic γ and in particularits length is constant along γ.The l.h.s of equation (3.245) defines a first–order elliptic differentialoperator T , the so–caled twistor operator. Equivalently one can describe aconformal Killing form as a form in the kernel of twistor operator T . Fromthis point of view conformal Killing forms are similar to Penrose’s twistorspinors in Lorentzian spin geometry. One shared property is the conformalinvariance of the defining equation. In particular, any form which is parallelfor some metric g, and thus a Killing form for trivial reasons, induces non–parallel conformal Killing forms for metrics conformally equivalent to g (bya non–trivial change of the metric) [Semmelmann (2002)].3.15.2 Conformal Killing Tensors and LaplacianSymmetryIn an nD Riemannian manifold (M, g), a Killing tensor–field (of order 2)is a symmetric tensor K ab satisfying (generalizing (3.244))K (ab;c) = 0. (3.246)A conformal Killing tensor–field (of order 2) is a symmetric tensor Q absatisfyingQ (ab;c) = q (a g bc) , with q a = (Q ,a + 2Q a;dd)/(n + 2), (3.247)where comma denotes partial derivative and Q = Q d d. When the associatedconformal vector q a is nonzero, the conformal Killing tensor will be calledproper and otherwise it is a (ordinary) Killing tensor. If q a is a Killingvector, Q ab is referred to as a homothetic Killing tensor. If the associatedconformal vector q a = q ,a is the gradient of some scalar field q, then Q abis called a gradient conformal Killing tensor. For each gradient conformal


444 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionKilling tensor Q ab there is an associated Killing tensor K ab given byK ab = Q ab − qg ab , (3.248)which is defined only up to the addition of a constant multiple of the inversemetric tensor g ab .Some authors define a conformal Killing tensor as a trace–free tensorP ab satisfying P (ab;c) = p (a g bc) . Note that there is no contradiction betweenthe two definitions: if P ab is a trace–free conformal Killing tensor then forany scalar field λ, P ab + λg ab is a conformal Killing tensor and converselyif Q ab is a conformal Killing tensor, its trace–free part Q ab − 1 n Qgab is atrace–free Killing tensor [Rani et. al. (2003)].Killing tensor–fields are of importance owing to their connection withquadratic first integrals of the geodesic equations: if p a is tangent to anaffinely parameterized geodesic (i.e., p a ;b pb = 0) it is easy to see thatK ab p a p b is constant along the geodesic. For conformal Killing tensorsQ ab p a p b is constant along null geodesics and here, only the trace–free partof Q ab contributes to the constants of motion. Both Killing tensors andconformal Killing tensors are also of importance in connection with the separabilityof the Hamiltonian–Jacobi equations [Conway and Hopf (1964)](as well as other PDEs).A Killing tensor is said to be reducible if it can be written as a constantlinear combination of the metric and symmetrized products of Killingvectors,K ab = a 0 g ab + a IJ ξ I(a ξ |J|b) , (3.249)where ξ I for I = 1 . . . N are the Killing vectors admitted by the manifold(M, g) and a 0 and a IJ for 1 ≤ I ≤ J ≤ N are constants. Generally one isinterested only in Killing tensors which are not reducible since the quadraticconstant of motion associated with a reducible Killing tensor is a constantlinear combination of p a p a and of pairwise products of the linear constantsof motion ξ Ia p a [Rani et. al. (2003)].More generally, any linear differential operator on a Riemannian manifold(M, g) may be written in the form [Eastwood (1991); Eastwood (2002)]D = V bc···d ∇ b ∇ c · · · ∇ d + lower order terms,where V bc···d is symmetric in its indices, and ∇ a = ∂/∂x a (differentiation incoordinates). This tensor is called the symbol of D. We shall write φ (ab···c)for the symmetric part of φ ab···c .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 445Now, a conformal Killing tensor on (M, g) is a symmetric trace–freetensor–field, with s indices, satisfyingor, equivalently,the trace–free part of ∇ (a V bc···d) = 0, (3.250)for some tensor–field T c···d or, equivalently,∇ (a V bc···d) = g (ab T c···d) , (3.251)∇ (a V bc···d) =sn+2s−2 g(ab ∇ e V c···d)e , (3.252)where ∇ a = g ab ∇ b (the standard convention of raising and lowering indiceswith the metric tensor g ab ). When s = 1, these equations define a conformalKilling vector.M. Eastwood proved the following Theorem: any symmetry D of theLaplacian ∆ = ∇ a ∇ a on a Riemannian manifold (M, g) is canonicallyequivalent to one whose symbol is a conformal Killing tensor [Eastwood(1991); Eastwood (2002)].3.15.3 Application: Killing Vector and Tensor Fields inMechanicsRecall from subsection 3.15 above, that on a Riemannian manifold (M, g)with the system’s kinetic energy metric tensor g = (g ij ), for any pair ofvectors V and T , the following relation holds 9∂ s 〈V, T 〉 = 〈∇ s V, T 〉 + 〈V, ∇ s T 〉, (3.253)where 〈V, T 〉 = g ij V i T j . If the curve γ(s) is a geodesic, for a generic vectorX we havewhere∂ s 〈X, ˙γ〉 = 〈∇ s X, ˙γ〉 + 〈X, ∇ s ˙γ〉 = 〈∇ s X, ˙γ〉 ≡ 〈∇ ˙γ X, ˙γ〉, (3.254)so that in components it reads(∇ ˙γ X) i = ∂ s x l ∂ x lX i + Γ i jk∂ s x j X k ,∂ s (X i ẋ i ) = ẋ i ∇ i (X j ẋ j ).9 In this subsection, the overdot denotes the derivative upon the arc–length parameters, namely () ˙ ≡ ∂ s ≡ d/ds, while ∇ s is the covariant derivative along a curve γ(s).


446 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionUsing the fact that X j ẋ i ∇ i ẋ j = X j ∇ ˙γ ˙γ j = 0, as well as the auto–parallelism of the geodesics, this can be rewritten as∂ s (X i ẋ i ) = 1 2ẋj ẋ i (∇ i X j + ∇ j X i ),(i, j = 1, ..., N).This means that the conservation of X i ẋ i along a geodesic, i.e., ∂ s (X i ẋ i ) =0, is guaranteed by (see [Clementi and Pettini (2002)])∇ (i X j) ≡ ∇ i X j + ∇ j X i = 0. (3.255)If such a field exists on a manifold, it is the Killing vector–field. Recallthat (3.255) is equivalent to L X g = 0, where L is the Lie derivative. Onthe biodynamical manifolds (M, g), being the unit vector ˙q i – tangent to ageodesic – proportional to the canonical momentum p i = ∂L∂ ˙q= ˙q i , the existenceof a Killing vector–field X implies that the corresponding momentumimap (see subsection 3.12.3.5 above),J(q, p) = X k (q)∂ s q k =1√ X k (q) ˙q k 1= √ X k (q)p k ,2(E − V (q)) 2T (q)(3.256)is a constant of motion along the geodesic flow. Thus, for an NDOF Hamiltoniansystem, a physical conservation law, involving a conserved quantitylinear in the canonical momenta, can always be related with a symmetryon the manifold (M, g) due to the action of a Killing vector–field on themanifold. These are the Noether conservation laws. The equation (3.255)is equivalent to the vanishing of the Poisson brackets,( ∂H ∂J{H, J} =∂q i − ∂H )∂J∂p i ∂p i ∂q i = 0, (3.257)which is the standard definition of a constant of motion J(q, p) (see, e.g.,[Abraham and Marsden (1978)]).However, if a 1–1 correspondence is to exist between conserved physicalquantities along a Hamiltonian flow and suitable symmetries of thebiodynamical manifolds (M, g), then integrability will be equivalent to theexistence of a number of symmetries at least equal to the number of DOF,which is equal to dim(M). If a Lie group G acts on the phase–space manifoldthrough completely canonical transformations, and there exists anassociated momentum map, then every Hamiltonian having G as a symmetrygroup, with respect to its action, admits the momentum map asthe constant of motion. These symmetries are usually referred to as hidden


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 447symmetries because, even though their existence is ensured by integrability,they are not easily recognizable [Clementi and Pettini (2002)].Let us now extend what has been presented so far about Killing vector–fields, trying to generalize the form of the conserved quantity along ageodesic flow from J = X i ẋ i to J = K j1j 2...j rẋ j1 ẋ j2 . . . ẋ jr , with K j1j 2...j ra tensor of rank r. Thus, we look for the conditions that entail∂ s (K j1j 2...j rẋ j1 ẋ j2 . . . ẋ jr ) = ẋ j ∇ j (K j1j 2...j rẋ j1 ẋ j2 . . . ẋ jr ) = 0. (3.258)In order to work out from this equation a condition for the existence ofa suitable tensor K j1j 2...j r, which is called a Killing tensor–field, let usfirst consider the 2r rank tensor K j1j 2...j r ẋ i1 ẋ i2 . . . ẋ ir and its covariantderivative along a geodesic [Clementi and Pettini (2002)]ẋ j ∇ j (K j1j 2...j r ẋ i1 ẋ i2 . . . ẋ ir ) = ẋ i1 ẋ i2 . . . ẋ ir ẋ j ∇ j K j1j 2...j r, (3.259)where we have again used ẋ j ∇ j ẋ i k= 0 along a geodesic, and a standardcovariant differentiation formula (see 3.10.1 above). Now, by contraction onthe indices i k and j k the 2r−rank tensor in (3.259) gives a new expressionfor (3.258), which reads∂ s (K j1j 2...j r ẋ j1 ẋ j2 . . . ẋ jr ) = ẋ j1 ẋ j2 . . . ẋ jr ẋ j ∇ (j K j1j 2...j r), (3.260)where ∇ (j K j1j 2...j r) = ∇ j K j1j 2...j r+ ∇ j1 K jj2...j r+ · · · + ∇ jr K j1j 2...j r−1j.The vanishing of (3.260), entailing the conservation of K j1j 2...j r ẋ j1 ẋ j2 . . . ẋ jralong a geodesic flow, is therefore guaranteed by the existence of a tensor–field fulfilling the conditions [Clementi and Pettini (2002)]∇ (j K j1j 2...j r) = 0. (3.261)These equations generalize (3.255) and give the definition of a Killingtensor–field on a Riemannian biodynamical manifold (M, g). These N r+1equations in (N + r − 1)!/r!(N − 1)! unknown independent components ofthe Killing tensor constitute an overdetermined system of equations. Thus,a‘priori, we can expect that the existence of Killing tensor–fields has to berather exceptional.If a Killing tensor–field exists on a Riemannian manifold, then the scalarK j1j 2...j r ˙q j1 ˙q j2 . . . ˙q jris a constant of motion for the geodesic flow on the same manifold. Withthe only difference of a more tedious combinatorics, also in this case it turns


448 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionout that the equations (3.261) are equivalent to the vanishing of the Poissonbrackets of J(q, p), that is{H, J} = 0 is equivalent to ∇ (j K j1j 2...j r) = 0.Thus, the existence of Killing tensor–fields, obeying (3.261), on a biodynamicalmanifold (M, g) give the rephrasing of integrability of Newtonian equationsof motion or, equivalently, of standard Hamiltonian systems, withinthe Riemannian geometrical framework.The first natural question to address concerns the existence of a Killingtensor–field, on any biodynamical manifold (M, g), to be associated withtotal energy conservation. Such a Killing tensor–field actually exists andcoincides with the metric tensor g, in fact it satisfies by definition (3.261).One of the simplest case of integrable system is represented by a decoupledsystem described by a generic HamiltonianN∑[ ]p2 N∑H = i2 + V i(q i ) = H i (q i , p i )i=1for which all the energies E i of the subsystems H i , i = 1, . . . , N, are conserved.On the associated biodynamical manifold, N second–order Killingtensor–fields exist, they are given byi=1K (i)jk = δ jk{V i (q i )[E − V (q i )] + δ i j[E − V (q i )] 2 }.In fact, these tensor–fields fulfil (3.261), which explicitly reads [Clementiand Pettini (2002)]∇ k K (i)lm + ∇ lK (i)mk + ∇ mK (i)kl= ∂ q kK (i)lm + ∂ q lK(i) mk + ∂ q mK(i) kl− 2Γ j kl K(i) jm − 2Γj km K(i) jl− 2Γ j lm K(i) jk = 0.The conserved quantities J (i) (q, p) are then get by saturation of the tensorsK (i) with the velocities ˙q i ,J (i) (q, p) = K (i)jk ˙qj ˙q k = E i .3.16 Application: Lax–Pair Tensors in GravitationRecall that many problems in general relativity require an understandingof the global structure of the space–time. Currently discussed global problemsinclude the occurrence of naked singularities [Ori and Piran (1987)]and universality in gravitational collapse situations [Choptuik (1993)]. The


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 449study of global properties of space–times relies to a large extent on theability to integrate the geodesic equations. In the absence of exact solutionsnumerical integration is often used to get a quantitative picture.However, in the quest for a deeper understanding the exact and numericalapproaches should be viewed as complementary tools. To perform an exactinvestigation of the global properties of a given space–time, not onlymust the space–time itself be an exact solution of the Einstein equations,but in addition the geodesic equations must be integrable. Usually, in ad−dimensional space, integrability of the geodesic equations is connectedwith the existence of at least d − 1 mutually commuting Killing vectorfields which span a hypersurface in the space–time. There are exceptionshowever. The most well–known example is the Kerr space–time which hasonly two commuting Killing vectors. In that case it is the existence ofan irreducible second rank Killing tensor which makes integration possible[Walker and Penrose (1970)]. Another example is given by Ozsvath’s classIII cosmologies [Ozsvath (1970)]. In that case the geodesic system was integratedusing the existence of a non–Abelian Lie algebra of Killing vectors[Rosquist (1980)]. In general integrability can only be guaranteed if thereis a set of d constants of the motion in involution (i.e. mutually Poissoncommuting). Since the metric itself always provides one constant of themotion corresponding to the squared length of the geodesic tangent vector,the geodesic system will be integrable by Liouville’s theorem if there ared − 1 additional Poisson commuting invariants.Exact solutions of Einstein’s equations typically admit a number ofKilling vector fields. Some of these Killing vector fields may be motivatedby physical considerations. For example if one is interested in static starsthe space–time must have a time–like Killing vector. For such systems itis also very reasonable to assume spherical spatial symmetry leading to atotal of four (non–commuting) Killing vectors. In most cases, the numberof Killing vectors is limited by the physics of the problem. In a sphericallysymmetric collapse situation, for example, the space–time admits exactlythree non–commuting Killing vector fields which form an isometry groupwith 2D orbits. That structure is not sufficient for an exact integrationof the geodesic equations. However, the physics of the problem does notimpose any a priori restrictions on higher rank (≥ 2) Killing tensors. AKilling vector field, ξ, plays a double role; it is both an isometry for the metric(L ξ g = 0) and a geodesic symmetry. This last property means that itcan be interpreted as a symmetry transformation for the geodesic equations.By contrast higher rank Killing tensors are only geodesic symmetries. They


450 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionhave no obvious geometric interpretation ([Rosquist (1989)]). Because ofthe isometry property of the Killing vector fields, such symmetries can beincorporated right from the start by assuming a particular form the metric.In this way the field equations are actually simplified by the assumptionof Killing vector symmetries. On the other hand, the higher rank Killingsymmetries can at present not be used to simplify the form of the fieldequations. Instead the Killing tensor equations must be imposed as extraconditions thereby increasing both the number of dependent variables andthe number of equations.The Lax tensors introduced in [Rosquist (1997)] provide a unifyingframework for Killing tensors of any rank and may lead to possibilitiesto incorporate the higher Killing symmetries in the field equations themselves.A single Lax tensor may generate Killing tensors of varying ranks.Lax tensors arise from a covariant formulation of the Lax pair equation [Lax(1968)] for Riemannian and pseudo–Riemannian geometries. The standardLax pair formulation involves a pair of matrices. In the covariant formulationon the other hand, the Lax pair is represented by two third ranktensors. The first Lax matrix corresponds exactly to the first Lax tensorwhile the second Lax matrix and the second Lax tensor differ by a termwhich coincides with the Levi–Civita connection. The derivative part ofthe tensorial Lax pair equation is identical to the Killing–Yano equation.Therefore Killing–Yano tensors are special cases of Lax tensors for whichthe second Lax tensor vanishes (the second Lax matrix however does not).However, whereas Killing–Yano tensors are by definition totally antisymmetricthe Lax tensors have no a priori symmetry restrictions.3.16.1 Lax–Pair TensorsIn this subsection we outline the approach to integrable geometries as givenin [Rosquist (1997)]. We consider a Riemannian or pseudo–Riemannianmanifold with metricds 2 = g µν dq µ dq ν .The geodesic equations can be represented by the HamiltonianH = 1 2 gµν p µ p ν , (3.262)


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 451together with the natural Poisson bracket (denoted by {, }) on the cotangentbundle. The geodesic system is given by˙q α = {q α , H} = g αµ p µ , ṗ α = {p α , H} = Γ µν α p µ p ν .The complete integrability of this system can be shown with the help ofa pair of matrices L and A with entries defined on the phase space (thecotangent bundle) and satisfying the Lax pair equation [Lax (1968)]˙L = {L, H} = [L, A]. (3.263)It follows from (3.263) that the quantities I k ≡ 1 k Lk are all constants of themotion. If in addition they commute with each other {I k , I j } = 0 (Liouvilleintegrability) then it is possible to integrate the system completely at leastin principle (see e.g., [Arnold (1989)]). The Lax representation (3.263) is notunique. In fact, the Lax pair equation is invariant under a transformationof the form˜L = ULU −1 , Ã = UAU −1 − ˙UU −1 . (3.264)We see that L transforms as a tensor while A transforms as a connection.As we will see, these statements acquire a more precise meaning in thegeometric formulation which we will now describe.Typically, the Lax matrices are linear in the momenta and in the geometricsetting they may also be assumed to be homogeneous. This motivatesthe introduction of two third rank geometrical objects, L α β γ and A α β γ ,such that the Lax matrices can be written in the form [Rosquist (1997)]L = (L α β) = (L α β µ p µ ) , A = (A α β) = (A α β µ p µ ).We will refer to L α β γ and A α β γ as the Lax tensor and the Lax connection,respectively. DefiningB = (B α β ) = (B α β µ p µ ) = A − Γ, where Γ = (Γ α β) = (Γ α β µ p µ )is the Levi–Civita connection with respect to g αβ , it then follows that theLax pair equation takes the covariant form (see [Rosquist (1997)] for details)L α β (γ;δ) = L α µ (γ B |µ| β δ) − B α µ (γ L |µ| β δ) ,where L α β γ and B α β γ are tensorial objects. Note that the right–hand sideof this equation is traceless, so that upon contracting over α and β we


452 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionget the Killing vector equation L µ µ(α;β) = 0. Splitting the Lax tensors insymmetric and antisymmetric parts with respect to the first two indices,S αβγ = L (αβ)γ , R αβγ = L [αβ]γ andP αβγ = B (αβ)γ , Q αβγ = B [αβ]γ ,the Lax pair equation can be written as the system [Rosquist and Goliath(1997)]S αβ (γ;δ) = −2S (α µ(γQ β)µ δ) + 2R (α µ(γP β)µ δ),R αβ (γ;δ) = −2R [α µ(γQ β]µ δ) + 2S [α µ(γP β]µ δ).It is evident that this system is coupled via P αβγ . Setting P αβγ = 0 givesthe two separate sets of equationsS αβ (γ;δ) = −2S (α µ(γQ β)µ δ),R αβ (γ;δ) = −2R [α µ(γQ β]µ δ).We will see below that the Lax tensors L αβγ and B αβγ in a geometrizedversion of the open Toda lattice are symmetric and antisymmetric respectivelyand therefore satisfy (3.265). If R αβγ is totally antisymmetric (withrespect to all three indices) and Q αβγ = 0, then the equations (3.265) areidentical to the third rank Killing–Yano equations [Yano (1952)]. Thereforethird rank Killing–Yano tensors are special cases of Lax tensors.It is possible but not necessary to identify the invariant I 2 with thegeodesic Hamiltonian (3.262). If such an identification is done then themetric is given by g αβ = L µ ν α L ν µ β . Defining matrices L µ with components(L µ ) α β = L α β µ , the metric components are given by g αβ = (L α L β ), whichsuggests using the components of the L µ (or some internal variables fromwhich the L µ are built) as the basic variables already in the formulation ofthe field equations much like in the Ashtekar variable formalism [Ashtekar(1988)].3.16.2 <strong>Geom</strong>etrization of the 3–Particle Open Toda LatticeIntegrable systems are usually discussed in the context of classical mechanics.Classical Hamiltonians typically consist of a flat positive–definitekinetic energy together with a potential energy term. They are thus superficiallyquite different from geometric Hamiltonians of the form (3.262).However, any classical Hamiltonian with a quadratic kinetic energy canbe transformed to a geometric representation. One such geometrization


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 453results in the Jacobi Hamiltonian [Lanczos (1986)]. Another closely relatedgeometrization was used in [Rosquist (1997)]. Both methods involvea re–parameterization of the independent variable. Usually we will refer tothe independent variable as the time, although its physical interpretationmay vary. As a consequence of this feature, the original Lax representationis not preserved. It is known how to transform the invariants themselvesunder the time re–parameterization [Rosquist and Pucacco (1995);Rosquist (1997)]. Given that the geometrized invariants are also in involution,the existence of a Lax representation is guaranteed [Babelon andViallet (1990)]. However, to actually find such a Lax representation isnon–trivial. Another geometrization scheme which does preserve the originalLax representation is to apply a suitable canonical transformation. Thisis however only possible for Hamiltonians with a potential of a special form.One such system that we will consider in this paper is the 3–particle openToda latticeH = 1 2(¯p1 2 + ¯p 2 2 + ¯p 32 ) + e 2(¯q1 −¯q 2) + e 2(¯q2 −¯q 3) . (3.265)Below we will discuss two canonical transformations which correspond to inequivalentgeometric representations of (3.265). For an explicit integrationof the Toda lattice, see e.g. [Perelomov (1990)]. The standard symmetricLax representation is [Perelomov (1990)]⎛ ⎞ ⎛⎞¯p 1 ā 1 00 ā 1 0L = ⎝ ā 1 ¯p 2 ā 2⎠ , A = ⎝ −ā 1 0 ā 2⎠ ,0 ā 2 ¯p 3 0 −ā 2 0where ā 1 = exp(¯q 1 − ¯q 2 ), ā 2 = exp(¯q 2 − ¯q 3 ).Note that the definitions of ā 1 and ā 2 differ from the ones used in [Rosquist(1997)]. The Hamiltonian (3.265) admits the linear invariant, I 1 = L = ¯p 1 +¯p 2 + ¯p 3 , corresponding to translational invariance. The Lax representationalso gives rise to the two invariants, I 2 = 1 2 L2 = H and I 3 = 1 3 L3 . We willassume that the tensorial Lax representation is linear and homogeneous inthe momenta. A homogeneous Lax representation can be obtained fromthe standard representation by applying a canonical transformation of thephase space.3.16.2.1 Tensorial Lax RepresentationHere we straightforwardly apply a simple canonical transformation that willgive a linear and homogeneous Lax representation [Rosquist and Goliath


454 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(1997)]The resulting Lax pair matrices are¯q 1 = q 1 + ln p 1 , ¯p 1 = p 1 ,¯q 2 = q 2 , ¯p 2 = p 2 ,¯q 3 = q 3 − ln p 3 , ¯p 3 = p 3 .⎛⎞ ⎛⎞p 1 a 1 p 1 00 a 1 p 1 0L = ⎝ a 1 p 1 p 2 a 2 p 3⎠ , A = ⎝ −a 1 p 1 0 a 2 p 3⎠ ,0 a 2 p 3 p 3 0 −a 2 p 3 0where a 1 = exp(q 1 − q 2 ), a 2 = exp(q 2 − q 3 ).The Hamiltonian is now purely kineticH = 1 2 L2 = 1 2Using (3.262) we identify a metric[(1 + 2a12 ) p 1 2 + p 2 2 + ( 1 + 2a 22 ) p 32 ] .ds 2 = g 11 (dq 1 ) 2 + (dq 2 ) 2 + g 33 (dq 3 ) 2 , whereg 11 = ( 21 + 2a ) −11 , g33 = ( 21 + 2a ) −12 .The non–zero Levi–Civita connection coefficients, Γ α βγ = Γα (βγ), of this metricareΓ 1 11 = −2a 2 1 g 11 , Γ 2 33 = 2a 2 2 (g 33 ) 2 ,Γ 1 12 = 2a 2 1 g 11 , Γ 3 23 = −2a 2 2 g 33 ,Γ 2 11 = −2a 2 1 (g 11 ) 2 , Γ 3 33 = 2a 2 2 g 33 .Following the arguments above, the homogeneous Lax matrix should correspondto a tensor with mixed indices L α β. It is a reasonable assumptionthat the covariant Lax formulation inherits the symmetries of the standardformulation we started with. We therefore expect L αβ and B αβ to have thesymmetriesL (αβ) = L αβ and B [αβ] = B αβ .Note that the symmetry properties are not imposed on the Lax matrices,L α β and Bα β, themselves. In fact, the required symmetries are not consistentwith the representation (3.266). We can however perform a similaritytransformation (3.264) of the Lax matrix, L → ˜L in such a way that ˜L αβ


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 455will be symmetric. Using the transformation matrixwill get a new Lax pair⎛1/ √ ⎞g 11 0 0U = ⎝ 0 1 00 0 1/ √ ⎠ ,g 33⎛p 1 a 1 / √ ⎞g 11 p 1 0L = ⎝√ √a 1 g11 p 1 p 2 a 2 g33 p 30 a 2 / √ ⎠ ,g 33 p 3 p 3⎛Γ 111 p 1 + Γ 121 p 2 a 1 / √ ⎞g 11 p 1 0A = ⎝√ √−a 1 g11 p 1 0 a 2 g33 p 30 −a 2 / √ ⎠ ,g 33 p 3 Γ 323 p 2 + Γ 333 p 3where L is such that L αβ is symmetric. Defining ˆL = (L αβ ) and  =(A αβ ) byˆL = gL,  = gA, where g = (g αβ ), we get⎛√ ⎞g 11 p 1 a 1 g11 p 1 0ˆL = ⎝√ √a 1 g11 p 1 p 2 a 2 g33 p 3⎠ , and√0 a 2 g33 p 3 g 33 p 3⎛Γ 1 11 p 1 + Γ 2 √ ⎞11 p 2 a 1 g11 p 1 0 = ⎝√ √−a 1 g11 p 1 0 a 2 g33 p 3⎠ .√0 −a 2 g33 p 3 Γ 2 33 p 2 + Γ 3 33 p 3Note that the upper triangular parts of ˆL and  coincide. This propertyis peculiar to the open Toda lattice. We also define the correspondingconnection matrix, ˆΓ = gΓ, given by⎛Γ 1 11 p 1 + Γ 2 11 p 2 2a 2 ⎞1 g 11 p 1 0ˆΓ = ⎝ −2a 2 1 g 11 p 1 0 2a 2 2 g 33 p 3⎠ .0 −2a 2 2 g 33 p 3 Γ 2 33 p 2 + Γ 3 33 p 3We see that the off–diagonal part of the matrix ˆΓ is antisymmetric likethat of  and furthermore that their off–diagonal components are relatedby the simple relationΓ αβ γ = 2(A αβ γ ) 2 ,(for α < β).


456 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis gives the following relation between the upper triangular parts of ˆLand ÂΓ αβ γ = 2(L αβ γ ) 2 ,(for α < β).Using the relation  = ˆΓ+ ˆB, where ˆB = gB, we find the following relationbetween the upper triangular components of ˆL and ˆBB αβ γ = L αβ γ − 2(L αβ γ ) 2 ,(for α < β).Finally expressing ˆL and ˆB in terms of ˆΓ we have for the upper triangularpartsL αβ γ =√12 Γ αβ γ , B αβ γ = −Γ αβ γ +√12 Γ αβ γ , (for α < β).Furthermore, the diagonal elements of  and ˆΓ are identical. This impliesthat ˆB is antisymmetric in agreement with our expectations.3.16.3 4D GeneralizationsRecall that we can get a 4D space–time simply by adding a time coordinateaccording to the prescription [Rosquist and Goliath (1997)](4) ds 2 = −(dq 0 ) 2 + ds 2 ,( )where ds 2 is a 3D positive–definite metric. It follows that (4) 0 0Γ = .0 ΓFor the cases obtained above this will lead to inequivalent space–times. Oneway to generalize the 3D Lax pair is( )( )i (4) p0 00 0L = ,(4) A = ,0 L0 Afor which(4) B =( 0 00 BThis Lax pair gives the geodesic Hamiltonian of the corresponding space–time metric as quadratic invariant.3.16.3.1 Case IAdding a time dimension to (3.266) we get the metric [Rosquist and Goliath(1997)]).ds 2 = −(dq 0 ) 2 + g 11 (dq 1 ) 2 + (dq 2 ) 2 + g 33 (dq 3 ) 2 ,where


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 457g 11 = ( 21 + 2a ) −11 , g33 = ( 21 + 2a ) −12 , anda 1 = exp(q 1 − q 2 ), a 2 = exp(q 2 − q 3 ).This space–time is of Petrov type I. The nonzero components of the energy–momentum tensor calculated in a Lorentz frame are (κ = 1)T 00 = −((g 11 ) 2 T 11 + T 22 + (g 33 ) 2 T 33 )= −4a 1 2 (g 11 ) 2 (a 1 2 − 1) + 4a 1 2 a 2 2 g 11 g 33 − 4a 2 2 (g 33 ) 2 (a 2 2 − 1),T 11 = 4a 1 2 (a 1 2 − 1), T 22 = −4a 1 2 a 2 2 g 11 g 33 ,T 33 = 4a 2 2 (a 2 2 − 1).3.16.3.2 Case IIHere we have the metric [Rosquist and Goliath (1997)]ds 2 = −(dq 0 ) 2 + g 11 (dq 1 ) 2 + (dq 2 ) 2 + (dq 3 ) 2 , whereg 11 = [ 1 + 2(a 2 1 + a 2 2 ) ] −1, a1 = exp( √ 1√3q 1 + 2 2 q2 ),a 2 = exp( 1 √2q 1 −√32 q2 ).This space–time is of Petrov type D. The nonzero components of theenergy–momentum tensor calculated in a Lorentz frame are (κ = 1)(T 00 = 12 e 2√ 2q 1 2g 11 4 − 2 sinh 2 ( √ 6q 2 ) + e −√ 2q 1 cosh( √ )6q 2 ) ,T 11 = −(g 11 ) 2 T 00 .3.16.3.3 Energy–Momentum TensorsIn both of the above cases, the energy–momentum tensor takes the form[Rosquist and Goliath (1997)]⎛ ⎞µ 0 0 0T αβ = ⎜ 0 p 1 0 0⎟⎝ 0 0 p 2 0 ⎠ ,0 0 0 p 3where µ ≡ T 00 is the energy density, and p i ≡ T ii , (i = 1, 2, 3) areanisotropic pressures. Such an energy–momentum tensor is physicallymeaningful if the weak energy condition [Hawking and Ellis (1973)]µ ≥ 0, µ + p i ≥ 0, (i = 1, 2, 3),


458 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis satisfied. For case I, there is an unbounded sub–domain of the spacecoordinates (q 1 , q 2 , q 3 ) for which the weak energy condition holds. For caseII, it is easily seen that the restrictions on the energy–momentum tensorare inconsistent, so that the weak energy condition never holds.3.17 <strong>Applied</strong> Unorthodox <strong>Geom</strong>etries3.17.1 Noncommutative <strong>Geom</strong>etryIn this subsection we give review of noncommutative geometry and its maingravitational applications. In the last section of the book, we will give itsapplications to string theory.3.17.1.1 Moyal Product and Noncommutative AlgebraNoncommutative geometry is concerned with the possible spatial interpretationsof algebraic structures for which the commutative law fails; that is,for which xy does not always equal yx. The challenge of the theory is toget around the lack of commutative multiplication, which is a requirementof previous geometric theories of such structures (see [Connes (1994)]).Recall that an ordinary differentiable manifold can be characterized bythe commutative algebra of smooth functions defined on it, and the spaceof smooth sections of its tangent bundle, cotangent bundle and other fiberbundles. All these spaces are modules over the commutative algebra ofsmooth functions. The concepts of exterior derivative, Lie derivative andcovariant derivative are also important elements in understanding derivationsover this algebra. In the noncommutative case, the algebras in questionare noncommutative. To handle differential forms, one must work withthe graded exterior algebra bundle of all p−forms under the wedge productand look at its algebra of smooth sections. A ‘differential’ is taken tobe an anti–derivation (or, something more general) on this algebra, whichincreases the grading by 1 and is quadratically nilpotent.Historically first noncommutative product was the Moyal product[Moyal (1949)], 10 that is an associative, noncommutative ‘star’–product.For any two functions f, g on a Poisson manifold M, the Moyal product ∗10 The Moyal product is also sometimes called Weyl–Moyal product.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 459is defined as:f ∗ g = fg +∞∑ n C n (f, g),n=1where each C n is a bi–differential operator of order n with the followingproperties:1. f ∗ g = fg + O() – deformation of the pointwise product.2. f ∗ g − g ∗ f = i{f, g} + O( 2 ) – deformation in the direction of thePoisson bracket.3. f ∗ 1 = 1 ∗ f = f – the 1 of the un–deformed algebra is the 1 in thenew algebra.4. f ∗ g = g ∗ f – the complex conjugate is an anti–linear anti–automorphism.Let the Poisson structure on a symplectic manifold M be defined byω = ω ij ∂ i ∧ ∂ j .If this structure is constant, that is, if ω ij do not depend on the localcoordinates on M 11 , then the Moyal product of two functions f, g ∈ M canbe defined asf ∗ g = fg + ω ij (∂ i f)(∂ j g) + 22 ωij ω km (∂ i ∂ k f)(∂ j ∂ m g) + . . .where is the (reduced) Planck constant.For example, in Weyl deformation quantization [Weyl (1927)], the symplecticphase–space of classical mechanics is deformed into a noncommutativephase–space generated by the position and momentum operators, usingthe Moyal product.3.17.1.2 Noncommutative Space–Time ManifoldsIn physical field theories one usually considers differential space–time manifolds.Now, in the noncommutative realm, the notion of a point is no longerwell–defined and we have to give up the concept of differentiable manifolds.However, the space of functions on a manifold forms an algebra. A generalizationof this algebra can be considered in the noncommutative case. Wetake the algebra freely generated by the noncommutative coordinates {ˆx i },which respects commutation relations of the type [Madore (1995)][ˆx µ , ˆx ν ] = C µν (ˆx) ≠ 0. (3.266)11 Such a form can always be found at least locally by Darboux’s Theorem


460 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWe can take the space of formal power series in the coordinates ˆx i anddivide by the ideal generated by the above relationsˆx = C〈〈ˆx 0 , . . . , ˆx n 〉〉/([ˆx µ , ˆx ν ] − C µν (ˆx)) ,where the function C µν (ˆx) is unknown. For physical reasons, C µν (ˆx) shouldbe a function that vanishes at large distances where we experience thecommutative world and may be determined by experiments. Nevertheless,one can consider its power–series expansion [Meyer (2005)]C µν (ˆx) = iθ µν + iC µν ρˆx ρ + (q ˆR µν ρσ − δ ν ρδ µ σ)ˆx ρˆx σ + . . . ,where θ µν , C µν ρ and q ˆR µν ρσ are constants, and study cases where the commutationrelations are constant, linear or quadratic in the ˆx i −coordinates.At very short distances, these cases provide a reasonable approximation forC µν (ˆx) and lead to the following three structures:(1) Canonical, or θ−deformed case:(2) Lie–algebra case:(3) Quantum spaces case:[ˆx µ , ˆx ν ] = iθ µν . (3.267)[ˆx µ , ˆx ν ] = iC µν ρˆx ρ . (3.268)ˆx µˆx ν = q ˆR µν ρσ ˆx ρˆx σ . (3.269)We denote by  the algebra generated by noncommutative coordinatesˆx µ which are subject to the relations (3.267) and call it the algebra ofnoncommutative functions. The algebra of the corresponding commutativefunctions will be denoted by A. We consider the θ−deformed case(3.267), but we note that the algebraic construction presented here can begeneralized to more complicated noncommutative structures of the abovetype, which possess the so–called Poincaré–Birkhoff–Witt property, whichstates that the space of polynomials in noncommutative coordinates of agiven degree is isomorphic to the space of polynomials in the commutativecoordinates. Such an isomorphism between polynomials of a fixed degreeis given by an ordering prescription. One example is the symmetric Weylordering, denoted W, which assigns to any monomial the totally symmetric


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 461ordered monomial [Meyer (2005)]W : A → Â, xµ ↦→ ˆx µ , x µ x ν ↦→ 1 2 (ˆxµˆx ν + ˆx ν ˆx µ ) · · · . (3.270)To study the dynamics of physical fields we need a differential calculuson the noncommutative algebra Â. Derivatives are maps on the deformedcoordinate space [Wess (2004)]ˆ∂ µ :  →  .This means that they have to be consistent with the commutation relationsof the coordinates. In the θ−constant case a consistent differential calculuscan be defined very easily by 12[ ˆ∂ µ , ˆx ν ] = δ ν µ( ˆ∂ µˆx ν ) = δ ν µ, [ ˆ∂ µ , ˆ∂ ν ] = 0. (3.271)It is the fully undeformed differential calculus. The above definitions yieldthe usual Leibniz-rule for the derivatives ˆ∂ µ( ˆ∂ µ ˆfĝ) = ( ˆ∂µ ˆf)ĝ + ˆf( ˆ∂µ ĝ). (3.272)This is a special feature of the fact that θ µν are constants. In the more complicatedexamples of noncommutative structures this undeformed Leibnizruleusually cannot be preserved but one has to consider deformed Leibnizrulesfor the derivatives [Wess and Zumino (1991)]. Note that (3.271) alsoimplies that( ˆ∂ µ ˆf) = ̂(∂µ f). (3.273)The Weyl ordering (3.270) can be formally implemented by the mapf ↦→ W (f) = 1 ∫d n k e ikµ ˆxµ ˜f(k)(2π) n 2where ˜f is the Fourier transform of f˜f(k) = 1 ∫d n x e −ikµxµ f(x).(2π) n 2This is due to the fact that the exponential is a fully symmetric function.Using the Baker–Campbell–Hausdorff formula one findse ikµ ˆxµ e ipν ˆxν = e i(kµ+pµ)ˆxµ − i 2 kµθµν p ν. (3.274)12 We use brackets to distinguish the action of a differential operator from the multiplicationin the algebra of differential operators.


462 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis immediately leads to the following observationˆfĝ = W (f)W (g) = 1 ∫(2π) n d n kd n p e ikµ ˆxµ e ipν ˆxν ˜f(k)˜g(p) = (3.275)∫1(2π) n d n kd n p e i(kµ+pµ)ˆxµ e − i 2 kµθµν ip ν ˜f(k)˜g(p) = W (µ ◦ e 2 θµν ∂ µ⊗∂ νf ⊗ g),where µ(f ⊗ g) := fg is the multiplication map. With (3.273) we deducefrom (3.275) the equationµ ◦ e − i 2 θµν ˆ∂µ⊗ ˆ∂ ν ˆf ⊗ ĝ = ̂fg. (3.276)The above formula shows us how the commutative and the noncommutativeproduct are related. It will be important for us later on.3.17.1.3 Symmetries and <strong>Diff</strong>eomorphisms onDeformed SpacesIn general, the commutation relations (3.266) are not covariant with respectto undeformed symmetries. For example, the canonical commutationrelations (3.267) break Lorentz symmetry if we assume that the noncommutativityparameters θ µν do not transform.The question arises whether we can deform the symmetry in such a waythat it acts consistently on the deformed space (i.e., leaves the deformedspace invariant) and such that it reduces to the undeformed symmetry inthe commutative limit [Meyer (2005)]. The answer is yes: Lie algebras canbe deformed in the category of Hopf algebras. 13 Important examples of suchdeformations are q−deformations: there exists a q−deformation of the universalenveloping algebra of an arbitrary semisimple Lie algebra. 14 Modulealgebras of this q−deformed universal enveloping algebras are noncommutativespaces with commutation relations of type (3.269). There exists alsoa so–called κ−deformation of the Poincaré algebra (see [Dimitrijevic et al.(2003); Dimitrijevic et al. (2004)]), which leads to a noncommutative spaceof the Lie type (3.268).Quantum group symmetries lead to new features of field theories onnoncommutative spaces. Because of its simplicity, θ−deformed spaces arevery well–suited to study them. For results on the consequences of theθ−deformed Poincaré algebra, see [Chaichian et. al. (2004); Aschieri et. al.(2005)].13 Hopf algebras coming from a Lie algebra are also called Quantum groups.14 It is called q−deformation since it is a deformation in terms of a parameter q.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 463Now, recall that gravity is a theory invariant with respect to diffeomorphisms.However, to generalize the Einstein formalism to noncommutativespaces in order to establish a noncommutative gravity theory, it is importantto first understand that diffeomorphisms possess more geometricalstructure than the algebraic one: They are naturally equipped with a Hopfalgebra structure.Recall that diffeomorphisms are generated by vector–fields ξ. Actingon functions, vector–fields are represented as linear differential operatorsξ = ξ µ ∂ µ . Vector–fields form a Lie algebra Ξ over the field C with the Liebracket given by[ξ, η] = ξ × η,where ξ × η is defined by its action on functions(ξ × η)(f) = (ξ µ (∂ µ η ν )∂ ν − η µ (∂ µ ξ ν )∂ ν )(f).The Lie algebra of infinitesimal diffeomorphisms Ξ can be embedded intoits universal enveloping algebra which we want to denote by U(Ξ) . Theuniversal enveloping algebra is an associative algebra and possesses a naturalHopf algebra structure. It is given by the following structure maps[Meyer (2005)]:• An algebra homomorphism called coproduct defined by∆ : U(Ξ) → U(Ξ) ⊗ U(Ξ), Ξ ∋ ξ ↦→ ∆(ξ) := ξ ⊗ 1 + 1 ⊗ ξ. (3.277)• An algebra homomorphism called counit defined byε : U(Ξ) → C, Ξ ∋ ξ ↦→ ε(ξ) = 0. (3.278)• An anti–algebra homomorphism called antipode defined byS : U(Ξ) → U(Ξ), Ξ ∋ ξ ↦→ S(ξ) = −ξ. (3.279)For a precise definition and more details on Hopf algebras we refer thereader to the text–books [Chari and Presley (1995); Klimyk and Schmüdgen(1997)]. For our purposes it shall be sufficient to note that the coproductimplements how the Hopf algebra acts on a product in a representationalgebra. It is now possible to study deformations of U(Ξ) in the categoryof Hopf algebras. This leads to a deformed version of diffeomorphisms,which is the fundamental building block of our approach to noncommutativegravity theory.


464 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall that scalar fields are defined by their transformation propertywith respect to infinitesimal coordinate transformations:δ ξ φ = −ξφ = −ξ µ (∂ µ φ). (3.280)The product of two scalar fields is transformed using the Leibniz ruleδ ξ (φψ) = (δ ξ φ)ψ + φ(δ ξ ψ) = −ξ µ (∂ µ φψ), (3.281)such that the product of two scalar fields transforms again as a scalar. Theabove Leibniz rule can be understood in mathematical terms as follows[Meyer (2005)]: The Hopf algebra U(Ξ) is represented on the space of scalarfields by infinitesimal coordinate transformations δ ξ . On scalar fields, theaction of δ ξ is explicitly given by the differential operator -ξ µ ∂ µ . Recallthat the space of scalar fields is not only a vector space, as it possesses alsoan algebra structure, such as U(Ξ) is not only an algebra, but also a Hopfalgebra, as it possesses in addition the co–structure maps defined above.We say that a Hopf algebra H acts on an algebra A if A is a module withrespect to the algebra H and if in addition for all h ∈ H and a, b ∈ Ah(ab) = µ ◦ ∆h(a ⊗ b),h(1) = ε(h).Here µ is the multiplication map defined by µ(a ⊗ b) = ab. In our concreteexample where H = U(Ξ) and A is the algebra of scalar fields we indeedhave that the algebra of scalar fields is a U(Ξ)−module algebra. This canbe seen easily if we rewrite (3.281) using (3.277) for the generators ξ ∈ Ξfor U(Ξ):It is also evident thatδ ξ (φψ) = (δ ξ φ)ψ + φ(δ ξ ψ) = µ ◦ ∆ξ(φ ⊗ ψ).δ ξ 1 = 0 = ε(ξ)1.Now we are in the right mathematical framework: We study a Lie algebra(here infinitesimal diffeomorphisms Ξ) and embed it in its universal envelopingalgebra (here U(Ξ)). This universal enveloping algebra is a Hopfalgebra via a natural Hopf structure induced by (3.277,3.278,3.279).Physical quantities live in representations of this Hopf algebras. Forinstance, the algebra of scalar fields is a U(Ξ)−module algebra. The actionof U(Ξ) on scalar fields is given in terms of infinitesimal coordinatetransformations δ ξ .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 465Similarly one studies tensor representations of U(Ξ). For example vectorfields are introduced by the transformation propertyδ ξ V α = −ξ µ (∂ µ V α ) − (∂ α ξ µ )V µ , δ ξ V α = −ξ µ (∂ µ V α ) + (∂ µ ξ α )V µ .The generalization to arbitrary tensor fields is straight forward:δ ξ T µ 1···µ nν 1···ν n= −ξ µ (∂ µ T µ 1···µ nν 1···ν n) + (∂ µ ξ µ 1)T µ···µ nν 1···ν n+ · · · + (∂ µ ξ µ n)T µ 1···µν 1···ν n−(∂ ν1 ξ ν )T µ 1···µ nν···ν n− · · · − (∂ νn ξ ν )T µ 1···µ nν 1···ν .As for scalar fields, we also find that the product of two tensors transformslike a tensor. Summarizing, we have seen that scalar fields, vectorfields and tensor fields are representations of the Hopf algebra U(Ξ), theuniversal enveloping algebra of infinitesimal diffeomorphisms. The Hopfalgebra U(Ξ) acts via infinitesimal coordinate transformations δ ξ which aresubject to the relations:[δ ξ , δ η ] = δ ξ×η ε(δ ξ ) = 0, ∆δ ξ = δ ξ ⊗ 1 + 1 ⊗ δ ξ S(δ ξ ) = −δ ξ . (3.282)The transformation operator δ ξ is explicitly given by differential operatorswhich depend on the representation under consideration. In case of scalarfields this differential operator is given by -ξ µ ∂ µ .3.17.1.4 Deformed <strong>Diff</strong>eomorphismsThe above concepts can be deformed in order to establish a consistenttensor calculus on the noncommutative space–time algebra (3.267). In thiscontext it is necessary to account the full Hopf algebra structure of theuniversal enveloping algebra U(Ξ).In our setting the algebra  possesses a noncommutative product definedby[ˆx µ , ˆx ν ] = iθ µν . (3.283)We want to deform the structure maps (3.282) of the Hopf algebra U(Ξ) insuch a way that the resulting deformed Hopf algebra which we denote byU(ˆΞ) consistently acts on Â. In the language introduced in the previoussection this means that we want  to be a U(ˆΞ)−module algebra. We claimthat the following deformation of U(Ξ) does the job. Let U(ˆΞ) be generatedas algebra by elements ˆδ ξ , ξ ∈ Ξ. We leave the algebra relation undeformedand demand [Meyer (2005)][ˆδ ξ , ˆδ η ] = ˆδ ξ×η (3.284)


466 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbut we deform the co–sectorwhere∆ˆδ ξ = e − i 2 hθρσ ˆ∂ρ⊗ ˆ∂ σ(ˆδ ξ ⊗ 1 + 1 ⊗ ˆδ ξ )e i 2 hθρσ ˆ∂ρ⊗ ˆ∂ σ,(3.285)[ ˆ∂ ρ , ˆδ ξ ] = ˆδ (∂ρξ).The deformed coproduct (3.285) reduces to the undeformed one (3.282) inthe limit θ → 0. Antipode and counit remain undeformedS(ˆδ ξ ) = −ˆδ ξ ε(ˆδ ξ ) = 0.We have to check whether the above deformation is a good one in thesense that it leads to a consistent action on Â. First we need a differentialoperator acting on fields in  which represents the algebra (3.284). Let usconsider the differential operatorˆX ξ :=∞∑n=01n! (− i 2 )n θ ρ 1 σ1 · · · θ ρ n σn ( ˆ∂ ρ1 · · · ˆ∂ ρnˆξµ ) ˆ∂µ ˆ∂σ1 · · · ˆ∂ σn . (3.286)This is to be understood like that: A vector–field ξ = ξ µ ∂ µ is determined byits coefficient functions ξ µ . Before we have seen that there is a vectorspaceisomorphism W from the space of commutative to the space of noncommutativefunctions which is given by the symmetric ordering prescription. Theimage of a commutative function f under the isomorphism W is denotedby ˆfW : f ↦→ W (f) = ˆf.In (3.286) ˆξ µ is therefore to be interpreted as the image of ξ µ with respectto W . Then indeed we haveTo see this we use result (3.276) to rewrite ( ˆX ξˆφ) :( ˆX ξˆφ) = ∞ ∑∞∑n=0n=0[ ˆX ξ , ˆX η ] = ˆX ξ×η . (3.287)1n! (− i 2 )n θ ρ 1 σ1 · · · θ ρ n σn ( ˆ∂ ρ1 · · · ˆ∂ ρnˆξµ )( ˆ∂µ ˆ∂σ1 · · · ˆ∂ σnˆφ) = (3.288)1n! (− i 2 )n θ ρ 1 σ1 · · · θ ρ n σn ( ˆ∂ ρ1 · · · ˆ∂ ρnˆξµ )( ˆ∂σ1 · · · ˆ∂ σn̂∂µ φ) = ̂ ξ µ (∂ µ φ) =From (3.288) it follows( ˆX ξ ( ˆX ηˆφ)) − ( ˆXη ( ˆX ξˆφ)) = ̂ ([ξ, η]φ) = ( ˆXξ×ηˆφ),̂ (ξφ).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 467which amounts to (3.287). It is therefore reasonable to introduce scalarfields ˆφ ∈  by the transformation propertyˆδξˆφ = −( ˆXξˆφ).The next step is to work out the action of the differential operators ˆX ξ onthe product of two fields. A calculation [Aschieri et. al. (2005)] shows that( ˆX ξ (ˆφˆψ)) = µ ◦ (e − i 2 hθρσ ˆ∂ρ⊗ ˆ∂ σ( ˆX ξ ⊗ 1 + 1 ⊗ ˆX ξ )e i 2 hθρσ ˆ∂ρ⊗ ˆ∂ σ ˆφ ⊗ ˆψ).This means that the differential operators ˆX ξ act via a deformed Leibnizrule on the product of two fields. Comparing with (3.285) we see thatthe deformed Leibniz rule of the differential operator ˆX ξ is exactly the oneinduced by the deformed coproduct (3.285):ˆδξ (ˆφˆψ) = e − i 2 hθρσ ˆ∂ρ⊗ ˆ∂ σ(ˆδ ξ ⊗ 1 + 1 ⊗ ˆδ ξ )e i 2 hθρσ ˆ∂ρ⊗ ˆ∂ σ(ˆφˆψ) = − ˆX ξ ⊲ (ˆφˆψ).Hence, the deformed Hopf algebra U(ˆΞ) is indeed represented on scalarfields ˆφ ∈  by the differential operator ˆX ξ . The scalar fields form aU(ˆΞ)−module algebra.In analogy to the previous section we can introduce vector and tensorfields as representations of the Hopf algebra U(ˆΞ). The transformationproperty for an arbitrary tensor readsˆδξ ˆTµ 1···µ rν 1···ν s= −( ˆX µξ ˆT 1···µ nν 1···ν n) + ( ˆX ˆT µ···µ (∂µξ µ n1 ) ν 1···ν n) + · · · + ( ˆX µ(∂µξ µ n ) ˆT1···µν 1···ν n)−( ˆX ˆT µ 1···µ (∂ν1 ξ ν n) ν···ν n) − · · · − ( ˆX ˆT µ 1···µ (∂νn ξ ν n) ν 1···ν ).3.17.1.5 Noncommutative Space–Time <strong>Geom</strong>etryThe deformed algebra of infinitesimal diffeomorphisms and the tensor calculuscovariant with respect to it is the fundamental building-block for thedefinition of a noncommutative geometry on θ−deformed spaces. In thissection we sketch the important steps towards a deformed Einstein-Hilbertaction [Aschieri et. al. (2005)]. A first ingredient is the covariant derivativeˆD µ . Algebraically, it can be defined by demanding that acting on avector-field it produces a tensor–field [Meyer (2005)]ˆδξ ˆDµ ˆVν = −( ˆX ξ ˆDµ ˆVν ) − ( ˆX (∂µξ α ) ˆD α ˆVν ) − ( ˆX (∂νξ α ) ˆD µ ˆVα ) (3.289)The covariant derivative is given by a connection ˆΓ µνρˆD µ ˆVν = ˆ∂ µ ˆVν − ˆΓ µνρ ˆVρ .


468 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFrom (3.289) it is possible to deduce the transformation property of ˆΓ µνρˆδξ ˆΓµν ρ = ( ˆX ξ ˆΓµν ρ )−( ˆX (∂µξ α )ˆΓ αν ρ )−( ˆX (∂νξ α )ˆΓ µα ρ )+( ˆX (∂αξ ρ )ˆΓ µν α )−( ˆ∂ µ ˆ∂νˆξρ ).The metric Ĝµν is defined as a symmetric tensor of rank two. It can beobtained for example by a set of vector–fields ʵ a , a = 0, . . . , 3, where a isto be understood as a mere label. These vector–fields are called vierbeins.Then the symmetrized product of those vector–fields is indeed a symmetrictensor of rank twoĜ µν := 1 2 (ʵ a Ê ν b + Êν b Ê µ a )η ab .Here η ab stands for the usual flat Minkowski space metric with signature(− + ++). Let us assume that we can choose the vierbeins ʵ a such thatthey reduce in the commutative limit to the usual vierbeins e µ a . Then alsothe metric Ĝµν reduces to the usual, undeformed metric g µν .The inverse metric tensor we denote by upper indicesĜ µν Ĝ νρ = δ ρ µ.We use Ĝµν respectively Ĝµν to raise and lower indices.The curvature and torsion tensors are obtained by taking the commutatorof two covariant derivatives 15which leads to the expressions[ ˆD µ , ˆD ν ] ˆV ρ = ˆR µνρα ˆVα + ˆT µνα ˆDα ˆVρˆR µνρ σ = ˆ∂ ν ˆΓµρ σ − ˆ∂ µˆΓνρ σ + ˆΓ νρβ ˆΓµβ σ − ˆΓ µρβ ˆΓνβσˆT µν α = ˆΓ νµ α − ˆΓ νµ α .If we assume the torsion–free case, i.e.,ˆΓ µν σ = ˆΓ νµ σ ,we find a unique expression for the metric connection (Christoffel symbols)defined (by means of ˆDα Ĝ βγ = 0 ) in terms of the metric and its inverseˆΓ αβ σ = 1 2 ( ˆ∂ α Ĝ βγ + ˆ∂ β Ĝ αγ − ˆ∂ γ Ĝ αβ )Ĝγσ .15 The generalization of covariant derivatives acting on tensors is straight forward [Aschieriet. al. (2005)].


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 469From the Riemann curvature tensor ˆR µνρ σ we get the Ricci curvaturescalar by contracting the indicesˆR := Ĝµν ˆRνµρ ρ .ˆR indeed transforms as a scalar which may be checked explicitly by takingthe deformed coproduct (3.285) into account.To get an integral which is invariant with respect to the Hopf algebraof deformed infinitesimal diffeomorphisms we need a measure function Ê.We demand the transformation propertyˆδξ Ê = − ˆX ξ Ê − ˆX (∂µξ µ)Ê. (3.290)Then it follows with the deformed coproduct (3.285) that for any scalarfield Ŝˆδξ ÊŜ = − ˆ∂ µ ( ˆX ξ µ(ÊŜ)).Hence, transforming the product of an arbitrary scalar field with a measurefunction Ê we get a total derivative which vanishes under the integral. Asuitable measure function with the desired transformation property (3.290)is for instance given by the determinant of the vierbein ʵ aÊ = det(ʵ a ) := 1 4! εµ 1···µ 4 εa1···a 4Ê µ1a 1Ê µ2a 2Ê µ3a 3Ê µ4a 4.That Ê transforms correctly can be shown by using that the product offour ʵ a iitransforms as a tensor of fourth rank and some combinatorics.Now we have all ingredients to write down the Einstein–Hilbert action.Note that having chosen a differential calculus as in (3.271), the integral isuniquely determined up to a normalization factor by requiring 16 [Douglasand Nekrasov (2001)]∫ˆ∂ µ ˆf = 0for all ˆf ∈ Â. Then we define the Einstein–Hilbert action on  as∫Ŝ EH := det(ʵ a ) ˆR + complex conjugate.16 We consider functions that “vanish at infinity”.


470 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIt is by construction invariant with respect to deformed diffeomorphismsmeaning thatˆδξ Ŝ EH = 0.In this section we have presented the fundamentals of a noncommutativegeometry on the algebra  and defined an invariant Einstein–Hilbert action.There is however one important step missing which is subject of thefollowing section: We want to make contact of the noncommutative gravitytheory with Einstein’s gravity theory. This we achieve by introducing the⋆−product formalism.3.17.1.6 Star–Products and Expanded Einstein–Hilbert ActionTo express the noncommutative fields in terms of their commutative counterpartswe first observe that we can map the whole algebraic constructionof the previous sections to the algebra of commutative functions via thevector space isomorphism W introduced above. By equipping the algebraof commutative functions with a new product denoted by ⋆ be can renderW an algebra isomorphism. We define [Meyer (2005)]f ⋆ g : = W −1 (W (f)W (g)) = W −1 ( ˆfĝ), (3.291)and get (A, ⋆) ∼ = Â.The ⋆−product corresponding to the symmetric ordering prescription W isthen given explicitly by the Moyal product 17f ⋆ g = µ ◦ e i 2 θµν ∂ µ⊗∂ νf ⊗ g = fg + i 2 θµν (∂ µ f)(∂ ν g) + O(θ 2 ).It is a deformation of the commutative point–wise product to which itreduces in the limit θ → 0.In virtue of the isomorphism W we can map all noncommutative fieldsto commutative functions in AˆF ↦→ W −1 ( ˆF ) ≡ F.We then expand the image F in orders of the deformation parameter θF = F (0) + F (1) + F (2) + O(θ 3 ),17 This is an immediate consequence of (3.275).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 471where the zeroth order always corresponds to the undeformed quantity.Products of functions in  are simply mapped to ⋆−products of the correspondingfunctions in A. The same can be done for the action of thederivative ˆ∂ µ and consequently for an arbitrary differential operator actingon  [Aschieri et. al. (2005)].The fundamental dynamical field of our gravity theory is the vierbeinfield ʵ a . All other quantities such as metric, connection and curvaturecan be expressed in terms of it. Its image with respect to W −1 is denotedby E a µ . In the first approximation, we study the case E a µ = e a µ , wheree a µ is the usual vierbein field. Then for instance the metric is given up tosecond order in θ byG µν = 1 2 (E µ a ⋆ E ν b + E ν b ⋆ E µ a )η ab = 1 2 (e µ a ⋆ e ν b + e ν b ⋆ e µ a )η ab= g µν − 1 8 θα1β 1θ α2β 2(∂ α1 ∂ α2 e µ a )(∂ β1 ∂ β2 e ν b )η ab + . . . ,where g µν is the usual, undeformed metric. For the Christoffel symbol onefinds up to the second order [Meyer (2005)]:(i) the 0th order is the undeformed expressionΓ (0)ρµν = 1 2 [p µg νγ + p ν g µγ − p γ g µν ]g γρ ;(ii) the first order readsΓ (1)µν ρ = i 2 θαβ (∂ α Γ (0)σµν )g στ (∂ β g τρ ); and(iii) the second order isΓ (2)µν ρ = − 1 8 θα1β 1θ α2β 2((∂ α1 ∂ α2 Γ (0)µνσ)(∂ β1 ∂ β2 g σρ )−2(∂ α1 Γ (0)µνσ)∂ β1 ((∂ α2 g στ )(∂ β2 g τξ )g ξρ )−Γ (0)µνσ((∂ α1 ∂ α2 g στ )(∂ β1 ∂ β2 g τξ ) + g στ (∂ α1 ∂ α2 e τ a )(∂ β1 ∂ β2 e ξ b )η ab (3.292)−2∂ α1 ((∂ α2 g στ )(∂ β2 g τλ )g λκ )(∂ β1 g κξ ))g ξρ + 1 2 (∂ µ((∂ α1 ∂ α2 e aν )(∂ β1 ∂ β2 e bσ ))+ ∂ ν ((∂ α1 ∂ α2 e aσ )(∂ β1 ∂ β2 e bµ )) − ∂ σ ((∂ α1 ∂ α2 e aµ )(∂ β1 ∂ β2 e bν )))η ab g σρ ),where Γ (0)µνσ = Γ (0)ρµν g ρσ .


472 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe expressions for the curvature tensor now readR (1)µνρ σ =− i 2 θκλ ((∂ κ R (0)µνρ τ )(∂ λ g τγ )g γσ − (∂ κ Γ (0)−Γ (0)νρ β )(Γ (0)µβµτ σ (∂ λ g βγ )g γτ + ∂ µ ((∂ λ g βγ )g γσ ) + (∂ λ Γ (0) σ ))+(∂ κ Γ (0)µρ β )(Γ (0)νβµβτ (∂ λ g τγ )g γσ − Γ (0)ντ σ (∂ λ g βγ )g γττ (∂ λ g τγ )g γσ+∂ ν [(∂ λ g βγ )g γσ ] + (∂ λ Γ (0) σ νβ))), (3.293)R (2)µνρ σ =∂ ν Γ (2)µρ σ + Γ (2)νρ γ Γ (0)µγ σ + Γ (0)νρ γ Γ (2)µγ σ+ i 2 θαβ [(∂ α Γ (1)νρ γ )(∂ β Γ (0)µγ σ ) + (∂ α Γ (0)νρ γ )(∂ β Γ (1)µγ σ )]− 1 8 θα1β 1θ α2β 2(∂ α1 ∂ α2 Γ (0)νρ γ )(∂ β1 ∂ β2 Γ (0)µγ σ ) − (µ ↔ ν), (3.294)where the second order is given implicitly in terms of the Christoffel symbols.The deformed Einstein–Hilbert action is given byS EH = 1 ∫d 4 x det ⋆ e a µ ⋆ R + c.c.2= 1 ∫d 4 x det ⋆ e a µ ⋆ (R +2¯R) = 1 ∫d 4 x det ⋆ e a µ (R +2¯R)∫= S (0)EH + d 4 x (dete a µ )R (2) + (det ⋆ e a µ ) (2) R (0) , (3.295)where we used that the integral together with the Moyal product (by partialintegration) has the property∫∫ ∫d 4 x f ⋆ g = d 4 x fg = d 4 x g ⋆ f.In (3.295) det ⋆ e µ a is the ⋆−determinantdet ⋆ e µ a = 1 4! εµ 1···µ 4 εa1···a 4e µ1a 1⋆e µ2a 2⋆e µ3a 3⋆e µ4a 4= dete µ a +(det ⋆ e µ a ) (2) +. . . ,where(det ⋆ ) (2) = − 1 18 4! θα1β 1θ α2β 2ε µ 1 ...µ 4 a εa1...a 4[(∂ α1 ∂ α2 e 1 aµ1 )(∂ β1 ∂ β2 e 2 aµ2 )e 3 aµ3 e 4µ4 +a∂ α1 ∂ α2 (e 1 aµ1 e 2 aµ2 )(∂ β1 ∂ β2 e 3 aµ3 )e 4 aµ4 + ∂ α1 ∂ α2 (e 1 aµ1 e 2 aµ2 e 3 aµ3 )(∂ β1 ∂ β2 e 4µ4 )].The odd orders of θ vanish in (3.295) but the even orders of θ give nontrivialcontributions.


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 4733.17.2 Synthetic <strong>Diff</strong>erential <strong>Geom</strong>etryThe sense in which we understand the word ‘synthetic’ in this context isthat we place ourselves in the certain category E of manifold–like objectswhere everything is smooth. A main assumption about our category E isthat it is cartesian closed, meaning that functional spaces as well as methodsof classical functional analysis are available.Recall that distributions are usually thought of as very non–smoothfunctions, like the Heaviside function, or the Dirac δ−function. These so–called generalized functions are commonly presented following the Sobolev–Schwartz functional analysis, usually including the integral transforms ofFourier, Laplace, Mellin, Hilbert, Cauchy–Bochner and Poisson. The mainapplication of the theory of generalized functions is the solution of classicalequations of mathematical physics (see e.g., [Vladimirov (1971); Vladimirov(1986)]).On the other hand, there is a viewpoint, firstly stressed by Lawvere[Lawvere (1979)], and fully elaborated by A. Kock [Kock (1981); Kock andReyes (2003)], that distributions are extensive quantities, where functionsare intensive quantities. This viewpoint also makes it quite natural toformulate partial equations of mathematical physics (like classical wave andheat equations) – as ODEs describing the evolution over time of any initialdistributions. For example, the main construction in the theory of the waveequation is the construction of the fundamental solution: the descriptionof the evolution of a point δ−distribution over time.To say that distributions are extensive quantities implies that theytransform covariantly (in a categorical sense). To say that functions areintensive quantities implies that they transform contravariantly. Distributionsare here construed, as linear functionals on the space of (smooth) functions.However, since all functions in the synthetic context are smooth, aswell as continuous, there is no distinction between distributions and Radonmeasures.In the category E one can define the vector space D c(M) ′ of distributionsof compact support on M, for each manifold–like object M ∈ E, namelythe object of −linear maps R M → R. We shall assume that elementarydifferential calculus for functions R → R is available, as in all models ofsynthetic differential geometry (SDG, see [Kock (1981); Moerdijk and Reyes(1991)]). Following Kock [Kock (1981)], we shall also assume some integralcalculus, but only in the weakest possible sense, namely we assume thatfor every ψ : R → R, there is a unique Ψ : R → R with Ψ ′ = ψ and with


474 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionΨ(0) = 0. Using this result, intervals will be construed as distributions:for a, b ∈ R, [a, b] denotes the distribution3.17.2.1 Distributionsψ ↦→∫ baψ(x) dx = Ψ(b) − Ψ(a).Let us make the formula for covariant functorality D ′ c explicit. Let f : M →N be a map. Then the corresponding map f ∗ = D ′ c(f) : D ′ c(M) → D ′ c(N)is described by declaring< f ∗ (T ), φ >=< T , φ ◦ f >, (3.296)where T is a distribution on M, and φ is a function on N. The bracketsdenote evaluation of distributions on functions. If we similarly denote thevalue of the contravariant functor M ↦→ R M on a map f by f ∗ , the definingequation for f ∗ reads< f ∗ (T ), φ >=< T , f ∗ (φ) > .D ′ c(M) is an R−linear space, and all maps f ∗ : D ′ c(M) → D ′ c(N) areR−linear. Also D ′ c(M) is a Euclidean vector space V , meaning that thebasic differential calculus in available.For any distribution T of compact support on M, one has its Total,which is just the number < T , 1 >∈ R, where 1 denotes the function on Mwith constant value 1. Since f ∗ (1) = 1 for any map f, it follows that f ∗preserves Totals.Recall that a distribution T on M may be multiplied by any functiong : M → R, by the rule< g · T , φ >=< T , g · φ > . (3.297)A basic result from single–variable calculus, ‘integration by substitution’,in a pure ‘distribution’ form reads: Given any function g : R → R,and given a, b ∈ R, we haveg ∗ (g ′ · [a, b]) = [g(a), g(b)].Let ψ be a test function, and let Ψ be a primitive of it, Ψ ′ = ψ. Then< [g(a), g(b)], ψ >= Ψ(g(b)) − Ψ(g(a)).


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 475On the other hand, by the chain rule, Ψ ◦ g is a primitive of g ′ · (ψ ◦ g), soΨ(g(a))−Ψ(g(b)) =< [a, b], g ′·(ψ◦g) >=< g ′·[a, b], ψ◦g >=< g ∗ (g ′·[a, b]), ψ > .The external product of distributions of compact support is defined as follows.If P is a distribution on M, and Q a distribution on N, we get adistribution P × Q on M × N, by< P × Q, ψ >=< P, [m ↦→< Q, ψ(m, −) >] > .However, if [a, b] and [c, d] are intervals (viewed as distributions on R,as described above), [a, b] × [c, d] = [a, b]×[c, d], as distributions on R 2 , byan application of Fubini’s Theorem, (which holds in our context here as aconsequence of equality of mixed partial derivatives). Distributions arisingin this way on R 2 , are called rectangles. The obvious generalizations tohigher dimensions are called boxes. We have< [a, b] × [c, d], ψ >=∫ b ∫ dacψ(x, y) dy dx,in traditional notation. Here, we can define the boundary of the box [a, b] ×[c, d] as the obvious distribution on R 2 ,(p 2 c) ∗ [a, b] + (p 1 b) ∗ [c, d] − (p 2 d) ∗ [a, b] − (p 1 a) ∗ [c, d],where p 2 c(x) = (x, c), p 1 b(y) = (b, y), etc.By a singular box in a manifold–like object M, we understand the dataof a map γ : R 2 → M and a box [a, b] × [c, d] in R 2 , and similarly forsingular intervals and singular rectangles. Such a singular box gives rise toa distribution on M, namely g ∗ ([a, b] × [c, d]).By differential operator on an object M, we here understand just anR−linear map D : R M → R M . If D is such an operator, and T is adistribution on M, we define D(T ) by< D(T ), ψ >=< T , D(ψ) >,and in this way, D becomes a linear operator D ′ c(M) → D ′ c(M).In particular, if X is a vector–field on M, one defines the directional(i.e., Lie) derivative D X (T ) of a distribution T on M by the formula< D X (T ), ψ >=< T , D X (ψ) > . (3.298)This in particular applies to the vector–field ∂/∂x on R, and reads here< T ′ , ψ >=< T , ψ ′ >,


476 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere ψ ′ denotes the ordinary derivative of the function ψ.The following Proposition is an application of the covariant functoralityof the functor D ′ c, which will be used in connection with the wave (andheat) equations in dimension 2. We consider the (orthogonal) projectionp : R 3 → R 2 onto the xy−plane; ∆ denotes the Laplace operator in therelevant R n , so for R 3 , ∆ is ∂ 2 /∂x 2 +∂ 2 /∂y 2 +∂ 2 /∂z 2 . For any distributionS (of compact support) on R 3 ,p ∗ (∆(S)) = ∆(p ∗ (S)).This follows from the fact that for any ψ : R 2 → R,namely, ∂ 2 ψ/∂x 2 + ∂ 2 ψ/∂y 2 .∆(p ∗ ψ)) = p ∗ (∆(ψ)),3.17.2.2 Synthetic Calculus in Euclidean SpacesRecall that a vector space E (which in the present context means anR−module) is called Euclidean if differential and integral calculus for functionsR → E is available (see [Kock (1981); Moerdijk and Reyes (1991)]).The coordinate vector spaces are Euclidean, but so are also the vector spacesR M , and D c(M) ′ for any M. To describe for instance the ‘time–derivative’f˙of a function f : R → D c(M), ′ we put< ˙ f(t), ψ >= d dt < f(t), ψ > .Similarly, from the integration Axiom for R, one immediately proves thatD c(M) ′ satisfies the integration Axiom, in the sense that for any h : R →D c(M), ′ there exists a unique H : R → D c(M) ′ satisfying H(0) = 0 andH∫ ′ (t) = h(t) for all t. In particular, if h : R → D c(M), ′ the ‘integral’bh(u) du = H(b) − H(a) makes sense, and the Fundamental Theorem ofaCalculus holds.As a particular case of special importance, we consider a linear vector–field on a Euclidean R−module V . To say that the vector field is linear isto say that its principal–part formation V → V is a linear map, Γ, say. Wehave then the following version of a classical result. By a formal solutionfor an ordinary differential equation, we mean a solution defined on the setD ∞ of nilpotent elements in R (these form a subgroup of (R, +)).Let a linear vector–field on a Euclidean vector space V be given by thelinear map Γ : V → V . Then the unique formal solution of the correspond-


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 477ing differential equation,˙ F (t) = Γ(F (t)),with initial position v, is the map D ∞ × V → V given by [Kock (1981);Kock and Reyes (2003)](t, v) ↦→ e t·Γ (v), (3.299)where the r.h.s here means the sum of the following ‘series’ (which has onlyfinitely many non–vanishing terms, since t is assumed nilpotent):v + tΓ(v) + t22! Γ2 (v) + t33! Γ3 (v) + . . .where Γ 2 (v) means Γ(Γ(v)), etc.There is an analogous result for second order differential equations ofthe form¨F (t) = Γ(F (t)).The formal solution of this second order differential equation with initialposition v and initial velocity w, is given by [Kock (1981); Kock and Reyes(2003)]F (t) = v + t · w + t2 t3 t4Γ(v) + Γ(w) +2! 3! 4! Γ2 (v) + t55! Γ2 (w) + ....Also, given f : R → V , where V is a Euclidean vector space, and giveng : R → R. Then for any a, b ∈ R,∫ baf(g(x)) · g ′ (x) dx =∫ g(b)g(a)f(u) du.Linear maps between Euclidean vector spaces preserve differentiation andintegration of functions R → V ; we shall explicitly need the following particularassertion: Let F : V → W be a linear map between Euclidean vectorspaces. Then for any f : R → V ,F (∫ baf(t) dt) =∫ baF (f(t)) dt.


478 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.17.2.3 Spheres and Balls as DistributionsFor a, b ∈ R, we let [a, b] denote the distribution f ↦→ ∫ bf(x) dx. Recallathat such distributions on the line we call intervals; the length of an interval[a, b] is defined to be b − a. Let [a 1 , b 1 ] and [a 2 , b 2 ] be two such intervals.They are equal as distributions if and only if they have same length, b 1 −a 1 = b 2 − a 2 (= l, say), and l · (a 1 − a 2 ) = 0 (this then also impliesl · (b 1 − b 2 ) = 0).We shall also consider such ‘balls’ in dimension 2 and 3, where, however,t cannot in general be recovered from the distribution, unless t is strictlypositive.We fix a positive integer n. We shall consider the sphere S t of radiust, and the ball B t of radius t, for any t ∈ R, as distributions on R n (ofcompact support), in the following sense:∫< S t , ψ >= ψ(x)dx = t∫S n−1 ψ(t · u) du,t S∫1< B t , ψ >= ψ(x)dx = t∫B n ψ(t · u) du,t B 1where du refers to the surface element of the unit sphere S 1 in the firstequation and to the volume element of the unit ball B 1 in the second. Theexpressions involving ∫ S tand ∫ B tare to be understood symbolically, unlesst > 0; if t > 0, they make sense literally as integrals over sphere and ball,respectively, of radius t, with dx denoting surface-, resp. volume element.The expression on the right in both equations makes sense for any t, and sothe distributions S t and B t are defined for all t; in particular, for nilpotentones.It is natural to consider also the following distributions S t and B t onR n (again of compact support):∫< S t , ψ >= ψ(t · u) du,S 1∫< B t , ψ >= ψ(t · u) du.B 1For t > 0, they may, modulo factors of the type 4π, be considered as‘average over S t ’ and ‘average over B t ’, respectively, since S t differs fromS t by a factor t n−1 , which is just the surface area of S t (modulo the factorof type 4π), and similarly for B t .


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 479Note that S 1 = S 1 and B 1 = B 1 . And also note that the definition ofS t and B t can be formulated asS t = H t (S 1 ) , B t = H t (B 1 ),where H t : R n → R n is the homothetic transformation u ↦→ t · u, and wherewe are using the covariant functoriality of distributions of compact support.For low dimensions, we shall describe the distributions S t , B t , S t andB t explicitly:Dimension 1:Dimension 2:Dimension 3:< S t , ψ >=< B t , ψ >=< S t , ψ >=< B t , ψ >=< S t , ψ >= ψ(−t) + ψ(t), < B t , ψ >=< S t , ψ >= ψ(−t) + ψ(t), < B t , ψ >=< S t , ψ >=< B t , ψ >=< S t , ψ >=< B t , ψ >=∫ π ∫ 2π0 0∫ t ∫ π ∫ 2π00∫ π ∫ 2π∫ 2π0∫ t ∫ 2π0 0∫ 2π0∫ 1 ∫ 2π0ψ(t cos θ, t sin θ) t dθ,∫ t−t∫ 1−1ψ(s cos θ, s sin θ) s dθ ds,ψ(t cos θ, t sin θ) dθ,0ψ(s) ds,ψ(t · s) ds.ψ(ts cos θ, t s sin θ) s dθ ds.ψ(t cos θ sin φ, t sin θ sin φ, t cos φ)t 2 sin φ dθ dφ,00 0∫ 1 ∫ π ∫ 2π00ψ(s cos θ sin φ, s sin θ sin φ, s cos φ) s 2 sin φ dθ dφ ds,ψ(t cos θ sin φ, t sin θ sin φ, t cos φ) sin φ dθ dφ,0ψ(ts cos θ sin φ, ts sin θ sin φ, ts cos φ) s 2 sin φ dθ dφ ds.These formulas make sense for all t, whereas set–theoretically S t and B t(as point sets) only make good sense for t > 0.


480 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3.17.2.4 Stokes Theorem for Unit SphereRecall that the main Theorem of vector calculus is the Stokes Theorem:∫ ∫ω = dω,∂γfor ω an (n−1)−form, γ a suitable n−dimensional figure (with appropriatemeasure on it) and ∂γ its geometric boundary. In the synthetic context,the Theorem holds at least for any singular cubical chain γ : I n → M (I nthe n−dimensional coordinate cube), because the Theorem may then bereduced to the fundamental Theorem of calculus, which is the only wayintegration enters in the elementary synthetic context; measure theory notbeing available therein (see [Moerdijk and Reyes (1991)] for details). Below,we shall apply the result not only for singular cubes, but also for singularboxes, like the usual (γ : R 2 → R 2 , [0, 2π] × [0, 1]), ‘parameterizing the unitdisk B by polar coordinates’,γγ(θ, r) = (r cos θ, r sin θ). (3.300)We shall need from vector calculus the Green–Gauss–Ostrogradsky DivergenceTheorem∫flux of F over ∂γ = (divergence of F),with F a vector–field, for the geometric ‘figure’ γ = the unit ball in R n . Forthe case of the unit ball in R n , the reduction of the Divergence Theoremto Stokes’ Theorem is a matter of the differential calculus of vector fieldsand differential forms. For the convenience of the reader, we recall the casen = 2.Given a vector–field F(x, y) = (F (x, y), G(x, y)) in R 2 , apply Stokes’Theorem to the differential formω = −G(x, y)dx + F (x, y)dy,for the singular rectangle γ given by (3.300) above.We also need that the trigonometric functions cos and sin should bepresent. We assume that they are given as part of the data, and that theysatisfy cos 2 + sin 2 = 1, and cos ′ = − sin, sin ′ = cos. Also as part of thedata, we need specified an element π ∈ R so that cos π = −1, cos 0 = 1.γ


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 481Then we haveSince⎧⎨ γ ∗ (dx) = cos θdr − r sin θdθγ ∗ (dy) = sin θdr + r cos θdθ⎩γ ∗ (dx ∧ dy) = r (dr ∧ dθ).dω = (∂G/∂y + ∂F/∂x) dx ∧ dy = div (F) dx ∧ dy,thenγ ∗ (dω) = div (F) r (dr ∧ dθ).On the other hand,γ ∗ ω = (F sin θ − G cos θ)dr + (F r cos θ + G r sin θ) dθ, (3.301)(all F , G, and F to be evaluated at (r cos θ, r sin θ)). Therefore∫ ∫ 2π ∫ 1dω = div(F) r dr dθ;γ00this is ∫ B 1div (F) dA. On the other hand, by Stokes’ Theorem ∫ ∫γ dω =ω, which is a curve integral of the 1–form (3.301) around the boundary∂γof the rectangle [0, 2π] × [0, 1]. This curve integral is a sum of four termscorresponding to the four sides of the rectangle. Two of these (correspondingto the sides θ = 0 and θ = 2π) cancel, and the term corresponding tothe side where r = 0 vanishes because of the r in r (dr∧dθ), so only the sidewith r = 1 remains, and its contribution is, with the correct orientation,∫ 2π∫(F (cos θ, sin θ) cos θ + G(cos θ, sin θ) sin θ) dθ = F · n ds ,0S 1where n is the outward unit normal of the unit circle. This expression isthe flux of F over the unit circle, which thus equals the divergence integralcalculated above.3.17.2.5 Time Derivatives of Expanding SpheresWe now combine vector calculus with the calculus of the basic ball– andsphere–distributions, to get the following result [Kock and Reyes (2003)]:In R n (for any n), we have, for any t,ddt St = t · ∆(B t ),


482 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere ∆ denotes the Laplacian operator (see [Kock (2001)] for ‘synthetic’analysis of the Laplacian).We collect some further information about t−derivatives of some of thet−parameterized distributions considered:In dimension 1, we haveforwhereasddt (B t) = S t . (3.302)ddt (S t) = ∆(B t ), (3.303)ddt < S t, ψ >= d dt < ψ(t) + ψ(−t) >= ψ′ (t) − ψ ′ (−t),< ∆B t , ψ >=< B t , ψ ′′ >=∫ t−tψ ′′ (t) dt,and the result follows from the Fundamental Theorem of calculus.The equation (3.303) implies the following equation if n = 1, while in[Kock and Reyes (2003)] it was proved that it also holds if n ≥ 2:3.17.2.6 The Wave Equationt · ddt (S t) = (n − 1)S t + t · ∆(B t ). (3.304)Let ∆ denote the Laplacian operator ∑ ∂ 2 /∂x 2 i on Rn (see [Kock (2001)]).We shall consider the wave equation (WE) in R n , (for n = 1, 2, 3),d 2Q(t) = ∆Q(t) (3.305)dt2 as a second order ordinary differential equation on the Euclidean vectorspace D ′ c(R n ) of distributions of compact support; in other words, we arelooking for functionsQ : R → D ′ c(R n ),so that for all t ∈ R, ¨Q(t) = ∆(Q(t)), viewing ∆ as a map D′c (R n ) →D ′ c(R n ) [Kock and Reyes (2003)].Consider a function f : R → V , where V is a Euclidean vector space(we are interested in V = D ′ c(R n )). Then we call the pair of vectors in V


<strong>Applied</strong> Manifold <strong>Geom</strong>etry 483consisting of f(0) and ˙ f(0) the initial state of f. We can now, for each ofthe cases n = 1, n = 3, and n = 2 describe fundamental solutions to thewave equations. By fundamental solutions, we mean solutions whose initialstate is either a constant times (δ(0, 0)), or a constant times (0, δ(0).In dimension 1 : The function R → D ′ c(R) given byt ↦→ S t (= S t )is a solution of the WE; its initial state is 2(δ(0), 0).The function R → D ′ c(R) given byt ↦→ B tis a solution of the WE with initial state 2(0, δ(0)).In dimension 3: The function R → D ′ c(R 3 ) given byt ↦→ S t + t 2 ∆(B t )is a solution of the WE with initial state 4πδ(0), 0). The function R →D ′ c(R 3 ) given byt ↦→ t · S tis a solution of the WE with initial state 4π(0, δ(0)).In dimension 2: The function R → D ′ c(R 2 ) given byt ↦→ p ∗ (S t + t 2 ∆(B t ))is a fundamental solution of the WE in dimension 2; its initial state is4π(δ(0), 0). The function R → D ′ c(R 2 ) given byt ↦→ p ∗ (t · S t )is a fundamental solution of the WE in dimension 2; its initial state is4π(0, δ(0)).


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Chapter 4<strong>Applied</strong> Bundle <strong>Geom</strong>etry4.1 Intuition Behind a Fibre BundleRecall that tangent and cotangent bundles, T M and T ∗ M, are specialcases of a more general geometrical object called fibre bundle, where theword fiber V of a map π : Y −→ X denotes the preimage π −1 (x) of anelement x ∈ X. It is a space which locally looks like a product of twospaces (similarly as a manifold locally looks like Euclidean space), but maypossess a different global structure. To get a visual intuition behind thisfundamental geometrical concept, we can say that a fibre bundle Y is ahomeomorphic generalization of a product space X × V (see Figure 4.1),where X and V are called the base and the fibre, respectively. π : Y → Xis called the projection, Y x = π −1 (x) denotes a fibre over a point x ofthe base X, while the map f = π −1 : X → Y defines the cross–section,producing the graph (x, f(x)) in the bundle Y (e.g., in case of a tangentbundle, f = ẋ represents a velocity vector–field) (see [Steenrod (1951)]).Fig. 4.1 A sketch of a fibre bundle Y ≈ X × V as a generalization of a product spaceX × V ; left – main components; right – a few details (see text for explanation).The main reason why we need to study fibre bundles is that all dy-485


486 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionnamical objects (including vectors, tensors, differential forms and gaugepotentials) are their cross–sections, representing generalizations of graphsof continuous functions.4.2 Definition of a Fibre BundleLet M denote an n−manifold with an atlas Ψ M consisting of local coordinatesx α ∈ M, (α = 1, ..., dim M), given byΨ M = {U ξ , φ ξ }, φ ξ (x) = x α e α , (for all x ∈ U ξ ⊂ M),where {e α } is a fixed basis of R m . Its tangent and cotangent bundles,T M and T ∗ M, respectively, admit atlases of induced coordinates (x α , ẋ α )and (x α , ẋ α ), relative to the holonomic fibre bases {∂ α } and {dx α }, respectively.For all elements (i.e., points) p ∈ T M and p ∗ ∈ T ∗ M, we have(see [Sardanashvily (1993); Sardanashvily (1995); Giachetta et. al. (1997);Mangiarotti and Sardanashvily (2000a); Sardanashvily (2002a)])p = ẋ α ∂ α , p ∗ = ẋ α dx α , ∂ α ⌋dx α = δ α α, (α = 1, ..., dim M).Also, we will use the notationω = dx 1 ∧ · · · ∧ dx n , ω α = ∂ α ⌋ω, ω µα = ∂ µ ⌋∂ α ⌋ω. (4.1)If f : M → M ′ is a smooth manifold map, we define the induced tangentmap T f over f, given byT f : T M −→ T M ′ ,ẋ ′ α ◦ T f =∂f α∂x α ẋα . (4.2)Given a manifold product M × N, π 1 and π 2 denote the natural projections(i.e., canonical surjections),π 1 : M × N → M, π 2 : M × N → N.Now, as a homeomorphic generalization of a product space, a fibre bundlecan be viewed either as a topological or a geometrical (i.e., coordinate)construction. As a topological construction, a fibre bundle is a class of moregeneral fibrations. To have a glimpse of this construction, let I = [0, 1]. Amap π : Y → X is said to have the homotopy lifting property (HLP, see[Switzer (1975)]) with respect to a topological space Z if for every mapf : Z → Y and homotopy H : Z × I → X of π ◦ f there is a homotopyV : Z × I → Y with f = V 0 and π ◦ V = H. V is said to be a lifting of H. π


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 487is called a fibration if it has the HLP for all spaces Z and a weak fibrationif it has the HLP for all disks D n , (n ≥ 0). If x ∈ X is the base point, thenV = π −1 (x) is called the fibre of the fibration π. The projection onto thefirst factor, π 1 : X × V → X, is clearly a fibration and is called the trivialfibration over X with the fibre V .However, for the sake of applying differential and integral dynamicson fibre bundles, we will rather use Steenrod’s coordinate bundle definition(see [Steenrod (1951)]), which defines fibre bundle Y as a sextuple(Y, X, π, V, G, Ψ Y ), with:(1) a space Y called the total space, bundle space (or simply bundle),(2) a space X called the base space,(3) a surjection π : Y −→ X called the projection,(4) a space V ⊂ Y called the fibre,(5) an effective topological (or Lie) transformation group G of V called thegroup of the bundle, and(6) a bundle atlas Ψ Y .Some standard examples of fibre bundles include any Cartesian productX × V → X (which is a bundle over X with fibre V ), the Möbius strip(which is a nontrivial fibre bundle over the circle S 1 with fibre given by theunit interval I = [0, 1]; the corresponding trivial bundle is a cylinder), theKlein bottle (which can be viewed as a ‘twisted’ circle bundle over anothercircle; thus, the corresponding trivial bundle is a torus, S 1 ×S 1 ), a 3–sphereS 3 (which is a bundle over S 2 with fibre S 1 ; more generally, a sphere bundleis a fiber bundle whose fiber is an n−sphere), while a covering space is afiber bundle whose fiber is a discrete space.Main properties of graphs of functions f : X → V carry over to fibrebundles. A graph of such a function, (x, f(x)), sits in the product spaceX × V, or in its homeomorphic generalization bundle. A graph is always1–1 and projects onto the base X.A special class of fibre bundle is the vector bundle, in which the fibre isa vector space. Special cases of fibre bundles that we will use in dynamicsof complex systems are: vector, affine, and principal bundles.A fibre bundle also comes with a group G action on its fibre V , so itcan also be called a G−bundle. This group action represents the differentways the fibre V can be viewed as equivalent (e.g., the group G might bethe group of homeomorphisms (topological group) or diffeomorphisms (Liegroup) of the fibre V ; or, the group G on a vector bundle is the group ofinvertible linear maps, which reflects the equivalent descriptions of a vector


488 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionspace using different vector–space bases). A principal bundle is G−bundlewhere the fiber can be identified with the group G itself and where there isa right action of G on the bundle space which is fiber preserving.Fibre bundles are not always used to generalize functions. Sometimesthey are convenient descriptions of interesting manifolds. A common exampleis a torus bundle on the circle.More specifically, a fibre bundle, or fibre bundle Y over an nD base Xis defined as a manifold surjectionπ : Y → X, (4.3)where Y admits an atlas Ψ Yof fibre coordinates(x α , y i ), x α → x ′ α (x µ ), y i → y ′ i (x µ , y j ), (4.4)compatible with the fibration (4.3), i.e., such that x α are coordinates on thebase X,π : Y ∋ (x α , y i ) ↦→ x α ∈ X.This condition is equivalent to π being a submersion, which means that itstangent map T π : T Y → T X is a surjection. This also implies that π is anopen map.A fibre bundle Y → X is said to be trivial if it is equivalent to theCartesian product of manifolds, Y ∼ = X × V , i.e., defined as π 1 : X × V →X. 1 A fibre bundle over a contractible base is always trivial [Steenrod(1951)].A fibre bundle Y → X is said to be locally trivial if there exists a fibredcoordinate atlas Ψ Y over an open covering {π −1 (U ξ )} ∈ Y of the bundlespace Y where {U ξ } ∈ X is an open covering of the base space X. Inother words, all points of the same fibre Y x = π −1 (x) of a bundle Y can becovered by the same fibred coordinate chart ψ ξ ∈ Ψ Y , so that we have thestandard fibre–manifold V for all local bundle splittingsψ ξ : π −1 (U ξ ) → U ξ × V.For the purpose of our general dynamics, the most important fibre bundlesare those which are at the same time smooth manifolds. A fibre bundleY → X is said to be smooth (C ∞ ) if there exist a typical fibre–manifold1 A trivial fibre bundle admits different trivializations Y ∼ = X × V that differ fromeach other in surjections Y → V .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 489V and an open covering {U ξ } of X such that Y is locally diffeomorphic tothe splittingsψ ξ : π −1 (U ξ ) → U ξ × V, (4.5)glued together by means of smooth transition functionsρ ξζ = ψ ξ ◦ ψ −1ζ: U ξ ∩ U ζ × V → U ξ ∩ U ζ × V (4.6)on overlaps U ξ ∩ U ζ . It follows that fibres Y x = π −1 (x), (for all x ∈ X), ofa fibre bundle are its closed imbedded submanifolds. Transition functionsρ ξζ fulfil the cocycle conditionρ ξζ ◦ ρ ζι = ρ ξι (4.7)on all overlaps U ξ ∩ U ζ ∩ U ι . Trivialization charts (U ξ , ψ ξ ) together withtransition functions ρ ξζ (4.6) constitute a bundle atlasΨ Y = {(U ξ , ψ ξ ), ρ ξζ } (4.8)of a fibre bundle Y → X. Two bundle atlases are said to be equivalent iftheir union is also a bundle atlas, i.e., there exist unique transition functionsbetween trivialization charts of different atlases. A fibre bundle Y → X isuniquely defined by a bundle atlas, and all its atlases are equivalent. Everysmooth fibre bundle Y → X admits a bundle atlas Ψ Y over a finite covering{U ξ } of X.If Y → X is a fibre bundle, the fibre coordinates (x α , y i ) ∈ Y areassumed to be bundle coordinates associated with a bundle atlas Ψ Y , thatis,y i (y) = (v i ◦ π 2 ◦ ψ ξ )(y), (π(y) ∈ U ξ ⊂ X), (4.9)where v i ∈ V ⊂ Y are coordinates of the standard fibre V of Y .Maps of fibre bundles (or, bundle maps), by definition, preserve theirfibrations, i.e., send a fibre to a fibre. Namely, a bundle map of a fibrebundle π : Y → X to a fibre bundle π ′ : Y ′ → X ′ is defined as a pair (Φ, f)of manifold maps such that the following diagram commutesΦY ✲ Y ′ππ ′❄X ✲ ❄Xf′


490 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioni.e., Φ is a fibrewise map over f which sends a fibre Y x , (for all x ∈ X),to a fibre Y ′ f(x), (for all f(x) ∈ X ′ ). A bundle diffeomorphism is calledan automorphism if it is an isomorphism to itself. In field theory, anyautomorphism of a fibre bundle is treated as a gauge transformation.Given a bundle Y → X, every map f : X ′ → X induces a bundleY ′ = f ∗ Y over X ′ which is called the pull–back of the bundle Y by f, suchthat the following diagram commutesY ✛ f ∗Y ′ππ ′❄X ✛❄Xf′In particular, the product Y × Y ′ over X of bundles π : Y → X and π ′ :Y ′ → X is the pull–backY × Y ′ = π ∗ Y ′ = π ′ ∗ Y.Classical fields are described by sections of fibre bundles. A (global)section of a fibre bundle Y → X is defined as a π−inverse manifold injections : X → Y, s(x) ↦→ Y x , such that π ◦ s = Id X . That is, a section s sendsany point x ∈ X into the fibre Y x ⊂ Y over this point. A section sis an imbedding, i.e., s(X) ⊂ Y is both a submanifold and a topologicalsubspace of Y . It is also a closed map, which sends closed subsets of X ontoclosed subsets of Y . Similarly, a section of a fibre bundle Y → X over asubmanifold of X is defined. Given a bundle atlas Ψ Y and associated bundlecoordinates (x α , y i ), a section s of a fibre bundle Y → X is represented bycollections of local functions {s i = y i ◦ψ ξ ◦s} on trivialization sets U ξ ⊂ X.A fibre bundle Y → X whose typical fibre is diffeomorphic to an Euclideanspace R m has a global section. More generally, its section over aclosed imbedded submanifold (e.g., a point) of X is extended to a globalsection [Steenrod (1951)].In contrast, by a local section is usually meant a section over an opensubset of the base X. A fibre bundle admits a local section around eachpoint of its base, but need not have a global section.For any n ≥ 1 the normal bundle NS n of the n−sphere S n is the fibrebundle (S n , p ′ , E ′ , R 1 ), where E ′ = {(x, y) ∈ R n+1 × R n+1 : ‖x‖ = 1, y =λx, λ ∈ R 1 } and p ′ : E ′ → S n is defined by p ′ (x, y) = x [Switzer (1975)].


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 4914.3 Vector and Affine BundlesThe most important fibre bundles are vector and affine bundles, whichgive a standard framework in both classical and quantum dynamics andfield theory (e.g., matter fields are sections of vector bundles, while gaugepotentials are sections of affine bundles).Recall that both the tangent bundle (T M, π M , M) and the cotangentbundle (T ∗ M, π ∗ M , M) are examples of a more general notion of vectorbundle (E, π, M) of a manifold M, which consists of manifolds E (the totalspace) and M (the base), as well as a smooth map π : E → M (theprojection) together with an equivalence class of vector bundle atlases (see[Kolar et al. (1993)]). A vector bundle atlas (U α , φ α ) α∈Afor (E, π, M) is aset of pairwise compatible vector bundle charts (U α , φ α ) such that (U α ) α∈Ais an open cover of M. Two vector bundle atlases are called equivalent, iftheir union is again a vector bundle atlas.On each fibre E m = π −1 (m) corresponding to the point m ∈ M there isa unique structure of a real vector space, induced from any vector bundlechart (U α , φ α ) with m ∈ U α . A section u of (E, π, M) is a smooth mapu : M → E with π ◦ u = Id M .Let (E, π M , M) and (F, π N , N) be vector bundles. A vector bundlehomomorphism Φ : E → F is a fibre respecting, fibre linear smooth mapinduced by the smooth map ϕ : M → N between the base manifolds Mand N, i.e., the following diagram commutes:EΦ✲ Fπ M❄Mϕπ N❄✲ NWe say that Φ covers ϕ. If Φ is invertible, it is called a vector bundleisomorphism.All smooth vector bundles together with their homomorphisms form acategory VB.If (E, π, M) is a vector bundle which admits a vector bundle atlas(U( α , φ α ) α∈Awith the given open cover, then, we have φ α ◦ φ −1β(m, v) =m, φαβ (m)v ) for C k −transition functions φ αβ : U αβ = U α ∩ U β → GL(V )(where we have fixed a standard fibre V ). This family of transition maps


492 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsatisfies the cocycle condition{φαβ (m) · φ βγ (m) = φ αγ (m) for each m ∈ U αβγ = U α ∩ U β ∩ U γ ,φ αα (m) = e for all m ∈ U α .The family (φ αβ ) is called the cocycle of transition maps for the vectorbundle atlas (U α , φ α ) .Now, let us suppose that the same vector bundle (E, π, M) is describedby an equivalent vector bundle atlas (U α , ψ α ) α∈Awith the same open cover(U α ). Then the vector bundle charts (U α , φ α ) and (U α , ψ α ) are compatiblefor each α, so ψ α ◦ φ −1β (m, v) = (m, τ α(m)v) for some τ α : U α → GL(V ).We getτ α (m) φ αβ (m) = φ αβ (m) τ β (m) for all m ∈ U αβ ,and we say that the two cocycles (φ αβ ) and (ψ αβ ) of transition maps overthe cover (U α ) are cohomologous. If GL(V ) is an Abelian group, i.e., if thestandard fibre V is of real or complex dimension 1, then the cohomologyclasses of cocycles (φ αβ ) over the open cover (U α ) form a usual cohomologygroup H 1 (M, GL(V )) with coefficients in the sheaf GL(V ) [Kolar et al.(1993)].Let (E, π, M) be a vector bundle and let ϕ : N → M be a smooth mapbetween the base manifolds N and M. Then there exists the pull–backvector bundle (ϕ ∗ E, ϕ ∗ π, ϕ ∗ N) with the same typical fibre and a vectorbundle homomorphism, given by the commutative diagram [Kolar et al.(1993)]:ϕ ∗ πϕ ∗ E❄Nπ ∗ ϕϕ✲ Eπ❄✲ MThe vector bundle (ϕ ∗ E, ϕ ∗ π, ϕ ∗ N) is constructed as follows. Let E =V B(φ αβ ) denote that E is described by a cocycle (φ αβ ) of transition mapsover an open cover (U α ) of M. Then (φ αβ ◦ ϕ) is a cocycle of transitionmaps over the open cover ( ϕ −1 (U α ) ) of N and the bundle is given byϕ ∗ E = V B(φ αβ ◦ ϕ).In other words, a vector bundle is a fibre bundle which admits an atlasof linear bundle coordinates. Typical fibres of a smooth vector bundleπ : Y → X are vector spaces of some finite dimension (called the fibre


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 493dimension, fdim Y of Y ), and Y admits a bundle atlas Ψ Y (4.8) wheretrivialization maps ψ ξ (x) and transition functions ρ ξζ (x) are linear isomorphismsof vector spaces. The corresponding bundle coordinates (y i ) obeya linear coordinate transformation lawy ′ i = ρij (x)y j .We have the decomposition y = y i e i (π(y)), where{e i (x)} = ψ −1ξ (x){v i}, x = π(y) ∈ U ξ ,are fibre bases (or frames) for fibres Y x of Y and {v i } is a fixed basis forthe typical fibre V of Y .There are several standard constructions of new vector bundles from oldones:• Given two vector bundles Y and Y ′ over the same base X, their Whitneysum Y ⊕ Y ′ is a vector bundle over X whose fibres are the direct sumsof those of the vector bundles Y and Y ′ .• Given two vector bundles Y and Y ′ over the same base X, their tensorproduct Y ⊗ Y ′ is a vector bundle over X whose fibres are the tensorproducts of those of the vector bundles Y and Y ′ . In a similar way theexterior product Y ∧Y of vector bundles is defined, so that the exteriorbundle of Y is defined as∧Y = X × R ⊕ Y ⊕ ∧ 2 Y ⊕ · · · ⊕ ∧ m Y, (m = fdim Y ).• Let Y → X be a vector bundle. By Y ∗ → X is denoted the dual vectorbundle whose fibres are the duals of those of Y . The interior product(or contraction) of Y and Y ∗ is defined as a bundle map⌋ : Y ⊗ Y ∗ → X × R.Given a linear bundle map Φ : Y ′ → Y of vector bundles over X, itskernel Ker Φ is defined as the inverse image Φ −1 (̂0(X)) of the canonicalzero section ̂0(X) of Y . If Φ is of constant rank, its kernel Ker Φ and itsimage Im Φ are subbundles of the vector bundles Y ′ and Y , respectively.For example, monomorphisms and epimorphisms of vector bundles fulfilthis condition. If Y ′ is a subbundle of the vector bundle Y → X, the factorbundle Y/Y ′ over X is defined as a vector bundle whose fibres are thequotients Y x /Y ′ x, x ∈ X.


494 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionConsider the short exact sequence of vector bundles over X,j0 → Y ′ i−→ Y −→Y ′′ → 0, (4.10)which means that i is a bundle monomorphism, j is a bundle epimorphism,and Ker j = Im i. Then Y ′′ is the factor bundle Y/Y ′ . One says thatthe short exact sequence (4.10) admits a splitting if there exists a bundlemonomorphism s : Y ′′ → Y such that j ◦ s = Id Y ′′, i.e.,Y = i(Y ′ ) ⊕ s(Y ′′ ) ∼ = Y ′ ⊕ Y ′′ .Vector bundles of rank 1 are called line bundles.The only two vector bundles with base space B a circle and 1D fibreF are the Mőbius band and the annulus, but the classification of all thedifferent vector bundles over a given base space with fibre of a given dimensionis quite difficult in general. For example, when the base space isa high–dimensional sphere and the dimension of the fibre is at least three,then the classification is of the same order of difficulty as the fundamentalbut still largely unsolved problem of computing the homotopy groups ofspheres [Hatcher (2002)].Now, there is a natural direct sum operation for vector bundles overa fixed base space X, which in each fibre reduces just to direct sum ofvector spaces. Using this, one can get a weaker notion of isomorphism ofvector bundles by defining two vector bundles over the same base space Xto be stably isomorphic if they become isomorphic after direct sum withproduct vector bundles X ×R n for some n, perhaps different n’s for the twogiven vector bundles. Then it turns out that the set of stable isomorphismclasses of vector bundles over X forms an Abelian group under the directsum operation, at least if X is compact Hausdorff. The traditional notationfor this group is ˜KO(X). It is the basis for K–theory (see below). In thecase of spheres the groups ˜KO(S n ) have the quite unexpected propertyof being periodic in n. This is called Bott Periodicity, and the values of˜KO(S n ) are given by the following table [Hatcher (2002)]:n mod 8 1 2 3 4 5 6 7 8˜KO(S n ) Z 2 Z 2 0 Z 0 0 0 ZFor example, ˜KO(S 1 ) is Z 2 , a cyclic group of order two, and a generatorfor this group is the Mőbius bundle. This has order two since the directsum of two copies of the Mőbius bundle is the product S 1 × R 1 , as one cansee by embedding two Mőbius bands in a solid torus so that they intersect


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 495orthogonally along the common core circle of both bands, which is also thecore circle of the solid torus.The complex version of ˜KO(X), called ˜K(X), is constructed in the sameway as ˜KO(X) but using vector bundles whose fibers are vector spaces overC rather than R. The complex form of Bott Periodicity asserts that ˜K(S n )is Z for n even and 0 for n odd, so the period is two rather than eight.The groups ˜K(X) and ˜KO(X) for varying X share certain formal propertieswith the cohomology groups studied in classical algebraic topology[Hatcher (2002)]. Using a more general form of Bott periodicity, it is infact possible to extend the groups ˜K(X) and ˜KO(X) to a full cohomologytheory, families of Abelian groups ˜K n (X) and ˜KO n (X) for n ∈ Z thatare periodic in n of period two and eight, respectively. However, thereis more algebraic structure here than just the additive group structure.Namely, tensor products of vector spaces give rise to tensor products ofvector bundles, which in turn give product operations in both real andcomplex K−theory similar to cup product in ordinary cohomology. Furthermore,exterior powers of vector spaces give natural operations withinK−theory (for more development, see next section, below).4.3.1 The Second Vector Bundle of the Manifold MLet (E, π, M) be a vector bundle over the biodynamical manifold M withfibre addition + E : E × M E → E and fibre scalar multiplication m E t : E →E. Then (T E, π E , E), the tangent bundle of the manifold E, is itself avector bundle, with fibre addition denoted by + T E and scalar multiplicationdenoted by m T tE . The second vector bundle structure on (T E, T π, T M), isthe ‘derivative’ of the original one on (E, π, M). In particular, the space{Ξ ∈ T E : T π.Ξ = 0 ∈ T M} = (T p) −1 (0) is denoted by V E and is calledthe vertical bundle over E. Its main characteristics are vertical lift andvertical projection (see [Kolar et al. (1993)] for details).All of this is valid for the second tangent bundle T 2 M = T T M of amanifold, but here we have one more natural structure at our disposal.The canonical flip or involution κ M : T 2 M → T 2 M is defined locally by(T 2 φ ◦ κ M ◦ T 2 φ −1 )(x, ξ; η, ζ) = (x, η; ξ, ζ).where (U, φ) is a local chart on M (this definition is invariant under changesof charts). The flip κ M has the following properties (see [Kolar et al.(1993)]):


496 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(1) κ M ◦ T 2 f = T 2 f ◦ κ M for each f ∈ C k (M, N);(2) T (π M ) ◦ κ M = π T M ;(3) π T M ◦ κ M = T (π M );(4) κ −1M = κ M ;(5) κ M is a linear isomorphism from the bundle (T T M, T (π M ), T M) to(T T M, π T M , T M), so it interchanges the two vector bundle structureson T T M;(6) κ M is the unique smooth map T T M → T T M which, for each γ : R →M, satisfies∂ t ∂ s γ(t, s) = κ M ∂ t ∂ s γ(t, s).In a similar way the second cotangent bundle of a manifold M can bedefined. Even more, for every manifold there is a geometrical isomorphismbetween the bundles T T ∗ M = T (T ∗ M) and T ∗ T M = T ∗ (T M) [Modugnoand Stefani (1978)].4.3.2 The Natural Vector BundleIn this section we mainly follow [Michor (2001); Kolar et al. (1993)].A vector bundle functor or natural vector bundle is a functor F whichassociates a vector bundle (F(M), π M , M) to each n−manifold M and avector bundle homomorphismF(M)F(ϕ)✲ F(N)π M❄Mϕπ N❄✲ Nto each ϕ : M → N in M, which covers ϕ and is fiberwise a linear isomorphism.Two common examples of the vector bundle functor F are tangentbundle functor T and cotangent bundle functor T ∗ (see section 3.5).The space of all smooth sections of the vector bundle (E, π M , M) is denotedby Γ (E, π M , M). Clearly, it is a vector space with fiberwise additionand scalar multiplication.Let F be a vector bundle functor on M. Let M be a smooth manifoldand let X ∈ X (M) be a vector–field on M. Then the flow F t of X for fixed


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 497t, is a diffeomorphism defined on an open subset of M. The mapF(F t )F(M)✲ F(M)π M❄MF tπ M❄✲ Mis then a vector bundle isomorphism, defined over an open subset of M.We consider a tensor–field τ (3.6), which is a section τ ∈ Γ (F(M)) ofthe vector bundle (F(M), π M , M) and we define for t ∈ RF ∗ t τ = F(F −t ) ◦ τ ◦ F t ,a local section of the bundle F(M). For each point m ∈ M the valueFt ∗ τ(x) ∈ F(M) m is defined, if t is small enough (depending on x). So,in the vector space F(M) m the expression d dt | t=0 Ft ∗ τ(x) makes sense andtherefore the sectionL X τ = d dt | t=0 F ∗ t τis globally defined and is an element of Γ (F(M)). It is called the Liederivative of the tensor–field τ along a vector–field X ∈ X (M) (see section3.7, for details on Lie derivative).In this situation we have:(1) F ∗ t F ∗ r τ = F ∗ t+rτ, whenever defined.(2) d dt F ∗ t τ = F ∗ t L X τ = L X (F ∗ t τ), so[L X , F ∗ t ] = L X ◦ F ∗ t − F ∗ t ◦ L X = 0, whenever defined.(3) F ∗ t τ = τ for all relevant t iff L X τ = 0.Let F 1 and F 2 be two vector bundle functors on M. Then the (fiberwise)tensor product (F 1 ⊗ F 2 ) (M) = F 1 (M) ⊗ F 2 (M) is again a vector bundlefunctor and for τ i ∈ Γ (F i (M)) with i = 1, 2, there is a section τ 1 ⊗ τ 2 ∈Γ (F 1 ⊗ F 2 ) (M), given by the pointwise tensor product.Also in this situation, for X ∈ X (M) we haveL X (τ 1 ⊗ τ 2 ) = L X τ 1 ⊗ τ 2 + τ 1 ⊗ L X τ 2 .In particular, for f ∈ C k (M, R) we have L X (f τ) = df(X) τ + f L X τ.For any vector bundle functor F on M and X, Y ∈ X (M) we have:[L X , L Y ] = L X ◦ L Y − L Y ◦ L X = L [X,Y ] : Γ (F(M)) → Γ (F(M)) .


498 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction4.3.3 Vertical Tangent and Cotangent Bundles4.3.3.1 Tangent and Cotangent Bundles RevisitedRecall (from section 3.5 above) that the most important vector bundles arefamiliar tangent and cotangent bundles. The fibres of the tangent bundleπ M : T M → M of a manifold M are tangent spaces to M. The peculiarityof the tangent bundle T M in comparison with other vector bundles overM lies in the fact that, given an atlas Ψ M = {(U ξ , φ ξ )} of a manifoldM, the tangent bundle of M admits the holonomic atlas Ψ = {(U ξ , ψ ξ =T φ ξ )}, where by T φ ξ is denoted the tangent map to φ ξ . Namely, givencoordinates x α on a manifold M, the associated bundle coordinates on T Mare holonomic coordinates (ẋ α ) with respect to the holonomic frames {∂ α }for tangent spaces T x M, x ∈ M. Their transition functions readẋ ′α = ∂x′α∂x µ ẋµ .Every manifold map f : M → M ′ induces the linear bundle map over f ofthe tangent bundles (4.2).The cotangent bundle of a manifold M is the dual π ∗M : T ∗ M →M of the tangent bundle T M → M. It is equipped with the holonomiccoordinates (x α , ẋ α ) with respect to the coframes {dx α } for T ∗ M whichare the duals of {∂ α }. Their transition functions readẋ ′ α = ∂xµ∂x ′ α ẋµ.Recall that a tensor product of tangent and cotangent bundles over M,T = (⊗ m T M) ⊗ (⊗ k T ∗ M), (m, k ∈ N), (4.11)is called a tensor bundle. Given two vector bundles Y and Y ′ over the samebase X, their tensor product Y ⊗ Y ′ is a vector bundle over X whose fibresare the tensor products of those of the vector bundles Y and Y ′ .Tangent, cotangent and tensor bundles belong to the category BUN ofnatural fibre bundles which admit the canonical lift of any diffeomorphismf of a base to a bundle automorphism, called the natural automorphism[Kolar et al. (1993)]. For example, the natural automorphism of the tangentbundle T M over a diffeomorphism f of its base M is the tangent map T f(4.2) over f. In view of the expression (4.2), natural automorphisms arealso called holonomic transformations or general covariant transformations(in gravitation theory).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 499Let T Y → Y be the tangent bundle of a bundle Y → X. The followingdiagram commutesπ YT Y❄YT ππ✲ T Xπ X❄✲ Xwhere T π : T Y → T X is a fibre bundle. Note that T π is still the bundlemap of the bundle T Y → Y to T X over π and the fibred map of the bundleT Y → X to T X over X. There is also the canonical surjectionπ T : T Y −→ T X −→ Y, given by π T = π X ◦ T π = π ◦ π Y .Now, given the fibre coordinates (x α , y i ) of a fibre bundle Y , the correspondinginduced coordinates of T Y are(x α , y i , ẋ α , ẏ i ), ẏ ′ i =∂y ′ i∂y j ẏj .This expression shows that the tangent bundle T Y → Y of a fibre bundleY has the vector subbundleV Y = Ker T πwhere T π is regarded as the fibred map of T Y → X to T X over X. Thesubbundle V Y consists of tangent vectors to fibres of Y . It is called thevertical tangent bundle of Y and provided with the induced coordinates(x α , y i , ẏ i ) with respect to the fibre bases {∂ i }.The vertical cotangent bundle V ∗ Y → Y of a fibre bundle Y → X isdefined as the dual of the vertical tangent bundle V Y → Y . Note that itis not a subbundle of the cotangent bundle T ∗ Y , but there is the canonicalsurjectionζ : T ∗ Y −→ V ∗ Y, ẋ α dx α + ẏ i dy i ↦→ ẏ i dy i , (4.12)where {dy i } are the bases for the fibres of V ∗ Y which are duals of theholonomic frames {∂ i } for the vertical tangent bundle V Y .With V Y and V ∗ Y , we have the following short exact sequences ofvector bundles over a fibre bundle Y → X:0 → V Y ↩→ T Y → Y × T X → 0, (4.13)0 → Y × T ∗ X ↩→ T ∗ Y → V ∗ Y → 0 (4.14)


500 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionEvery splittingof the exact sequence (4.13) andY × T X ↩→ T Y, ∂ α ↦→ ∂ α + Γ i α(y)∂ i ,V ∗ Y → T ∗ Y, d i ↦→ dy i − Γ i α(y)dx α ,of the exact sequence (4.14), by definition, corresponds to a certain connectionon the bundle Y → X, and vice versa.Let Φ be a fibred map of a bundle Y → X to a bundle Y ′ → X ′ overf : X → X ′ . The tangent map T Φ : T Y → T Y ′ to Φ reads(ẋ ′ α , ẏ′i ) ◦ T Φ = (∂µ f α ẋ µ , ∂ µ Φ i ẋ µ + ∂ j Φ i ẏ j ). (4.15)It is both the linear bundle map over Φ, given by the commutativity diagramT ΦT Y ✲ T Y ′π Yπ Y ′❄Y ✲ ❄YΦ′as well as the fibred map over the tangent map T f to f, given by thecommutativity diagramT ΦT Y ✲ T Y ′4.3.4 Affine Bundles❄T X ✲ ❄T XT f′Given a vector bundle Y → X, an affine bundle modelled over Y is afibre bundle Y → X whose fibres Y x , (for all x ∈ X), are affine spacesmodelled over the corresponding fibres Y x of the vector bundle Y , and Yadmits a bundle atlas Ψ Y (4.8) whose trivialization morphisms ψ ξ (x) andtransition functions functions ρ ξζ (x) are affine maps. The correspondingbundle coordinates (y i ) possess an affine coordinate transformation lawy ′ i = ρij (x α )y j + ρ i (x α ).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 501In other words, an affine bundle admits an atlas of affine bundle coordinates(x α , y i ) such thatr : (x α , y i ) × (x α , y i ) ↦→ (x α , y i + y i )where (x α , y i ) are linear bundle coordinates of the vector bundle Y . Inparticular, every vector bundle Y has the canonical structure of an affinebundle modelled on Y itself by the mapr : (y, y ′ ) ↦→ y + y ′ .Every affine bundle has a global section.One can define a direct sum Y ⊕ Y ′ of a vector bundle Y ′ → X and anaffine bundle Y → X modelled over a vector bundle Y → X, as is an affinebundle modelled over the Whitney sum of vector bundles Y ′ ⊕ Y .Affine bundles are subject to affine bundle maps which are affine fibrewisemaps. Any affine bundle map Φ : Y → Y ′ from an affine bundle Ymodelled over a vector bundle Y to an affine bundle Y ′ modelled over avector bundle Y ′ , induces the linear bundle map of these vector bundlesΦ : Y → Y ′ ,y ′ i ∂Φ i◦ Φ =∂y j yj . (4.16)4.4 Application: Semi–Riemannian <strong>Geom</strong>etrical MechanicsIn this subsection we develop a Finsler–like approach to semi–Riemanniangeometrical dynamics.4.4.1 Vector–Fields and ConnectionsLet M be an nD smooth manifold. Recall that a smooth (C ∞ ) vector–fieldX on M defines the flowẋ = X(x). (4.17)By definition, a semi–Riemannian metric g on M is a smooth symmetrictensor–field of type (0, 2) which assigns to each point x ∈ M a nondegenerateinner product g(x) on the tangent space T x M of signature (r, s). Thepair (M, g) is called a semi–Riemannian manifold.The vector–field X and the semi–Riemannian metric g determine theenergy f : M → R,given by f = 1 2g(X, X). The vector–field X (and itsflow) on (M, g) is called [Udriste (2000)]:


502 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(1) time–like, if f < 0;(2) nonspacelike or causal, if f ≤ 0;(3) null or lightlike, if f = 0;(4) space–like, if f > 0.Let ∇ be the Levi–Civita connection of (M, g). Using the semi–Riemannian version of the covariant derivative operator (3.157), we getthe prolongation∇dtẋ = ∇ ẋX (4.18)of the differential system (4.17) or of any perturbation of the system (4.17)get adding to the second member X a parallel vector–field Y with respect tothe covariant derivative ∇. The prolongation by derivation represents thegeneral dynamics of the flow. The vector–field Y can be used to illustratea progression from stable to unstable flows, or converse.The vector–field X, the metric g, and the connection ∇ determine theexternal (1, 1)−tensor–fieldF = ∇X − g −1 ⊗ g(∇X), F j i = ∇ j X i − g ih g kj ∇ h X k ,(with i, j, h, k = 1, ..., n), which characterizes the helicity of vector–field Xand its flow.First we write the differential system (4.18) in the equivalent form∇dtẋ = g−1 ⊗ g(∇X) (ẋ) + F (ẋ) . (4.19)Successively we modify the differential system (4.19) as follows [Udriste(2000)]:∇dtẋ = g−1 ⊗ g(∇X)(X) + F (ẋ) , (4.20)∇dtẋ = g−1 ⊗ g(∇X) (ẋ) + F (X), (4.21)∇dtẋ = g−1 ⊗ g(∇X)(X) + F (X). (4.22)Obviously, the second–order systems (4.20), (4.21), (4.22) are prolongationsof the first–order system (4.17). Each of them is connected either to thedynamics of the field X or to the dynamics of a particle which is sensitiveto the vector–field X. Sinceg −1 ⊗ g(∇X)(X) = grad f,


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 503we shall show that the prolongation (4.20) describes a conservative dynamicsof the vector–field X or of a particle which is sensitive to the vector–fieldX. The physical phenomenon produced by (4.21) or (4.22) was not yetstudied [Udriste (2000)].In the case F = 0, the kinematic system (4.17) prolongs to a potentialdynamical system with n degrees of freedom, namely∇= grad f. (4.23)dtẋIn the case F ≠ 0, the kinematic system (4.17) prolongs to a non–potential dynamical system with n degrees of freedom, namely∇= grad f + F (ẋ) . (4.24)dtẋLet us show that the dynamical systems (4.23) and (4.24) are conservative.To simplify the exposition we identity the tangent bundle T Mwith the cotangent bundle T ∗ M using the semi–Riemann metric g [Udriste(2000)]. The trajectories of the dynamical system (4.23) are the extremalsof the LagrangianL = 1 g (ẋ, ẋ) + f(x).2The trajectories of the dynamical system (4.24) are the extremals of theLagrangianL = 1 2 g (ẋ − X, ẋ − X) = 1 g (ẋ, ẋ) − g (X, ẋ) + f(x).2The dynamical systems (4.23) and (4.24) are conservative, the Hamiltonianbeing the same for both cases, namelyH = 1 g (ẋ, ẋ) − f(x).2The restriction of the Hamiltonian H to the flow of the vector–field X iszero.4.4.2 Hamiltonian Structures on the Tangent BundleLet (N, ω) be a 2nD symplectic phase–space manifold, and H : N → R bea C ∞ real function. We define the Hamiltonian gradient X H as being thevector–field which satisfiesω p (X H (x), v) = dH(x)(v),(for all v ∈ T x N),


504 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand the Hamiltonian equations asẋ = X H (x).Let (M, g) be a semi–Riemann nD manifold. Let X be a C ∞ vector–field on M, and ω = g ◦ F the two–form associated to the tensor–fieldF = ∇X − g −1 ⊗ g(∇X) via the metric g.The tangent bundle is usually equipped with the Sasakian metric G,induced by g,G = g ij dx i ⊗ dx j + g ij δy i ⊗ δy j .If (x i , y i ) are the coordinates of the point (x, y) ∈ T M and Γ i jkare thecomponents of the connection induced by g ij , then we have the followingdual frames [Udriste (2000)]( δδx i =(dx j ,∂∂x i − Γh ijy j ∂∂y h ,)∂∂y i ⊂ X (T M),δy j = dy j + Γ j hk yk dx h ) ⊂ X ∗ (T M).The dynamical system (4.23) lifts to T M as a Hamiltonian dynamicalsystem with respect to theandHamiltonian H = 1 g(ẋ, ẋ) − f(x), and2symplectic two–form Ω 1 = g ij dx i ωδy j .This can be verified by putting η 1 = g ij y i dx j , and dη 1 = −Ω 1 .The dynamical system (4.24) lifts to T M as a Hamiltonian dynamicalsystem with respect to the above Hamiltonian function and the symplectictwo–formΩ 2 = 1 2 ω ijdx i ωdx j + g ij dx i ωδy j .This can be verified by putting η 2 = −g ij X i dx j +g ij y i dx j , and dη 2 = −Ω 2 .In the remainder of this subsection, we give three examples in Euclideanspaces, so we can put all indices down (still summing over repeated indices).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 505Pendulum <strong>Geom</strong>etryWe use the Riemannian manifold (R 2 , δ ij ). The small oscillations of a planependulum are described as solutions of the following differential systemgiving the plane pendulum flow,ẋ 1 = −x 2 , ẋ 2 = x 1 . (4.25)In this case, the set {x 1 (t) = 0, x 2 (t) = 0, (t ∈ R)} is the equilibriumpoint andx 1 (t) = c 1 cos t + c 2 sin t,x 2 (t) = c 1 sin t − c 2 cos tis the general solution, which is a family of circles with a common center.Let X = (X 1 , X 2 ), X 1 (x 1 , x 2 ) = −x 2 , X 2 (x 1 , x 2 ) = x 1 ,f(x 1 , x 2 ) = 1 2 (x2 1 + x 2 2), curl X = (0, 0, 2), div X = 0.The pendulum flow conserves the areas. The prolongation by derivationof the kinematic system (4.25) is [Udriste (2000)]ẍ i = ∂X iẋ j ,∂x j(i, j = 1, 2)or ẍ 1 = −ẋ 2 , ẍ 2 = ẋ 1 .This prolongation admits a family of circles as the general solutionx 1 (t) = a 1 cos t+a 2 sin t+h, x 2 (t) = a 1 sin t−a 2 cos t+k, (t ∈ R).The pendulum geometrodynamics is described byẍ i = ∂f ( ∂Xi+ − ∂X )jẋ j , (i, j = 1, 2),∂x i ∂x j ∂x ior ẍ 1 = x 1 − 2ẋ 2 , ẍ 2 = x 2 + 2ẋ 1 , (4.26)with a family of spirals as the general solutionx 1 (t) = b 1 cos t + b 2 sin t + b 3 t cos t + b 4 t sin t,x 2 (t) = b 1 sin t − b 2 cos t + b 3 t sin t − b 4 t cos t,(t ∈ R).


506 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionUsingL = 1 [(ẋ 1 ) 2 + (ẋ 2 ) 2] + x 2 ẋ 1 − x 1 ẋ 2 + f,2H = 1 [(ẋ 1 ) 2 + (ẋ 2 ) 2] − f, g ij = (H + f)δ ij ,2N j i = −F j i = −δ ih F jh ,F ij = ∂X j∂x i− ∂X i∂x j, (i, j, h = 1, 2),the solutions of the differential system (4.26) are horizontal pregeodesics ofthe Riemann–Jacobi–Lagrangian manifold (R 2 \ {0}, g ij , N j i ).<strong>Geom</strong>etry of the Lorenz FlowWe use the Riemannian manifold (R 3 , δ ij ). The Lorenz flow is a first dissipativemodel with chaotic behavior discovered in numerical experiment.Its state equations are (see [Lorenz (1963); Sparrow (1982)])ẋ 1 = −σx 1 + σx 2 , ẋ 2 = −x 1 x 3 + rx 1 − x 2 , ẋ 3 = x 1 x 2 − bx 3 ,where σ, r, b are real parameters. Usually σ, b are kept fixed whereas r isvaried. Atr > r 0 =σ(σ + b + 3)σ − b − 1chaotic behavior is observed [Udriste (2000)].Let X = (X 1 , X 2 , X 3 ), X 1 (x 1 , x 2 , x 3 ) = −σx 1 + σx 2 ,X 2 (x 1 , x 2 , x 3 ) = −x 1 x 3 + rx 1 − x 2 , X 3 (x 1 , x 2 , x 3 ) = x 1 x 2 − bx 3 ,f = 1 2 [(−σx 1 + σx 2 ) 2 + (−x 1 x 3 + rx 1 − x 2 ) 2 + (x 1 x 2 − bx 3 ) 2 ],curl X = (2x 1 , −x 2 , r − x 3 − σ).The Lorenz dynamics is described byẍ i = ∂f ( ∂Xi+ − ∂X )jẋ j , (i, j = 1, 2, 3), or∂x i ∂x j ∂x iẍ 1 = ∂f∂x 1+ (σ + x 3 − r)ẋ 2 − x 2 ẋ 3 ,ẍ 2 = ∂f∂x 2+ (r − x 3 − σ)ẋ 1 − 2x 1 ẋ 3 ,ẍ 3 = ∂f∂x 3+ x 2 ẋ 1 + 2x 1 ẋ 2 . (4.27)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 507Using L = 1 23∑(ẋ i ) 2 −i=13∑X i ẋ i + f, H = 1 2g ij = (H + f)δ ij , N j i = −F j i = −δ ih F jh , F ij = ∂X j∂x ii=13∑(ẋ i ) 2 − f,i=1− ∂X i∂x j,(i, j, h = 1, 2, 3), the solutions of the differential system (4.27) are horizontalpregeodesics of the Riemann–Jacobi–Lagrangian manifold (R 3 \E, g ij , N j i ), where E is the set of equilibrium points.<strong>Geom</strong>etry of the ABC FlowWe use the Riemannian manifold (R 3 , δ ij ). One example of a fluid velocitythat contains exponential stretching and hence instability is the ABC flow,named after Arnold, Beltrami and Childress,ẋ 1 = A sin x 3 +C cos x 2 , ẋ 2 = B sin x 1 +A cos x 3 , ẋ 3 = C sin x 2 +B cos x 1 .For nonzero values of the constants A, B, C the preceding system is notglobally integrable. The topology of the flow lines is very complicated andcan only be investigated numerically to reveal regions of chaotic behavior.The ABC flow conserves the volumes since the ABC field is solenoidal.The ABC geometrodynamics is described by [Udriste (2000)]ẍ i = ∂f ( ∂Xi+ − ∂X )jẋ j , (i, j = 1, 2, 3).∂x i ∂x j ∂x iSince f = 1 2 (A + B + C + 2AC sin x 3 cos x 2 + 2BA sin x 1 cos x 3 +2CB sin x 2 cos x 1 ), and curl X = X, the ABC geometrodynamics is givenby the system,ẍ 1 = AB cos x 1 cos x 3 − BC sin x 1 sin x 2− (B cos x 1 + C sin x 2 )ẋ 2 + (B sin x 1 + A cos x 3 )ẋ 3 ,ẍ 2 = −AC sin x 2 sin x 3 + BC cos x 1 cos x 2+ (B cos x 1 + C sin x 2 )ẋ 1 − (A sin x 3 + C cos x 2 )ẋ 2 ,ẍ 3 = AC cos x 3 cos x 2 − BA sin x 1 sin x 3− (B sin x 1 + A cos x 3 )ẋ 1 + (C cos x 2 + A sin x 3 )ẋ 2 .


508 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionUsingL = 1 2ẋiẋ i − X i ẋ i + f, H = 1 2ẋiẋ i − f, (i, j, h = 1, 2, 3),g ij = (H + f)δ ij , N j i = −F j i = −δ ih F jh , F ij = ∂X j∂x i− ∂X i∂x j,the solutions of the above differential system are horizontal pregeodesics ofthe Riemann–Jacobi–Lagrangian manifold (R 3 \ E, g ij , N i j ), where E isthe set of equilibrium points, which is included in the surface of equationsin x 1 sin x 2 sin x 3 + cos x 1 cos x 2 cos x 3 = 0.4.5 K−Theory and Its ApplicationsRecall from [Dieudonne (1988)] that the 1930s were the decade of the developmentof the cohomology theory, as several research directions grew togetherand the de Rham cohomology, that was implicit in Poincaré’s work,became the subject of definite theorems. The development of algebraictopology from 1940 to 1960 was very rapid, and the role of homology theorywas often as ‘baseline’ theory, easy to compute and in terms of whichtopologists sought to calculate with other functors. The axiomatization ofhomology theory by Eilenberg and Steenrod (celebrated Eilenberg–SteenrodAxioms) revealed that what various candidate homology theories had incommon was, roughly speaking, some exact sequences (in particular, theMayer–Vietoris Theorem and the Dimension Axiom that calculated thehomology of the point).4.5.1 Topological K−TheoryNow, K–theory is an extraordinary cohomology theory, which consists oftopological K−theory and algebraic K−theory. The topological K–theorywas founded to study vector bundles on general topological spaces, bymeans of ideas now recognisee as (general) K−theory that were introducedby Alexander Grothendieck. The early work on topological K−theory wasdue to Michael Atiyah and Friedrich Hirzebruch.Let X be a compact Hausdorff space and k = R or k = C. Then K k (X)is the Grothendieck group of the commutative monoid 2 which elements are2 Recall that a monoid is an algebraic structure with a single, associative binaryoperation and an identity element; a monoid whose operation is commutative is called acommutative monoid (or, an Abelian monoid); e.g., every group is a monoid and every


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 509the isomorphism classes of finite dimensional k−vector bundles on X withthe operation[E ⊕ F ] := [E] ⊕ [F ]for vector bundles E, F . 3 Usually, K k (X) is denoted KO(X) in real caseand KU(X) in the complex case.More precisely, the stable equivalence, i.e., the equivalence relation onbundles E and F on X of defining the same element in K(X), occurs whenthere is a trivial bundle G, so that E ⊕G ∼ = F ⊕G. Under the tensor productof vector bundles, K(X) then becomes a commutative ring. The rankof a vector bundle carries over to the K−group define the homomorphism:K(X) → Ȟ0 (X, Z), where Ȟ0 (X, Z) is the 0−group of the Chech cohomologywhich is equal to group of locally constant functions with values inZ.The constant map X −→ {x 0 }, x 0 ∈ X defines the reduced K−group(of reduced homology)˜K(X) = Coker(K(X) −→ {x 0 }).In particular, when X is a connected space, then˜K(X) ∼ = Ker(K(X) → Ȟ0 (X, Z) = Z).4.5.1.1 Bott Periodicity TheoremAn important property in the topological K−theory is the Bott PeriodicityTheorem [Bott (1959)] 4 , which can be formulated this way:Abelian group a commutative monoid.3 The Grothendieck group construction in abstract algebra constructs an Abeliangroup from a commutative monoid ‘in the best possible way’.4 The Bott Periodicity Theorem is a result from homotopy theory discovered by RaoulBott during the latter part of the 1950s, which proved to be of foundational significancefor much further research, in particular in K−theory of stable complex vector bundles,as well as the stable homotopy groups of spheres. Bott periodicity can be formulatedin numerous ways, with the periodicity in question always appearing as a period 2phenomenon, with respect to dimension, for the theory associated to the unitary group.The context of Bott periodicity is that the homotopy groups of spheres, which would beexpected to play the basic part in algebraic topology by analogy with homology theory,have proved elusive (and the theory is complicated). The subject of stable homotopytheory was conceived as a simplification, by introducing the suspension (smash productwith a circle) operation, and seeing what (roughly speaking) remained of homotopytheory once one was allowed to suspend both sides of an equation, as many times asone wished. The stable theory was still hard to compute with, in practice. What Bottperiodicity offered was an insight into some highly non-trivial spaces, with central status


510 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(1) K(X ×S 2 ) = K(X)⊗K(S 2 ), and K(S 2 ) = [H]/(H −1) 2 , where H isthe class of the tautological bundle on the S 2 = P 1 , i.e., the Riemannsphere as complex projective line;(2) ˜K n+2 (X) = ˜K n (X);(3) Ω 2 BU ≃ BU × Z.In real K−theory there is a similar periodicity, but modulo 8.4.5.2 Algebraic K−TheoryOn the other hand, the so–called algebraic K–theory is an advanced partof homological algebra concerned with defining and applying a sequenceK n (R) of functors from rings to Abelian groups, for n = 0, 1, 2, .... Here,for traditional reasons, the cases of K 0 and K 1 are thought of in somewhatdifferent terms from the higher algebraic K−groups K n for n ≥ 2. Infact K 0 generalizes the construction of the ideal class group, using projectivemodules; and K 1 as applied to a commutative ring is the unit groupconstruction, which was generalized to all rings for the needs of topology(simple homotopy theory) by means of elementary matrix theory. Thereforethe first two cases counted as relatively accessible; while after thatthe theory becomes quite noticeably deeper, and certainly quite hard tocompute (even when R is the ring of integers).Historically, the roots of the theory were in topological K–theory (basedon vector bundle theory); and its motivation the conjecture of Serre 5 thatnow is the Quillen–Suslin Theorem. 6Applications of K−groups were found from 1960 onwards in surgeryin topology because of the connection of their cohomology with characteristic classes,for which all the (unstable) homotopy groups could be calculated. These spaces are the(infinite, or stable) unitary, orthogonal and symplectic groups U, O and Sp.5 Jean–Pierre Serre used the analogy of vector bundles with projective modules tofound in 1959 what became algebraic K−theory. He formulated the Serre’s Conjecture,that projective modules over the ring of polynomials over a field are free modules; thisresisted proof for 20 years.6 The Quillen–Suslin Theorem is a Theorem in abstract algebra about the relationshipbetween free modules and projective modules. Projective modules are modules that arelocally free. Not all projective modules are free, but in the mid–1950s, Jean–Pierre Serrefound evidence that a limited converse might hold. He asked the question: Is everyprojective module over a polynomial ring over a field a free module? A more geometricvariant of this question is whether every algebraic vector bundle on affine space is trivial.This was open until 1976, when Daniel Quillen and Andrei Suslin independently provedthat the answer is yes. Quillen was awarded the Fields Medal in 1978 in part for hisproof.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 511theory for manifolds, in particular; and numerous other connections withclassical algebraic problems were found. A little later a branch of the theoryfor operator algebras was fruitfully developed. It also became clear thatK−theory could play a role in algebraic cycle theory in algebraic geometry:here the higher K−groups become connected with the higher codimensionphenomena, which are exactly those that are harder to access. The problemwas that the definitions were lacking (or, too many and not obviouslyconsistent). A definition of K 2 for fields by John Milnor, for example, gavean attractive theory that was too limited in scope, constructed as a quotientof the multiplicative group of the field tensored with itself, with someexplicit relations imposed; and closely connected with central extensions[Milnor and Stasheff (1974)].Eventually the foundational difficulties were resolved (leaving a deepand difficult theory), by a definition of D. Quillen:K n (R) = π n (BGL(R) + ).This is a very compressed piece of abstract mathematics. Here π n is an nthhomotopy group, GL(R) is the direct limit of the general linear groups overR for the size of the matrix tending to infinity, B is the classifying spaceconstruction of homotopy theory, and the + is Quillen’s plus construction.4.5.3 Chern Classes and Chern CharacterAn important properties in K–theory are the Chern classes and Cherncharacter [Chern (1946)]. The Chern classes are a particular type of characteristicclasses (topological invariants, see [Milnor and Stasheff (1974)]).associated to complex vector bundles of a smooth manifold. Recall thata characteristic class is a way of associating to each principal bundle on atopological space X a cohomology class of X. The cohomology class measuresthe extent to which the bundle is ‘twisted’ – particularly, whether itpossesses sections or not. In other words, characteristic classes are globalinvariants which measure the deviation of a local product structure from aglobal product structure. They are one of the unifying geometric conceptsin algebraic topology, differential geometry and algebraic geometry. 77 Recall that characteristic classes are in an essential way phenomena of cohomologytheory – they are contravariant functors, in the way that a section is a kind of functionon a space, and to lead to a contradiction from the existence of a section we do needthat variance. In fact cohomology theory grew up after homology and homotopy theory,which are both covariant theories based on mapping into a space; and characteristicclass theory in its infancy in the 1930s (as part of obstruction theory) was one major


512 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIf we describe the same vector bundle on a manifold in two differentways, the Chern classes will be the same, i.e., if the Chern classes of apair of vector bundles do not agree, then the vector bundles are different(the converse is not true, though). In topology, differential geometry,and algebraic geometry, it is often important to count how many linearlyindependent sections a vector bundle has. The Chern classes offer someinformation about this through, for instance, the Riemann–Roch Theoremand the Atiyah-Singer Index Theorem. Chern classes are also feasible tocalculate in practice. In differential geometry (and some types of algebraicgeometry), the Chern classes can be expressed as polynomials in the coefficientsof the curvature form.In particular, given a complex hermitian vector bundle V of complexrank n over a smooth manifold M, a representative of each Chern class(also called a Chern form) c k (V ) of V are given as the coefficients of thecharacteristic polynomial( ) itΩdet2π + I = c k (V )t k ,reason why a ‘dual’ theory to homology was sought. The characteristic class approachto curvature invariants was a particular reason to make a theory, to prove a generalGauss–Bonnet Theorem.When the theory was put on an organized basis around 1950 (with the definitionsreduced to homotopy theory) it became clear that the most fundamental characteristicclasses known at that time (the Stiefel–Whitney class, the Chern class, and the Pontryaginclass) were reflections of the classical linear groups and their maximal torusstructure. What is more, the Chern class itself was not so new, having been reflected inthe Schubert calculus on Grassmannians, and the work of the Italian school of algebraicgeometry. On the other hand there was now a framework which produced families ofclasses, whenever there was a vector bundle involved.The prime mechanism then appeared to be this: Given a space X carrying a vectorbundle, implied in the homotopy category a mapping from X to a classifying spaceBG, for the relevant linear group G. For the homotopy theory, the relevant informationis carried by compact subgroups such as the orthogonal groups and unitary groups ofG. Once the cohomology H ∗ (BG) was calculated, once and for all, the contravarianceproperty of cohomology meant that characteristic classes for the bundle would be definedin H ∗ (X) in the same dimensions. For example, the Chern class is really one class withgraded components in each even dimension.This is still the classic explanation, though in a given geometric theory it is profitableto take extra structure into account. When cohomology became ‘extra–ordinary’ withthe arrival of K−theory and Thom’s cobordism theory from 1955 onwards, it was reallyonly necessary to change the letter H everywhere to say what the characteristic classeswere.Characteristic classes were later found for foliations of manifolds; they have (in amodified sense, for foliations with some allowed singularities) a classifying space theoryin homotopy theory.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 513of the curvature form Ω of V , which is defined asΩ = dω + 1 [ω, ω],2with ω the connection form and d the exterior derivative, or via the sameexpression in which ω is a gauge form for the gauge group of V . Thescalar t is used here only as an indeterminate to generate the sum fromthe determinant, and I denotes the n × n identity matrix. To say that theexpression given is a representative of the Chern class indicates that ‘class’here means up to addition of an exact differential form. That is, Chernclasses are cohomology classes in the sense of de Rham cohomology. It canbe shown that the cohomology class of the Chern forms do not depend onthe choice of connection in V .For example, let CP 1 be the Riemann sphere: a 1D complex projectivespace. Suppose that z is a holomorphic local coordinate for the Riemannsphere. Let V = T CP 1 be the bundle of complex tangent vectors having theform a∂/∂z at each point, where a is a complex number. In the followingwe prove the complex version of the Hairy Ball Theorem: V has no sectionwhich is everywhere nonzero.For this, we need the following fact: the first Chern class of a trivialbundle is zero, i.e., c 1 (CP 1 × C) = 0. This is evinced by the fact that atrivial bundle always admits a flat metric. So, we will show that c 1 (V ) ≠ 0.Consider the Kähler metrich =dzd¯z(1 + |z| 2 ) ..One can show that the curvature 2–form is given byΩ =2dz ∧ d¯z(1 + |z| 2 ) 2 .Furthermore, by the definition of the first Chern classc 1 =i2π Ω..We need to show that the cohomology class of this is non–zero. It sufficesto compute its integral over the Riemann sphere:∫c 1 = i ∫dz ∧ d¯zπ (1 + |z| 2 ) 2 = 2,


514 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionafter switching to polar coordinates. By Stokes Theorem, an exact formwould integrate to 0, so the cohomology class is nonzero. This proves thatT CP 1 is not a trivial vector bundle.An important special case occurs when V is a line bundle. Then theonly nontrivial Chern class is the first Chern class, which is an elementof H 2 (X; Z)−the second cohomology group of X. As it is the top Chernclass, it equals the Euler class of the bundle. The first Chern class turnsout to be a complete topological invariant with which to classify complexline bundles. That is, there is a bijection between the isomorphism classesof line bundles over X and the elements of H 2 (X; Z), which associates to aline bundle its first Chern class. Addition in the second cohomology groupcoincides with tensor product of complex line bundles.In algebraic geometry, this classification of (isomorphism classes of)complex line bundles by the first Chern class is a crude approximationto the classification of (isomorphism classes of) holomorphic line bundlesby linear equivalence classes of divisors. For complex vector bundles ofdimension greater than one, the Chern classes are not a complete invariant.The Chern classes can be used to construct a homomorphism of ringsfrom the topological K−theory of a space to (the completion of) its rationalcohomology. For line bundles V , the Chern character ch is defined bych(V ) = exp[c 1 (V )].For sums of line bundles, the Chern character is defined by additivity. Forarbitrary vector bundles, it is defined by pretending that the bundle is asum of line bundles; more precisely, for sums of line bundles the Cherncharacter can be expressed in terms of Chern classes, and we use the sameformulas to define it on all vector bundles. For example, the first few termsarech(V ) = dim(V ) + c 1 (V ) + c 1 (V )2/2 − c 2 (V ) + ...If V is filtered by line bundles L 1 , ..., L k having first Chern classes x 1 , ..., x k ,respectively, thench(V ) = e x1 + · · · + e x k..If a connection is used to define the Chern classes, then the explicit formof the Chern character is[ ( )] iΩch(V ) = Tr exp ,2π


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 515where Ω is the curvature of the connection.The Chern character is useful in part because it facilitates the computationof the Chern class of a tensor product. Specifically, it obeys thefollowing identities:ch(V ⊕ W ) = ch(V ) + ch(W ), ch(V ⊗ W ) = ch(V )ch(W ).Using the Grothendieck Additivity Axiom for Chern classes, the first ofthese identities can be generalized to state that ch is a homomorphism ofAbelian groups from the K−theory K(X) into the rational cohomology ofX. The second identity establishes the fact that this homomorphism alsorespects products in K(X), and so ch is a homomorphism of rings. TheChern character is used in the Hirzebruch–Riemann–Roch Theorem.The so–called twisted K–theory a particular variant of K−theory, inwhich the twist is given by an integral 3D cohomology class. In physics,it has been conjectured to classify D−branes, Ramond–Ramond fieldstrengths and in some cases even spinors in type II string theory.4.5.4 Atiyah’s View on K−TheoryAccording to Michael Atiyah [Atiyah and Anderson (1967); Atiyah (2000)],K–theory may roughly be described as the study of additive (or, Abelian)invariants of large matrices. The key point is that, although matrix multiplicationis not commutative, matrices which act in orthogonal subspacesdo commute. Given ‘enough room’ we can put matrices A and B into theblock form( ) A 0,0 1( ) 1 0,0 Bwhich obviously commute. Examples of Abelian invariants are traces anddeterminants.The prime motivation for the birth of K−theory came from Hirzebruch’sgeneralization of the classical Riemann–Roch Theorem (see [Hirzebruch(1966)]). This concerns a complex projective algebraic manifold X anda holomorphic (or algebraic) vector bundle E over X. Then one has thesheaf cohomology groups H q (X, E), which are finite–dimensional vectorspaces, and the corresponding Euler characteristicsχ(X, E) =n∑(−1) q dim H q (X, E),q=0


516 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere n is the complex dimension of X. One also has topological invariantsof E and of the tangent bundle of X, namely their Chern classes. Fromthese one defines a certain explicit polynomial T (X, E) which by evaluationon X becomes a rational number. Hirzebruch’s Riemann–Roch Theoremasserts the equality: χ(X, E) = T (X, E).It is an important fact, easily proved, that both χ and T are additivefor exact sequences of vector bundles:0 → E ′ → E → E ′′ → 0,χ(X, E) = χ(X, E ′ ) + χ(X, E ′′ ), T (X, E) = T (X, E ′ ) + T (X, E ′′ ).This was the starting point of the Grothendieck’s generalization.Grothendieck defined an Abelian group K(X) as the universal additiveinvariant of exact sequences of algebraic vector bundles over X, so that χand T both gave homomorphisms of K(X) into the integers (or rationals).More precisely, Grothendieck defined two different K−groups, one arisingfrom vector bundles (denoted by K 0 ) and the other using coherentsheaves (denoted by K 0 ). These are formally analogous to cohomology andhomology respectively. Thus K 0 (X) is a ring (under tensor product) whileK 0 (X) is a K 0 (X)−module. Moreover, K 0 is contravariant while K 0 iscovariant (using a generalization of χ). Finally, Grothendieck establishedthe analogue of Poincaré duality. While K 0 (X) and K 0 (X) can be definedfor an arbitrary projective variety X, singular or not, the natural mapK 0 (X) → K 0 (X) is an isomorphism if X is non–singular.The Grothendieck’s Riemann–Roch Theorem concerns a morphism f :X → Y and compares the direct image of f in K−theory and cohomology.It reduces to the Hirzeburch’s version when Y is a point.Topological K−theory started with the famous Bott Periodicity Theorem[Bott (1959)], concerning the homotopy of the large unitary groupsU(N) (for N → ∞). Combining Bott’s Theorem with the formalism ofGrothendieck, Atiyah and Hirzebruch, in the late 1950’s, developed a K−theorybased on topological vector bundles over a compact space [Atiyahand Hirzebruch (1961)]. Here, in addition to a group K 0 (X), they alsointroduced an odd counterpart K 1 (X), defined as the group of homotopyclasses of X into U(N), for N large. Putting these together,K ∗ (X) = K 0 (X) ⊕ K 1 (X),they obtained a periodic ‘generalized cohomology theory’.Over the ratio-


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 517nals, the Chern character gave an isomorphism:ch : K ∗ (X) ⊗ Q ∼ = H ∗ (X, Q).But, over the integers, K−theory is much more subtle and it has had manyinteresting topological applications. Most notable was the solution of thevector fields on spheres problem by Frank Adams, using real K−theory(based on the orthogonal groups) [Adams (1962)].An old generalisation of K−theory is related to projective bundles[Atiyah and Anderson (1967); Atiyah (2000)]. Given a vector bundle Vover a space X, we can form the bundle P (V ) whose fibre at x ∈ X is theprojective space P (V x ). In terms of groups and principal bundles, this isthe passage from GL(n, C) to P GL(n, C), or from U(n) to P U(n). We havetwo exact sequences of groups:1 → U(1) → U(n) → P U(n) → 11 → Z n → SU(n) → P U(n) → 1 .The first gives rise to an obstruction α ∈ H 3 (X, Z) to lifting a projectivebundle to a vector bundle, while the second gives an obstructionβ ∈ H 2 (X, Z n ) to lifting a projective bundle to a special unitary bundle.They are related byα = δ(β), where δ : H 2 (X, Z n ) → H 3 (X, Z)is the coboundary operator. This shows that nα = 0. In fact, one canshow that any α ∈ H 3 (X, Z) of order dividing n arises in this way.Can we define an appropriate K−theory for projective bundles withα ≠ 0 ? The answer is yes. For each fixed α of finite order we candefine an Abelian group K α (X). Moreover this is a K(X) module. We willnow indicate how these twisted K–groups (i.e., twisted K−theory) can bedefined.Note first that, for any vector space V, End V = V ⊗ V ∗ depends onlyP (V ). Hence, given a projective bundle P over X, we can define the associatedbundle E(P ) of endomorphism (matrix) algebras. The sections ofE(P ) form a non–commutative C ∗ −algebra and one can define its K−groupby using finitely–generated projective modules. This K−group turns outto depend not on P but only on its obstruction class α ∈ H 3 (X, Z) and socan be denoted by K α (X).


518 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn addition to the K(X)−module structure of K α (X) there are multiplicationsK α (X) ⊗ K β (X) → K α+β (X).Today, there are many variants and generalizations of K−theory, somethingwhich is not surprising given the universality of linear algebra andmatrices [Atiyah and Anderson (1967); Atiyah (2000)]. In each case thereare specific features and techniques relevant to the particular area.First, as already mentioned, is the real K−theory based on real vectorbundles and the Bott periodicity theorems for the orthogonal groups: herethe period is 8 rather than 2.Next there is equivariant theory K G (X), where G is a a compact Liegroup acting on the space X. If X is a point, we just get the representationor character ring R(G) of the group G. In general K G (X) is a moduleover R(G) and this can be exploited in terms of the fixed–point sets in Xof elements of G.If we pass from the space X to the ring C(X) of continuous complex–valued functions on X then K(X) can be defined purely algebraically interms of finitely–general projective modules over X. This then lends itselfto a major generalization if we replace C(X), which is a commutativeC ∗ −algebra, by a non–commutative C ∗ −algebra. This has become a richtheory linked to many basic ideas in functional analysis, in particular tothe von Neumann dimension theory.4.5.5 Atiyah–Singer Index TheoremWe shall recall here very briefly some essential results of Atiyah–SingerIndex Theory. The reader who is not familiar with the topological and analyticproperties of the index of elliptic operators is urged to gain some familiaritywith the Atiyah–Singer Index Theorem [Atiyah and Singer (1963);Atiyah and Singer (1968)] 8 (for technical details, see also [Boos and Bleecker8 In the geometry of manifolds and differential operators, the Atiyah—Singer IndexTheorem is an important unifying result that connects topology and analysis. It dealswith elliptic differential operators (such as the Laplacian) on compact manifolds. It findsnumerous applications, including many in theoretical physics. When Michael Atiyah andIsadore Singer were awarded the Abel Prize by the Norwegian Academy of Science andLetters in 2004, the prize announcement explained the Atiyah—Singer Index Theoremin these words:“Scientists describe the world by measuring quantities and forces that vary overtime and space. The rules of nature are often expressed by formulas, called


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 519(1985)]).A differential operator of order m, mapping the smooth sections of avector bundle E over a compact manifold Y to those of another such bundleF , can be described in local coordinates and local trivializations of thebundles asD = ∑a α (x)D α ,|α|≤mwith α = (α 1 , . . . , α n ). The coefficients a α (x) are matrices of smoothfunctions that represent elements of Hom(E, F ) locally; and D α =∂ ∂∂x α 1 · · ·∂x . αn1nThe principal symbol associated to the operator D is the expressionσ m (D)(x, p) = ∑a α (x)p α .|α|=mGiven the differential operator D : Γ(Y, E) → Γ(Y, F ), the principalsymbol with the local expression above defines a global map σ m : π ∗ (E) →π ∗ (F ), where T ∗ Y π → Y is the cotangent bundle; that is, the variables (x, p)are local coordinates on T ∗ Y .Consider bundles E i , i = 1 . . . k, over a compact nD manifold Y suchthat there is a complex Γ(Y, E) formed by the spaces of sections Γ(Y, E i )and differential operators d i of order m:0 → Γ(Y, E 1 ) → d1· · · dk−1→ Γ(Y, E k ) → 0.Construct the principal symbols σ m (d i ); these determine an associatedsymbol complex0 → π ∗ (E 1 ) σm(d1)→· · · σm(d k−1)→ π ∗ (E k ) → 0.The complex Γ(Y, E) is elliptic iff the associated symbol complex isexact.differential equations, involving their rates of change. Such formulas may havean ‘index’, the number of solutions of the formulas minus the number of restrictionsthat they impose on the values of the quantities being computed. TheAtiyah–Singer index Theorem calculated this number in terms of the geometryof the surrounding space. A simple case is illustrated by a famous paradoxicaletching of M. C. Escher, ‘Ascending and Descending’, where the people, goinguphill all the time, still manage to circle the castle courtyard. The indexTheorem would have told them this was impossible.”


520 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn the case of just one operator, this means that σ m (d) is an isomorphismoff the zero section.Recall that the Hodge Theorem states that the cohomology of the complexΓ(Y, E) coincides with the harmonic forms, i.e.,H i (E) = Ker(d i)Im(d i−1 ) ∼ = Ker(∆ i ), where ∆ i = d ∗ i d i + d i−1 d ∗ i−1.Without loss of generality, by passing to the assembled complexE + = E 1 ⊕ E 3 ⊕ · · · , E − = E 2 ⊕ E 4 ⊕ · · · ,we can always think of one elliptic operatorD : Γ(Y, E + ) → Γ(Y, E − ),D = ∑ i(d 2i−1 + d ∗ 2i).The Index Theorem states: Consider an elliptic complex over a compact,orientable, even dimensional manifold Y without boundary. The index ofD, which is given byInd(D) = dim[Ker(D)]− dim[Coker(D)] = ∑ i(−1) i dim[Ker ∆ i ] = −χ(E),χ(E) being the Euler characteristic of the complex, can be expressed interms of characteristic classes as:〈 ∑〉ch(Ind(D) = (−1) n/2 i (−1)i [E i ])td(T Y C ), [Y ] .e(Y )Here, ch is the Chern character, e is the Euler class of the tangent bundleof Y , td(T Y C ) is the Todd class of the complexified tangent bundle. TheAtiyah–Singer Index Theorem, which computes the index of of a family ofelliptic differential operators, is naturally formulated in terms of K−theoryand is an extension of the Riemann–Roch Theorem.4.5.6 The Infinite–Order CaseTopological K−theory turned out to have a very natural link with the theoryof operators in quantum Hilbert space. If H is an infinite–dimensionalcomplex Hilbert space and B(H) the space of bounded operators on H withthe uniform norm, then one defines the subspace F(H) ⊂ B(H) of Fredholmoperators T, by the requirement that both Ker(T ) and Coker(T ) have


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 521finite–dimensions.The Atiyah–Singer index is then defined byInd(T ) = dim[Ker(T )] − dim[Coker(T )],and it has the key property that it is continuous, and therefore constant oneach connected component of F(H). Moreover, Ind : F → Z identifiesthe components of F.This has a generalization to any compact space X. To any continuousmap X → F (i.e., a continuous family of Fredholm operators parametrizedby X) one can assign an index in K(X). Moreover one gets in this wayan isomorphism Ind : [X, F] → K(X), where [X, F] denotes the set ofhomotopy classes of maps of X into F. A notable example of a Fredholmoperator is an elliptic differential operator on a compact manifold (theseare turned into bounded operators by using appropriate Sobolev norms).Now, in the quantum–physical situation, one meets infinite–order elementsα ∈ H 3 (X, Z) and the question arises of whether one can still definea ‘twisted’ group K α (X). It turns out that it is possible to do this and oneapproach is being developed by [Atiyah and Segal (1971)].Since an α of order n arises from an obstruction problem involvingnD vector bundles, it is plausible that, for α of infinite order, we need toconsider bundles of Hilbert spaces H. But here we have to be careful notto confuse the ‘small’ unitary groupU(∞) = limN→∞ U(N)with the ‘large’ group U(H) of all unitary operators in Hilbert space. Thesmall unitary group has interesting homotopy groups given by Bott’s periodicityTheorem, but U(H) is contractible, by Kuiper’s Theorem. Thismeans that all Hilbert space bundles (with U(H)) as structure group) aretrivial. This implies the following homotopy equivalences:P U(H) = U(H)/U(1) ∼ CP ∞ = K(Z, 2), BP U(H) ∼ K(Z, 3),where B denotes here the classifying space and on the right we have theEilenberg–MacLane spaces. It follows that P (H)−bundles over X are classifiedcompletely by H 3 (X, Z). Thus, for each α ∈ H 3 (X, Z), there is anessentially unique bundle P α over X with fibre P (H).As in finite dimensions. B(H) depends only on P (H), we can define abundle B α of algebras over X. We now let F α ⊂ B α be the correspondingbundle of Fredholm operators. Finally we defineK α (X) = Homotopy classes of sections of F α .


522 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis definition works for all α. If α is of finite order, then P α contains afinite–dimensional sub–bundle, but if α is of infinite order this is not true.Thus we are essentially in an infinite–dimensional analytic situation.To get the twisted odd groups we recall that F 1 ⊂ F, the space of self–adjoint Fredholm operators, is a classifying space for K 1 and so we take F 1 α⊂ F α to defineK 1 α(X) = Homotopy classes of sections of F 1 α.One peculiar feature of the infinite–order case is that all sections of F alie in the zero–index component, or equivalently that the restriction mapK α (X) → K α (point)is zero.Now, what can we say about the relation between twisted K−groupsand cohomology? Over the rationals, if α is of finite order, nothing muchchanges [Atiyah and Anderson (1967); Atiyah (2000)]. In particular theChern character induces an isomorphism. However if α is of infinite ordersomething new happens. We can now consider the operation u → αu onH ∗ (X, Q) as a differential d α (α 2 = 0 since α has odd dimension). We canthen form the cohomology with respect to this differentialH a = Ker d α / Im d α .One can then prove that there is an isomorphismK ∗ α(X) ⊗ Q ∼ = H α .In the usual Atiyah–Hirzebruch spectral sequence [Atiyah and Hirzebruch(1961)], relating K−theory to integral cohomology, all differentials are offinite order and so vanish over Q. In particular, d 3 = Sq 3 , the Steenrodoperation. However for K α one findsd 3 u = Sq 3 u + αuand this explains why an α of infinite–order gives the isomorphism aboveover the rationals.Chern classes over the integers are a more delicate matter. One canproceed as follows. In F there are various subspaces F r,s (of finite codimension)wheredim[Ker] = r, dim[Coker] = s,


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 523and these lie in the component of Ind(r − s).Since the F r,s ⊂ F are invariant under the action of U(H), it followsthat they can be defined fibrewise and this shows that the classes c r,s canbe defined for K α (X). However the classes for r = s (and so of index zero)are not sufficient to generate all Chern classes. It is a not unreasonableconjecture that the c r,r are the only integral characteristic classes for thetwisted K−theories [Atiyah and Anderson (1967); Atiyah (2000)].While the use of Hilbert spaces H and the corresponding projectivespaces P (H) may not come naturally to a topologist, these are perfectlynatural in physics. Recall that P (H) is the space of quantum states. Bundlesof such arise naturally in quantum field theory.4.5.7 Twisted K−Theory and the Verlinde AlgebraTwistings of cohomology theories are most familiar for ordinary cohomology[Freed (2001); Freed et. al. (2003)]. Let M be a smooth manifold.Then a flat real vector bundle E → M determines twisted real cohomologygroups H • (M; E). In differential geometry these cohomology groups aredefined by extending the de Rham complex to forms with coefficients in Eusing the flat connection. The sorts of twistings of K−theory we considerare 1D, so analogous to the case when E is a line bundle. There are also1D twistings of integral cohomology, determined by a local system Z → M.This is a bundle of groups isomorphic to Z, so is determined up to isomorphismby an element of H 1 (M; Aut(Z)) ∼ = H 1 (M; Z mod 2), since the onlynontrivial automorphism of Z is multiplication by −1. The twisted integralcohomology H • (M; Z) may be thought of as sheaf cohomology, or definedusing a cochain complex. We give a Čech description as follows. Let {U i}be an open covering of M andg ij : U i ∩ U j −→ {±1} (4.28)a cocycle defining the local system Z. Then an element of H q (M; Z) isrepresented by a collection of q−cochains a i ∈ Z q (U i ) which satisfya j = g ij a i on U ij = U i ∩ U j . (4.29)We can use any model of co–chains, since the group Aut(Z) ∼ = {±1} alwaysacts. In place of co–chains we represent integral cohomology classesby maps to an Eilenberg–MacLane space K(Z, q). The cohomology group isthe set of homotopy classes of maps, but here we use honest maps as representatives.The group Aut(Z) acts on K(Z, q). One model of K(Z, 0) is the


524 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionintegers, with −1 acting by multiplication. The circle is a model for K(Z, 1),and −1 acts by reflection. Using the action of Aut(Z) on K(Z, q) and thecocycle (4.28) we build an associated bundle H q → M with fiber K(Z, q).Equation (4.29) says that twisted cohomology classes are represented bysections of H q → M; the twisted cohomology group H q (M; Z) is the set ofhomotopy classes of sections of H q → M.Twistings may be defined for any generalized cohomology theory; ourinterest is in complex K−theory [Freed (2001); Freed et. al. (2003)]. Inhomotopy theory one regards K as a marriage of a ring and a space (moreprecisely, spectrum), and it makes sense to ask for the units in K, denotedGL 1 (K). In the previous paragraph we used the units in integralcohomology, the group Z mod 2. For complex K−theory there is a richergroup of unitsGL 1 (K) ∼ Zmod2 × CP ∞ × BSU. (4.30)In our problem the last factor does not enter and all the interest is in thefirst two, which we denote GL 1 (K) ′ . As a first approximation, view Kas the category of all finite dimensional Z mod 2–graded complex vectorspaces. Then CP ∞ is the subcategory of even complex lines, and it is agroup under tensor product. It acts on K by tensor product as well. Thenontrivial element of Z mod 2 in (4.30) acts on K by reversing the parityof the grading. This model is deficient since there is not an appropriatetopology. One may consider instead complexes of complex vector spaces,or spaces of operators as we do below. Of course, there are good topologicalmodels of CP ∞ , for example the space of all 1D subspaces of a fixed complexHilbert space H. For a manifold M the twistings of K−theory of interestare classified up to isomorphism byH 1 (M; GL 1 (K) ′ ) ∼ = H 1 (M; Zmod2) × H 3 (M; Z).In this paper we will not encounter twistings from the first factor and willfocus exclusively on the second. These twistings are represented by co–cycles g ij with values in the space of lines, in other words by complex linebundles L ij → U ij which satisfy a cocycle condition. This is the data oftengiven to define a gerbe. 99 Recall that a gerbe is a construct in homological algebra. It is defined as a stackover a topological space which is locally isomorphic to the Picard groupoid of that space.Recall that the Picard groupoid on an open set U is the category whose objects are linebundles on U and whose morphisms are isomorphisms. A stack refers to any categoryacting like a moduli space with a universal family (analogous to a classifying space)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 525Now, let X = G be a compact Lie group and, for simplicity, we shallassume that it is simply connected, though the theory works in the generalcase. We consider G as G−space, the group acting on itself by conjugation.Since H 3 (G, Z) ∼ = Z we can construct twisted K−theories for eachinteger k. Moreover, we can also do this equivariantly, thus obtainingAbelian groups KG,k ∗ (G). These will all be R(G)−modules.Now, the group multiplication map µ : G × G → G is compatible withconjugation and so is a G−map. In addition, to the pull back µ ∗ , we canalso consider the push–forward µ ∗ . This depends on Poincaré duality forK−theory and it works also, when appropriately formulated, in the presentcontext.If dim(G) is even, this gives us a commutative multiplication onKG,k 0 (G), while for dim(G) odd, our multiplication is on K1 G,k(G). In eithercase we get a ring.The claim of [Freed (2001); Freed et. al. (2003)] is that this ring (accordingto the parity of dim(G) is naturally isomorphic to the Verlindealgebra of G at level k − h (where h is the Coxeter number). The Verlindealgebra is a key tool in certain quantum field theories and it has been muchstudied by physicists, topologists, group theorists and algebraic geometers.The K−theory approach is totally new and much more direct than mostother ways.The Verlinde algebra is defined in the theory of loop groups. Let G bea compact Lie group. There is a version of the Theorem for any compactgroup G, but here for the most part we focus on connected, simply connected,and simple groups—G = SU 2 is the simplest example. In this casea central extension of the free loop group LG is determined by the level,which is a positive integer k. There is a finite set of equivalence classesof positive energy representations of this central extension; let V k (G) denotethe free Abelian group they generate. One of the influences of 2Dconformal field theory on the theory of loop groups is the construction ofan algebra structure on V k (G), the fusion product. This is the Verlindealgebra [Verlinde (1988)].More precisely, let G act on itself by conjugation. Then with our assumptionsthe equivariant cohomology group HG 3 (G) is free of rank one. Leth(G) be the dual Coxeter number of G, and define ζ(k) ∈ HG 3 (G) to bek + h(G) times a generator. We will see that elements of H 3 may be usedto twist K−theory, and so elements of equivariant H 3 twist equivariantparameterizing a family of related mathematical objects such as schemes or topologicalspaces, especially when the members of these families have nontrivial automorphisms.


526 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionK−theory.The Freed–Hopkins–Teleman Theorem [Freed (2001); Freed et. al.(2003)] states: There is an isomorphism of algebrasV k (G) ∼ = Kdim G+ζ(k)G(G),where the right hand side is the ζ(k)−twisted equivariant K−theory indegree dim(G). The group structure on the right–hand side is inducedfrom the multiplication map G × G → G.For an arbitrary compact Lie group G the level k is replaced by a classin H 4 (BG; Z) and the dual Coxeter number h(G) is pulled back from auniversal class in H 4 (BSO; Z) via the adjoint representation. The twistingclass is obtained from their sum by transgression.4.5.8 Application: K−Theory in String TheoryIn string theory, K−theory has been conjectured to classify the allowedRamond–Ramond field strengths; 10 and also the charges of stableD−branes.4.5.8.1 Classification of Ramond–Ramond FluxesIn the classical limit of type II string theory, which is type II supergravity,the Ramond–Ramond (RR) field strengths are differential forms. In thequantum theory the well–definedness of the partition functions of D−branesimplies that the RR–field strengths obey Dirac quantization conditionswhen space–time is compact, or when a spatial slice is compact and one considersonly the (magnetic) components of the field strength which lie alongthe spatial directions. This led twentieth century physicists to classify RRfield strengths using cohomology with integral coefficients.However, some authors have argued that the cohomology of space–timewith integral coefficients is too big. For example, in the presence of Neveu–Schwarz (NS) H−flux, or non–spin cycles, some RR–fluxes dictate the presenceof D−branes. In the former case this is a consequence of the super-10 Recall that Ramond–Ramond (RR) fields are differential–form fields in the 10Dspace–time of type II supergravity theories, which are the classical limits of type IIstring theory. The ranks of the fields depend on which type II theory is considered. AsJoe Polchinski argued in 1995, D−branes are the charged objects that act as sources forthese fields, according to the rules of p−form electrodynamics. It has been conjecturedthat quantum RR fields are not differential forms, but instead are classified by twistedK−theory.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 527gravity equation of motion which states that the product of a RR–flux withthe NS 3–form is a D−brane charge density. Thus the set of topologicallydistinct RR–field strengths that can exist in brane–free configurations isonly a subset of the cohomology with integral coefficients.This subset is still too big, because some of these classes are related bylarge gauge transformations. In QED there are large gauge transformationswhich add integral multiples of 2π to Wilson loops. 1111 Recall that in gauge theory, a Wilson loop (named after Ken Wilson) is a gauge–invariant observable obtained from the holonomy of the gauge connection around a givenloop. In the classical theory, the collection of all Wilson loops contains sufficient informationto reconstruct the gauge connection, up to gauge transformation. In quantum fieldtheory, the definition of Wilson loops observables as bona fide operators on Fock space(actually, Haag’s Theorem states that Fock space does not exist for interacting QFTs)is a mathematically delicate problem and requires regularization, usually by equippingeach loop with a framing. The action of Wilson loop operators has the interpretation ofcreating an elementary excitation of the quantum field which is localized on the loop. Inthis way, Faraday’s ”flux tubes” become elementary excitations of the quantum electromagneticfield.Wilson loops were introduced in the 1970s in an attempt at a non–perturbative formulationof quantum chromodynamics (QCD), or at least as a convenient collection of variablesfor dealing with the strongly–interacting regime of QCD. The problem of confinement,which Wilson loops were designed to solve, remains unsolved to this day. The factthat strongly–coupled quantum gauge field theories have elementary non–perturbativeexcitations which are loops motivated Alexander Polyakov to formulate the first stringtheories, which described the propagation of an elementary quantum loop in spacetime.Wilson loops played an important role in the formulation of loop quantum gravity, butthere they are superseded by spin networks, a certain generalization of Wilson loops.In particle physics and string theory, Wilson loops are often called Wilson lines, especiallyWilson loops around non–contractible loops of a compact manifold.A Wilson line W C is a quantity defined by a path–ordered exponential of a gauge fieldA µIW C = Tr P exp i A µdx µ .CHere, C is a contour in space, P is the path–ordering operator, and the trace Tr guaranteesthat the operator is invariant under gauge transformations. Note that the quantitybeing traced over is an element of the gauge Lie group and the trace is really the characterof this element with respect to an irreducible representation, which means thereare infinitely many traces, one for each irrep.Precisely because we’re looking at the trace, it doesn’t matter which point on the loopis chosen as the initial point. They all give the same value.Actually, if A is viewed as a connection over a principal G−bundle, the equation abovereally ought to be ‘read’ as the parallel transport of the identity around the loop whichwould give an element of the Lie group G.Note that a path–ordered exponential is a convenient shorthand notation commonin physics which conceals a fair number of mathematical operations. A mathematicianwould refer to the path–ordered exponential of the connection as ‘the holonomy of theconnection’ and characterize it by the parallel–transport differential equation that itsatisfies.


528 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe p− form potentials in type II supergravity theories also enjoythese large gauge transformations, but due to the presence of Chern–Simons terms in the supergravity actions these large gauge transformationstransform not only the p−form potentials but also simultaneously the(p + 3)−form field strengths. Thus to get the space of inequivalent fieldstrengths from the forementioned subset of integral cohomology we mustquotient by these large gauge transformations.The Atiyah–Hirzebruch spectral sequence constructs twisted K−theory,with a twist given by the NS 3−form field strength, as a quotient of asubset of the cohomology with integral coefficients. In the classical limit,which corresponds to working with rational coefficients, this is preciselythe quotient of a subset described above in supergravity. The quantumcorrections come from torsion classes and contain mod 2 torsion correctionsdue to the Freed–Witten anomaly.Thus twisted K−theory classifies the subset of RR–field strengths thatcan exist in the absence of D−branes quotiented by large gauge transformations.4.5.8.2 Classification of D−BranesNow, in many applications one wishes to add sources for the RR fields.These sources are called D−branes. As in classical electromagnetism, onemay add sources by including a coupling CpJ 10−p of the p−form potentialto a (10 − p)−form current J 10−p in the Lagrangian (density). The usualconvention in the string theory literature appears to be to not write thisterm explicitly in the action.The current J 10−p modifies the equation of motion that comes fromthe variation of Cp. As is the case with magnetic monopoles in electromagnetism,this source also invaliditates the dual Bianchi identity as it isa point at which the dual field is not defined. In the modified equationof motion J p+2 appears on the left hand side of the equation of motioninstead of zero. For simplicity, we will also interchange p and 7 − p, thenthe equation of motion in the presence of a source isJ 9−p = d 2 C 7−p = dG 8−p = dF 8−p + H ∧ G p−1 .The (9−p)−form J 9−p is the Dp−brane current, which means that it isPoincaré dual to the world–volume of a (p + 1)−D extended object calledIn finite temperature QCD, the expectation value of the Wilson line distinguishesbetween the ‘confined phase’ and the ‘deconfined phase’ of the theory.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 529a Dp−brane. The discrepency of one in the naming scheme is historicaland comes from the fact that one of the p + 1 directions spanned by theDp−brane is often time–like, leaving p spatial directions.The above Bianchi identity is interpreted to mean that the Dp−braneis, in analogy with magnetic monopoles in electromagnetism, magneticallycharged under the RR p−form C7−p. If instead one considers this Bianchiidentity to be a field equation for Cp + 1, then one says that the Dp−braneis electrically charged under the (p + 1)−form Cp + 1.The above equation of motion implies that there are two ways to derivethe Dp−brane charge from the ambient fluxes. First, one may integratedG 8−p over a surface, which will give the Dp−brane charge intersected bythat surface. The second method is related to the first by Stoke’s Theorem.One may integrate G 8−p over a cycle, this will yield the Dp−brane chargelinked by that cycle. The quantization of Dp−brane charge in the quantumtheory then implies the quantization of the field strengths G, but not of theimproved field strengths F .It has been conjectured that twisted K−theory classifies classifiesD−branes in noncompact space–times, intuitively in space–times in whichwe are not concerned about the flux sourced by the brane having nowhereto go. While the K−theory of a 10D space–time classifies D−branes assubsets of that space–time, if the space–time is the product of time anda fixed 9−manifold then K−theory also classifies the conserved D−branecharges on each 9D spatial slice. While we were required to forget aboutRR potentials to get the K−theory classification of RR field strengths,we are required to forget about RR field strengths to get the K−theoryclassification of D−branes.We will continue exposition on K−theory applications to string theoryin the last section of the book.4.6 Principal BundlesRecall that a principal bundle is a special case of a fibre bundle where thefibre is a group G. More specifically, G is usually a Lie group. A principalbundle is a total bundle space Y along with a surjective map π : Y → Xto a base manifold X. Any fibre π −1 (x) is a space isomorphic to G. Morespecifically, G acts freely and transitively without fixed point on the fibers,and this makes a fibre into a homogeneous space. It follows that the orbitsof the G−action are precisely the fibers of π : Y → X and the orbit space


530 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionY/G is homeomorphic the base space X. To say that G acts freely andtransitively on the fibers means that the fibers take on the structure ofG−torsors. 12For example, in the case of a circle bundle (G = S 1 ≡ {e it }), thefibers are circles, which can be rotated, although no point in particularcorresponds to the identity. Near every point, the fibers can be given thegroup structure of G in the fibers over a neighborhood by choosing anelement in each fibre to be the identity element. However, the fibers cannotbe given a group structure globally, except in the case of a trivial bundle.An important principal bundle is the frame bundle on a Riemannianmanifold. This bundle reflects the different ways to give an orthonormalbasis for tangent vectors.In general, any fibre bundle corresponds to a principal bundle where thegroup (of the principal bundle) is the group of isomorphisms of the fibre (ofthe fibre bundle). Given a principal bundle π : Y → X and an action of Gon a space V , which could be a group representation, this can be reversedto give an associated fibre bundle.A trivialization of a principal bundle, an open set U in X such that thebundle π −1 (U) over U, is expressed as U × G, has the property that thegroup G acts on the left and transition functions take values in G, actingon the fibers by right multiplication (so that the action of G on a fibre Vis independent of coordinate chart).More precisely, a principal bundle π P : P → Q of a configuration manifoldQ, with a structure Lie group G, is a general affine bundle modelled onthe right on the trivial group bundle Q × G where the group G acts freelyand transitively on P on the right,R G : P × G → P, R g : p↦→pg, (p ∈ P, g ∈ G). (4.31)We call P a principal G−bundle. A typical fibre of a principal G−bundleis isomorphic to the group space of G, and P/G = Q. The structure groupG acts on the typical fibre by left multiplications which do not preservethe group structure of G. Therefore, the typical fibre of a principal bundleis only a group space, but not a group. Since the left action of transitionfunctions on the typical fibre G commutes with its right multiplications,a principal bundle admits the global right action (4.31) of the structuregroup.12 A G−torsor is a space which is homeomorphic to G but lacks a group structure sincethere is no preferred choice of an identity element.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 531A principal G−bundle P is equipped with a bundle atlaswhose trivialization mapsobey the equivariance conditionΨ P = {(U α , ψ P α , ρ αβ )}, (4.32)ψ P α : π −1P(U α) → U α × G(π 2 ◦ ψ P α )(pg) = (π 2 ◦ ψ P α )(p)g, (g ∈ G, p ∈ π −1P (U α)). (4.33)Due to this property, every trivialization map ψ P α determines a unique localsection z α of P over U α such thatπ 2 ◦ ψ P α ◦ z α = 1,where 1 is the unit element of G. The transformation rules for z α readz β (q) = z α (q)ρ αβ (q), (q ∈ U α ∩ U β ), (4.34)where ρ αβ (q) are G−valued transition functions of the atlas Ψ P . Conversely,the family {(U α , z α )} of local sections of P with the transitionfunctions (4.34) determines a unique bundle atlas of P . In particular, itfollows that only trivial principal bundles have global sections.Note that the pull–back of a principal bundle is also a principal bundlewith the same structure group.The quotient of the tangent bundle T P → P and that of the verticaltangent bundle V P of P by the tangent prolongation T R G of the canonicalaction R G (4.31) are vector bundlesT G P = T P/G, V G P = V P/G (4.35)over Q. Sections of T G P → Q are naturally identified with G-invariantvector–fields on P , while those of V G P → Q are G-invariant vertical vector–fields on P . Therefore, the Lie bracket of G-invariant vector–fields on Pgoes to the quotients (4.35), and induces the Lie brackets of their sections.Let us write these brackets in an explicit form.Owing to the equivariance condition (4.33), any bundle atlas (4.32)of P induces the associated bundle atlases {U α , T ψ P α /G)} of T G P and{U α , V ψ P α /G)} of V G P . Given a basis {ε p } for the right Lie algebra g r ,let {∂ α , e p } and {e p }, where e p = (ψ P α /G) −1 (ε p ), be the corresponding localfibre bases for the vector bundles T G P and V G P , respectively. Relative


532 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionto these bases, the Lie bracket of sectionsξ = ξ α ∂ α + ξ p e p ,η = η µ ∂ µ + η q e qof the vector bundle T G P → Q reads[ξ, η] = (ξ µ ∂ µ η α − η µ ∂ µ ξ α )∂ α + (ξ α ∂ α η r − η α ∂ α ξ r + c r pqξ p η q )e r . (4.36)Putting ξ α = 0 and η µ = 0, we get the Lie bracket[ξ, η] = c r pqξ p η q e r (4.37)of sections of the vector bundle V G → P .A principal bundle P is also the general affine bundle modelled on theleft on the associated group bundle ˜P with the standard fibre G on which thestructure group G acts by the adjoint representation. The correspondingbundle map reads˜P × P −→ P,(˜p, p) ↦→ ˜pp.Note that the standard fibre of the group bundle ˜P is the group G, whilethat of the principal bundle P is the group space of G on which the structuregroup G acts on the left.A principal bundle P → Q with a structure Lie group G possesses thecanonical trivial vertical splittingα : V P → P × g l , π 2 ◦ α ◦ e m = J m ,where {J m } is a basis for the left Lie algebra g l and e m denotes the correspondingfundamental vector–fields on P . Given a principal bundle P → Q,the bundle T P → T Q is a principal bundleT P × T (Q × G) → T Pwith the structure group T G = G × g l where g l is the left Lie algebra ofleft–invariant vector–fields on the group G.If P → Q is a principal bundle with a structure group G, the exactsequence (4.13) can be reduced to the exact sequence0 → V G P ↩→ T G P → T Q → 0, (4.38)where T G P = T P/G, V G P = V P/Gare the quotients of the tangent bundle T P of P and the vertical tangentbundle V P of P respectively by the canonical action (4.31) of G on P on theright. The bundle V G P → Q is called the adjoint bundle. Its standard fibre


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 533is the right Lie algebra g r of the right–invariant vector–fields on the groupG. The group G acts on this standard fibre by the adjoint representation.4.7 Distributions and Foliations on ManifoldsLet M be an nD smooth manifold. A smooth distribution T of codimensionk on M is defined as a subbundle of rank n − k of the tangent bundle T M.A smooth distribution T is called the involutive distribution if [u, u ′ ] is asection of T whenever u and u ′ are sections of T .Let T be a k−codimensional distribution on M. Its annihilator T ∗ isa kD subbundle of T ∗ M called the Pfaffian system. It means that, on aneighborhood U of every point x ∈ M, there exist k linearly independentsections s 1 , . . . , s k of T ∗ such thatT x | U = ∩ j Ker s j .Let C(T ) be the ideal of ∧(M) generated by sections of T ∗ .A smooth distribution T is involutive iff the ideal C(T ) is differential,that is, dC(T ) ⊂ C(T ).Given an involutive k−codimensional distribution T on M, the quotientT M/T is a kD vector bundle called the transversal bundle of T . There isthe exact sequence0 → T ↩→ T M → T M/T → 0. (4.39)Given a bundle Y → X, its vertical tangent bundle V Y exemplifies aninvolutive distribution on Y .A submanifold N of M is called the integral manifold of a distribution Ton M if the tangent spaces to N coincide with the fibres of this distributionat each point of N.Let T be a smooth involutive distribution on M. For any point x ∈ M,there exists a maximal integral manifold of T passing through x [Kamberand Tondeur (1975)]. In view of this fact, involutive distributions are alsocalled completely integrable distributions.Every point x ∈ M has an open neighborhood U which is a domain ofa coordinate chart (x 1 , . . . , x n ) such that the restrictions of T and T ∗ to U∂∂x n−k∂are generated by the n − k vector–fields∂x, . . . , and the k Pfaffian1forms dx n−k+1 , . . . , dx n respectively.In particular, it follows that integral manifolds of an involutive distributionconstitute a foliation. Recall that a k−codimensional foliation on


534 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionan nD manifold M is a partition of M into connected leaves F ι with thefollowing property: every point of M has an open neighborhood U which isa domain of a coordinate chart (x α ) such that, for every leaf F ι , the componentsF ι ∩ U are described by the equations x n−k+1 = const,..., x n = const[Kamber and Tondeur (1975)]. Note that leaves of a foliation fail to beimbedded submanifolds in general.For example, every projection π : M → X defines a foliation whoseleaves are the fibres π −1 (x), for all x ∈ X. Also, every nowhere vanishingvector–field u on a manifold M defines a 1D involutive distribution onM. Its integral manifolds are the integral curves of u. Around each pointx ∈ M, there exist local coordinates (x 1 , . . . , x n ) of a neighborhood of xsuch that u is given by u =∂∂x i .4.8 Application: Nonholonomic MechanicsLet T M = ∪ x∈M T x M, be the tangent bundle of a smooth nD mechanicalmanifold M. Recall (from the subsection 4.7 above) that sub–bundle V =∪ x∈M V x , where V x is a vector subspace of T x M, smoothly dependent onpoints x ∈ M, is called the distribution. If the manifold M is connected,dim V x is called the dimension of the distribution. A vector–field X onM belongs to the distribution V if X(x) ⊂ V x . A curve γ is admissiblerelatively to V , if the vector–field ˙γ belongs to V . A differential systemis a linear space of vector–fields having a structure of C ∞ (M) – module.Vector–fields which belong to the distribution V form a differential systemN(V ). A kD distribution V is integrable if the manifold M is foliated to kDsub–manifolds, having V x as the tangent space at the point x. Accordingto the Frobenius Theorem, V is integrable iff the corresponding differentialsystem N(V ) is involutive, i.e., if it is a Lie sub–algebra of the Lie algebraof vector–fields on M. The flag of a differential system N is a sequence ofdifferential systems: N 0 = N, N 1 = [N, N], . . . , N l = [N l−1 , N], . . . .The differential systems N i are not always differential systems of somedistributions V i , but if for every i, there exists V i , such that N i = N(V i ),then there exists a flag of the distribution V : V = V 0 ⊂ V 1 . . . . Suchdistributions, which have flags, will be called regular. Note that the sequenceN(V i ) is going to stabilize, and there exists a number r such thatN(V r−1 ) ⊂ N(V r ) = N(V r+1 ). If there exists a number r such thatV r = T M, the distribution V is called completely nonholonomic, and minimalsuch r is the degree of non–holonomicity of the distribution V .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 535Now, let us see the mechanical interpretation of these geometrical objects.Consider a nonholonomic mechanical system corresponding to a Riemannianmanifold (M, g), where g is a metric defined by the system’s kineticenergy [Dragovic and Gajic (2003)]. Suppose that the distribution Vis defined by (n − m) one–forms ω α ; in local coordinates q = (q 1 , ..., q n ) onMω ρ (q)( ˙q) = a ρi (q) ˙q i = 0,(ρ = m + 1, . . . , n; i = 1, . . . , n).A virtual displacement is a vector–field X on M, such that ω ρ (X) = 0, i.e.,X belongs to the differential system N(V ).<strong>Diff</strong>erential equations of motion of a given mechanical system followfrom the D’Alambert–Lagrangian principle: trajectory γ of the given systemis a solution of the equation〈∇ ˙γ ˙γ − Q, X〉 = 0, (4.40)where X is an arbitrary virtual displacement, Q a vector–field of internalforces, and ∇ is the affine Levi–Civita connection for the metric g.The vector–field R(x) on M, such that R(x) ∈ Vx ⊥ , Vx⊥ ⊕ V x = T x M, iscalled reaction of ideal nonholonomic connections. (4.40) can be rewrittenas∇ ˙γ ˙γ − Q = R, ω α ( ˙γ) = 0. (4.41)If the system is potential, by introducing L = T − U, where U is thepotential energy of the system (Q = − grad U), then in local coordinates qon M, equations (4.41) becomes the forced Lagrangian equation:ddt L ˙q − L q = ˜R, ω α ( ˙q) = 0.Now ˜R is a one–form in (V ⊥ ), and it can be represented as a linear combinationof one–forms ω m+1 , . . . , ω n which define the distribution, ˜R = λ α ω α .Suppose e 1 , . . . , e n are the vector–fields on M, such that e 1 (x), . . . , e n (x)form a base of the vector space T x M at every point x ∈ M, and e 1 , . . . , e mgenerate the differential system N(V ). Express them through the coordinatevector–fields:e i = A j i (q)∂ q j ,(i, j = 1, . . . , n).Denote by p a projection p : T M → V orthogonal according to themetric g. Corresponding homomorphism of C ∞ −modules of sections of


536 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionT M and V isp∂ q i = p a i e a , (a = 1, . . . , m, i = 1, . . . , n).Projecting by p the equations (4.41), from R(x) ∈ V ⊥ (x), we get p(R) = 0,and denoting p(Q) = ˜Q we get˜∇ ˙γ ˙γ = ˜Q,where ˜∇ is the projected connection [DragovicandGajic(2003)]. A relationshipbetween standard Christoffel symbols Γ k ij and coefficients ˜Γ c ab ofthe connection ˜∇, defined by˜∇ ea e b = ˜Γ c abe c , is given by ˜Γc ab = Γ k ijA i aA j b pc k + A i a ∂ q iA j b pc j.If the motion takes place under the inertia (Q = ˜Q = 0), the trajectoriesof nonholonomic mechanical problem are the geodesics for ˜∇.Now, let V be a distribution on M. Denote a C ∞ (M)−module ofsections on V by Γ(V ). A nonholonomic connection on the sub–bundle Vof T M is a map ∇ : Γ(V ) × Γ(V ) → Γ(V ) with the properties:∇ X (Y + Z) = ∇ X Y + ∇ X Z, ∇ X (f · Y ) = X(f)Y + f∇ X Y ,∇ fX+gY Z = f∇ X Z + g∇ Y Z, (X, Y, Z ∈ Γ(V ); f, g ∈ C ∞ (M)).Having a morphism of vector bundles p 0 : T M → V , formed by theprojection on V , denote by q 0 = 1 T M − p 0 the projection on W , V ⊕ W =T M.The tensor–field T ∇ : Γ(V ) × Γ(V ) → Γ(V ) defined byT ∇ (X, Y ) = ∇ X Y − ∇ Y X − p 0 [X, Y ], (X, Y ∈ Γ(V )),is called the torsion tensor for the connection ∇.Suppose there is a positively defined metric tensor g = g ij on V . Givena distribution V , with p 0 and g, there exists a unique nonholonomic connection∇ with the properties [Dragovic and Gajic (2003)]∇ X g(Y, Z) = X(g(Y, Z)) − g(∇ X Y, Z) − g(Y, ∇ X Z) = 0, T ∇ = 0.These conditions can be rewritten in the form:∇ X Y = ∇ Y X + p 0 [X, Y ], Z(g(X, Y )) = g(∇ Z X, Y ) + g(X, ∇ Z Y ).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 537By cyclic permutation of X, Y, Z and summing we get:g(∇ X Y, Z) = 1 {X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y )) (4.42)2+ g(Z, p 0 [X, Y ]) + g(Y, p 0 [Z, X]) − g(X, p 0 [Y, Z]}.Let q i , (i = 1, . . . , n) be local coordinates on M, such that the first mcoordinate vector–fields ∂ q j are projected by projection p 0 into vector–fields e a , (a = 1, . . . , m), generating the distribution V : p 0 ∂ q j = p a i (q)e a.Vector–fields e a can be expressed in the basis ∂ q j as e a = B i a∂ q j , withB i ap b i = δb a. Now we give coordinate expressions for the coefficients of theconnection Γ c ab , defined as ∇ e ae b = Γ c ab e c. From (4.42) we getΓ c ab = { c ab} + g ae g cd Ω e bd + g be g cd Ω e ad − Ω c ab,where Ω is get from p 0 [e a , e b ] = −2Ω c ab e c as2Ω c ab = p c ie a (B i b) − p c ie b (B i a),and { c ab } = 1 2 gce (e a (g be ) + e b (g ae ) − e e (g ab )).4.9 Application: <strong>Geom</strong>etrical Nonlinear Control4.9.1 Introduction to <strong>Geom</strong>etrical Nonlinear ControlIn this section we give a brief introduction to geometrical nonlinear controlsystems. Majority of techniques developed under this name consider the so–called affine nonlinear MIMO–systems of the form (see [Isidori (1989); Nijmeijerand van der Schaft (1990); Lewis (1995); Lewis and Murray (1997);Lewis (1998)])ẋ(t) = f 0 (x(t)) + u i (t)f i (x(t)), (i = 1, ..., m) (4.43)where t ↦→ x(t) is a curve in a system’s state manifold M. The vector–fieldf 0 is called the drift vector–field, describing the dynamics of the systemin the absence of controls, and the vector–fields f 1 , ..., f m are the inputvector–fields or control vector–fields, indicating how we are able to actuatethe system. The vector–fields f 0 , f 1 , ..., f m are assumed to be real analytic.We do not ask for any sort of linear independence of the control vector–fields f 1 , ..., f m . We shall suppose that the controls u : [0, T ] → U arelocally integrable with U some subset of R m . We allow the length T of theinterval on which the control is defined to be arbitrary. It is convenientto denote by τ(u) the right endpoint of the interval for a given control


538 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionu. For a fixed U we denote by U the collection of all measurable controlstaking their values in U. To be concise about this, a control affine systemis a triple Σ = (M, F = {f 0 , f 1 , ..., f m }, U), with all objects as definedabove. A controlled trajectory for Σ is a pair (c, u), where u ∈ U and wherec : [0, τ(u)] → M is defined so thatċ(t) = f 0 (c(t)) + u i (t)f i (c(t)).One can show that for admissible controls, the curve c will exist at least forsufficiently small times, and that the initial condition c(0) = x 0 uniquelydefines c on its domain of definition.For x ∈ M and T > 0 we define several types of reachable sets as:R Σ (x, T ) = {c(T ) :(c, u) is a controlled trajectory for Σ with τ(u) = T and c(0) = x},R Σ (x, ≤ T ) = ∪ t∈[0,T ] R Σ (x, t),R Σ (x) = ∪ t≥0 R Σ (x, t),that allow us to give several definitions of controllability as follows. LetΣ = (M, F, U) be a control affine system and let x ∈ M. We say that:(1) Σ is accessible from x if int(R Σ (x)) ≠ 0.(2) Σ is strongly accessible from x if int(R Σ (x, T )) ≠ 0 for each T > 0.(3) Σ is locally controllable from x if x ∈ int(R Σ (x)).(4) Σ is small–time locally controllable (STLC) from x if there exists T > 0so that x ∈ int(R Σ (x, ≤ T )) for each t ∈ [0, T ].(5) Σ is globally controllable from x if (R Σ (x)) = M.For example, a typical simple system that is accessible but not controllableis given by the following data:M = R 2 , m = 1, U = [−1, 1],ẋ = u, ẏ = x 2 .This system is (not obviously) accessible from (0, 0), but is (obviously) notlocally controllable from that same point. Note that although R Σ ((0, 0), ≤T ) has nonempty interior, the initial point (0, 0) is not in that interior.Thus this is a system that is not controllable in any sense. Note that thesystem is also strongly accessible.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 5394.9.2 Feedback LinearizationRecall that the core of control theory is the idea of the feedback. In case ofnonlinear control, this implies feedback linearization.Exact Feedback LinearizationThe idea of feedback linearization is to algebraically transform the nonlinearsystem dynamics into a fully or partly linear one so that the linearcontrol techniques can be applied. Note that this is not the same as a conventionallinearization using Jacobians. In this subsection we will presentthe modern, geometrical, Lie–derivative based techniques for exact feedbacklinearization of nonlinear control systems.The Lie Derivative and Lie Bracket in Control Theory. Recall(see (3.7) above) that given a scalar function h(x) and a vector–field f(x),we define a new scalar function, L f h = ∇hf, which is the Lie derivativeof h w.r.t. f, i.e., the directional derivative of h along the direction of thevector f. Repeated Lie derivatives can be defined recursively:L 0 f h = h, L i f h = L f(L i−1f) (h = ∇L i−1f)h f, (for i = 1, 2, ...)Or given another vector–field, g, then L g L f h(x) is defined asL g L f h = ∇ (L f h) g.For example, if we have a control systemẋ = f(x),y = h(x),with the state x = x(t) and the output y, then the derivatives of the outputare:ẏ = ∂h∂xẋ = L f h, and ÿ = ∂L f h∂x ẋ = L2 f h.Also, recall that the curvature of two vector–fields, g 1 , g 2 , gives a non–zero Lie bracket, [g 1 , g 2 ] ( (3.7.2) see Figure 4.2). Lie bracket motions cangenerate new directions in which the system can move.In general, the Lie bracket of two vector–fields, f(x) and g(x), is definedby[f, g] = Ad f g = ∇gf − ∇fg = ∂g∂x f − ∂f∂x g,


540 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 4.2 The so–called ‘Lie bracket motion’ is possible by appropriately modulating thecontrol inputs (see text for explanation).where ∇f = ∂f/∂x is the Jacobian matrix. We can define Lie bracketsrecursively,Ad 0 f g = g,Ad i f g = [f, Ad i−1fg], (for i = 1, 2, ...)Lie brackets have the properties of bilinearity, skew–commutativity andJacobi identity.For example, if( cos x2f =x 1), g =( )x1,1then we have( ) ( ) ( ) ( ) ( )1 0 cos x2 0 − sin x2 x1 cos x2 + sin x[f, g] =−=2.0 0 x 1 1 0 1−x 1Input/Output Linearization.(SISO) systemGiven a single–input single–outputẋ = f(x) + g(x) u, y = h(x), (4.44)we want to formulate a linear–ODE relation between output y and a newinput v. We will investigate (see [Isidori (1989); Sastri and Isidori (1989);Wilson (2000)]):• How to generate a linear input/output relation.• What are the internal dynamics and zero–dynamics associated with theinput/output linearization?• How to design stable controllers based on the I/O linearization.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 541This linearization method will be exact in a finite domain, rather thantangent as in the local linearization methods, which use Taylor series approximation.Nonlinear controller design using the technique is called exactfeedback linearization.Algorithm for Exact Feedback Linearization. We want to finda nonlinear compensator such that the closed–loop system is linear (seeFigure 4.3). We will consider only affine SISO systems of the type (4.44),i.e, ẋ = f(x) + g(x) u, y = h(x), and we will try to construct a control lawof the formu = p(x) + q(x) v, (4.45)where v is the setpoint, such that the closed–loop nonlinear systemẋ = f(x) + g(x) p(x) + g(x) q(x) v,y = h(x),is linear from command v to y.Fig. 4.3Feedback linearization (see text for explanation).The main idea behind the feedback linearization construction is to finda nonlinear change of coordinates which transforms the original system intoone which is linear and controllable, in particular, a chain of integrators.The difficulty is finding the output function h(x) which makes this constructionpossible.We want to design an exact nonlinear feedback controller. Given thenonlinear affine system, ẋ = f(x) + g(x), y = h(x),.we want to find thecontroller functions p(x) and q(x). The unknown functions inside our con-


542 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiontroller (4.45) are given by:(−p(x) =q(x) =L r f h(x) + β 1L r−1f)h(x) + ... + β r−1 L f h(x) + β r h(x)L g L r−1fh(x)1L g L r−1fh(x) , (4.46)which are comprised of Lie derivatives, L f h(x). Here, the relative order, r,is the smallest integer r such that L g L r−1fh(x) ≠ 0. For linear systems r isthe difference between the number of poles and zeros.To get the desired response, we choose the r parameters in the β polynomialto describe how the output will respond to the setpoint, v (pole–placement).d r ydt r + β d r−1 y1dt r−1 + ... + β dyr−1dt + β ry = v.Here is the proposed algorithm [Isidori (1989); Sastri and Isidori (1989);Wilson (2000)]):(1) Given nonlinear SISO process, ẋ = f(x, u), and output equation y =h(x), then:(2) Calculate the relative order, r.(3) Choose an rth order desired linear response using pole–placement technique(i.e., select β). For this could be used a simple rth order low–passfilter such as a Butterworth filter.(4) Construct the exact linearized nonlinear controller (4.46), using Liederivatives and perhaps a symbolic manipulator (Mathematica orMaple).(5) Close the loop and get a linear input–output black–box (see Figure4.3).(6) Verify that the result is actually linear by comparing with the desiredresponse.Relative DegreeA nonlinear SISO system,ẋ = f(x) + g(x) u,y = h(x),is said to have relative degree r at a point x o if (see [Isidori (1989); Nijmeijerand van der Schaft (1990)])


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 543(1) L g L k f h(x) = 0 for all x in a neighborhood of x o and all k < r − 1; and(2) L g L r−1fh(x o ) ≠ 0.For example, controlled Van der Pol oscillator has the state–space form[] [ ]xẋ = f(x) + g(x) u =202ωζ (1 − µx 2 1) x 2 − ω 2 + u.x 1 1Suppose the output function is chosen as y = h(x) = x 1 . In this case wehaveL g h(x) = ∂h∂x g(x) = [ 1 0 ] [ ]0= 0, and1L f h(x) = ∂h∂x f(x) = [ 1 0 ] [ ]x 22ωζ (1 − µx 2 1) x 2 − ω 2 = x 2 .x 1MoreoverL g L f h(x) = ∂(L f h)∂x g(x) = [ 0 1 ] [ ]0= 1,1and thus we see that the Vand der Pol oscillator system has relative degree2 at any point x o .However, if the output function is, for instance y = h(x) = sin x 2 , thenL g h(x) = cos x 2 . The system has relative degree 1 at any point x o , providedthat (x o ) 2 ≠ (2k + 1) π/2. If the point x o is such that this condition isviolated, no relative degree can be defined.As another example, consider a linear system in the state–space formẋ = A x + B u, y = C x.In this case, since f(x) = A x, g(x) = B, h(x) = C x, it can be seen thatL k f h(x) = C A k x,L g L k f h(x) = C A k B.and therefore,Thus, the integer r is characterized by the conditionsC A k B = 0, for all k < r − 1C A r−1 B ≠ 0,otherwise.It is well–known that the integer satisfying these conditions is exactly equalto the difference between the degree of the denominator polynomial and the


544 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondegree of the numerator polynomial of the transfer functionof the system.H(s) = C (sI − A) −1 BApproximative Feedback LinearizationConsider a SISO systemẋ = f(x) + g(x) u, (4.47)where f and g are smooth vector–fields defined on a compact contractibleregion M of R n containing the origin. (Typically, M is a closed ball in R n .)We assume that f(0) = 0, i.e., that the origin is an equilibrium for ẋ = f(x).The classical problem of feedback linearization can be stated as follows: findin a neighborhood of the origin a smooth change of coordinates z = Φ(x)(a local diffeomorphism) and a smooth feedback law u = k(x) + l(x) u newsuch that the closed–loop system in the new coordinates with new controlis linear,ż = Az + B u new ,and controllable (see [Banaszuk and Hauser (1996)]). We usually requirethat Φ(0) = 0. We assume that the system (4.47) has the linear controllabilitypropertydim(span{g, Ad f g, ..., Ad n−1fg}) = n, for all x ∈ M (4.48)(where Ad i fare iterated Lie brackets of f and g). We define the characteristicdistribution for (4.47)D = span{g, Ad f g, ..., Ad n−2fg},which is an (n − 1)D smooth distribution by assumption of linear controllability(4.48). We call any nowhere vanishing 1−form ω annihilating D acharacteristic 1−form for (4.47). All the characteristic 1−forms for (4.47)can be represented as multiples of some fixed characteristic 1−form ω 0 by asmooth nowhere vanishing function (zero–form) β. Suppose that there is anon–vanishing β so that βω 0 is exact, i.e., βω 0 = dα for some smooth functionα, where d denotes the exterior derivative. Then ω 0 is called integrableand is called an integrating factor for ω 0 . The following result is standardin nonlinear control: Suppose that the system (4.47) has the linear controllabilityproperty (4.48) on M. Let D be the characteristic distribution


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 545and ω 0 be a characteristic 1−form for (4.47). The following statements areequivalent:(1) Equation (4.47) is feedback linearizable in a neighborhood of the originin M;(2) D is involutive in a neighborhood of the origin in M; and(3) ω 0 is integrable in a neighborhood of the origin in M.As is well known, a generic nonlinear system is not feedback linearizablefor n > 2. However, in some cases, it may make sense to considerapproximate feedback linearization.Namely, if one can find a feedback linearizable system close to (4.47),there is hope that a control designed for the feedback linearizable systemand applied to (4.47) will give satisfactory performance if the feedbacklinearizable system is close enough to (4.47). The first attempt in thisdirection goes back to [Krener (1984)], where it was proposed to apply to(4.47) a change of variables and feedback that yield a system of the formż = Az + B u new + O(z, u new ),where the term O(z, u new ) contains higher–order terms. The aim was tomake O(z, u new ) of as high order as possible. Then we can say that thesystem (4.47) is approximately feedback linearized in a small neighborhoodof the origin. Later [Hunt and Turi (1993)] introduced a new algorithm toachieve the same goal with fewer steps.Another idea has been investigated in [Hauser et al. (1992)]. Roughlyspeaking, the idea was to neglect nonlinearities in (4.47) responsible forthe failure of the involutivity condition in above Theorem. This approachhappened to be successful in the ball–and–beam system, whenneglect of centrifugal force acting on ball yielded a feedback linearizablesystem. Application of a control scheme designed for the system with centrifugalforce neglected to the original system gave much better resultsthan applying a control scheme based on classical Jacobian linearization.This approach has been further investigated in [Xu and Hauser (1994);Xu and Hauser (1995)] for the purpose of approximate feedback linearizationabout the manifold of constant operating points. However, a generalapproach to deciding which nonlinearities should be neglected to get thebest approximation has not been set forth.All of the above–mentioned work dealt with applying a change of coordinatesand a preliminary feedback so that the resulting system looks like


546 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionlinearizable part plus nonlinear terms of highest possible order around anequilibrium point or an equilibrium manifold. However, in many applicationsone requires a large region of operation for the nonlinearizable system.In such a case, demanding the nonlinear terms to be neglected to be of highestpossible order may, in fact, be quite undesirable. One might prefer thatthe nonlinear terms to be neglected be small in a uniform sense over theregion of operation. In tis section we propose an approach to approximatefeedback linearization that uses a change of coordinates and a preliminaryfeedback to put a system (4.47) in a perturbed Brunovsky form,ż = Az + B u new + P (z) + Q(z) u new ), (4.49)where P (z) and Q(z) vanish at z = 0 and are ‘small’ on M. We get upperbounds on uniform norms of P and Q (depending on some measures ofnoninvolutivity of D) on any compact, contractible M.A different, indirect approach was presented in [Banaszuk and Hauser(1996)]. In this section, the authors present an approach for finding feedbacklinearizable systems that approximate a given SISO nonlinear systemon a given compact region of the state–space. First, they it is shown thatif the system is close to being involutive, then it is also close to being linearizable.Rather than working directly with the characteristic distributionof the system, the authors work with characteristic 1−forms, i.e., with the1−forms annihilating the characteristic distribution. It is shown that homotopyoperators can be used to decompose a given characteristic 1−forminto an exact and an antiexact part. The exact part is used to define achange of coordinates to a normal form that looks like a linearizable partplus nonlinear perturbation terms. The nonlinear terms in this normalform depend continuously on the antiexact part, and they vanish wheneverthe antiexact part does. Thus, the antiexact part of a given characteristic1−form is a measure of nonlinearizability of the system. If the nonlinearterms are small, by neglecting them we get a linearizable system approximatingthe original system. One can design control for the original systemby designing it for the approximating linearizable system and applying itto the original one. We apply this approach for design of locally stabilizingfeedback laws for nonlinear systems that are close to being linearizable.Let us start with approximating characteristic 1−forms by exact formsusing homotopy operators (compare with equation (3.44) above). Namely,on any contractible region M one can define a linear operator H that sat-


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 547isfiesω = d(Hω) + Hdω (4.50)for any form ω. The homotopy identity (4.50) allows to decompose anygiven 1−form into the exact part d(Hω) and an ‘error part’ ɛ = Hdω,which we call the antiexact part of ω. For given ω 0 annihilating D anda scaling factor β we define α β = Hβw 0 and ɛ β = Hdβw 0 . The 1−formɛ β measures how exact ω β = βw 0 is. If it is zero, then ω β is exact andthe system (4.47) is linearizable, and the zero–form α β and its first n − 1Lie derivatives along f are the new coordinates. In the case that ω 0 isnot exactly integrable, i.e., when no exact integrating factor β exists, wechoose β so that dβw 0 is smallest in some sense (because this also makesɛ β small). We call this β an approximate integrating factor for ω 0 . Weuse the zero–form α β and its first n − 1 Lie derivatives along f as the newcoordinates as in the linearizable case. In those new coordinates the system(4.47) is in the formż = Az + Bru + Bp + Eu,where r and p are smooth functions, r ≠ 0 around the origin, and the termE (the obstruction to linearizablity) depends linearly on ɛ β and some of itsderivatives. We choose u = r −1 (u new − p), where u new is a new controlvariable. After this change of coordinates and control variable the systemis of the form (4.49) with Q = r −1 E, P = −r −1 pE. We get estimates onthe uniform norm of Q and P (via estimates on r, p, and E) in terms of theerror 1−form ɛ β , for any fixed β, on any compact, contractible manifoldM. Most important is that Q and P depend in a continuous way on ɛ β andsome of its derivatives, and they vanish whenever ɛ does (see [Banaszukand Hauser (1996)]).4.9.3 Nonlinear ControllabilityLinear ControllabilityRecall that a system is said to be controllable if the set of all states itcan reach from initial state x 0 = x(0) at the fixed time t = T contains aball B around x 0 . Again, a system is called small time locally controllable(STLC) iff the ball B for t ≤ T contains a neighborhood of x 0 . 1313 The above definition of controllability tells us only whether or not something canreach an open neighborhood of its starting point, but does not tell us how to do it. That


548 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn the case of a linear system in the standard state–space form (seesubsection (3.13.4.3) above)ẋ = Ax + Bu, (4.51)where A is the n × n state matrix and B is the m × n input matrix, allcontrollability definitions coincide, i.e.,0 → x(T ), x(0) → 0, x(0) → x(T ),where T is either fixed or free.Rank condition states: System (4.51) is controllable iff the matrixW n = ( B AB ... A n−1 B )has full rank.In the case of nonlinear systems the corresponding result is get using theformalism of Lie brackets, as Lie algebra is to nonlinear systems as matrixalgebra is to linear systems.Nonlinear ControllabilityNonlinear MIMO–systems are generally described by differential equationsof the form (see [Isidori (1989); Nijmeijer and van der Schaft (1990);Goodwine (1998)]):ẋ = f(x) + g i (x) u i , (i = 1, ..., n), (4.52)defined on a smooth n−manifold M, where x ∈ M represents the state ofthe control system, f(x) and g i (x) are vector–fields on M and the u i arecontrol inputs, which belong to a set of admissible controls, u i ∈ U. Thesystem (4.52) is called driftless, or kinematic, or control linear if f(x) isidentically zero; otherwise, it is called a system with drift, and the vector–field f(x) is called the drift term. The flow φ g t (x 0 ) represents the solution ofthe differential equation ẋ = g(x) at time t starting from x 0 . <strong>Geom</strong>etricalway to understand the controllability of the system (4.52) is to understandthe geometry of the vector–fields f(x) and g i (x).Example: Car–Parking Using Lie Brackets In this popular example,the driver has two different transformations at his disposal. He/she canturn the steering wheel, or he/she can drive the car forward or back. Here,we specify the state of a car by four coordinates: the (x, y) coordinatesis the point of the trajectory generation.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 549of the center of the rear axle, the direction θ of the car, and the angle φbetween the front wheels and the direction of the car. L is the constantlength of the car. Therefore, the configuration manifold of the car is 4D,M = (x, y, θ, φ).Using (4.52), the driftless car kinematics can be defined as:ẋ = g 1 (x) u 1 + g 2 (x) u 2 , (4.53)with two vector–fields g 1 , g 2 ∈ X k (M).The infinitesimal transformations will be the vector–fields⎛g 1 (x) ≡ drive = cos θ ∂∂x + sin θ ∂ ∂y + tan φ ∂L ∂θ ≡ ⎜⎝and⎛ ⎞0g 2 (x) ≡ steer = ∂∂φ ≡ ⎜ 0⎟⎝ 0 ⎠ .1cos θsin θ1L tan φ0⎞⎟⎠ ,Now, steer and drive do not commute; otherwise we could do allyour steering at home before driving of on a trip. Therefore, we have a Liebracket1 ∂[g 2 , g 1 ] ≡ [steer, drive] =L cos 2 φ ∂θ ≡ rotate.The operation [g 2 , g 1 ] ≡ rotate ≡ [steer,drive] is the infinitesimal versionof the sequence of transformations: steer, drive, steer back, and driveback, i.e.,{steer, drive, steer −1 , drive −1 }.Now, rotate can get us out of some parking spaces, but not tight ones: wemay not have enough room to rotate out. The usual tight parking spacerestricts the drive transformation, but not steer. A truly tight parkingspace restricts steer as well by putting your front wheels against the curb.Fortunately, there is still another commutator available:[g 1 , [g 2 , g 1 ]] ≡ [drive, [steer, drive]] = [[g 1 , g 2 ], g 1 ] ≡(1[drive, rotate] =L cos 2 sin θ ∂φ ∂x − cos θ ∂ )≡ slide.∂y


550 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe operation [[g 1 , g 2 ], g 1 ] ≡ slide ≡ [drive,rotate] is a displacement atright angles to the car, and can get us out of any parking place. We justneed to remember to steer, drive, steer back, drive some more, steer, driveback, steer back, and drive back:{steer, drive, steer −1 , drive, steer, drive −1 , steer −1 , drive −1 }.We have to reverse steer in the middle of the parking place. This is notintuitive, and no doubt is part of the problem with parallel parking.Thus from only two controls u 1 and u 2 we can form the vector–fieldsdrive ≡ g 1 , steer ≡ g 2 , rotate ≡ [g 2 , g 1 ], and slide ≡ [[g 1 , g 2 ], g 1 ],allowing us to move anywhere in the configuration manifold M. The carkinematics ẋ = g 1 u 1 + g 2 u 2 is thus expanded as:⎛ ⎞⎛ẋ⎜ẏ⎟⎝ ˙θ ⎠ = drive · u 1 + steer · u 2 ≡ ⎜⎝˙φcos θsin θ1L tan φ0⎞ ⎛ ⎞0⎟⎠ · u 1 + ⎜ 0⎟⎝ 0 ⎠ · u 2 .1The parking Theorem says: One can get out of any parking lot that islarger than the car.Fig. 4.4Classical unicycle problem (see text for explanation).The Unicycle Example. Now, consider the unicycle example (seeFigure 4.4). Here we have⎛ ⎞cos x 3⎛ ⎞0⎛sin x 3⎞g 1 = ⎝ sin x 3⎠ , g 2 = ⎝ 0 ⎠ , [g 1 , g 2 ] = ⎝ − cos x 3⎠ .010The unicycle system is full rank and therefore controllable.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 551Controllability ConditionNonlinear controllability is an extension of linear controllability. Thenonlinear MIMO systemẋ = f(x) + g(x) uis controllableif the set of vector–fields {g, [f, g], ..., [f n−1 , g]} is independent.For example, for the kinematic car system of the form (4.53), the nonlinearcontrollability criterion reads: If the Lie bracket tree:g 1 , g 2 , [g 1 , g 2 ], [[g 1 , g 2 ], g 1 ], [[g 1 , g 2 ], g 2 ], [[[g 1 , g 2 ], g 1 ], g 1 ],[[[g 1 , g 2 ], g 1 ], g 2 ], [[[g 1 , g 2 ], g 2 ], g 1 ], [[[g 1 , g 2 ], g 2 ], g 2 ], ...– has full rank then the system is controllable [Isidori (1989); Nijmeijer andvan der Schaft (1990); Goodwine (1998)]. In this case the combined input⎧(1, 0), t ∈ [0, ε]⎪⎨(0, 1), t ∈ [ε, 2ε](u 1 , u 2 ) =(−1, 0), t ∈ [2ε, 3ε]⎪⎩(0, −1), t ∈ [3ε, 4ε]gives the motion x(4ε) = x(0) + ε 2 [g 1 , g 2 ] + O(ε 3 ), with the flow given by(see (3.49) below)() nF [g1,g2]t = lim F√ −g2 F√ −g1 F√ g2F√ g1.n→∞ t/n t/n t/n t/nDistributionsIn control theory, the set of all possible directions in which the systemcan move, or the set of all points the system can reach, is of obviousfundamental importance. <strong>Geom</strong>etrically, this is related to distributions.Recall from subsection 4.7 above that a distribution ∆ ⊂ X k (M) on asmooth nD manifold M is a subbundle of its tangent bundle T M, whichassigns a subspace of the tangent space T x M to each point x ∈ M in asmooth way. The dimension of ∆(x) over R at a point x ∈ M is called therank of ∆ at x.A distribution ∆ is involutive if, for any two vector–fields X, Y ∈ ∆,their Lie bracket [X, Y ] ∈ ∆.A function f ∈ C ∞ (M) is called an integral of ∆ if df(x) ∈ ∆ 0 (x) foreach x ∈ M. An integral manifold of ∆ is a submanifold N of M suchthat T x N ⊂ ∆(x) for each x ∈ N. A distribution ∆ is integrable if, forany x ∈ M, there is a submanifold N ⊂ M, whose dimension is the same


552 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionas the rank of ∆ at x,.containing x such that the tangent bundle, T N, isexactly ∆ restricted to N, i.e., T N = ∆| N . Such a submanifold is calledthe maximal integral manifold through x.It is natural to consider distributions generated by the vector–fieldsappearing in the sequence of flows (3.48). In this case, consider the distributiondefined by∆ = span{f; g 1 ...g m },where the span is taken over the set of smooth real–valued functions. Denoteby ¯∆ the involutive closure of the distribution ∆, which is the closureof ∆ under bracketing. Then, ¯∆ is the smallest subalgebra of X k (M) whichcontains {f; g 1 ...g m }. We will often need to ‘add’ distributions. Since distributionsare, pointwise, vector spaces, define the sum of two distributions,Similarly, define the intersection(∆ 1 + ∆ 2 )(x) = ∆ 1 (x) + ∆ 2 (x).(∆ 1 ∩ ∆ 2 )(x) = ∆ 1 (x) ∩ ∆ 2 (x).More generally, we can arrive at a distribution via a family of vector–fields, which is a subset V ⊂ X k (M). Given a family of vector–fields V, wemay define a distribution on M by∆ V (x) = 〈X(x)|X ∈ V〉 R .Since X k (M) is a Lie algebra, we may ask for the smallest Lie subalgebraof X k (M) which contains a family of vector–fields V. It will be denoted asLie(V), and will be represented by the set of vector–fields on M generatedby repeated Lie brackets of elements in V. Let V (0) = V and then iterativelydefine a sequence of families of vector–fields byV (i+1) = V (i) ∪ {[X, Y ]|X ∈ V (0) = V and Y ∈ V (i) }.Now, every element of Lie(V) is a linear combination of repeated Lie bracketsof the formwhere Z i ∈ V for i = 1, ..., k.[Z k , [Z k−1 , [· · ·, [Z 2 , Z 1 ] · ··]]]


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 553FoliationsRecall that related to integrable distributions are foliations.The Frobenius Theorem asserts that integrability and involutivity areequivalent, at least locally. Thus, associated with an involutive distributionis a partition Φ of M into disjoint connected immersed submanifolds calledleaves. This partition Φ is called a foliation. More precisely, a foliation Fof a smooth manifold M is a collection of disjoint immersed submanifoldsof M whose disjoint union equals M. Each connected submanifold of Fis called a leaf of the foliation. Given an integrable distribution ∆, thecollection of maximal integral manifolds for ∆ defines a foliation on M,denoted by F D .A foliation F of M defines an equivalence relation on M whereby twopoints in M are equivalent if they lie in the same leaf of F. The set ofequivalence classes is denoted M/F and is called the leaf space of F. Afoliation F is said to be simple if M/F inherits a manifold structure so thatthe projection from M to M/F is a surjective submersion.In control theory, foliation leaves are related to the set of points that acontrol system can reach starting from a given initial condition. A foliationΦ of M defines an equivalence relation on M whereby two points in M areequivalent if they lie in the same leaf of Φ. The set of equivalence classesis denoted M/Φ and is called the leaf space of Φ.Philip Hall BasisasGiven a set of vector–fields {g 1 ...g m }, define the length of a Lie productl(g i ) = 1, l([A, B]) = l(A) + l(B), (for i = 1, ..., m),where A and B may be Lie products. A Philip Hall basis is an ordered setof Lie products H = {B i } satisfying:(1) g i ∈ H, (i = 1, ..., m);(2) If l(B i ) < l(B j ), then B i < B j ; and(3) [B i , B j ] ∈ H iff(a) B i , B j ∈ H and B i < B j , and(b) either B j = g k for some k or B j = [B l , B r ] with B l , B r ∈ Hand B l ≤ B i .Essentially, the ordering aspect of the Philip Hall basis vectors accountsfor skew symmetry and Jacobi identity to determine a basis.


554 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction4.9.4 <strong>Geom</strong>etrical Control of Mechanical SystemsMuch of the existing work on control of mechanical systems has reliedon the presence of specific structure. The most common examples of thetypes of structure assumed are symmetry (conservation laws) and constraints.While it may seem counter–intuitive that constraints may helpin control theory, this is sometimes in fact the case. The reason is thatthe constraints give extra forces (forces of constraint) which can be usedto advantage. probably, the most interesting work is done from the Lagrangian(respectively Hamiltonian) perspective where we study systemswhose Lagrangians are ‘kinetic energy minus potential energy’ (resp. ‘kineticenergy plus potential energy’). For these simple mechanical controlsystems, the controllability questions are different than those typically askedin nonlinear control theory. In particular, one is often more interested inwhat happens to configurations rather than states, which are configurationsand velocities (resp. momenta) for these systems (see [Lewis (1995);Lewis and Murray (1997)]).4.9.4.1 Abstract Control SystemIn general, a nonlinear control system Σ can be represented as a triple(Σ, M, f), where M is the system’s state–space manifold with the tangentbundle T M and the general fibre bundle E, and f is a smooth map, suchthat the following bundle diagram commutes [Manikonda (1998)]ψE✲ T M❅❅ ππ M❅❅❘ ✠Mwhere ψ : (x, u) ↦→ (x, f(x, u)), π M is the natural projection of T M onM, the projection π : E → M is a smooth fibre bundle, and the fibers ofE represent the input spaces. If one chooses fibre–respecting coordinates(x, u) for E, then locally this definition reduces to ψ : (x, u) ↦→ (x, ψ(x, u)),i.e.,ẋ = ψ(x, u).The specific form of the map ψ, usually used in nonlinear control, is


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 555ψ : (x, u) ↦→ (x, f(x) + g(x, u)), with g(x, 0) = 0, producing standardnonlinear system equationẋ = f(x) + g(x, u).4.9.4.2 Global Controllability of Linear ControlSystemsConsider a linear biodynamical control system:ẋ(t) = Ax(t) + Bu(t), (4.54)where x ∈ R n , u ∈ R m , A ∈ L(R n , R n ), and B ∈ L(R m , R n ). One shouldthink of t ↦→ u(t) as being a specified input signal, i.e., a function on thecertain time interval, [0, T ]. Now, control theory wants to design the signalto make the state t ↦→ x(t) do what we want. What this is may vary,depending on the situation at hand. For example, one may want to steerfrom an initial state x i to a final state x f , perhaps in an optimal way. Or,one may wish to design u : R n → R m so that some state, perhaps x = 0,is stable for the dynamical system ẋ(t) = Ax + Bu(x), which is called statefeedback (often one asks that u be linear). One could also design u to be afunction of both x and t, etc.One of the basic control questions is controllability, which comes in manyguises. Basically we are asking for ‘reachable’ points. In particular,R(0) = span R {[B|AB|...|A n−1 B]},which is the smallest A−invariant subspace containing Im(B), denotes theset of points reachable from 0 ∈ R n . For the linear system (4.54), the basiccontrollability questions have definite answers. We want to do somethingsimilar for a class of simple mechanical systems [Lewis (1995); Lewis andMurray (1997)].4.9.4.3 Local Controllability of Affine Control SystemsThe nonlinear control system that we most often consider in humanoidrobotics (see next section) has state–space M, a smooth n−manifold, andis affine in the controls. Thus it has the form (see [Lewis (1995); Lewis andMurray (1997)])ẋ = f(x) + u a g a (x), (x ∈ M), (4.55)


556 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere f, g 1 , ..., g m are vector–fields on M. The drift vector–field f = f(x)describes how the system would evolve in the absence of any inputs. Eachof the control vector–fields g 1 , ..., g m specifies a direction in which onecan supply actuation. To fully specify the control system properly, oneshould also specify the type of control action to be considered. Herewe consider our controls to be taken from the set: U = {u : R →R m | u is piecewise constant}. This class of controls is sufficient to deal withall analytic control systems. More generally, one may wish to consider measurablefunctions which take their values in a subset of R m .Given an affine control system (4.55), it is possible to define a family ofvector–fields on M by: V Σ = {f + u a g a | u ∈ R m }.A solution of the system (4.55) is a pair (γ, u), where γ : [0, T ] → M isa piecewise smooth curve on M and u ∈ U such that˙γ(t) = f(γ(t)) + u a (t) g a (γ(t)), for each t ∈ [0, T ].The reachable set from x 0 in time T isR(x 0 , T ) = {x|∃γ : [0, T ] → Mandu : [0, T ] → R m satisfying (4.55)with γ(0) = x 0 and γ(T ) = x}.Note that since the system has drift f, when we reach the point γ(T ) wewill not remain there if this is not an equilibrium point for f. Also, wehave, R(x 0 , ≤ T ) = ∪ 0 0. We say thatequation (4.55) represents a locally accessible system at x 0 if R(x 0 , ≤ T )contains an open subset of M for each V and for each T sufficiently small.Furthermore, we say that the system (4.55) is small–time local controllability(STLC, see [Sussmann (1983); Sussmann (1987)]), if it is locallyaccessible and if x 0 is in the interior of R(x 0 , ≤ T ) for each V and for eachT sufficiently small.4.9.4.4 Lagrangian Control SystemsSimple Mechanical Control SystemsAs a motivation/prototype of a simple mechanical control system, considera simple robotic leg (see Figure 4.5), in which inputs are: (1) aninternal torque F 1 moving the leg relative to the body and (2) a force F 2extending the leg. This system is ‘controllable’ in the sense that, starting


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 557from rest, one can reach any configuration from a given initial configuration.However, as a traditional control system, it is not controllablebecause of conservation of angular momentum. If one asks for the states(i.e., configurations and velocities) reachable from configurations with zeroinitial velocity, one finds that not all states are reachable. This is a consequenceof the fact that angular momentum is conserved, even with inputs.Thus if one starts with zero momentum, the momentum will remain zero(this is what enables one to treat the system as nonholonomic). Nevertheless,all configurations are accessible. This suggests that the question ofcontrollability is different depending on whether one is interested in configurationsor states. We will be mainly interested in reachable configurations.Considering the system with just one of the two possible inputforces is also interesting. In the case where we are just allowed to useF 2 , the possible motions are quite simple; one can only move the ball onthe leg back and forth. With just the force F 1 available, things are abit more complicated. But, for example, one can still say that no matterhow you apply the force, the ball with never move ‘inwards’ [Lewis (1995);Lewis and Murray (1997)].Fig. 4.5A simple robotic leg (see text for explanation).In general, simple mechanical control systems are characterized by:• An nD configuration manifold M;• A Riemannian metric g on M;• A potential energy function V on M; and• m linearly independent 1−forms, F 1 , ..., F m on M (input forces; e.g.,in the case of the simple robotic leg, F 1 = dθ − dψ and F 2 = dr).When we say these systems are not amenable to liberalization–basedmethods, we mean that their liberalizations at zero velocity are not controllable,and that they are not feedback linearizable. This makes simple


558 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmechanical control systems a non–trivial class of nonlinear control systems,especially from the point of view of control design.As a basic example to start with, consider a planar rigid body (see Figure4.6), with coordinates (x, y, θ). Inputs are (1) force pointing towards centerof mass, F 1 = cos θdx + sin θdy, (2) force orthogonal to line to center ofmass, F 2 = − sin θdx + cos θdy − hdθ, and (3) torque at center of massF 3 = dθ. The planar rigid body, although seemingly quite simple, can beactually interesting. Clearly, if one uses all three inputs, the system is fullyactuated, and so boring for investigating reachable configurations. But ifone takes various combinations of one or two inputs, one gets a pretty nicesampling of what can happen for these systems. For example, all possiblecombinations of two inputs allow one to reach all configurations. UsingF 1 or F 3 alone give simple, 1D reachable sets, similar to using F 2 for therobotic leg (as we are always starting with zero initial velocity). However,if one is allowed to only use F 2 , then it is not quite clear what to expect,at least just on the basis of intuition.Fig. 4.6Coordinate systems of a planar rigid body.It turns out that our simplifying assumptions, i.e., zero initial velocityand restriction of our interest to configurations (i.e., as all problem data ison M, we expect answers to be describable using data on M), makes ourtask much simpler. In fact, the computations without these assumptionshave been attempted, but have yet to yield coherent answers.Now, we are interested in how do the input 1−forms F 1 , ..., F m interactwith the unforced mechanics of the system as described by the kineticenergy Riemannian metric. That is, what is the analogue of linear system’s‘the smallest A−invariant subspace containing Im(B)’ – for simplemechanical control systems?


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 559Motion and Controllability in Affine ConnectionsIf we start with the local Riemannian metric form g ↦−→ g ij (q) dq i dq j ,then we have a kinetic energy Lagrangian L(q, v) = g ij (q) ˙q i ˙q j , and consequentlythe Euler–Lagrangian equations read(ddt ∂ ˙q iL−∂ q iL ≡ g ij ¨q j + ∂ q kg ij − 1 )2 ∂ q ig jk ˙q j ˙q k = u a Fi a , (i = 1, ..., n).Now multiply this by g li and take the symmetric part of the coefficientof ˙q j ˙q k to get ¨q l + Γ l jk ˙qj ˙q k = u a Ya, l (l = 1, ..., n,), where Γ i jkare theChristoffel symbols (3.143) for the Levi–Civita connection ∇ (see (3.10.1.1)above). So, the equations of motion an be rewritten∇ ˙γ(t) ˙γ(t) = u a (t) Y a (γ(t)) ,(a = 1, ..., m),where Y a = (F a ) ♯ , while ♯ : T ∗ M → T M is the ‘sharp’–isomorphism associatedwith the Riemannian metric g.Now, there is nothing to be gained by using a Levi–Civita connection,or by assuming that the vector–fields come from 1−forms. At this point,perhaps the generalization to an arbitrary affine connection seems like asenseless abstraction. However, as we shall see, this abstraction allows usto include another large class of mechanical control systems. So we willstudy the control system∇ ˙γ(t) ˙γ(t) = u a (t) Y a (γ(t)) [+Y 0 (γ(t))] , (4.56)with ∇ a general affine connection on M, and Y 1 ..., Y m linearly independentvector–fields on M. The ‘optional’ term Y 0 = Y 0 (γ(t)) in (4.56) indicateshow potential energy may be added. In this case Y 0 = − grad V (however,one looses nothing by considering a general vector–field instead of agradient) [Lewis (1998)].A solution to (4.56) is a pair (γ, u) satisfying (4.56) where γ : [0, T ] → Mis a curve and u : [0; T ] → R m is bounded and measurable.Let U be a neighborhood of q 0 ∈ M and denote by R U M (q 0, T ) thosepoints in M for which there exists a solution (γ, u) with the following properties:(1) γ(t) ∈ U for t ∈ [0, T ];(2) ˙γ(0) = 0 q ; and(3) γ(T ) ∈ T q M.


560 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAlso R U M (q 0, ≤ T ) = ∪ 0≤t≤T R U M (q 0, t). Now, regarding the local controllability,we are only interested in points which can be reached withouttaking ‘large excursions’. Control problems which are local in this way havethe advantage that they can be characterized by Lie brackets. So, we wantto describe our reachable set R U M (q, ≤ T ) for the simple mechanical controlsystem (4.56). The system (4.56) is locally configuration accessible (LCA)at q if there exists T > 0 so that R U M (q, ≤ t) contains a non–empty opensubset of M for each neighborhood U of q and each t ∈]0, T ]. Also, (4.56)is locally configuration controllable (LCC) at q if there exists T > 0 so thatR U M (q, ≤ t) contains a neighborhood of q for each neighborhood U of qand each t ∈]0, T ]. Although sound very similar, the notions of local configurationaccessibility and local configuration controllability are genuinelydifferent (see Figure 4.7). Indeed, one need only look at the example ofthe robotic leg with the F 1 input. In this example one may show that thesystem is LCA, but is not LCC [Lewis (1998)].Fig. 4.7 <strong>Diff</strong>erence between the notions of local configuration accessibility (a), and localconfiguration controllability (b).Local Configuration AccessibilityThe accessibility problem is solved by looking at Lie brackets. For thiswe need to recall the definition of the vertical lift [Lewis (1998)]:verlift(Y (v q )) = d dt∣ (v q + tY (q)),t=0in local coordinates, if Y = Y i ∂ q i, then verlift(Y ) = Y i ∂ v i. Now we canrewrite (4.56) in the first–order form:˙v = Z(v) + u a verlift(Y a (v)),


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 561where Z is the geodesic spray for ∇.We evaluate all brackets at 0 q (recall that T 0q T M ≃ T q M ⊕T q M). Here,the first component we think of as being the ‘horizontal’ bit which is tangentto the zero section in T M, and we think of the second component as beingthe ‘vertical’ bit which is the tangent space to the fibre of τ M : T M → M.To get an answer to the local configuration accessibility problem, weemploy standard nonlinear control techniques involving Lie brackets. Doingso gives us our first look at the symmetric product, 〈X : Y 〉 = ∇ X Y +∇ Y X.Our sample brackets suggest that perhaps the only things which appear inthe bracket computations are symmetric products and Lie brackets of theinput vector–fields Y 1 , ..., Y m .Here are some sample brackets:(i) [Z, verlift(Y a )](0 q ) = (−Y a (q), 0);(ii) [verlift(Y a ), [Z, verlift(Y b )]](0 q ) = (0, 〈Y a : Y b 〉 (q));(iii) [[Z, verlift(Y a )], [Z, verlift(Y b )]](0 q ) = ([Y a , Y b ](q), 0).Now, let C ver be the closure of span{Y 1 , ..., Y m } under symmetric product.Also, let C hor be the closure of C ver under Lie bracket. So, we assumeC ver and C hor to be distributions (i.e., of constant rank) on M. The closureof span{Z, verlift(Y 1 ), ..., verlift(Y m )} under Lie bracket, when evaluated at0 q , is then the distributionq ↦→ C hor (q) ⊕ C ver (q) ⊂ T q M ⊕ T q M.Proving that the involutive closure of span{Z, verlift(Y 1 ), ..., verlift(Y m )} isequal at 0 q to C hor (q) ⊕ C ver (q) is a matter of computing brackets, samplesof which are given above, and seeing the patterns to suggest an inductiveproof. The brackets for these systems are very structured. For example,the brackets of input vector–fields are identically zero. Many other bracketsvanish identically, and many more vanish when evaluated at 0 q .C hor is integrable: let Λ q be the maximal integral manifold throughq ∈ M. Then, R U M (q, ≤ T ) is contained in Λ q, and R U M (q, ≤ T ) containsa non–empty open subset of Λ q . In particular, if rank(C hor ) = n then(4.56) is LCA [Lewis (1995); Lewis and Murray (1997)]. This Theoremgives a ‘computable’ description of the reachable sets (in the sense that wecan calculate Λ q by solving some over–determined nonlinear PDE’s). But itdoes not give the kind of insight that we had with the ‘smallest A−invariantsubspace containing Im(B)’.Recall that a submanifold N of M is totally geodesic if every geodesic


562 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwith initial velocity tangent to N remains on N. This can be weakened todistributions: a distribution D on M is geodesically invariant if for everygeodesic γ : [0, T ] → M, ˙γ(0) ∈ D γ(0) implies ˙γ(t) ∈ D γ(t) for t ∈]0, T ].D is geodesically invariant i it is closed under symmetric product [Lewis(1998)]. This Theorem says that the symmetric product plays for geodesicallyinvariant distributions the same role the Lie bracket plays for integrabledistributions. This result was key in providing the geometricaldescription of the reachable configurations.An integrable distribution is geodesically generated distribution if it isthe involutive closure of a geodesically invariant distribution. This basicallymeans that one may reach all points on a leaf with geodesics lying in somesubdistribution. The picture one should have in mind with the geometry ofthe reachable sets is a foliation of M by geodesically generated (immersed)submanifolds onto which the control system restricts if the initial velocity iszero. The idea is that when we start with zero velocity we remain on leavesof the foliation defined by C hor [Lewis and Murray (1997); Lewis (2000a)].Note that for cases when the affine connection possesses no geodesicallyinvariant distributions, the system (4.56) is automatically LCA. This istrue, for example, of S 2 with the affine connection associated with its roundmetric.Clearly C ver is the smallest geodesically invariant distribution containingspan{Y 1 , ..., Y m }. Also, C hor is geodesically generated by spanspan{Y 1 , ..., Y m }. Thus R U M is contained in, and contains a non–emptyopen subset of, the distribution geodesically generated by span{Y 1 , ..., Y m }.Note that the pretty decomposition we have for systems with no potentialenergy does not exist at this point for systems with potential energy.Local Configuration ControllabilityThe problem of configuration controllability is harder than the one ofconfiguration accessibility. Following [Lewis and Murray (1999); Lewis(2000a)], we will call a symmetric product in {Y 1 , ..., Y m } bad if it containsan even number of each of the input vector–fields. Otherwise we willcall it good. The degree is the total number of vector–fields. For example,〈〈Y a : Y b 〉 : 〈Y a : Y b 〉〉 is bad and of degree 4, and 〈Y a : 〈Y b : Y b 〉〉 is good andof degree 3. If each bad symmetric product at q is a linear combination ofgood symmetric products of lower degree, then (4.56) is LCC at q.Now, the single–input case can be solved completely: The system (4.56)with m = 1 is LCC iff dim(M) = 1 [Lewis and Murray (1999)].


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 563Systems With Nonholonomic ConstraintsLet us now add to the data a distribution D defining nonholonomic constraints.One of the interesting things about this affine connection approachis that we can easily integrate into our framework systems with nonholonomicconstraints. As a simple example, consider a rolling disk (see Figure4.8), with two inputs: (1) a ‘rolling’ torque, F 1 = dθ and (2) a ‘spinning’torque, F 2 = dφ. It can be analyzed as a nonholonomic system (see [Lewis(1999); Lewis (2000a)]).Fig. 4.8Rolling disk problem (see text for explanation).The control equations for a simple mechanical control system with constraintsare:∇ ˙γ(t) ˙γ(t) = λ(t) + u a (t) Y a (γ(t)) [− grad V (γ(t))] , ˙γ(t) ∈ D γ(t) ,where λ(t) ∈ Dγ(t) ⊥ are Lagrangian multipliers.Examples1. Recall that for the simple robotic leg (Figure 4.5) above, Y 1 wasinternal torque and Y 2 was extension force. Now, in the following threecases:(i) both inputs active – this system is LCA and LCC (satisfies sufficientcondition);(ii) Y 1 only, it is LCA but not LCC; and(iii) Y 2 only, it is not LCA.In theses three cases, C hor is generated by the following linearly independentvector–fields:(i) both inputs: {Y 1 , Y 2 , [Y 1 , Y 2 ]};(ii) Y 1 only: {Y 1 , 〈Y 1 : Y 1 〉 , 〈Y 1 : 〈Y 1 : Y 1 〉〉}; and(iii) Y 2 only: 〈Y 2 〉.


564 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall that with both inputs the system was not accessible in T M as a consequenceof conservation of angular momentum. With the input Y 2 only,the control system behaves very simply when given zero initial velocity.The ball on the end of the leg just gets moved back and forth. This reflectsthe foliation of M by the maximal integral manifolds of C hor , which areevidently 1D in this case. With the Y 1 input, recall that the ball will alwaysgo ‘outwards’ no matter what one does with the input. Thus the systemis not LCC. But apparently (since rank(C hor ) = dim(M)) one can reach anon–empty open subset of M. The behavior exhibited in this case is typicalof what one can expect for single–input systems with no potential energy.2. For the planar rigid body (Figure 4.6) above, we have the followingfive cases:(i) Y 1 and Y 2 active, this system is LCA and LCC (satisfies sufficient condition);(ii) Y 1 and Y 3 , it is LCA and LCC (satisfies sufficient condition);(iii) Y 1 only or Y 3 only, not LCA;(iv) Y 2 only, LCA but not LCC; and(v) Y 2 and Y 3 : LCA and LCC (fails sufficient condition).Now, with the inputs Y 1 or Y 3 alone, the motion of the system is simple.In the first case the body moves along the line connecting the point ofapplication of the force and the center of mass, and in the other case thebody simply rotates. The equations in (x, y, θ) coordinates areẍ = cos θm u1 − sin θm u2 ,ÿ = sin θm u1 + cos θ1 (m u2 , ¨θ = u 3 − hu 2) ,Jwhich illustrates that the θ−equation decouples when only Y 3 is applied.We make a change of coordinates for the case where we have only Y 1 :(ξ, η, ψ) = (x cos θ + y sin θ, −x sin θ + y cos θ, θ). In these coordinates wehave¨ξ − 2 ˙η ˙ψ − ξ ˙ψ2 =1m u1 , ¨η + 2˙ξ ˙ψ − η ˙ψ 2 = 0, ˙ψ = 0,which illustrates the decoupling of the ξ−equation in this case.C hor has the following generators:(i) Y 1 and Y 2 : {Y 1 , Y 2 , [Y 1 , Y 2 ]};(ii) Y 1 and Y 3 : {Y 1 , Y 3 , [Y 1 , Y 3 ]};(iii) Y 1 only or Y 3 only: {Y 1 } or {Y 3 };(iv) Y 2 only: {Y 2 , 〈Y 2 : Y 2 〉 , 〈Y 2 : 〈Y 2 : Y 2 〉〉};(v) Y 2 and Y 3 {Y 2 , Y 3 , [Y 2 , Y 3 ]}.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 5653. Recall that for the rolling disk (Figure 4.8) above, Y 1 was ‘rolling’input and Y 2 was ‘spinning’ input. Now, in the following three cases:(i) Y 1 and Y 2 active, this system is LCA and LCC (satisfies sufficient condition);(ii) Y 1 only: not LCA; and(iii) Y 2 only: not LCA.In theses three cases, C hor has generators:(i) Y 1 and Y 2 : {Y 1 , Y 2 , [Y 1 , Y 2 ], [Y 2 , [Y 1 , Y 2 ]]};(ii) Y 1 only: {Y 1 }; and(iii) Y 2 only: {Y 2 }.The rolling disk passes the good/bad symmetric product test. Another wayto show that it is LCC is to show that the inputs allow one to follow anycurve which is admitted by the constraints. Local configuration controllabilitythen follows as the constraint distribution for the rolling disk has aninvolutive closure of maximal rank [Lewis (1999)].Categorical Structure of Control Affine SystemsControl affine systems make a category CAS (see [Elkin (1999)]). Thecategory CAS has the following data:• An object in CAS is a pair ∑ = (M, F = {f 0 , f 1 , ..., f m }) where F is afamily of vector–fieldsẋ(t) = f 0 (x(t)) + u a (t)f a (x(t))on the manifold M.• A morphism sending ∑ = (M, F = {f 0 , f 1 , ..., f m }) to ∑′ = (M ′ , F ′ ={f ′ 0, f ′ 1, ..., f ′ m ′}) is a triple (ψ, λ 0, Λ) where ψ : M → M ′ , λ 0 : M →R m′ , and Λ : M → L(R m , R m′ ) are smooth maps satisfying:(1) T x ψ(f a (x)) = Λ α a (x)f ′ α(ψ(x)), a ∈ {1, ..., m}, and(2) T x ψ(f 0 (x)) = f ′ 0(ψ(x)) + λ α 0 f ′ α(ψ(x)).This corresponds to a change of state–input by(x, u) ↦−→ (ψ(x), λ 0 (x) + Λ(x)u).Elkin [Elkin (1999)] discusses equivalence, inclusion, and factorizationin the category CAS. Using categorical language, he considers local equivalencefor various classes of nonlinear control systems, including single–input systems, systems with involutive input distributions, and systemswith three states and two inputs.


566 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction4.9.4.5 Lie–Adaptive ControlIn this subsection we develop the concept of machine learning in the frameworkof Lie derivative control formalism (see (4.9.2) above). Consider annD, SISO system in the standard affine form (4.44), rewritten here forconvenience:ẋ(t) = f(x) + g(x) u(t), y(t) = h(x), (4.57)As already stated, the feedback control law for the system (4.57) can bedefined using Lie derivatives L f h and L g h of the system’s output h alongthe vector–fields f and g.If the SISO system (4.57) is a relatively simple (quasilinear) system withrelative degree r = 1 it can be rewritten in a quasilinear formẋ(t) = γ i (t) f i (x) + d j (t) g j (x) u(t), (4.58)where γ i (i = 1, ..., n) and d j (j = 1, ..., m) are system’s parameters, whilef i and g j are smooth vector–fields.In this case the feedback control law for tracking the reference signaly R = y R (t) is defined as (see [Isidori (1989); Nijmeijer and van der Schaft(1990)])u = −L f h + ẏ R + α (y R − y), (4.59)L g hwhere α denotes the feedback gain.Obviously, the problem of reference signal tracking is relatively simpleand straightforward if we know all the system’s parameters γ i (t) and d j (t)of (4.58). The question is can we apply a similar control law if the systemparameters are unknown?Now we have much harder problem of adaptive signal tracking. However,it appears that the feedback control law can be actually cast in a similarform (see [Sastri and Isidori (1989); Gómez (1994)]):û = − ̂L f h + ẏ R + α (y R − y), (4.60)̂L g hwhere Lie derivatives L f h and L g h of (4.59) have been replaced by theirestimates ̂L f h and ̂L g h, defined respectively aŝL f h = ̂γ i (t) L fi h, ̂Lg h = ̂d j (t) L gi h,in which ̂γ i (t) and ̂d j (t) are the estimates for γ i (t) and d j (t).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 567Therefore, we have the straightforward control law even in the uncertaincase, provided that we are able to estimate the unknown system parameters.Probably the best known parameter update law is based on the so–calledLyapunov criterion (see [Sastri and Isidori (1989)]) and given by˙ψ = −γ ɛ W, (4.61)where ψ = {γ i − ̂γ i , d j − ̂d j } is the parameter estimation error, ɛ = y − y Ris the output error, and γ is a positive constant, while the matrix W isdefined as:W = [ W1 T W2T ] T, with⎡ ⎤⎡ ⎤W 1 =⎢⎣L f1 h.L fn h⎥⎦ , W 2 =⎢⎣L g1 h⎥. ⎦ · − ̂L f h + ẏ R + α (y R − y).̂L g hL gm hThe proposed adaptive control formalism (4.60–4.61) can be efficientlyapplied wherever we have a problem of tracking a given signal with anoutput of a SISO–system (4.57–4.58) with unknown parameters.4.9.5 Hamiltonian Optimal Control and MaximumPrinciple4.9.5.1 Hamiltonian Control SystemsHamiltonian control system on a symplectic manifold (P, ω) is defined as anaffine control system whose drift and control vector–fields are Hamiltonian.It can be written asṗ = X H (p) + u a X a (p),where the vector–fields X a are assumed to be Hamiltonian with HamiltonianH a for a = 1, ..., m. Examples of systems which are (at least locally)Hamiltonian control systems are those which evolve on the symplectic manifoldT ∗ M and where the control Hamiltonians are simply coordinate functionson M.Alternatively, Hamiltonian control systems can be defined on Poissonmanifolds. However, for the purposes of this subsection, it will be morenatural to work within the Poisson context. Recall that given a smoothHamiltonian function h : M → R, on the Poisson manifold M, the Poissonbracket {, } : C ∞ (M) × C ∞ (M) → C ∞ (M) (such that {f, g} = −{g, f},


568 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction{f, {g, h}}}+{g, {h, f}}+{h, {f, g}} = 0, and {fg, h} = {f, h}g +f{g, h})allows us to get a Hamiltonian vector–field X h with Hamiltonian h throughthe equalityL Xh f = {f, h}, for all f ∈ C ∞ (M),where L Xh f is the Lie derivative of f along X h . Note that the vector–fieldX h is well defined since the Poisson bracket verifies the Leibniz rule andtherefore defines a derivation on C ∞ (M) (see [Marsden and Ratiu (1999)]).Furthermore C ∞ (M) equipped with a Poisson bracket is a Lie algebra,called a Poisson algebra. Also, we say that the Poisson structure on M isnondegenerate if the {, }−associated map B # : T ∗ M → T M defined bydg(B # (x)(df)) = B(x)(df, dg),(where df denotes the exterior derivative of f) is an isomorphism for everyx ∈ M.An affine Hamiltonian control system Σ = (U, M, h) consists of asmooth manifold U (the input space), a Poisson manifold M with nondegeneratePoisson bracket (the state–space), and a smooth function H :M × U → R (the controlled Hamiltonian). Furthermore, H is locally ofthe form H = h 0 + h i u i (i = 1, ..., n), with h i locally defined smooth realvalued maps and u i local coordinates for U [Tabuada and Pappas (2001)].Using the controlled Hamiltonian and the Poisson structure on M wecan recover the familiar system map F : M × U → T M, locally given byF = X h0 + X hi u i ,and defines an affine distribution on M given byD M (x) = X h0 (x) + span{X h1 (x), X h2 (x), ..., X hn (x)}.This distribution captures all the possible directions of motion availableat a certain point x, and therefore describes a control system, up to aparametrization by control inputs. This affine distribution will is our mainobject of interest here, and we will assume that the rank of D M does notchange with x. Furthermore, we denote an affine distribution D M by X+∆,where X is a vector–field and ∆ a distribution. When this affine distributionis defined by a Hamiltonian control system we have X = X h0 and ∆ =span{X h1 (x), X h2 (x), ..., X hn (x)}. A similar reasoning is possible at thelevel of Hamiltonians. Locally, we can define the following affine space of


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 569smooth mapswhich defines D M by the equalityH M = h 0 + span R {h 1 , h 1 , ..., h n },D M = B # (dH M ),where we used the notation dH M to denote the set ∪ h∈HM dh. We also usethe notation H M = h 0 + H ∆ for an affine space of smooth maps where h 0is a smooth map and H ∆ a linear space of smooth maps.Having defined Hamiltonian control systems we turn to their trajectoriesor solutions: A smooth curve γ : I → M, I ⊆ R + 0 is called a trajectory ofcontrol system Σ = (U, M, H), iff there exists a curve γ U : I → U satisfying[Tabuada and Pappas (2001)]ẏ(t) = F (γ(t), γ U (t)), for every t ∈ I.Now, given a Hamiltonian control system and a desired property, an abstractedHamiltonian system is a reduced system that preserves the propertyof interest while ignoring modelling detail (see [Tabuada and Pappas(2001)]). Property preserving abstractions of control systems are importantfor reducing the complexity of their analysis or design. From an analysisperspective, given a large scale control system and a property to be verified,one extracts a smaller abstracted system with equivalent properties.Checking the property on the abstraction is then equivalent to checking theproperty on the original system. From a design perspective, rather thandesigning a controller for the original large scale system, one designs a controllerfor the smaller abstracted system, and then refines the design to theoriginal system while incorporating modelling detail.This approach critically depends on whether we are able to constructhierarchies of abstractions as well as characterize conditions under whichvarious properties of interest propagate from the original to the abstractedsystem and vice versa. In [Pappas et al. (2000)], hierarchical abstractionsof linear control systems were extracted using computationally efficient constructions,and conditions under which controllability of the abstracted systemimplied controllability of the original system were obtained. This ledto extremely efficient hierarchical controllability algorithms. In the samespirit, abstractions of nonlinear control affine systems were considered in[Pappas and Simic (2002)], and the canonical construction for linear systemswas generalized to nonlinear control affine systems.


570 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn [Tabuada and Pappas (2001)], abstractions of Hamiltonian controlsystems are considered, which are control systems completely specified bycontrolled Hamiltonians. This additional structure allows to simplify theabstraction process by working with functions instead of vector–fields ordistributions as is the case for general nonlinear systems [Pappas and Simic(2002)]. This is possible since the controlled Hamiltonian contains all therelevant information that must be captured by the abstracted system. Onthe other hand, to be able to relate the dynamics induced by the controlledHamiltonians, we need to restrict the class of abstracting maps to those thatpreserve the Hamiltonian structure. More precisely, given a Hamiltoniancontrol system on a Poisson manifold M, and a (quotient) Poisson map φ :M → N, one presents a canonical construction that extracts an abstractedHamiltonian control system on N. One then characterizes abstracting mapsfor which the original and abstracted system are equivalent from a localaccessibility point of view [Tabuada and Pappas (2001)].4.9.5.2 Pontryagin’s Maximum PrincipleRecall that the Pontryagin Maximum Principle (PMP, see [Pontryagin et al.(1986); Iyanaga and Kawada (1980)]) applies to a general optimizationproblem called a Bolza problem. To apply PMP to optimal control, weneed to define Hamiltonian function:H(ψ, x, u) = (ψ, f(x, u)) = ψ i f i (x, u), (i = 1, ..., n). (4.62)Then in order for a control u(t) and a trajectory x(t) to be optimal, it isnecessary that there exist a nonzero absolutely continuous vector functionψ(t) = (ψ 0 (t), ψ 1 (t), ..., ψ n (t)) corresponding to the functions u(t) and x(t)such that:(1) The function H(ψ(t), x(t), u(t)) attains its maximum at the point u =u(t) almost everywhere in the interval t 0 ≤ t ≤ T ,H(ψ(t), x(t), u(t)) = max H(ψ(t), x(t), u(t)).u∈U(2) At the terminal time T , the following relations are satisfied:ψ 0 (T ) ≤ 0 and H(ψ(T ), x(T ), u(T )) = 0.PMP states the following algorithm: To maximize the set of steeringfunctions γ i x i (t) (with n constants γ i ) for controlling the changes in the


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 571state variablesẋ i (t) = f i (x i , u k ), (i = 0, 1, ..., n, k = 1, ..., m),we maximize at each instant the Hamiltonian function (4.62), where˙ψ i = −ψ j∂f j4.9.5.3 Affine Control Systems∂x i and ψ i (T ) = γ i .Now, let us look at PMP as applied to the affine control system (see [Lewis(2000b)])ẏ(t) = f 0 (γ(t)) + u a (t) f a (γ(t)),with γ(t) ∈ M, u taking values in U ⊂ R m , and objective function L(x, u).We need to have the control Hamiltonian on U × T ∗ M:H(α x , u) = α x (f 0 (x))} {{ }H 1+ α x (u a f a (x))} {{ }− L(x, u)} {{ }H 2H 3.One of several consequences of the PMP is that if (u, γ) is a minimizerthen there exists a 1−form field λ along γ with the property that t ↦→ λ(t) isan integral curve for the time–dependent Hamiltonian (α x , u) ↦→ H(α x , u).The Hamiltonian H(α x , u) is a sum of three terms, and so too will be theHamiltonian vector–field.Let us look at the first term, that with (old) Hamiltonian H 1 =α x (f 0 (x)). In local coordinates X H1 is written asẋ i = f i 0(x),ṗ i = − ∂f j 0 (x)∂x i p j . (4.63)X H1 is the cotangent lift of f 0 and, following [Lewis (2000b)], we denote itf T ∗0 . So we want to understand f T ∗0 on T M with f 0 = Z.Let f 0 be a vector–field on a general manifold N with f0T its tangent liftdefined byf0 T (v x ) = d dt∣ T x F t (v x ),t=0where F t denotes the flow of f 0 . Therefore, f T 0 is the ‘linearization’ of f 0and in local coordinates it is given by (compare with (4.63))ẋ i = f i 0(x),˙v i = − ∂f i 0(x)∂x j v j .


572 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe flow of f0T measures how the integral curves of f 0 change as we changethe initial condition in the direction of v x .Now, perhaps we can understand Z T on T M with f 0 = Z in the discussionof tangent lift. Let γ(t) be a geodesic. By varying the initial conditionfor the geodesic we generate an ‘infinitesimal variation’ which satisfies theextended Jacobi equation,∇ 2 ẏ(t)ξ(t) + R(ξ(t), ẏ(t)) ẏ(t) + ∇ẏ(t) (T (ξ(t), ẏ(t))) = 0. (4.64)To make the ‘connection’ between Z T and the Jacobi equation, we performconstructions on the tangent bundle using the spray Z. ∇ comes from alinear connection on M which induces an Ehresmann connection on τ M :T M → M. Thus we may write T vq T M ≃ T q M ⊕ T q M. Now, if I M :T T M → T T M is the canonical involution then I ∗ M ZT is a spray. We useI ∗ M ZT to induce an Ehresmann connection on τ T M : T T M → T M. Thus,T Xvq T T M ≃ T vq T M ⊕ T vq T M ≃T q M ⊕ T q M} {{ }⊕ T q M ⊕ T q M} {{ }.geodesic equations variation equationsOne represents Z T in this splitting and determines that the Jacobi equationsits ‘inside’ one of the four components. Now one applies similar constructionsto T ∗ T M and Z T ∗to derive a 1−form version of the Jacobi equation(4.64), the so–called adjoint Jacobi equation [Lewis (2000b)]:∇ 2 ẏ(t)λ(t) + R ∗ (λ(t), ẏ(t)) ẏ(t) − T ∗ ( ∇ẏ(t) λ(t), ẏ(t) ) = 0, (4.65)where we have used 〈R ∗ (α, u)v; ω〉 = 〈α; R(ω, u)v〉, and 〈T ∗ (α, u); ω〉 =〈α; T (ω, u)〉 .The adjoint Jacobi equation forms the backbone of a general statementof the PMP for affine connection control systems. When objective functionis the Lagrangian L(u, v q ) = 1 2 g(v q, v q ), when ∇ is the Levi–Civita connectionfor the Riemannian metric g, and when the system is fully actuated,then we recover the equation of [Noakes et al. (1989)]∇ 3 ẏ(t) ẏ(t) + R ( ∇ẏ(t) ẏ(t), ẏ(t) ) = 0.Therefore, the adjoint Jacobi equation (4.65) captures the interestingpart of the Hamiltonian vector–field Z T ∗ , which comes from the PMP, interms of affine geometry, i.e., from Z T ∗follows∇ẏ(t) ẏ(t) = 0, ∇ 2 ẏ(t)λ(t) + R ∗ (λ(t), ẏ(t)) ẏ(t) − T ∗ ( ∇ẏ(t) λ(t), ẏ(t) ) = 0.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 573The geometry of Z on TM gives a way of globally pulling out the adjointJacobi equation from the PMP in an intrinsic manner, which is not generallypossible in the PMP [Lewis (2000b)].4.9.6 Brain–Like Control Functor in BiodynamicsIn this final section we propose our most recent model [<strong>Ivancevic</strong> and Beagley(2005)] of the complete biodynamical brain–like control functor. Thisis a neurodynamical reflection on our covariant force law, F i = mg ij a j ,and its associated covariant force functor F ∗ : T T ∗ M → T T M (see section3.13.4.1 above).Recall that traditional hierarchical robot control (see, e.g., [Vukobratovicand Stokic (1982); Vukobratovic et al. (1990)]) consists of three levels:the executive control–level (at the bottom) performs tracking of nominaltrajectories in internal–joint coordinates, the strategic control–level (atthe top) performs ‘planning’ of trajectories of an end–effector in external–Cartesian coordinates, and the tactical control–level (in the middle) connectsother two levels by means of inverse kinematics.The modern version of the hierarchical robot control includes decision–making done by the neural (or, neuro–fuzzy) classifier to adapt the (manipulator)control to dynamically changing environment.On the other hand, the so–called ‘intelligent’ approach to robot controltypically represents a form of function approximation, which is itself basedon some combination of neuro–fuzzy–genetic computations. Many specialissues and workshops focusing on physiological models for robot controlreflect the increased attention for the development of cerebellar models [vander Smagt (1999); Schaal and Atkeson (1998); Schaal (1999); Schaal (1998);Arbib (1998)] for learning robot control with functional decomposition,where the main result could be formulated as: the cerebellum is more thenjust the function approximator.In this section we try to fit between these three approaches for humanoidcontrol, emphasizing the role of muscle–like robot actuators. Wepropose a new, physiologically based, tensor–invariant, hierarchical forcecontrol (FC, for short) for the physiologically realistic biodynamics. Weconsider the muscular torque one–forms F i as the most important componentof human–like motion; therefore we propose the sophisticated hierarchicalsystem for the subtle F i −-control: corresponding to the spinal, thecerebellar and cortical levels of human motor control. F i are first set–upas testing input–signals to biodynamics, and then covariantly updated as


574 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfeedback 1−forms u i on each FC–level. On the spinal FC–level the nominaljoint–trajectory tracking is proposed in the form of affine Hamiltoniancontrol; here the driving torques are given corrections by spinal–reflexcontrols. On the cerebellar FC–level, the relation is established betweencanonical joint coordinates q i , p i and gradient neural–image coordinatesx i , y i , representing bidirectional, self–organized, associative memory machine;here the driving torques are given the cerebellar corrections. On thecortical FC–level the topological ‘hyper–joystick’ is proposed as the centralFC command–space, selector, with the fuzzy–logic feedback–control mapdefined on it, giving the cortical corrections to the driving torques.The model of the spinal FC–level formulated here resembles autogeneticmotor servo, acting on the spinal–reflex level of the human locomotor control.The model of the cerebellar FC–level formulated here mimics theself–organizing, associative function of the excitatory granule cells and theinhibitory Purkinje cells of the cerebellum [Houk et al. (1996)]. The modelof the cortical FC–level presented in this section mimics the synergisticregulation of locomotor conditioned reflexes by the cerebellum [Houk et al.(1996)].We believe that (already mentioned) extremely high order of the drivingforce redundancy in biodynamics justifies the formulation of the three–levelforce control system. Also, both brain–like control systems can be easilyextended to give SE(3)−based force control for moving inverse kinematics(IK) chains of legs and arms.4.9.6.1 Functor Control MachineIn this subsection we define the functor control–machine (compare withsection (3.13.4.2) above), for the learning control with functional decomposition,by a two–step generalization of the Kalman’s theory of linearMIMO–feedback systems. The first generalization puts the Kalman’s theoryinto the pair of mutually dual linear categories Vect and Vect ∗ of vectorspaces and linear operators, with a ‘loop–functor’ representing the closed–loop control, thus formulating the unique, categorical formalism valid bothfor the discrete and continual MIMO–systems.We start with the unique, feedforward continual–sequential state equationẋ(t + 1) = Ax(t) + Bu(t), y(t) = Cx(t), (4.66)where the nD vector spaces of state X ∋ x, input U ∋ u, and output


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 575Y ∋ y have the corresponding linear operators, respectively A : X → X,B : U → X, and C : X → Y . The modular system theory comprises thesystem dynamics, given by a pair (X, A), together with a reachability mape : U → X of the pair (B, A), and an observability map m : X → Y ofthe pair (A, C). If the reachability map e is surjection the system dynamics(X, A) is called reachable; if the observability map m is injection the systemdynamics (X, A) is called observable. If the system dynamics (X, A) is bothreachable and observable, a composition r = m ◦ e : U → Y defines thetotal system’s response, which is given by solution of equation (4.66). If theunique solution to the continual–sequential state equation exists, it givesthe answer to the (minimal) realization problem: find the system S thatrealizes the given response r = m ◦ e : U → Y (in the smallest number ofdiscrete states and in the shortest time).The inverse map r −1 = e −1 ◦ m −1 : Y → U of the total system’sresponse r : U → Y defines the linear feedback operator K : Y → U, givenby standard feedback equationu(t) = Ky(t). (4.67)In categorical language, the feedforward system dynamics in the categoryVect is a pair (X, A), where X ∈ Ob(Vect) is an object in Vectand A : X → X ∈ Mor(Vect) is a Vect−-morphism. A feedforward decomposablesystem in Vect is such a sixtuple S ≡ (X, A, U, B, Y, C) that(X, A) is the system dynamics in Vect, a Vect–morphism B : U → X isan input map, and a Vect–morphism C : X → Y is an output map. Anyobject in Vect is characterized by mutually dual notions of its degree (anumber of its input morphisms) and its codegree (a number of its outputmorphisms). Similarly, any decomposable system S in Vect has a reachabilitymap given by an epimorphism e = A ◦ B : U → X and its dualobservability map given by a monomorphism m = C ◦ A : X → Y ; theircomposition r = m ◦ e : U → Y in Mor(Vect) defines the total system’sresponse in Vect given by the unique solution of the continual–sequentialstate equation (4.66) [<strong>Ivancevic</strong> and Snoswell (2001)].The dual of the total system’s response, defined by the feedback equation(4.67), is the feedback morphism K = e −1 ◦ m −1 : Y → U belonging tothe dual category Vect ∗ .In this way, the linear, closed–loop, continual–sequential MIMO–system(4.66–4.67) represents the linear iterative loop functor L : Vect ⇒ Vect ∗ .Our second generalization represents a natural system process Ξ[L], thattransforms the linear loop functor L : Vect ⇒ Vect ∗ – into the nonlinear


576 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionloop functor N L : CAT ⇒ CAT ∗ between two mutually dual nonlinearcategories CAT and CAT ∗ . We apply the natural process Ξ, separately(1) To the feedforward decomposable systemS ≡ (X, A, U, B, Y, C) in Vect, and(2) To the feedback morphism K = e −1 ◦ m −1 : Y → U in Vect ∗ .Under the action of the natural process Ξ, the linear feedforward systemdynamics (X, A) in Vect transforms into a nonlinear feedforward Ξ−dynamics(Ξ[X], Ξ[A]) in CAT , represented by a nonlinear feedforward decomposablesystem, Ξ[S] ≡ (Ξ[X], Ξ[A], Ξ[U], Ξ[B], Ξ[Y ], Ξ[C]).The reachability map transforms into the input process Ξ[e] = Ξ[A] ◦Ξ[B] : Ξ[U] −→ Ξ[X], while its dual, observability map transforms into theoutput process Ξ[m] = Ξ[C] ◦ Ξ[A] : Ξ[X] −→ Ξ[Y ]. In this way the totalresponse of the linear system r = m ◦ e : U → Y in Mor(Vect) transformsinto the nonlinear system behavior, Ξ[r] = Ξ[m] ◦ Ξ[e] : Ξ[U] −→ Ξ[Y ] inMor(CAT ). Obviously, Ξ[r], if exists, is given by a nonlinear Ξ−-transformof the linear state equations (4.66–4.67).Analogously, the linear feedback morphism K = e −1 ◦ m −1 : Y → Uin Mor(Vect ∗ ) transforms into the nonlinear feedback morphism Ξ[K] =Ξ[e −1 ] ◦ Ξ[m −1 ] : Ξ[Y ] → Ξ[U] in Mor(CAT ∗ ).In this way, the natural system process Ξ : L ⇛ N L is established.That means that the nonlinear loop functor L = Ξ[L] : CAT ⇒ CAT ∗ isdefined out of the linear, closed–loop, continual–sequential MIMO–system(4.66).In this section we formulate the nonlinear loop functor L = Ξ[L] :CAT ⇒ CAT ∗ for various hierarchical levels of muscular–like FC.4.9.6.2 Spinal Control LevelOur first task is to establish the nonlinear loop functor L = Ξ[L] : EX ⇒EX ∗ on the category EX of spinal FC–level.Recall that our dissipative, driven δ−Hamiltonian biodynamical systemon the configuration manifold M is, in local canonical–symplectic coordinatesq i , p i ∈ U p on the momentum phase–space manifold T ∗ M, given by


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 577autonomous equations˙q i = ∂H 0∂p iṗ i = F i − ∂H 0∂q i+ ∂R∂p i, (i = 1, . . . , N) (4.68)+ ∂R∂q i , (4.69)q i (0) = q i 0, p i (0) = p 0 i , (4.70)including contravariant equation (4.68) – the velocity vector–field, and covariantequation (4.69) – the force 1−form, together with initial joint anglesq i 0 and momenta p 0 i . Here the physical Hamiltonian function H 0 : T ∗ M →R represents the total biodynamical energy function, in local canonical coordinatesq i , p i ∈ U p on T ∗ M given byH 0 (q, p) = 1 2 gij p i p j + V (q),where g ij = g ij (q, m) denotes the contravariant material metric tensor.Now, the control Hamiltonian function H γ : T ∗ M → R of FC is in localcanonical coordinates on T ∗ M defined by [Nijmeijer and van der Schaft(1990)]H γ (q, p, u) = H 0 (q, p) − q i u i , (i = 1, . . . , N) (4.71)where u i = u i (t, q, p) are feedback–control 1−forms, representing the spinalFC–level u−corrections to the covariant torques F i = F i (t, q, p).Using δ−Hamiltonian biodynamical system (4.68–4.70) and the controlHamiltonian function (4.71), control γ δ −Hamiltonian FC–system can bedefined as˙q i = ∂H γ(q, p, u) ∂R(q, p)+ ,∂p i ∂p iṗ i = F i − ∂H γ(q, p, u)∂q i +∂R(q, p)∂q i ,o i = − ∂H γ(q, p, u)∂u i, (i = 1, . . . , N)q i (0) = q i 0, p i (0) = p 0 i ,where o i = o i (t) represent FC natural outputs which can be different fromcommonly used joint angles.If nominal reference outputs o i R = oi R (t) are known, the simple PDstiffness–servo [Whitney (1987)] could be formulated, via error function


578 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductione(t) = o j − o j R, in covariant formu i = K o δ ij (o j − o j R ) + K ȯδ ij (ȯ j − ȯ j R), (4.72)where Ks are the control–gains and δ ij is the Kronecker tensor.If natural outputs o i actually are the joint angles and nominal canonicaltrajectories ( qR i = qi R (t), pR i = p R i (t)) are known, then the stiffness–servo(4.72) could be formulated in canonical form asu i = K q δ ij (q i − q i R) + K p (p i − p R i ).Now, using the fuzzified µ−Hamiltonian biodynamical system withfuzzy system numbers (i.e, imprecise segment lengths, masses and momentsof inertia, joint dampings and muscular actuator parameters)˙q i = ∂H 0(q, p, σ µ )∂p iṗ i = ¯F i − ∂H 0(q, p, σ µ )∂q i+ ∂R∂p i, (4.73)+ ∂R∂q i , (4.74)q i (0) = ¯q i 0, p i (0) = ¯p 0 i , (i = 1, . . . , N), (4.75)(see 3.13.4.3 above) and the control Hamiltonian function (4.71),γ µ −Hamiltonian FC–system can be defined as˙q i = ∂H γ(q, p, u, σ µ ) ∂R(q, p)+ ,∂p i ∂p iṗ i = ¯F i − ∂H γ(q, p, u, σ µ )∂q i +∂R(q, p)∂q i ,ō i = − ∂H γ(q, p, u, σ µ )∂u i, q i (0) = ¯q i 0, p i (0) = ¯p 0 i ,where ō i = ō i (t) represent the fuzzified natural outputs.Finally, applying stochastic forces (diffusion fluctuations B ij [q i (t), t] anddiscontinuous jumps in the form of ND Wiener process W j (t)), i.e., usingthe fuzzy–stochastic [µσ]−Hamiltonian biodynamical system(dq i ∂H0 (q, p, σ µ )=+ ∂R )dt, (4.76)∂p i ∂p idp i = B ij [q i (t), t] dW j (t) +(¯F i − ∂H 0(q, p, σ µ )∂q i+ ∂R∂q i )dt, (4.77)q i (0) = ¯q i 0, p i (0) = ¯p 0 i . (4.78)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 579(see 3.13.4.3 above), and the control Hamiltonian function (4.71),γ µσ −Hamiltonian FC–system can be defined asdq i =( )∂Hγ (q, p, u, σ µ ) ∂R(q, p)+ dt,∂p i ∂p idp i = B ij [q i (t), t] dW j (t) +(¯F i − ∂H )γ(q, p, u, σ µ ) ∂R(q, p)∂q i +∂q i dt,dō i = − ∂H γ(q, p, u, σ µ )∂u idt, (i = 1, . . . , N)q i (0) = ¯q i 0, p i (0) = ¯p 0 i .If we have the case that not all of the configuration joints on the configurationmanifold M are active in the specified robot task, we can introducethe coupling Hamiltonians H j = H j (q, p), j = 1, . . . , M ≤ N,corresponding to the system’s active joints, and we come to affine Hamiltonianfunction H a : T ∗ M → R, in local canonical coordinates on T ∗ Mgiven as [Nijmeijer and van der Schaft (1990)]H a (q, p, u) = H 0 (q, p) − H j (q, p) u j . (4.79)Using δ−Hamiltonian biodynamical system (4.68–4.70) and the affineHamiltonian function (4.79), affine a δ −Hamiltonian FC–system can be definedas˙q i = ∂H 0(q, p)− ∂Hj (q, p)∂p i ∂p iṗ i = F i − ∂H 0(q, p)∂q i + ∂Hj (q, p)∂q iu j + ∂R∂p i, (4.80)u j + ∂R∂q i , (4.81)o i = − ∂H a(q, p, u)∂u i= H j (q, p), (4.82)q i (0) = q i 0, p i (0) = p 0 i , (4.83)(i = 1, . . . , N; j = 1, . . . , M ≤ N).Using the Lie–derivative exact feedback linearization (see (4.9.2) above),and applying the constant relative degree r (see [Isidori (1989); Sastri andIsidori (1989)]) to all N joints of the affine a δ −Hamiltonian FC–system(4.80–4.83), the control law for asymptotic tracking the reference outputs


580 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiono j Rcould be formulated asu j = ȯ(r)j R− L(r) fHj + ∑ rs=1 γ s−1(o (s−1)jR− L (s−1)fH j ),L g L (r−1)fH jwhere standard MIMO–vector–fields f and g are given by( ∂H0f = , − ∂H ))0∂p i ∂q i , g =(− ∂Hj , ∂Hj∂p i ∂q iand γ s−1 are the coefficients of linear differential equation of order r forthe error function e(t) = o j − o j Re (r) + γ r−1 e (r−1) + · · · + γ 1 e (1) + γ 0 e = 0.Using the fuzzified µ−Hamiltonian biodynamical system (4.73–4.75)and the affine Hamiltonian function (4.79), affine a µ −Hamiltonian FC–system can be defined as˙q i = ∂H 0(q, p, σ µ )− ∂Hj (q, p, σ µ ) ∂R(q, p)u j + ,∂p i ∂p i ∂p iṗ i = ¯F i − ∂H 0(q, p, σ µ )∂q i + ∂Hj (q, p, σ µ ) ∂R(q, p)∂q i u j +∂q i ,ō i = − ∂H a(q, p, u, σ µ )∂u i= H j (q, p, σ µ ),q i (0) = ¯q i 0, p i (0) = ¯p 0 i , (i = 1, . . . , N; j = 1, . . . , M ≤ N).Using the fuzzy–stochastic [µσ]−Hamiltonian biodynamical system(4.76–4.78) and the affine Hamiltonian function (4.79), affinea µσ −Hamiltonian FC–system can be defined as( )dq i ∂H0 (q, p, σ µ )=− ∂Hj (q, p, σ µ ) ∂R(q, p)u j + dt,∂p i ∂p i ∂p idp i = B ij [q i (t), t] dW j (t) +(¯F i − ∂H )0(q, p, σ µ )∂q i + ∂Hj (q, p, σ µ ) ∂R(q, p)∂q i u j +∂q i dt,dō i = − ∂H a(q, p, u, σ µ )dt = H j (q, p, σ µ ) dt,∂u iq i (0) = ¯q 0, i p i (0) = ¯p 0 i , (i = 1, . . . , N; j = 1, . . . , M ≤ N).Being high–degree and highly nonlinear, all of these affine control systemsare extremely sensitive upon the variation of parameters, inputs,


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 581and initial conditions. The sensitivity function S of the affine HamiltonianH a (q, p, u) upon the parameters β i (representing segment lengths L i ,masses m i , moments of inertia J i and joint dampings b i , see [<strong>Ivancevic</strong>and Snoswell (2001); <strong>Ivancevic</strong> (1991)]), is in the case of a δ −HamiltonianFC–system defined asS(H, β) =β i ∂H a (q, p, u),H a (q, p, u) ∂β iand similarly in other two a µ − and a µσ − cases.The three affine FC–level systems a δ , a µ and a µσ , resemble (in a fuzzy–stochastic–Hamiltonian form), Houk’s autogenetic motor servo of musclespindle and Golgi tendon proprioceptors [Houk (1979)], correcting the covariantdriving torques F i = F i (t, q, p) by local ‘reflex controls’ u i (t, q, p).They form the nonlinear loop functor L = Ξ[L] : EX ⇒ EX ∗ .4.9.6.3 Cerebellar Control LevelOur second task is to establish the nonlinear loop functor L = Ξ[L] : T A ⇒T A ∗ on the category T A of the cerebellar FC–level. Here we propose an oscillatoryneurodynamical (x, y, ω)−-system (adapted from [<strong>Ivancevic</strong> et al.(1999a)]), a bidirectional, self–organized, associative–memory machine, resemblingthe function of a set of excitatory granule cells and inhibitoryPurkinje cells in the middle layer of the cerebellum (see [Eccles et al. (1967);Houk et al. (1996)]). The neurodynamical (x, y, ω)−-system acts onneural–image manifold Mim N of the configuration manifold M N as a pair ofsmooth, ‘1 − 1’ and ‘onto’ maps (Ψ, Ψ −1 ), where Ψ : M N → Mim N representsthe feedforward map, and Ψ −1 : Mim N → M N represents the feedbackmap. Locally, it is defined in Riemannian neural coordinates x i , y i ∈ V y onMim N , which are in bijective correspondence with symplectic joint coordinatesq i , p i ∈ U p on T ∗ M.The (x, y, ω)−-system is formed out of two distinct, yet nonlinearly–coupled neural subsystems, with A i (q) (4.86) and B i (p) (4.87) as systeminputs, and the feedback–control 1−forms u i (4.92) as system outputs:(1) Granule cells excitatory (contravariant) and Purkinje cells inhibitory(covariant) activation (x, y)−-dynamics (4.84–4.87), defined respectivelyby a vector–field x i = x i (t) : M → T M, representing a cross–section of the tangent bundle T M, and a 1−form y i = y i (t) : M →T ∗ M, representing a cross–section of the cotangent bundle T ∗ M; and


582 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(2) Excitatory and inhibitory unsupervised learning (ω)–dynamics (4.87–4.89) generated by random differential Hebbian learning process (4.90–4.92), defined respectively by contravariant synaptic tensor–field ω ij =ω ij (t) : M → T T M N im and covariant synaptic tensor–field ω ij = ω ij (t) :M → T ∗ T ∗ M, representing cross–sections of contravariant and covarianttensor bundles, respectively.The system equations are defined asẋ i = A i (q) + ω ij f j (y) − x i , (4.84)ẏ i = B i (p) + ω ij f j (x) − y i , (4.85)A i (q) = K q (q i − q i R), (4.86)B i (p) = K p (p R i − p i ), (4.87)˙ω ij = −ω ij + I ij (x, y), (4.88)˙ω ij = −ω ij + I ij (x, y), (4.89)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.90)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.91)u i = 1 2 (δ ij x i + y i ), (i, j = 1, . . . , N). (4.92)Here ω is a symmetric 2nd order synaptic tensor–field; I ij = I ij (x, y, σ)and I ij = I ij (x, y, σ) respectively denote contravariant–excitatory andcovariant–inhibitory random differential Hebbian innovation–functions withtensorial Gaussian noise σ (in both variances); fs and fs ˙ denote sigmoidactivation functions (f = tanh(.)) and corresponding signal velocities( f ˙ = 1 − f 2 ), respectively in both variances;A i (q) and B i (p) are contravariant–excitatory and covariant–inhibitoryneural inputs to granule and Purkinje cells, respectively; u i are the correctionsto the feedback–control 1−forms on the cerebellar FC–level.Nonlinear activation (x, y)−-dynamics (4.84–4.87), describes a two–phase biological neural oscillator field, in which excitatory neural fieldexcites inhibitory neural field, which itself reciprocally inhibits the excitatoryone. (x, y)−-dynamics represents a nonlinear extension of a linear,Lyapunov–stable, conservative, gradient system, defined in local neural coordinatesx i , y i ∈ V y on T ∗ M asẋ i = − ∂Φ∂y i= ω ij y j − x i ,ẏ i = − ∂Φ∂x i = ω ijx j − y i . (4.93)The gradient system (4.93) is derived from scalar, neuro-synaptic action


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 583potential Φ : T ∗ M → R, given by a negative, smooth bilinear form inx i , y i ∈ V y on T ∗ M as− 2Φ = ω ij x i x j + ω ij y i y j − 2x i y i , (i, j = 1, . . . , N), (4.94)which itself represents a Ψ−-image of the Riemannian metrics g : T M → Ron the configuration manifold M.The nonlinear oscillatory activation (x, y)−-dynamics (4.84–4.87) is getfrom the linear conservative dynamics (4.93) by adding configura-tion–dependent inputs A i and B i , as well as sigmoid activation functions f jand f j , respectively. It represents an interconnected pair of excitatory andinhibitory neural fields.Both variant–forms of learning (ω)−-dynamics (4.88–4.89) are given bygeneralized unsupervised (self–organizing) Hebbian learning scheme (see[Kosko (1992)]) in which ˙ω ij (resp. ˙ω ij ) denotes the new–update value, -ω ij(resp. -ω ij ) corresponds to the old value and I ij (x i , y j ) (resp. I ij (x i , y j ))is the innovation function of the symmetric 2nd order synaptic tensor-fieldω. The nonlinear innovation functions I ij and I ij are defined by randomdifferential Hebbian learning process (4.90–4.91). As ω is symmetric andzero-trace coupling synaptic tensor, the conservative linear activation dynamics(4.93) is equivalent to the rule that the state of each neuron (in bothneural fields) is changed in time iff the scalar action potential Φ (4.94), islowered. Therefore, the scalar action potential Φ represents the monotonicallydecreasing Lyapunov function (such that ˙Φ ≤ 0) for the conservativelinear dynamics (4.93), which converges to a local minimum or ground stateof Φ. That is to say, the system (4.93) moves in the direction of decreasingthe scalar action potential Φ, and when both ẋ i = 0 and ẏ i = 0 for alli = 1, . . . , N, the steady state is reached.In this way, the neurodynamical (x, y, ω)−system acts as tensor–invariant self–organizing (excitatory / inhibitory) associative memory machine,resembling the set of granule and Purkinje cells of cerebellum [Houket al. (1996)].The feedforward map Ψ : M → M is realized by the inputs A i (q) andB i (p) to the (x, y, ω)−-system, while the feedback map Ψ −1 : M → M isrealized by the system output, i.e., the feedback–control 1−forms u i (x, y).These represent the cerebellar FC–level corrections to the covariant torquesF i = F i (t, q, p).The tensor–invariant form of the oscillatory neurodynamical (x, y, ω)−system(4.84–4.92) resembles the associative action of the granule and Purkinjecells in the tunning of the limb cortico–rubro–cerebellar recurrent net-


584 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwork [Houk et al. (1996)], giving the cerebellar correction u i (x, y) to thecovariant driving torques F i = F i (t, q, p). In this way (x, y, ω)−-systemforms the nonlinear loop functor L = Ξ[L] : T A ⇒ T A ∗ .4.9.6.4 Cortical Control LevelOur third task is to establish the nonlinear loop functor L = Ξ[L] : ST ⇒ST ∗ on the category ST of the cortical FC–level.Recall that for the purpose of cortical control, the purely rotationalbiodynamical manifold M could be firstly reduced to N−-torus and subsequentlytransformed to N−-cube (‘hyper–joystick’), using the followinggeometrical techniques (see (3.8.4.3) above).Denote by S 1 the constrained unit circle in the complex plane. This isan Abelian Lie group. We have two reduction homeomorphismsSO(3) SO(2) ✄ SO(2) ✄ SO(2), and SO(2) ≈ S 1 ,where ‘✄’ denotes the noncommutative semidirect product.Next, let I N be the unit cube [0, 1] N in R N and ‘∼’ an equivalencerelation on R N get by ‘gluing’ together the opposite sides of I N , preservingtheir orientation. Therefore, M can be represented as the quotient space ofR N by the space of the integral lattice points in R N , that is a constrainedtorus T N :R N /Z N = I N / ∼ ∼ =N∏Si 1 ≡ {(q i , i = 1, . . . , N) : mod 2π} = T N .i=1In the same way, the momentum phase–space manifold T ∗ M can be representedby T ∗ T N .Conversely by ‘ungluing’ the configuration space we get the primaryunit cube. Let ‘∼ ∗ ’ denote an equivalent decomposition or ‘ungluing’ relation.By the Tychonoff product–topology Theorem, for every such quotientspace there exists a ‘selector’ such that their quotient models are homeomorphic,that is, T N / ∼ ∗ ≈ A N / ∼ ∗ . Therefore IqN represents a ‘selector’for the configuration torus T N and can be used as an N−-directional ‘ˆq−command–space’for FC. Any subset of DOF on the configuration torus T Nrepresenting the joints included in the general biodynamics has its simple,rectangular image in the rectified ˆq−-command space – selector Iq N , andany joint angle q i has its rectified image ˆq i .In the case of an end–effector, ˆq i reduces to the position vector inexternal–Cartesian coordinates z r (r = 1, . . . , 3). If orientation of the end–


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 585effector can be neglected, this gives a topological solution to the standardinverse kinematics problem.Analogously, all momenta ˆp i have their images as rectified momentaˆp i in the ˆp−-command space – selector I N p . Therefore, the total momentumphase–space manifold T ∗ T N gets its ‘cortical image’ as the ̂(q, p)−commandspace, a trivial 2ND bundle Iq N × Ip N .Now, the simplest way to perform the feedback FC on the cortical̂(q, p)−-command space IqN × Ip N , and also to mimic the cortical–like behavior[1,2], is to use the 2ND fuzzy–logic controller, in pretty muchthe same way as in popular ‘inverted pendulum’ examples [Kosko (1992);Kosko (1996)].We propose the fuzzy feedback–control map Ξ that maps all the rectifiedjoint angles and momenta into the feedback–control 1−formsΞ : (ˆq i (t), ˆp i (t)) ↦→ u i (t, q, p), (4.95)so that their corresponding universes of discourse, ˆM i = (ˆq max i − ˆq min i ),ˆP i = (ˆp maxi − ˆp mini ) and U i = (u maxi − u mini ), respectively, are mapped asΞ :N∏i=1N∏ N∏ˆMM i × ˆP i → U i . (4.96)i=1 i=1The 2N−-D map Ξ (4.95–4.96) represents a fuzzy inference system,defined by (adapted from [<strong>Ivancevic</strong> et al. (1999b)]):(1) Fuzzification of the crisp rectified and discretized angles, momenta andcontrols using Gaussian–bell membership functionsµ k (χ) = exp[− (χ − m k) 22σ k], (k = 1, 2, . . . , 9),where χ ∈ D is the common symbol for ˆq i , ˆp i and u i (q, p) andD is the common symbol for M i , ˆP i and i ; the mean values m k ofthe seven partitions of each universe of discourse D are defined asm k = λ k D + χ min , with partition coefficients λ k uniformly spanningthe range of D, corresponding to the set of nine linguistic variablesL = {NL, NB, NM, NS, ZE, P S, P M, P B, P L}; standard deviationsare kept constant σ k = D/9. Using the linguistic vector L, the 9 × 9FAM (fuzzy associative memory) matrix (a ‘linguistic phase–plane’), isheuristically defined for each human joint, in a symmetrical weighted


586 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionformµ kl = ϖ kl exp{−50[λ k + u(q, p)] 2 }, (k, l = 1, 2, . . . , 9)with weights ϖ kl ∈ {0.6, 0.6, 0.7, 0.7, 0.8, 0.8, 0.9, 0.9, 1.0}.(2) Mamdani inference is used on each FAM–matrix µ kl for all humanjoints:(i) µ(ˆq i ) and µ(ˆp i ) are combined inside the fuzzy IF–THEN rules usingAND (Intersection, or Minimum) operator,µ k [ū i (q, p)] = minl{µ kl (ˆq i ), µ kl (ˆp i )}.(ii) the output sets from different IF–THEN rules are then combined usingOR (Union, or Maximum) operator, to get the final output, fuzzy–covariant torques,µ[u i (q, p)] = max {µ k[ū i (q, p)]}.k(3) Defuzzification of the fuzzy controls µ[u i (q, p)] with the ‘center of gravity’methodu i (q, p) =∫µ[ui (q, p)] du i∫dui,to update the crisp feedback–control 1−forms u i = u i (t, q, p). Theserepresent the cortical FC–level corrections to the covariant torques F i =F i (t, q, p).Operationally, the construction of the cortical ̂(q, p)−-command spaceI N q × I N p and the 2ND feedback map Ξ (4.95–4.96), mimic the regulationof locomotor conditioned reflexes by the motor cortex [Houk et al. (1996)],giving the cortical correction to the covariant driving torques F i . Togetherthey form the nonlinear loop functor N L = Ξ[L] : ST ⇒ ST ∗ .A sample output from the leading human–motion simulator, HumanBiodynamics Engine (developed by the authors in Defence Science & TechnologyOrganisation, Australia), is given in Figure 4.9, giving the sophisticated264 DOF analysis of adult male running with the speed of 5 m/s.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 587Fig. 4.9 Sample output from the Human Biodynamics Engine: running with thespeed of 5 m/s.4.9.6.5 Open Liouville Neurodynamics and Biodynamical Self–SimilarityRecall (see [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]) that neurodynamics has itsphysical behavior both at the macroscopic, classical, inter–neuronal level,and at the microscopic, quantum, intra–neuronal level. At the macroscopiclevel, various models of neural networks (NNs, for short) have been proposedas goal–oriented models of the specific neural functions, like for instance,function–approximation, pattern–recognition, classification, or control(see, e.g., [Haykin (1994)]). In the physically–based, Hopfield–typemodels of NNs [Hopfield (1982); Hopfield (1984)] the information is storedas a content–addressable memory in which synaptic strengths are modifiedafter the Hebbian rule (see [Hebb (1949)]. Its retrieval is made when thenetwork with the symmetric couplings works as the point–attractor withthe fixed points. Analysis of both activation and learning dynamics ofHopfield–Hebbian NNs using the techniques of statistical mechanics [Domanyet al. (1991)], gives us with the most important information of storagecapacity, role of noise and recall performance.On the other hand, at the general microscopic intra–cellular level, en-


588 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionergy transfer across the cells, without dissipation, had been first conjecturedto occur in biological matter by [Frölich and Kremer (1983)]. Thephenomenon conjectured by them was based on their 1D superconductivitymodel: in 1D electron systems with holes, the formation of solitonic structuresdue to electron–hole pairing results in the transfer of electric currentwithout dissipation. In a similar manner, Frölich and Kremer conjecturedthat energy in biological matter could be transferred without dissipation,if appropriate solitonic structures are formed inside the cells. This idea haslead theorists to construct various models for the energy transfer across thecell, based on the formation of kink classical solutions (see [Satarić et al.(1993); Satarić et al. (1998)].The interior of living cells is structurally and dynamically organizedby cytoskeletons, i.e., networks of protein polymers. Of these structures,microtubules (MTs, for short) appear to be the most fundamental(see [Dustin (1984)]). Their dynamics has been studied by a numberof authors in connection with the mechanism responsible for dissipation–free energy transfer. Hameroff and Penrose [Hameroff (1987)] have conjecturedanother fundamental role for the MTs, namely being responsiblefor quantum computations in the human neurons. [Penrose (1989);Penrose (1994); Penrose (1997)] further argued that the latter is associatedwith certain aspects of quantum theory that are believed to occurin the cytoskeleton MTs, in particular quantum superposition and subsequentcollapse of the wave function of coherent MT networks. Theseideas have been elaborated by [Mavromatos and Nanopoulos (1995a);Mavromatos and Nanopoulos (1995b)] and [Nanopoulos (1995)], based onthe quantum–gravity EMN–language of [Ellis et al. (1992); Ellis et al.(1999)] where MTs have been physically modelled as non-critical (SUSY)bosonic strings. It has been suggested that the neural MTs are the micrositesfor the emergence of stable, macroscopic quantum coherent states,identifiable with the preconscious states; stringy–quantum space–time effectstrigger an organized collapse of the coherent states down to a specificor conscious state. More recently, [Tabony et al. (1999)] have presentedthe evidence for biological self–organization and pattern formation duringembryogenesis.Now, we have two space–time biophysical scales of neurodynamics. Naturallythe question arises: are these two scales somehow inter–related, isthere a space–time self–similarity between them?The purpose of this subsection is to prove the formal positive answerto the self–similarity question. We try to describe neurodynamics on both


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 589physical levels by the unique form of a single equation, namely open Liouvilleequation: NN–dynamics using its classical form, and MT–dynamicsusing its quantum form in the Heisenberg picture. If this formulation isconsistent, that would prove the existence of the formal neurobiologicalspace–time self–similarity.Hamiltonian FrameworkSuppose that on the macroscopic NN–level we have a conservativeHamiltonian system acting in a 2ND symplectic phase–space T ∗ Q ={q i (t), p i (t)}, (i = 1 . . . N) (which is the cotangent bundle of the NN–configuration manifold Q = {q i }), with a Hamiltonian function H =H(q i , p i , t) : T ∗ Q × R → R. The conservative dynamics is defined by classicalHamiltonian canonical equations (3.34). Recall that within the conservativeHamiltonian framework, we can apply the formalism of classicalPoisson brackets: for any two functions A = A(q i , p i , t) and B = B(q i , p i , t)their Poisson bracket is defined as( ∂A ∂B[A, B] =∂q i − ∂A )∂B∂p i ∂p i ∂q i .Conservative Classical SystemAny function A(q i , p i , t) is called a constant (or integral) of motion ofthe conservative system (3.34) ifȦ ≡ ∂ t A + [A, H] = 0, which implies ∂ t A = −[A, H] . (4.97)For example, if A = ρ(q i , p i , t) is a density function of ensemble phase–points (or, a probability density to see a state x(t) = (q i (t), p i (t)) of ensembleat a moment t), then equation∂ t ρ = −[ρ, H] = −iL ρ (4.98)represents the Liouville Theorem, where L denotes the (Hermitian) Liouvilleoperator( ∂H ∂iL = [..., H] ≡∂p i ∂q i − ∂H )∂∂q i = div(ρẋ),∂p iwhich shows that the conservative Liouville equation (4.98) is actuallyequivalent to the mechanical continuity equation∂ t ρ + div(ρẋ) = 0. (4.99)


590 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionConservative Quantum SystemWe perform the formal quantization of the conservative equation (4.98)in the Heisenberg picture: all variables become Hermitian operators (denotedby ‘∧’), the symplectic phase–space T ∗ Q = {q i , p i } becomes theHilbert state–space H = Hˆq i ⊗ Hˆpi (where Hˆq i = Hˆq 1 ⊗ ... ⊗ Hˆq N andHˆpi = Hˆp1 ⊗ ... ⊗ HˆpN ), the classical Poisson bracket [ , ] becomes thequantum commutator { , } multiplied by -i/[ , ] −→ −i{ , } ( = 1 in normal units) . (4.100)In this way the classical Liouville equation (4.98) becomes the quantumLiouville equation∂ tˆρ = i{ˆρ, Ĥ} , (4.101)where Ĥ = Ĥ(ˆqi , ˆp i , t) is the Hamiltonian evolution operator, whileˆρ = P (a)|Ψ a >< Ψ a |, with Tr(ˆρ) = 1,denotes the von Neumann density matrix operator, where each quantumstate |Ψ a > occurs with probability P (a); ˆρ = ˆρ(ˆq i , ˆp i , t) is closely relatedto another von Neumann concept: entropy S = − Tr(ˆρ[ln ˆρ]).Open Classical SystemWe now move to the open (nonconservative) system: on the macroscopicNN–level the opening operation equals to the adding of a covariant vectorof external (dissipative and/or motor) forces F i = F i (q i , p i , t) to (the r.h.sof) the covariant Hamiltonian force equation, so that Hamiltonian equationsget the open (dissipative and/or forced) form˙q i = ∂H∂p i,ṗ i = F i − ∂H∂q i . (4.102)In the framework of the open Hamiltonian system (4.102), dynamics of anyfunction A(q i , p i , t) is defined by the open evolution equation:∂ t A = −[A, H] + Φ,where Φ = Φ(F i ) represents the general form of the scalar force term.In particular, if A = ρ(q i , p i , t) represents the density function of ensemblephase–points, then its dynamics is given by the (dissipative/forced)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 591open Liouville equation:∂ t ρ = −[ρ, H] + Φ . (4.103)In particular, the scalar force term can be cast as a linear Poisson–bracket formΦ = F i [A, q i ] , with [A, q i ] = − ∂A∂p i. (4.104)Now, in a similar way as the conservative Liouville equation (4.98) resemblesthe continuity equation (4.99) from continuum dynamics, also theopen Liouville equation (4.103) resembles the probabilistic Fokker–Planckequation from statistical mechanics. If we have a ND stochastic processx(t) = (q i (t), p i (t)) defined by the vector Itô SDEdx(t) = f(x, t) dt + G(x, t) dW,where f is a ND vector function, W is a KD Wiener process, and G is aN ×KD matrix valued function, then the corresponding probability densityfunction ρ = ρ(x, t|ẋ, t ′ ) is defined by the ND Fokker–Planck equation (see,e.g., [Gardiner (1985)])∂ 2∂ t ρ = − div[ρ f(x, t)] + 1 (Q ij ρ) , (4.105)2 ∂x i ∂x jwhere Q ij = ( G(x, t) G T (x, t) ) . It is obvious that the Fokker–Planckijequation (4.105) represents the particular, stochastic form of our generalopen Liouville equation (4.103), in which the scalar force term is given bythe (second–derivative) noise term∂ 2Φ = 1 (Q ij ρ) .2 ∂x i ∂x jEquation (4.103) will represent the open classical model of our macroscopicNN–dynamics.Continuous Neural Network DynamicsThe generalized NN–dynamics, including two special cases of gradedresponse neurons (GRN) and coupled neural oscillators (CNO), can bepresented in the form of a stochastic Langevin rate equation˙σ i = f i + η i (t), (4.106)


592 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere σ i = σ i (t) are the continual neuronal variables of ith neurons(representing either membrane action potentials in case of GRN, or oscillatorphases in case of CNO); J ij are individual synaptic weights;f i = f i (σ i , J ij ) are the deterministic forces (given, in GRN–case, byf i = ∑ j J ij tanh[γσ j ] − σ i + θ i , with γ > 0 and with the θ i representinginjected currents, and in CNO–case, by f i = ∑ j J ij sin(σ j − σ i ) + ω i ,with ω i representing the natural frequencies of the individual oscillators);the noise variables are given as η i (t) = lim ∆→0 ζ i (t) √ 2T/∆ where ζ i (t)denote uncorrelated Gaussian distributed random forces and the parameterT controls the amount of noise in the system, ranging from T = 0(deterministic dynamics) to T = ∞ (completely random dynamics).More convenient description of the neural random process (4.106) isprovided by the Fokker–Planck equation describing the time evolution ofthe probability density P (σ i )∂ t P (σ i ) = − ∂ (f i P (σ i )) + T ∂2∂σ i ∂σ 2 P (σ i ). (4.107)iNow, in the case of deterministic dynamics T = 0, equation (4.107)can be put into the form of the conservative Liouville equation (4.98), bymaking the substitutions: P (σ i ) → ρ, f i = ˙σ i , and [ρ, H] = div(ρ ˙σ i ) ≡∑i∂∂σ i(ρ ˙σ i ), where H = H(σ i , J ij ). Further, we can formally identifythe stochastic forces, i.e., the second–order noise–term T ∑ ∂ 2iρ with∂σ 2 iF i [ρ, σ i ] , to get the open Liouville equation (4.103).Therefore, on the NN–level deterministic dynamics corresponds to theconservative system (4.98). Inclusion of stochastic forces corresponds tothe system opening (4.103), implying the macroscopic arrow of time.Open Quantum SystemBy formal quantization of equation (4.103) with the scalar force termdefined by (4.104), in the same way as in the case of the conservativedynamics, we get the quantum open Liouville equation∂ tˆρ = i{ˆρ, Ĥ} + ˆΦ, with ˆΦ = −i ˆFi {ˆρ, ˆq i }, (4.108)where ˆF i = ˆF i (ˆq i , ˆp i , t) represents the covariant quantum operator of externalfriction forces in the Hilbert state–space H = Hˆq i ⊗ Hˆpi .Equation (4.108) will represent the open quantum–friction model of ourmicroscopic MT–dynamics. Its system–independent properties are [Ellis


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 593et al. (1992); Ellis et al. (1999); Mavromatos and Nanopoulos (1995a);Mavromatos and Nanopoulos (1995b); Nanopoulos (1995)]:(1) Conservation of probability P∂ t P = ∂ t [Tr(ˆρ)] = 0.(2) Conservation of energy E, on the average(3) Monotonic increase in entropy∂ t 〈〈E〉〉 ≡ ∂ t [Tr(ˆρ E)] = 0.∂ t S = ∂ t [− Tr(ˆρ ln ˆρ)] ≥ 0,and thus automatically and naturally implies a microscopic arrow oftime, so essential in realistic biophysics of neural processes.Non–Critical Stringy MT–DynamicsIn EMN–language of non–critical (SUSY) bosonic strings, our MT–dynamics equation (4.108) reads∂ tˆρ = i{ˆρ, Ĥ} − iĝ ij{ˆρ, ˆq i }ˆ˙q j , (4.109)where the target–space density matrix ˆρ(ˆq i , ˆp i ) is viewed as a function ofcoordinates ˆq i that parameterize the couplings of the generalized σ−modelson the bosonic string world–sheet, and their conjugate momenta ˆp i , whileĝ ij = ĝ ij (ˆq i ) is the quantum operator of the positive definite metric in thespace of couplings. Therefore, the covariant quantum operator of externalfriction forces is in EMN–formulation given as ˆF i (ˆq i , ˆ˙q i ) = ĝ ij ˆ˙q j .Equation (4.109) establishes the conditions under which a large–scalecoherent state appearing in the MT–network, which can be considered responsiblefor loss–free energy transfer along the tubulins.Equivalence of Neurodynamic FormsIt is obvious that both the macroscopic NN–equation (4.103) and themicroscopic MT–equation (4.108) have the same open Liouville form, whichimplies the arrow of time. These proves the existence of the formal neuro–biological space–time self–similarity.In this way, we have described neurodynamics of both NN and MTensembles, belonging to completely different biophysical space–time scales,


594 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionby the unique form of open Liouville equation, which implies the arrow oftime. The existence of the formal neuro–biological self–similarity has beenproved.4.9.7 Brain–Mind Functorial MachinesIn this section we propose two models of the brain–mind functorial machines:the first one is a psychologically–motivated top–down machine,while the second one is physically–motivated bottom–up solitary machine.4.9.7.1 Neurodynamical 2−FunctorHere we define the goal–directed cognitive neurodynamics as an evolution2−functor E given byAf✲ BE(A)E(f)✲ E(B)CURRENT✲DESIREDh NEURAL g E ✲ E(h) NEURAL E(g)❄ STATE ❄❄ STATE ❄C✲ DE(C) ✲ E(D)kE(k)(4.110)In (4.110), E represents a projection map from the source 2−category ofthe current neural state, defined as a commutative square of small categoriesA, B, C, D, . . . of current neural ensembles and their causal interrelationsf, g, h, k, . . ., onto the target 2−category of the desired neuralstate, defined as a commutative square of small categories E(A), E(B),E(C), E(D), . . . of evolved neural ensembles and their causal interrelationsE(f), E(g), E(h), E(k).The evolution 2−functor E can be horizontally decomposed in the followingthree neurodynamic components (see [Lewin (1997); Aidman andLeontiev (1991)]):(1) Intention, defined as a 3−cell:Need 1 ∗ Need 3INTENTION ❘Motive 1,2 > Motive 3,4✒ Need 2 ∗ Need 4


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 595(2) Action, defined as a 1−cell:ACTION✲Initial(3) Locomotion, defined as a 2−cell:Sustain ❄✲◆✻✍MonitorLOCOMOTIONNow, each causal arrow in (4.110), say f : A → B, stands for a generic‘neuro–morphism’, representing a self–organized, oscillatory neurodynamicsystem. We define a generic neuro–morphism f to be a nonlinear tensor–field (x, y, ω)−-system (4.111–4.116), acting as a bidirectional associativememory machine on a ND Riemannian manifold M N of the human cortex.It is formed out of two distinct, yet nonlinearly–coupled neural subsystems:(1) Activation (x, y)−-dynamics (4.111–4.112), defined as an interplay ofan excitatory vector–field x i = x i (t) : M N → T M, representing across–section of the tangent bundle T M, and and an inhibitory 1−formy i = y i (t) : M N → T ∗ M, representing a cross–section of the cotangentbundle T ∗ M.(2) Excitatory and inhibitory unsupervised learning (ω)–dynamics (4.113–4.116) generated by random differential Hebbian learning process(4.115–4.116), defined respectively by contravariant synaptic tensor–field ω ij = ω ij (t) : M N → T T Mim N and covariant synaptic tensor–field ω ij = ω ij (t) : M N → T ∗ T ∗ M, representing cross–sections of contravariantand covariant tensor bundles, respectively.(x, y, ω)−-system is analytically defined as a set of N coupled neurodynamicequations:ẋ i = A i + ω ij f j (y) − x i , (4.111)ẏ i = B i + ω ij f j (x) − y i , (4.112)˙ω ij = −ω ij + I ij (x, y), (4.113)˙ω ij = −ω ij + I ij (x, y), (4.114)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.115)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.116)(i, j = 1, . . . , N).Here ω is a symmetric, second–order synaptic tensor–field; I ij = I ij (x, y, σ)and I ij = I ij (x, y, σ) respectively denote contravariant–excitatory and


596 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncovariant–inhibitory random differential Hebbian innovation–functions withtensorial Gaussian noise σ (in both variances); fs and fs ˙ denote sigmoidactivation functions (f = tanh(.)) and corresponding signal velocities( f ˙ = 1 − f 2 ), respectively in both variances; A i = A i (t) and B i = B i (t)are contravariant–excitatory and covariant–inhibitory neural inputs to thecorresponding cortical cells, respectively;Nonlinear activation (x, y)−-dynamics, describes a two–phase biologicalneural oscillator field, in which the excitatory neural field excites theinhibitory neural field, which itself reciprocally inhibits the excitatory one.(x, y)−-dynamics represents a nonlinear extension of a linear, Lyapunov–stable, conservative, gradient system, defined in local neural coordinatesx i , y i ∈ V y on T ∗ M asẋ i = − ∂Φ∂y i= ω ij y j − x i ,ẏ i = − ∂Φ∂x i = ω ijx j − y i . (4.117)The gradient system (4.117) is derived from scalar, neuro–synaptic actionpotential Φ : T ∗ M → R, given by a negative, smooth bilinear form inx i , y i ∈ V y on T ∗ M as−2Φ = ω ij x i x j + ω ij y i y j − 2x i y i ,(i, j = 1, . . . , N),which itself represents a Ψ−-image of the Riemannian metrics g : T M → Ron the configuration manifold M N .The nonlinear oscillatory activation (x, y)−-dynamics (4.111–4.114) isget from the linear conservative dynamics (4.117), by adding configurationdependent inputs A i and B i , as well as sigmoid activation functions f jand f j , respectively. It represents an interconnected pair of excitatory andinhibitory neural fields.Both variant–forms of learning (ω)−-dynamics (4.113–4.114) are givenby a generalized unsupervised (self–organizing) Hebbian learning scheme(see [Kosko (1992)]) in which ˙ω ij (resp. ˙ω ij ) denotes the new–updatevalue, -ω ij (resp. ω ij ) corresponds to the old value and I ij (x i , y j ) (resp.I ij (x i , y j )) is the innovation function of the symmetric 2nd order synaptictensor–field ω. The nonlinear innovation functions I ij and I ij are definedby random differential Hebbian learning process (4.115–4.116). As ω is asymmetric and zero–trace coupling synaptic tensor, the conservative linearactivation dynamics (4.117) is equivalent to the rule that ‘the state of eachneuron (in both neural fields) is changed in time if, and only if, the scalaraction potential Φ (52), is lowered’. Therefore, the scalar action potentialΦ represents the monotonically decreasing Lyapunov function (such that


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 597˙Φ ≤ 0) for the conservative linear dynamics (4.117), which converges toa local minimum or ground state of Φ. That is to say, the system (4.117)moves in the direction of decreasing the scalar action potential Φ, and whenboth ẋ i = 0 and ẏ i = 0 for all i = 1, . . . , N, the steady state is reached.4.9.7.2 Solitary ‘Thought Nets’ and ‘Emerging Mind’Synergetic ‘Thought Solitons’Recall that synergetics teaches us that order parameters (and theirspatio–temporal evolution) are patterns, emerging from chaos. In our opinion,the most important of these order parameters, both natural and manmade, are solitons, because of their self–organizing quality to create orderout of chaos. From this perspective, nonlinearity – the essential characteristicof nature – is the cause of both chaos and order. Recall that the solitaryparticle–waves, also called the ‘light bullets’, are localized space–time excitationsΨ(x, t), propagating through a certain medium Ω with constantvelocities v j . They describe a variety of nonlinear wave phenomena in onedimension and playing important roles in optical fibers, many branches ofphysics, chemistry and biology.To derive our solitary network we start with modelling the conservative‘thought solitons’, using the following three classical nonlinear equations,defining the time evolution of the spatio–temporal wave function Ψ(x, t)(which is smooth, and either complex–, or real–valued) (see [Novikov et al.(1984); Fordy (1990); Ablowitz and Clarkson (1991); <strong>Ivancevic</strong> and Pearce(2001a)]; also compare with (3.13.2) above):(1) Nonlinear Schrödinger (NS) equationiΨ t = 2µ|Ψ| 2 Ψ − Ψ xx , (4.118)where Ψ = Ψ(x, t) is a complex-valued wave function with initial conditionΨ(x, t)| t=0 = Ψ(x) and µ is a nonlinear parameter representingfield strength. In the linear limit (a = 0) NS becomes the ordinarySchrödinger equation for the wave function of the free 1D particle withmass m = 1/2. Its Hamiltonian functionH NS =∫ +∞−∞(µ|Ψ| 4 + |Ψ x | 2) dx,is equal to the total and conserved energy of the soliton. NS describes,


598 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor example, nonlinear Faraday resonance in a vertically oscillating water,an easy–plane ferromagnet with a combination of a stationary anda high–frequency magnetic fields, and the effect of phase–sensitive amplifierson solitons propagating in optical fibers.(2) Korteveg–de Vries (KdV) equationΨ t = 6ΨΨ x − Ψ xxx ,with Hamiltonian (total conserved energy) given byH KdV =∫ +∞−∞(Ψ 3 + 1 )2 Ψ2 x dx.KdV is related to the ordinary Schrödinger equation by the inversescattering method. KdV is a well–known model of 1D turbulence thatwas derived in various physical contexts, including chemical–reactionwaves, propagation of combustion fronts in gases, surface waves in afilm of a viscous liquid flowing along an inclined plane, patterns inthermal convection, rapid solidification, and others. Its discretizationgives the Lotka–Voltera equationẋ j (t) = x j (t) ( x j+1 (t) − x j−1 (t) ) ,which appears in a model of struggle for existence of biological species.(3) Sine–Gordon (SG) equationΨ tt = Ψ xx − sin Ψ,with Hamiltonian (total conserved enegy) given byH SG =∫ +∞−∞(Ψ2t + Ψ 2 x + cos Ψ ) dx.SG gives one of the simplest models of the unified field theory, can befound in the theory of dislocations in metals, in the theory of Josephsonjunctions and so on. It can be used also in interpreting certain biologicalprocesses like DNA dynamics. Its discretization gives a system ofcoupled pendulums.Discrete solitons exist also in the form of the soliton celular automata(SCA) [Park et al. (1986)]. SCA is a 1(space)+1(time)–dimensional ‘boxand ball system’ made of infinite number of zeros (or, boxes) and finite


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 599number of ones (or, balls). The value of the jth SCA cell a j t at a discretetime time t, is given as⎧⎨ 1, if a j ∑ j−1a j t = 0 andt+1 = i=−∞ ui t > ∑ j−1i=−∞ ai t+1 ,⎩0, otherwise,where a j t = 0 is assumed for |j| ≫ 1. Any state of the SCA consists purelyof solitons (particularly, KdV–solitons), possessing conserved quantities ofthe form of H KdV . All of these properties have motivated a number ofsuggestive applications for a new kind of computational architecture thatwill use these evolution patterns of SCA in order to give a ‘gateless’ implementationof logical operations.In practice, both SCA and KdV are usually approximated by the Todalattice equation,¨q i = e qi+1 −q i − e qi −q i−1 , (i = 1, ..., N) (4.119)with quasiperiodic q N+i (t) = q i (t) + c, or,fast–dacaying boundary conditionslimi→−∞ qi (t) = 0,limi→+∞ qi (t) = c.The Toda equation (4.119) is a gradient Newtonian equation of motion¨q i = −∂ i qV, V (q) =N∑e qi+1 −q i .Otherwise, the Toda equation represents a Hamiltonian systemi=1˙q i = p i , ṗ i = e qi+1 −q i − e qi −q i−1 ,with the phase–space P = R 2N with coordinates (p i , q i ), standard Poissonstructure{p i , p j } = {q i , q j } = 0, {p i , q j } = δ j i ,N∑and Hamiltonian function H = ( 1 2 p2 i + e qi+1 −q i ), (i, j = 1, ..., N).i=1Next, to make our conservative thought solitons open to the environment,we have to modify them by adding:1. Input from the senses, in the form of the Weber–Fechner’s law,S(t) = a r log s r (t), (r = 1, ..., 5), (4.120)


600 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere S = S(t) is the sensation, s r = s r (t) the vector of stimuli from thefive senses, and a r a constant vector; and2. Disturbances, in the form of additive, zero–mean Gaussian whitenoise η = η(t), independent from the main soliton–signal.In this way, we get the modified solitary equations:MNS : iΨ t = 2µ|Ψ| 2 Ψ − Ψ xx + a r log s r Ψ + η,MKdV : Ψ t = 6ΨΨ x − Ψ xxx + a r log s r Ψ + η,MSG : Ψ tt = Ψ xx − sin Ψ + a r log s r Ψ + η,representing the three different models of the thought units.Now we will form a single emerging order–parameter, the general factor,that we call the Mind. It behaves like an orchestrated ensemble of thoughtsolitons, defined as systems of trainable, coupled soliton equations. Theirtensor couplings perform self–organizing associative learning by trial anderror, similar to that of the neural ensemble.The dynamics of the soliton ensemble, representing our model of the‘mind’ can be described as one of the following three soliton systems; eachof them performs learning, growing and competing between each other, andcommunicates with environment through the sensory inputs and the heatingnoise:1. Coupled modified nonlinear Schrödinger equationsiΨ k t= −Ψ k xx + 2µ k ∑ j≠k|Ψ k | 2 W j k Sj (Ψ j )+ ν k Ψ k (1 − ɛ k Ψ k ) + a r log s r Ψ k + η k ,2. Coupled modified Korteveg–de Vries equationsΨ k t = 6Ψ k x Ψ k − Ψ k xxx + ∑ j≠kW j k Sj (Ψ j )+ ν k Ψ k (1 − ɛ k Ψ k ) + a r log s r Ψ k + η k ,3. Coupled modified Sine–Gordon equationsΨ k tt = Ψ k xx − sin Ψ k + ∑ j≠k+ν k Ψ k (1 − ɛ k Ψ k ) + a r log s r Ψ k + η k ,


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 601where Ψ k = Ψ k (x, t), (k = 1, ..., n) is the set of wave functions of thesolitary thoughts, S(·) represents the sigmoidal threshold functions, ν k andɛ k are growing and competition parameters.W j k = Wj k(Ψ) are tensorial learning couplings, evolving according tothe Hebbian learning scheme (see [Kosko (1992)]):Ẇ j k = −Wj k + Φj k (Ψk , Ψ j ),with innovation defined in tensor signal form (here Ṡ(·) = 1 − tanh(·))Φ j k = Sj (Ψ j ) S k (Ψ k ) + Ṡj (Ψ j ) Ṡk (Ψ k ).Emerging Categorical Structure: MAT T ER ⇒ LIF E ⇒ MINDThesolitarythought nets effectively simulate the following 3−categorical structure ofMIND, emerging from the 2−categorical structure of LIFE, which is itselfemerging from the 1−categorical structure of MATTER:fF(f)Aαψ ❘> βFB ❅ ✒g ❅❅❅❅❅❅❅❅❅❘G ◦ FMINDG(F(f))G(F(ψ))G(F(α))G(F(A))G(F(g))F(ψ)✲ F(A) F(α) > F(g)G✠ ❘> G(F(β))✒ G(F(B))❘F(β) F(B)✒


602 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAfα ❘ FB∨✒g ❅❅❅❅❅❅❅❘G ◦ FG(F (A))LIFEG(F (f))∨G(F (g))F (f)✲❘F (A)F (α)F (B)∨ ✒F (g) G✠ ❘G(F (α))G(F (B))✒Af✲ f(A)❅ MATTERg ◦ f ❅ ❅ g ❅❘ ✠g(f(A))4.9.8 <strong>Geom</strong>etrodynamics of Human CrowdIn this subsection we formulate crowd representation model as an emotion–field induced collective behavior of individual autonomous agents [<strong>Ivancevic</strong>and Snoswell (2000)].It is well–known that crowd behavior is more influenced by collectiveemotion than by cognition. Recall from previous subsection that accordingto Lewinian psychodynamics, human behavior is largely determined byunderlying forces (needs). For him, a force–field is defined as ‘the totalityof coexisting motivational facts which are conceived of as mutually interdependent’[Lewin (1997)]. He also stresses psychological direction andvelocity of behavior.On the other hand, a number of factor–analysis based studies showthat human emotion is not a single quantity, but rather a multidimensionalspace. For example, in [Skiffington (1998)], authors assessed emotions withsingle adjective descriptors using standard linear factor analysis, by examiningsemantic as well as cognitive, motivational, and intensity features ofemotions. The focus was on seven negative emotions common to severalemotion typologies: anger, fear, sadness, shame, pity, jealousy, and contempt.For each of these emotions, seven items were generated correspond-


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 603ing to cognitive appraisal about the self, cognitive appraisal about the environment,action tendency, action fantasy, synonym, antonym, and intensityrange of the emotion, respectively. These findings set the groundwork forthe construction of an instrument to assess emotions multicomponentially.4.9.8.1 Crowd HypothesisWe consider a human crowd C as a group of m autonomous agents A i(i = 1, ..., m), each of which carries its own nD motivational factor–structure. This nonlinear factor structure, which can be get using modernnonlinear factor analysis techniques (see [Yalcin and Amemiya (2001);Amemiya (1993); Wall and Amemiya (1998); Wall and Amemiya (2000)];also compare with subsection 3.11.4 below), is defined by n hypotheticalmotivational factor–coordinates q i = {q µ i }, (µ = 1, . . . , n), spanning thesmooth nD motivational factor manifold M i for each autonomous agentA i .We understand crowd representation as an environmental field–inducedcollective behavior of individual autonomous agents. To model it in a generalgeometrodynamical framework, we firstly define the behavior of eachagent A i as a motion π i along his motivational manifold M i , caused by hisown emotion field Φ i , which is an active (motor) subset of M i .Secondly, we formulate a collective geometrodynamical model for thecrowd, considered as a union C = ∪ i A i ,in the form of a divergence equationfor the total crowd’s SEM–tensor C. 144.9.8.2 <strong>Geom</strong>etrodynamics of Individual AgentsTo formulate individual agents’ geometrodynamics, we firstly derive twohigher geometrical structures from a motivational factor manifold M i correspondingto an agent A i : (i) the agent’s velocity phase–space, defined as atangent bundle T M i , and (ii) the agent’s momentum phase–space, definedas a cotangent bundle T ∗ M i .Now, the sections of T M i we call the agent’s vector–fields v i , which canbe expanded in terms of the basis vector–fields {e i µ ≡ ∂ qµ }, as v i = v µ i i ei µ.Similarly, the sections of T ∗ M i we call the agent’s one–forms α i , which canbe expanded in terms of the basis one–forms {ω µ i ≡ dq µ i }, as α i = α i µ ω µ i .Here d denotes the exterior derivative (such that dd = 0). In particular,14 Throughout the text we use the following index convention: we label individualagents using Latin indices, and individual motivational factors using Greek indices; summationconvention is applied only to Greek factor indices.


604 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe velocity vector–fields ˙q i are defined on T M i as ˙q i = ˙q µ i ei µ,while themomentum one–forms π i are defined on T ∗ M i as π i = π i µ ω µ i .Also, all factor–configuration manifolds M i are assumed to be Riemannian,admitting the metrics g i = 〈ω µ i , ωη i 〉, determined by kinetic energiesT i = 〈 ˙q µ i , ˙qη i 〉 of individual agents A i, each with the metric tensorg i = gµη i ω µ i ⊗ ωη i . This implies that all vector–fields v i and one–forms α iare related by g−induced scalar products 〈 ω µ i , 〉 ei η = δiµη .Emotional/Environmental Fields and Induced Agents’ MotionsNow, for each autonomous agent A i three additional geometrodynamicalobjects are defined as:(1) Emotional/environmental potential one–form α i = α i µ ω µ i , which is thegradient of some scalar function f i = f µ i (qµ i ) on M i,α i = df i , in components, α i µ = ∂ qµi f µ i ωµ i .(2) Psycho–physical current vector–field J i = J µ i ei µ on M i , defined throughits motivational charge e i asJ µ i = e i∫t i˙q µ i δn [q µ i (t i)] dt i ,where δ n = δ n [q µ i (t i)] denotes the nD impulse delta–function definedon M i .(3) Emotional/environmental psycho–physical field is a two–formΦ i = Φ i µη ω µ i ⊗ ωη i = 1 2 Φi µη ω µ i ∧ ωη ion M i , defined as the exterior derivative (i.e., curl) of theemotional/enviro-nmental potential α i ,Φ i = dα i , in components, Φ i µη = α i η;µ − α i µ;η.The emotional/environmental psycho–physical field Φ i is governed bytwo standard field equations:dΦ i = ddα i = 0, in components, Φ i [µη;ν] = 0, (4.121)anddiv Φ i = g i J i , in components, Φ i;ηµη = g i µη J η i , (4.122)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 605where [µη; ν] ≡ µη; ν + ην; µ + vµ; η. The first field equation (4.121) statesthat for each agent A i the motivational tension–field Φ i is curl–free, whilethe second field equation (4.122) states that the environmental psycho–physical field Φ i has its source in the psycho–physical current J i .Now, the behavioral equation for each agent A i , induced by his/herenvironmental psycho–physical field Φ i , reads˙π i = e i Φ i ˙q i , in components, ˙π i µ = e i Φ i µη ˙q η i .This equation states that the force of an individual agent’s motion, orhis/her behavior, equals the product of his/her psycho–physical charge,environmental psycho–physical field, and velocity (speed) of behavior.4.9.8.3 Collective Crowd <strong>Geom</strong>etrodynamicsNow we define the total crowd’s geometrodynamics as a union C = ∪ i A i ofall individual agents’ geometrodynamics, to model their emotion–inducedbehavior. For this we use the total crowd’s energy–momentum tensor(CEM) and its divergence equation of motion.As a union of individual Riemannian manifolds, the total crowd’s manifoldM = ∪ i M i is also Riemannian, with the metrics g = ∑ i gi equalto the sum of individual metrics g i = gµη i ω µ i ⊗ ω η i . Now, the crowd’sCEM tensor C = C µη ω µ M ⊗ ωη M = 1 2 C µη ω µ M ∧ ωη M (where ωµ M denotethe basis one–forms on M) has two parts, C = C (E) + C (B) , in components,C µη = C µη (E) + C µη (B) , corresponding to the total crowd’s emotion andbehavior, which we define respectively as follows:(1) The CEM’s emotional part C (E) is in components defined as a sum ofindividual agents’ motivational–tension fields,C (E)µη =M∑i=1(Φiµν Φ ν iη − gµηΦ i i νλΦ νλ )i ,so the equation of the crowd’s emotion is defined in the form of thedivergence equationC (E);ηµη =M∑i=1Φ i µηΦ η;νiνM = − ∑Φ i µη J η i .i=1This equation says that CEM’s emotional part C (E)motivational field with a sink.is a collective


606 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(2) The CEM’s behavioral part C (B) as a sum of individual agents’motivational–currents times momenta:M∑∫C µη (B) = gµηi π i η(t i ) ˙q η i (t i) δ n [q µ i (t i)] dt i ,t ii=1so the equation of the crowd’s behavior is defined in the form of thedivergence equationC (B);ηµη =M∑i=1i=1g i µη∫π i η(t i ) ˙q η i (t i) ∂ qη δn [q µ i i (t i)] dt it iM∑M∑= e i Φ∫t i µη ˙q η i δn [q µ i (t i)] dt i = Φ i µη J η i .ii=1This equation says that CEM’s behavioral part C (B)behavioral field with a source.is a collectiveTherefore, the divergence equation for the total crowd’s CEM tensor,represents the crowd’s motivation–behavior conservation law(div C = div C (E) + C (B)) = 0, in components, C µη (E);η +C µη (B);η = 0.This gives our basic representation of an isolated crowd as a conservativespatio–temporal dynamical system. Naturally, additional crowd’s energy–momentum sources and sinks can violate this basic motivational–behaviorconservation law.4.10 Multivector–Fields and Tangent–Valued FormsRecall that a vector–field on a manifold M is defined as a global sectionof the tangent bundle T M → M. The set V 1 (M) of vector–fields on M isa real Lie algebra with respect to the Lie bracket [Sardanashvily (1993);Sardanashvily (1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily(2000a); Sardanashvily (2002a)][v, u] = (v α ∂ α u µ − u α ∂ α v µ )∂ µ , v = v α ∂ α , u = u α ∂ α . (4.123)Every vector–field on a manifold M can be seen as an infinitesimal generatorof a local 1–parameter Lie group of diffeomorphisms of M as follows[Kobayashi and Nomizu (1963/9)]. Given an open subset U ⊂ M and an


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 607interval (−ɛ, ɛ) ∈ R, by a local 1–parameter group of diffeomorphisms of Mdefined on (−ɛ, ɛ) × U is denoted a mapG → M,(t, x) ↦→ G t (x)such that:(1) for each t ∈ (−ɛ, ɛ), the map G t is a diffeomorphism of U onto the opensubset G t (U) ⊂ M; and(2) G t+t ′(x) = (G t ◦ G t ′)(x) if t, t ′ , t + t ′ ∈ (−ɛ, ɛ) and G t ′(x), x ∈ U.Any local 1–parameter group of diffeomorphisms G on U ⊂ M definesa local vector–field u on U by setting u(x) to be the tangent vector to thecurve x(t) = G t (x) at t = 0. Conversely, if u is a vector–field on a manifoldM, there exists a unique local 1–parameter group G u of diffeomorphisms ona neighborhood of every point x ∈ M which defines u. We call G u a flow ofthe vector–field u. A vector–field u on a manifold M is called complete if itsflow is a 1–parameter group of diffeomorphisms of M. In particular, everyvector–field on a compact manifold is complete [Kobayashi and Nomizu(1963/9)].A vector–field u on a fibre bundle Y −→ X is an infinitesimal generatorof a local 1–parameter group G u of isomorphisms of Y −→ X iff it is aprojectable vector–field on Y . A vector–field u on a fibre bundle Y −→ Xis called projectable if it projects onto a vector–field on X, i.e., there existsa vector–field τ on X such that the following diagram commutes:πY❄Xuτ✲ T YT π❄✲ T XA projectable vector–field has the coordinate expressionu = u α (x µ )∂ α + u i (x µ , y j )∂ i ,where u α are local functions on X. A projectable vector–field is said to bevertical if it projects onto the zero vector–field τ = 0 on X, i.e., u = u i ∂ itakes its values in the vertical tangent bundle V Y .For example, in field theory, projectable vector–fields on fibre bundlesplay a role of infinitesimal generators of local 1–parameter groups of gaugetransformations.


608 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn general, a vector–field τ = τ α ∂ α on a base X of a fibre bundle Y → Xinduces a vector–field on Y by means of a connection on this fibre bundle.Nevertheless, every natural fibre bundle Y → X admits the canonical lift ˜τonto Y of any vector–field τ on X. For example, if Y is the tensor bundle(4.11), the above canonical lift reads˜τ = τ µ ∂ µ + [∂ ν τ α1 ẋ να2···αmβ 1···β k+ . . . − ∂ β1 τ ν ẋ α1···αmνβ 2···β k− . . .]∂. (4.124)∂ẋ α1···αmβ 1···β kIn particular, we have the canonical lift onto the tangent bundle T X,and another one onto the cotangent bundle T ∗ X,˜τ = τ µ ∂ µ + ∂ ν τ α ẋ ν ∂∂ẋ α (4.125)˜τ = τ µ ∂ µ − ∂ β τ ν ∂ ẋ ν . (4.126)∂ẋ βA multivector–field ϑ of degree r (or simply a r-vector–field) on a manifoldM, by definition, is a global section of the bundle ∧ r T M → M. It isgiven by the coordinate expressionϑ = ϑ α1...αr ∂ α1 ∧ · · · ∧ ∂ αr , |ϑ| = r,where summation is over all ordered collections (λ 1 , ..., λ r ).Similarly, an exterior r−form on a manifold M with local coordinatesx α , by definition, is a global section of the skew–symmetric tensor bundle(exterior product) ∧ r T ∗ M → M,φ = 1 r! φ α 1...α rdx α1 ∧ · · · ∧ dx αr , |φ| = r.The 1–forms are also called the Pfaffian forms.The vector space V r (M) of r−vector–fields on a manifold M admits theSchouten–Nijenhuis bracket (or, SN bracket)[., .] SN : V r (M)×V s (M) → V r+s−1 (M)which generalizes the Lie bracket of vector–fields (4.123). The SN–brackethas the coordinate expression:ϑ = ϑ α1...αr ∂ α1 ∧ · · · ∧ ∂ αr , υ = υ α1...αs ∂ α1 ∧ · · · ∧ ∂ αs ,[ϑ, υ] SN = ϑ ⋆ υ + (−1) |ϑ||υ| υ ⋆ ϑ,whereϑ ⋆ υ = ϑ µα1...αr−1 ∂ µ υ α1...αs ∂ α1 ∧ · · · ∧ ∂ αr−1 ∧ ∂ α1 ∧ · · · ∧ ∂ αs .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 609The following relations hold for the SN–bracket:[ϑ, υ] SN = (−1) |ϑ||υ| [υ, ϑ] SN ,[ν, ϑ ∧ υ] SN = [ν, ϑ] SN ∧ υ + (−1) |ν||ϑ|+|ϑ| ϑ ∧ [ν, υ] SN ,(−1) |ν||ϑ|+|ν| [ν, ϑ ∧ υ] SN + (−1) |ϑ||ν|+|ϑ| [ϑ, υ ∧ ν] SN+ (−1) |υ||ϑ|+|υ| [υ, ν ∧ ϑ] SN = 0.In particular, let w = w µν ∂ µ ∧ ∂ ν be a bivector–field. We have[w, w] SN = w µα1 ∂ µ w α2α3 ∂ α1 ∧ ∂ α2 ∧ ∂ α3 . (4.127)Every bivector–field w on a manifold M induces the ‘sharp’ bundle mapw ♯ : T ∗ M → T M defined byw ♯ (p)⌋q := w(x)(p, q), w ♯ (p) = w µν (x)p µ ∂ ν , (p, q ∈ T ∗ x M).(4.128)A bivector–field w whose bracket (4.127) vanishes is called the Poissonbivector–field.Let ∧ r (M) denote the vector space of exterior r−forms on a manifoldM. By definition, ∧ 0 (M) = C ∞ (M) is the ring of smooth real functionson M. All exterior forms on M constitute the N−graded exterior algebra∧ ∗ (M) of global sections of the exterior bundle ∧T ∗ M with respect to theexterior product ∧. This algebra admits the exterior differentiald : ∧ r (M) → ∧ r+1 (M),dφ = dx µ ∧ ∂ µ φ = 1 r! ∂ µφ α1...α rdx µ ∧ dx α1 ∧ · · · dx αr ,which is nilpotent, i.e., d ◦ d = 0, and obeys the relationd(φ ∧ σ) = d(φ) ∧ σ + (−1) |φ| φ ∧ d(σ).The interior product (or, contraction) of a vector–field u = u µ ∂ µ and anexterior r−form φ on a manifold M is given by the coordinate expressionu⌋φ ==r∑ (−1) k−1u α kφr!α1...α k ...α rdx α1 ∧ · · · ∧ ̂dx α k∧ · · · ∧ dx(4.129)αrk=11(r − 1)! uµ φ µα2...α rdx α2 ∧ · · · ∧ dx αr ,


610 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere the caret ̂· denotes omission. The following relations hold:φ(u 1 , . . . , u r ) = u r ⌋ · · · u 1 ⌋φ, (4.130)u⌋(φ ∧ σ) = u⌋φ ∧ σ + (−1) |φ| φ ∧ u⌋σ, (4.131)[u, u ′ ]⌋φ = u⌋d(u ′ ⌋φ) − u ′ ⌋d(u⌋φ) − u ′ ⌋u⌋dφ, (φ ∈ ∧ 1 (M)). (4.132)Recall from section 3.7 above, that the Lie derivative L u σ of an exteriorform σ along a vector–field u is defined by the Cartan relationIt satisfies the relationIn particular, if f is a function, thenL u σ = u⌋dσ + d(u⌋σ).L u (φ ∧ σ) = L u φ ∧ σ + φ ∧ L u σ.L u f = u(f) = u⌋df.It is important for dynamical applications that an exterior form φ is invariantunder a local 1–parameter group of diffeomorphisms G t of M (i.e.,G ∗ t φ = φ) iff its Lie derivative L u φ along the vector–field u, generating G t ,vanishes.Let Ω be a two–form on M. It defines the ‘flat’ bundle map Ω ♭ , asΩ ♭ : T M → T ∗ M, Ω ♭ (v) = −v⌋Ω(x), (v ∈ T x M). (4.133)In coordinates, if Ω = Ω µν dx µ ∧ dx ν and v = v µ ∂ µ , thenΩ ♭ (v) = −Ω µν v µ dx ν .One says that Ω is of constant rank k if the corresponding map (4.133) isof constant rank k (i.e., k is the greatest integer n such that Ω n is not thezero form). The rank of a nondegenerate two–form is equal to dim M. Anondegenerate closed two–form is called the symplectic form.Given a manifold map f : M → M ′ , any exterior k-form φ on M ′induces the pull–back exterior form f ∗ φ on M by the conditionf ∗ φ(v 1 , . . . , v k )(x) = φ(T f(v 1 ), . . . , T f(v k ))(f(x))for an arbitrary collection of tangent vectors v 1 , · · · , v k ∈ T x M. The followingrelations hold:f ∗ (φ ∧ σ) = f ∗ φ ∧ f ∗ σ,df ∗ φ = f ∗ (dφ).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 611In particular, given a fibre bundle π : Y → X, the pull–back onto Y ofexterior forms on X by π gives the monomorphism of exterior algebrasπ ∗ : ∧ ∗ (X) → ∧ ∗ (Y ).Elements of its image π ∗ ∧ ∗ (X) are called basic forms. Exterior formson Y such that u⌋φ = 0 for an arbitrary vertical vector–field u on Y aresaid to be horizontal forms. They are generated by horizontal 1–forms{dx α }. For example, basic forms are horizontal forms with coefficients inC ∞ (X) ⊂ C ∞ (Y ). A horizontal form of degree n = dim X is called adensity. For example, Lagrangians in field theory are densities.Elements of the tensor product ∧ r (M) ⊗ V 1 (M) are called the tangent–valued r−forms on M. They are sectionsof the tensor bundleφ = 1 r! φµ α 1...α rdx α1 ∧ · · · ∧ dx αr ⊗ ∂ µ∧ r T ∗ M ⊗ T M → M.Tangent-valued 1–forms are usually called the (1,1) tensor–fields.In particular, there is the 1–1 correspondence between the tangent–valued 1–forms on M and the linear bundle maps over M,φ : T M → T M, φ : T x M ∋ v ↦→ v⌋φ(x) ∈ T x M. (4.134)In particular, the canonical tangent–valued one–form θ M = dx α ⊗∂ α definesthe identity map of T M.Tangent-valued forms play a prominent role in jet formalism and theoryof connections on fibre bundles. In particular, tangent–valued 0-forms arevector–fields on M. Also, there is 1–1 correspondence between the tangent–valued 1–forms φ on a manifold M and the linear bundle endomorphismŝφ : T M → T M, ̂φ : Tx M ∋ v ↦→ v⌋φ(x) ∈ T x M, (4.135)̂φ ∗ : T ∗ M → T ∗ M, ̂φ∗ : T∗x M ∋ v ∗ ↦→ φ(x)⌋v ∗ ∈ T ∗ x M, (4.136)over M. For example, the canonical tangent–valued 1–form on M,θ M = dx α ⊗ ∂ α , (4.137)corresponds to the identity maps (4.135) and (4.136).


612 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWe shall deal with the following particular types of vector–fields anddifferential forms on a bundle Y −→ X [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a)]:• a projectable vector–field on Y ,u = u µ (x)∂ µ + u i (y)∂ i ,which covers a vector–field τ u = u µ (x)∂ µfollowing diagram commutes:on the base X such that theYπ❄Xuτ u✲ T YT π❄✲ T X• a vertical vector–field, u : Y → V Y, given by u = u i (y)∂ i , is aprojectable vector–field which covers τ u = 0;• an exterior horizontal form, φ : Y → ∧ r T ∗ X, given byφ = 1 r! φ α 1...α r(y)dx α1 ∧ · · · ∧ dx αr ;• a tangent–valued horizontal form, φ : Y → ∧ r T ∗ X ⊗ T Y, given byφ = 1 r! dxα1 ∧ · · · ∧ dx αr ⊗ [φ µ α 1...α r(y)∂ µ + φ i α 1...α r(y)∂ i ];• a vertical–valued horizontal form, φ : Y → ∧ r T ∗ X ⊗ V Y, given byφ = 1 r! φi α 1...α r(y)dx α1 ∧ · · · ∧ dx αr ⊗ ∂ i .• a vertical-valued soldering form, σ : Y → T ∗ X ⊗ V Y, given byσ = σ i α(y)dx α ⊗ ∂ i (4.138)and, in particular, the canonical soldering form on T X,θ X = dx α ⊗ ∂ α .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 613The pull–back–valued forms on a bundle Y → X are the following twomaps: 15Y → ∧ r T ∗ Y ⊗ T X, φ = 1 r! φµ α 1...α r(y)dx α1 ∧ · · · ∧ dx αr ⊗ ∂ µ ,and (4.139)Y → ∧ r T ∗ Y ⊗ V ∗ X, φ = 1 r! φ α 1...α ri(y)dx α1 ∧ · · · ∧ dx αr ⊗ dy i .The pull–back-valued forms (4.139) are exemplified by the canonicalbundle monomorphism∧ n T ∗ X ⊗ V ∗ Y ↩→ ∧ n+1 T ∗ Y, ω ⊗ dy i ↦→ ω∧dy i .All horizontal n−forms on a bundle Y −→ X are called horizontal densities.For any vector–field τ on X, we can define its pull–back on Y ,π ∗ τ = τ ◦ π : Y −→ T X.This is not a vector–field on Y , for the tangent bundle T X of X fails to bea subbundle of the tangent bundle T Y of Y . One needs a connection on Y−→ X in order to set the imbedding T X ↩→ T Y .The space ∧ ∗ (M) ⊗ V 1 (M) of tangent–valued forms admits theFrölicher–Nijenhuis bracket (or, FN bracket)[., .] F N : ∧ r (M) ⊗ V 1 (M) × ∧ s (M) ⊗ V 1 (M) → ∧ r+s (M) ⊗ V 1 (M),[φ, σ] F N = 1r!s! (φν α 1...α r∂ ν σ µ α r+1...α r+s− σ ν α r+1...α r+s∂ ν φ µ α 1...α r− (4.140)rφ µ α 1...α r−1ν∂ αr σ ν α r+1...α r+s+ sσ µ να r+2...α r+s∂ αr+1 φ ν α 1...α r)dx α1∧ · · · ∧ dx αr+s ⊗ ∂ µ .15 The forms (4.139) are not tangent–valued forms. The pull–backsφ = 1 r! φµ α 1 ...α r(x)dx α 1∧ · · · ∧ dx αr ⊗ ∂ µof tangent–valued forms on X onto Y by π exemplify the pull–back-valued forms (4.139).In particular, we shall refer to the pull–back π ∗ θ X of the canonical form θ X on the baseX onto Y by π. This is a pull–back-valued horizontal one–form on Y which we denoteby the same symbolθ X : Y → T ∗ X ⊗ T X, θ X = dx α ⊗ ∂ α.


614 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe following relations hold for the FN–bracket:[φ, ψ] F N = (−1) |φ||ψ|+1 [ψ, φ] F N , (4.141)[φ, [ψ, θ] F N ] F N = [[φ, ψ] F N , θ] F N + (−1) |φ||ψ| [ψ, [φ, θ] F N ] F N .Given a tangent–valued form θ, the Nijenhuis differential, d θ σ, along θon ∧ ∗ (M) ⊗ V 1 (M) is defined asd θ σ = [θ, σ] F N . (4.142)By virtue of the relation (4.141), it has the propertyd φ [ψ, θ] F N = [d φ ψ, θ] F N + (−1) |φ||ψ| [ψ, d φ θ] F N .In particular, if θ = u is a vector–field, the Nijenhuis differential becomesthe Lie derivative of tangent–valued formsL u σ = d u σ = [u, σ] F N = (u ν ∂ ν σ µ α 1...α s− σ ν α 1...α s∂ ν u µ (4.143)+ sσ µ να 2...α s∂ α1 u ν )dx α1 ∧ · · · ∧ dx αs ⊗ ∂ µ , (σ ∈ ∧ s (M) ⊗ V(M)).4.11 Application: <strong>Geom</strong>etrical Quantization4.11.1 Quantization of Hamiltonian MechanicsRecall that classical Dirac quantization states [Dirac (1982)]:{f, g} = 1i [ ˆf, ĝ],which means that the quantum Poisson brackets (i.e., commutators) havethe same values as the classical Poisson brackets. In other words, we canassociate smooth functions defined on the symplectic phase–space manifold(M, ω) of the classical biodynamic system with operators on a Hilbert spaceH in such a way that the Poisson brackets correspond. Therefore, there is afunctor from the category Symplec to the category Hilbert. This functoris called prequantization.Let us start with the simplest symplectic manifold (M = T ∗ R n , ω =dp i ∧ dq i ) and state the Dirac problem: A prequantization of (T ∗ R n , ω =dp i ∧ dq i ) is a map δ : f ↦→ δ f , taking smooth functions f ∈ C ∞ (T ∗ R n , R)to Hermitian operators δ f on a Hilbert space H, satisfying the Dirac conditions:(1) δ f+g = δ f + δ g , for each f, g ∈ C ∞ (T ∗ R n , R);


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 615(2) δ λf = λδ f , for each f ∈ C ∞ (T ∗ R n , R) and λ ∈ R;(3) δ 1R n = Id H ; and(4) [δ f , δ g ] = (δ f ◦ δ g − δ g ◦ δ f ) = iδ {f,g}ω , for each f, g ∈ C ∞ (T ∗ R n , R);The pair (H, δ), whereH = L 2 (R n , C); δ : f ∈ C ∞ (T ∗ R n , R) ↦→ δ f : H → H;δ f = −iX f − θ(X f ) + f; θ = p i dq i ,gives a prequantization of (T ∗ R n , dp i ∧ dq i ), or equivalently, the answer tothe Dirac problem is affirmative [Puta (1993)].Now, let (M = T ∗ Q, ω) be the cotangent bundle of an arbitrary manifoldQ with its canonical symplectic ( structure ω = dθ. The prequantizationof M is given by the pair L 2 (M, C),δ θ) , where for each f ∈ C ∞ (M, R),the operator δ θ f : L 2 (M, C) →L 2 (M, C) is given byδ θ f = −iX f − θ(X f ) + f.Here, symplectic potential θ is not uniquely determined by the conditionω = dθ; for instance θ ′ = θ +du has the same property for any real functionu on M. On the other hand, in the general case of an arbitrary symplecticmanifold (M, ω) (not necessarily the cotangent bundle) we can find onlylocally a 1–form θ such that ω = dθ.In general, a symplectic manifold (M, ω = dθ) is quantizable (i.e., we candefine the Hilbert representation space H and the prequantum operator δ fin a globally consistent way) if ω defines an integral cohomology class. Now,by the construction Theorem of a fiber bundle, we can see that this conditionon ω is also sufficient to guarantee the existence of a complex line bundleL ω = (L, π, M) over M, which has exp(i u ji /) as gauge transformationsassociated to an open cover U = {U i |i ∈ I} of M such that θ i is a symplecticpotential defined on U i (i.e., dθ i = ω and θ i = θ i + d u ji on U i ∩ U j ).In particular, for exact symplectic structures ω (as in the case of cotangentbundles with their canonical symplectic structures) an integral cohomologycondition is automatically satisfied, since then we have only one setU i = M and do not need any gauge transformations.Now, for each vector–field X ∈ M there exists an operator ∇ ω X on thespace of sections Γ(L ω ) of L ω ,∇ ω X : Γ(L ω ) → Γ(L ω ), given by ∇ ω Xf = X(f) − i θ(X)f,


616 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand it is easy to see that ∇ ω is a connection on L ω whose curvature is ω/i.In terms of this connection, the definition of δ f becomesδ f = −i∇ ω X f+ f.The complex line bundle L ω = (L, π, M) together with its compatibleconnection and Hermitian structure is usually called the prequantum bundleof the symplectic manifold (M, ω).If (M, ω) is a quantizable manifold then the pair (H, δ) defines its prequantization.ExamplesEach exact symplectic manifold (M, ω = dθ) is quantizable, for the cohomologyclass defined by ω is zero. In particular, the cotangent bundle, withits canonical symplectic structure is always quantizable.Let (M, ω = dθ) be an exact symplectic manifold. Then it is quantizablewith the prequantum bundle given by [Puta (1993)]:L ω = (M × C, pr 1 , M);Γ(L ω ) ≃ C ∞ (M, C);∇ ω Xf = X(f) − i θ(X)f;((x, z 1 ), (x, z 2 )) x = ¯z 1 z 2 ; δ f = −i[X f − i θ(X f )] + f.Let (M, ω) = (T ∗ R, dp ∧ dq). It is quantizable with [Puta (1993)]:L ω = (R 2 × C, pr 1 , R 2 );Γ(L ω ) = C ∞ (R 2 , C);Therefore,∇ ω Xf = X(f) − i pdq(X)f; ((x, z 1) , (x, z 2 )) x= ¯z 1 z 2 ;[ ∂f ∂δ f = −i∂p ∂q − ∂f ]∂− p ∂f∂q ∂p ∂p + f.δ q = i ∂ ∂p + q, δ p = −i ∂ ∂q ,which differs from the classical result of the Schrödinger quantization:δ q = q, δ p = −i ∂ ∂q .Let H be a complex Hilbert space and U t : H → H a continuous one–parameter unitary group, i.e., a homomorphism t ↦→ U t from R to the group


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 617of unitary operators on H such that for each x ∈ H the map t ↦→ U t (x) iscontinuous. Then we have the self–adjoint generator A of U t , defined byAx = 1 iddt U t(x) = 1 i limh→0U h (x) − x.hLet ( R 2 , ω = dp ∧ dq, H = 1 2 (p2 + q 2) be the Hamiltonian structure ofthe 1D harmonic oscillator.If we take θ = 1 2(pdq − qdp) as the symplectic(potential)of ω, thenthe spectrum of the prequantum operator δ H = i q ∂ ∂p − p ∂ ∂qis [Puta(1993)]Spec(δ H ) = {..., −2, −, 0, , 2, ...}, where each eigenvalue occurs withinfinite multiplicity.Let g be the vector space spanned by the prequantum operatorsδ q , δ p , δ H , given byδ q = i ∂ ∂p + q, δ p = −i ∂ ∂q , (δ H = i q ∂ ∂p − p ∂ ),∂qand Id. Then we have [Puta (1993)]:(1) g is a Lie algebra called the oscillator Lie algebra, given by:[δ p , δ q ] = iδ {p,q}ω = i Id,[δ H , δ q ] = iδ {H,q}ω = −iδ p ,[δ H , δ p ] = iδ {H,p}ω = iδ q ,(2) [g, g] is spanned by δ q , δ p , δ H and Id, or equivalently, it is a HeisenbergLie algebra.(3) The oscillator Lie algebra g is solvable.4.11.2 Quantization of Relativistic Hamiltonian MechanicsGiven a symplectic manifold (Z, Ω) and a Hamiltonian H on Z, a Diracconstraint system on a closed imbedded submanifold i N : N −→ Z of Z is definedas a Hamiltonian system on N admitting the pull–back presymplecticform Ω N = i ∗ N Ω and the pull–back Hamiltonian i∗ N H [Gotay et. al. (1978);Mangiarotti and Sardanashvily (1998); Muñoz and Román (1992)]. Its solutionis a vector–field γ on N which fulfils the equationγ⌋Ω N + i ∗ NdH = 0.


618 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet N be coisotropic. Then a solution exists if the Poisson bracket {H, f}vanishes on N whenever f is a function vanishing on N. It is the Hamiltonianvector–field of H on Z restricted to N [Sardanashvily (2003)].Recall that a configuration space of non–relativistic time–dependentmechanics (henceforth NRM) of m degrees of freedom is an (m+1)D smoothfibre bundle Q −→ R over the time axis R [Mangiarotti and Sardanashvily(1998); Sardanashvily (1998)]. It is coordinated by (q α ) = (q 0 , q i ), whereq 0 = t is the standard Cartesian coordinate on R. Let T ∗ Q be the cotangentbundle of Q equipped with the induced coordinates (q α , p α = ˙q α ) withrespect to the holonomic coframes {dq α }. The cotangent bundle T ∗ Q playsthe role of a homogeneous momentum phase–space of NRM, admitting thecanonical symplectic formΩ = dp α ∧ dq α . (4.144)Its momentum phase–space is the vertical cotangent bundle V ∗ Q of theconfiguration bundle Q −→ R, coordinated by (q α , q i ). A Hamiltonian Hof NRM is defined as a section p 0 = −H of the fibre bundle T ∗ Q −→ V ∗ Q.Then the momentum phase–space of NRM can be identified with the imageN of H in T ∗ Q which is the one-codimensional (consequently, coisotropic)imbedded submanifold given by the constraintH T = p 0 + H(q α , p k ) = 0.Furthermore, a solution of a non–relativistic Hamiltonian system with aHamiltonian H is the restriction γ to N ∼ = V ∗ Q of the Hamiltonian vector–field of H T on T ∗ Q. It obeys the equation γ⌋Ω N = 0 [Mangiarotti andSardanashvily (1998); Sardanashvily (1998)]. Moreover, one can show thatgeometrical quantization of V ∗ Q is equivalent to geometrical quantizationof the cotangent bundle T ∗ Q where the quantum constraint ĤT ψ = 0on sections ψ of the quantum bundle serves as the Schrödinger equation[Sardanashvily (2003)].A configuration space of relativistic mechanics (henceforth RM) is anoriented pseudo–Riemannian manifold (Q, g), coordinated by (t, q i ). Itsmomentum phase–space is the cotangent bundle T ∗ Q provided with thesymplectic form Ω (4.144). Note that one also considers another symplecticform Ω + F where F is the strength of an electromagnetic field [Sniatycki(1980)]. A relativistic Hamiltonian is defined as a smooth real functionH on T ∗ Q [Mangiarotti and Sardanashvily (1998); Sardanashvily (1998)].Then a relativistic Hamiltonian system is described as a Dirac constraint


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 619system on the subspace N of T ∗ Q given by the equationH T = g µν ∂ µ H∂ ν H − 1 = 0. (4.145)To perform geometrical quantization of NRM, we give geometrical quantizationof the cotangent bundle T ∗ Q and characterize a quantum relativisticHamiltonian system by the quantum constraintĤ T ψ = 0. (4.146)We choose the vertical polarization on T ∗ Q spanned by the tangent vectors∂ α . The corresponding quantum algebra A ⊂ C ∞ (T ∗ Q) consists of affinefunctions of momentaf = a α (q µ )p α + b(q µ ) (4.147)on T ∗ Q. They are represented by the Schrödinger operatorŝf = −ia α ∂ α − i 2 ∂ αa α − i 4 aα ∂ α ln(−g) + b, (g = det(g αβ )) (4.148)in the space C ∞ (Q) of smooth complex functions on Q.Note that the function H T (4.145) need not belong to the quantumalgebra A. Nevertheless, one can show that, if H T is a polynomial ofmomenta of degree k, it can be represented as a finite compositionH T = ∑ if 1i · · · f ki (4.149)of products of affine functions (4.147), i.e., as an element of the envelopingalgebra A of the Lie algebra A [Giachetta et. al. (2002b)]. Then it isquantizedH T ↦→ ĤT = ∑ îf 1i · · · ̂f ki (4.150)as an element of A. However, the representation (4.149) and, consequently,the quantization (4.150) fail to be unique.The space of relativistic velocities of RM on Q is the tangent bundle T Qof Q equipped with the induced coordinates (t, q i , ˙q α ) with respect to theholonomic frames {∂ α }. Relativistic motion is located in the subbundle W gof hyperboloids [Mangiarotti and Sardanashvily (1998); Mangiarotti andSardanashvily (2000b)]g µν (q) ˙q µ ˙q ν − 1 = 0 (4.151)


620 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionof T Q. It is described by a second–order dynamical equationon Q which preserves the subbundle (4.151), i.e.,¨q α = Ξ α (q µ , ˙q µ ) (4.152)( ˙q α ∂ α + Ξ α ˙∂α )(g µν ˙q µ ˙q ν − 1) = 0, ( ˙∂ α = ∂/∂ ˙q α ).This condition holds if the r.h.s. of the equation (4.152) takes the formΞ α = Γ α µν ˙q µ ˙q ν + F α ,where Γ α µν are Christoffel symbols of a metric g, while F α obey the relationg µν F µ ˙q ν = 0. In particular, if the dynamical equation (4.152) is a geodesicequation,¨q α = K α µ ˙q µwith respect to a (non-linear) connection on the tangent bundle T Q → Q,this connections splits into the sumK = dq α ⊗ (∂ α + K µ α ˙∂ µ ),K α µ = Γ α µν ˙q ν + F α µ (4.153)of the Levi–Civita connection of g and a soldering formF = g λν F µν dq µ ⊗ ˙∂ α , F µν = −F νµ .As was mentioned above, the momentum phase–space of RM on Q isthe cotangent bundle T ∗ Q provided with the symplectic form Ω (4.144).Let H be a smooth real function on T ∗ Q such that the map˜H : T ∗ Q −→ T Q, ˙q µ = ∂ µ H (4.154)is a bundle isomorphism. Then the inverse image N = ˜H −1 (W g ) of thesubbundle of hyperboloids W g (4.151) is a one-codimensional (consequently,coisotropic) closed imbedded subbundle of T ∗ Q given by the constraintH T = 0 (4.145). We say that H is a relativistic Hamiltonian if the Poissonbracket {H, H T } vanishes on N. This means that the Hamiltonian vector–fieldγ = ∂ α H∂ α − ∂ α H∂ α (4.155)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 621of H preserves the constraint N and, restricted to N, it obeys the Hamiltonianequationγ⌋Ω N + i ∗ N dH = 0 (4.156)of a Dirac constraint system on N with a Hamiltonian H.The map (4.154) sends the vector–field γ (4.155) onto the vector–fieldγ T = ˙q α ∂ α + (∂ µ H∂ α ∂ µ H − ∂ µ H∂ α ∂ µ H) ˙∂ αon T Q. This vector–field defines the second–order dynamical equation¨q α = ∂ µ H∂ α ∂ µ H − ∂ µ H∂ α ∂ µ H (4.157)on Q which preserves the subbundle of hyperboloids (4.151).The following is a basic example of relativistic Hamiltonian systems.PutH = 12m gµν (p µ − b µ )(p ν − b ν ),where m is a constant and b µ dq µ is a covector–field on Q. Then H T =2m −1 H − 1 and {H, H T } = 0. The constraint H T = 0 defines a closedimbedded one-codimensional subbundle N of T ∗ Q. The Hamiltonian equation(4.156) takes the form γ⌋Ω N = 0. Its solution (4.155) reads˙q α = 1 m gαν (p ν − b ν ),ṗ α = − 12m ∂ αg µν (p µ − b µ )(p ν − b ν ) + 1 m gµν (p µ − b µ )∂ α b ν .The corresponding second–order dynamical equation (4.157) on Q is¨q α = Γ α µν ˙q µ ˙q ν − 1 m gλν F µν ˙q µ , (4.158)Γ α µν = − 1 2 gλβ (∂ µ g βν + ∂ ν g βµ − ∂ β g µν ), F µν = ∂ µ b ν − ∂ ν b µ .It is a geodesic equation with respect to the affine connectionK α µ = Γ α µν ˙q ν − 1 m gλν F µνof type (4.153). For example, let g be a metric gravitational field and letb µ = eA µ , where A µ is an electromagnetic potential whose gauge holdsfixed. Then the equation (4.158) is the well–known equation of motion ofa relativistic massive charge in the presence of these fields.


622 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet us now perform the quantization of RM, following the standardgeometrical quantization of the cotangent bundle (see [Blattner (1983);Sniatycki (1980); Woodhouse (1992)]). As the canonical symplectic formΩ (4.144) on T ∗ Q is exact, the prequantum bundle is defined as a trivialcomplex line bundle C over T ∗ Q. Note that this bundle need no metaplecticcorrection since T ∗ X is with canonical coordinates for the symplecticform Ω. Thus, C is called the quantum bundle. Let its trivializationC ∼ = T ∗ Q × C (4.159)hold fixed, and let (t, q i , p α , c), with c ∈ C, be the associated bundle coordinates.Then one can treat sections of C (4.159) as smooth complexfunctions on T ∗ Q. Note that another trivialization of C leads to an equivalentquantization of T ∗ Q.Recall that the Kostant–Souriau prequantization formula associates toeach smooth real function f ∈ C ∞ (T ∗ Q) on T ∗ Q the first–order differentialoperator̂f = −i∇ ϑf + f (4.160)on sections of C, where ϑ f = ∂ α f∂ α − ∂ α f∂ α is the Hamiltonian vector–field of f and ∇ is the covariant differential with respect to a suitableU(1)−principal connection A on C. This connection preserves the Hermitianmetric g(c, c ′ ) = cc ′ on C, and its curvature form obeys the prequantizationcondition R = iΩ. For the sake of simplicity, let us assume that Qand, consequently, T ∗ Q is simply–connected. Then the connection A up togauge transformations isA = dp α ⊗ ∂ α + dq α ⊗ (∂ α + icp α ∂ c ), (4.161)and the prequantization operators (4.160) read̂f = −iϑ f + (f − p α ∂ α f). (4.162)Let us choose the vertical polarization on T ∗ Q. It is the vertical tangentbundle V T ∗ Q of the fibration π : T ∗ Q → Q. As was mentioned above, thecorresponding quantum algebra A ⊂ C ∞ (T ∗ Q) consists of affine functionsf (4.147) of momenta p α . Its representation by operators (4.162) is definedin the space E of sections ρ of the quantum bundle C of compact supportwhich obey the condition ∇ ϑ ρ = 0 for any vertical Hamiltonian vector–fieldϑ on T ∗ Q. This condition takes the form∂ α f∂ α ρ = 0,(f ∈ C ∞ (Q)).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 623It follows that elements of E are independent of momenta and, consequently,fail to be compactly supported, unless ρ = 0. This is the well–known problem of Schrödinger quantization which is solved as follows [Blattner(1983); Giachetta et. al. (2002b)].Let i Q : Q −→ T ∗ Q be the canonical zero section of the cotangent bundleT ∗ Q. Let C Q = i ∗ QC be the pull–back of the bundle C (4.159) over Q. Itis a trivial complex line bundle C Q = Q × C provided with the pull–backHermitian metric g(c, c ′ ) = cc ′ and the pull–backA Q = i ∗ QA = dq α ⊗ (∂ α + icp α ∂ c )of the connection A (4.161) on C. Sections of C Q are smooth complexfunctions on Q, but this bundle need metaplectic correction.Let the cohomology group H 2 (Q; Z 2 ) of Q be trivial. Then a metalinearbundle D of complex half-forms on Q is defined. It admits the canonicallift of any vector–field τ on Q such that the corresponding Lie derivative ofits sections readsL τ = τ α ∂ α + 1 2 ∂ ατ α .Let us consider the tensor product Y = C Q ⊗ D over Q. Since the Hamiltonianvector–fieldsϑ f = a α ∂ α − (p µ ∂ α a µ + ∂ α b)∂ αof functions f (4.147) are projected onto Q, one can assign to each elementf of the quantum algebra A the first–order differential operator̂f = (−i∇ πϑf + f) ⊗ Id + Id ⊗ L πϑf = −ia α ∂ α − i 2 ∂ αa α + bon sections ρ Q of Y . For the sake of simplicity, let us choose a trivial metalinearbundle D → Q associated to the orientation of Q. Its sections can bewritten in the form ρ Q = (−g) 1/4 ψ, where ψ are smooth complex functionson Q. Then the quantum algebra A can be represented by the operatorŝf (4.148) in the space C ∞ (Q) of these functions. It can be justified thatthese operators obey the Dirac condition[ ̂f, ̂f ′ ] = −i ̂ {f, f ′ }.One usually considers the subspace E Q ⊂ C ∞ (Q) of functions of compactsupport. It is a pre–Hilbert space with respect to the non–degenerate


624 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionHermitian form∫〈ψ|ψ ′ 〉 = ψψ ′ (−g) 1/2 d m+1 qQNote that ̂f (4.148) are symmetric operators ̂f = ̂f ∗ in E Q , i.e., 〈 ̂fψ|ψ ′ 〉 =〈ψ| ̂fψ ′ 〉. However, the space E Q gets no physical meaning in RM.As was mentioned above, the function H T (4.145) need not belong tothe quantum algebra A, but a polynomial function H T can be quantizedas an element of the enveloping algebra A by operators ĤT (4.150). Thenthe quantum constraint (4.146) serves as a relativistic quantum equation.Let us again consider a massive relativistic charge whose relativisticHamiltonian isIt defines the constraintH = 12m gµν (p µ − eA µ )(p ν − eA ν ).H T = 1 m 2 gµν (p µ − eA µ )(p ν − eA ν ) − 1 = 0. (4.163)Let us represent the function H Tfunctions of momenta,(4.163) as symmetric product of affineH T = (−g)−1/4m(p µ − eA µ )(−g) 1/4 g µν (−g) 1/4 (p ν − eA ν ) (−g)−1/4− 1.mIt is quantized by the rule (4.150), where(−g) 1/4 ◦ ̂∂ α ◦ (−g) −1/4 = −i∂ α .Then the well–known relativistic quantum equation(−g) −1/2 [(∂ µ − ieA µ )g µν (−g) 1/2 (∂ ν − ieA ν ) + m 2 ]ψ = 0. (4.164)is reproduced up to the factor (−g) −1/2 .4.12 Symplectic Structures on Fiber BundlesIn this section, following [Lalonde et al. (1998); Lalonde et al. (1999);Lalonde and McDuff (2002)], we analyze general symplectic structures onfiber bundles. We first discuss how to characterize Hamiltonian bundles andtheir automorphisms, and then describe their main properties, in particularderiving conditions under which the cohomology of the total space splits as


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 625a product. Finally we state some applications to the action of Ham(M) onM and to non–Hamiltonian symplectic bundles.4.12.1 Hamiltonian Bundles4.12.1.1 Characterizing Hamiltonian BundlesRecall that a fiber bundle M → P → B is said to be symplectic fibrebundle if its structural group reduces to the group of symplectomorphismsSymp(M, ω) of the closed symplectic manifold (M, ω). In this case, eachfiber M b = π −1 (b) is equipped with a well defined symplectic form ω b suchthat (M b , ω b ) is symplectomorphic to (M, ω). Our first group of results establishgeometric criteria for a symplectic bundle to be Hamiltonian, i.e., forthe structural group to reduce to Ham(M, ω). Quite often we simplify thenotation by writing Ham(M) and Symp 0 (M) (or even Ham and Symp 0 )instead of Ham(M, ω) and Symp 0 (M, ω) (see [Lalonde et al. (1998);Lalonde et al. (1999); Lalonde and McDuff (2002)]).Recall that the group Ham(M, ω) is a connected normal subgroup ofthe identity component Symp 0 (M, ω) of the group of symplectomorphisms,and fits into the exact sequence{Id} → Ham(M, ω) → Symp 0 (M, ω) F lux−→ H 1 (M, R)/Γ ω → {0},where Γ ω is the flux group. As Ham(M) is connected, every Hamiltonianbundle is symplectically trivial over the 1−skeleton of the base. The followingproposition was proved in [McDuff and Salamon (1998)]: A symplecticbundle π : P → B is Hamiltonian iff the following conditions hold:(i) the restriction of π to the 1−skeleton B 1 of B is symplectically trivial,and(ii) there is a cohomology class a ∈ H 2 (P, R) that restricts to [ω b ] onM b .There is no loss of generality in assuming that the bundle π : P → Bis smooth. Then recall from [Guillemin et. al. (1998)] that any 2−form τon P that restricts to ω b on each fiber M b defines a connection ▽ τ on Pwhose horizontal distribution Hor τ is just the τ−orthogonal complementof the tangent spaces to the fibers:Hor τ (x) = {v ∈ T x P : τ(v, w) = 0 for all w ∈ T x (M π(x) )}.Such forms τ are called connection forms. The closedness of τ is a sufficient(but not necessary) condition for the holonomy of ▽ τ to be sym-


626 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionplectic. A simple argument due to [McDuff and Salamon (1998)] showsthat the cohomological condition (ii) above is equivalent to the existenceof a closed extension τ of the forms ω b . Condition (i) is then equivalent torequiring that the holonomy of ▽ τ around any loop in B belongs to theidentity component Symp 0 (M) of Symp(M). Hence the above result canbe rephrased in terms of such closed extensions τ as follows: A symplecticbundle π : P → B is Hamiltonian iff the forms ω b on the fibers have aclosed extension τ such that the holonomy of ▽ τ around any loop in B liesin the identity component Symp 0 (M) of Symp(M).This is a slight extension of a result of [Guillemin et. al. (1998)], whocalled a specific choice of τ the coupling form. As we show below, theexistence of τ is the key to the good behavior of Hamiltonian bundles undercomposition.When M is simply connected, Ham(M) is the identity componentSymp 0 (M) of Symp(M), and so a symplectic bundle is Hamiltonian iffcondition (i) above is satisfied, i.e., iff it is trivial over the 1−skeleton B 1 .In this case, as observed by [Gotay et. al. (1983)], it is known that (i)implies (ii) for general topological reasons to do with the behavior of evaluationmaps. More generally, (i) implies (ii) for all symplectic bundles withfiber (M, ω) iff the flux group Γ ω vanishes.4.12.1.2 Hamiltonian StructuresThe question then arises as to what a Hamiltonian structure on a fiberbundle actually is [Lalonde et al. (1998); Lalonde et al. (1999); Lalondeand McDuff (2002)]. That is, how many Hamiltonian structures can oneput on a given symplectic bundle π : P → B? And, what does one meanby an automorphism of such a structure?In homotopy theoretic terms, a Hamiltonian structure on a symplecticbundle π : P → B is simply a lift ˜g to B Ham(M) of the classifying mapg : B → B Symp(M, ω) of the underlying symplectic bundle, i.e., it is ahomotopy commutative diagram


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 627B Ham(M)˜g1 ✒❄B ✲g B Symp(M)Hamiltonian structures are in bijective correspondence with homotopyclasses of such lifts. There are two stages to choosing the lift ˜g: one firstlifts g to a map ĝ into B Symp 0 (M, ω), where Symp 0 is the identity componentof Symp, and then to a map ˜g into B Ham(M, ω). As we will showbelow, choosing ĝ is equivalent to fixing the isotopy class of an identificationof (M, ω) with the fiber (M b0 , ω b0 ) over the base point b 0 . If B issimply connected, in particular if B is a single point, there is then a uniqueHamiltonian structure on P , i.e., a unique choice of lift ˜g. Before describingwhat happens in the general case, we discuss properties of the extensionsτ.Let τ ∈ Ω 2 (P ) be a closed extension of the symplectic forms on thefibers. Given a loop, γ : S 1 → B, based at b 0 , and a symplectic trivializationT γ : γ ∗ (P ) → S 1 × (M, ω) that extends the given identification of M b0 withM, push forward τ to a form (T γ ) ∗ τ on S 1 × (M, ω). Its characteristic flowround S 1 is transverse to the fibers and defines a symplectic isotopy φ t of(M, ω) = (M b0 , ω b0 ) whose flux, as a map from H 1 (M) → R, is equal to(T γ ) ∗ [τ]([S 1 ] ⊗ ·). This flux depends only on the cohomology class a of τ.Moreover, as we mentioned above, any extension a of the fiber class [ω] canbe represented by a form τ that extends the ω b . Thus, given T γ and anextension a = [τ] ∈ H 2 (P ) of the fiber symplectic class [ω], it makes senseto define the flux class f(T γ , a) ∈ H 1 (M, R) byf(T γ , a)(δ) = (T γ ) ∗ (a)(γ ⊗ δ)for all δ ∈ H 1 (M).The equivalence class [f(T γ , a)] ∈ H 1 (M, R)/Γ ω does not depend on thechoice of T γ : indeed two such choices differ by a loop φ in Symp 0 (M, ω)and so the differencef(T γ , a) − f(T ′ γ, a) = f(T γ , a) ◦ Tr φ = ω ◦ Tr φbelongs to Γ ω . The following lemma is elementary: If π : P → B is asymplectic bundle satisfying the above conditions, there is an extension a


628 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionof the symplectic fiber class that has a trivial flux[f(T γ , a)] = 0 ∈ H 1 (M, R)/Γ ω around each loop γ ∈ B.An extension a of the symplectic fiber class [ω b0 ] is normalized if itsatisfies the conclusions of the above lemma. Two such extensions a anda ′ are equivalent (in symbols, a ∼ a ′ ) iff they have equal restrictions toπ −1 (B 1 ), or, equivalently, iff a − a ′ ∈ π ∗ (H 2 (B)).We show below that Hamiltonian structures are in one-to-one correspondencewith symplectic trivializations of the 1−skeleton B 1 of B, withtwo such trivializations being equivalent iff they differ by Hamiltonianloops. If two trivializations T γ , T γ ′ differ by a Hamiltonian loop φ thenf(T γ , a)−f(T γ, ′ a) = 0. In terms of fluxes of closed extensions, we thereforeget the following theorem: Assume that a symplectic bundle π : P → B canbe symplectically trivialized over B 1 . Then a Hamiltonian structure existson P iff there is a normalized extension a of ω. Such a structure consists ofan isotopy class of symplectomorphisms (M, ω) → (M b0 , ω b0 ) together withan equivalence class {a} of normalized extensions of the fiber symplecticclass.In other words, with respect to a fixed trivialization over B 1 , Hamiltonianstructures are in one–to–one correspondence with homomorphismsπ 1 (B) → Γ ω , given by the fluxes f γ (T, a) of monodromies round the loopsof the base. We will call {a} the Hamiltonian extension class, and willdenote the Hamiltonian structure on P by the triple (P, π, {a}).We now turn to the question of describing automorphisms of Hamiltonianstructures [Lalonde et al. (1998); Lalonde et al. (1999)]. It is convenientto distinguish between symplectic and Hamiltonian automorphisms,just as we distinguish between Symp(M, ω) and Ham(M, ω) in the casewhen B = pt. Notice that if P → B is a symplectic bundle, there is a naturalnotion of symplectic automorphism. This is a fiberwise diffeomorphismΦ : P → P that covers the identity map on the base and restricts on eachfiber to an element Φ b of the group Symp(M b , ω b ). Because Ham(M, ω)is a normal subgroup of Symp(M, ω), it also makes sense to require thatΦ b ∈ Ham(M b , ω b ) for each b. Such automorphisms are called Hamiltonianautomorphisms of the symplectic bundle P → B. Let us write Symp(P, π)and Ham(P, π) for the groups of such automorphisms. Observe that thegroup Ham(P, π) may not be connected. Because the fibers of Hamiltonianbundles are identified with (M, ω) up to isotopy, we shall also needto consider the (not necessarily connected) group Symp 0 (P, π) of symplectomorphismsof (P, π) where Φ b ∈ Symp 0 (M b , ω b ) for one and hence all


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 629b.Let us consider automorphisms of Hamiltonian bundles [Lalonde andMcDuff (2002)]. As a guide note that in the trivial case when B = pt, aHamiltonian structure on P is an identification of P with M up to symplecticisotopy. Hence the group of automorphisms of this structure canbe identified with Symp 0 (M, ω). In general, if {a} is a Hamiltonian structureon (P, π) and Φ ∈ Symp 0 (P, π) then Φ ∗ ({a}) = {a} iff Φ ∗ (a) = a forsome a in the class {a}, because Φ induces the identity map on the baseand a − a ′ ∈ π ∗ (H 2 (B)) when a ∼ a ′ . We therefore make the followingdefinition.Let (P, π, {a}) be a Hamiltonian structure on the symplectic bundleP → B and let Φ ∈ Symp(P, π). Then Φ is an automorphism of theHamiltonian structure (P, π, {a}) if Φ ∈ Symp 0 (P, π) and Φ ∗ ({a}) = {a}.The group formed by these elements is denoted by Aut(P, π, {a}).The following result is not hard to prove, but is easiest to see in thecontext of a discussion of the action of Ham(M) on H ∗ (M).Let P → B be a Hamiltonian bundle and Φ ∈ Symp 0 (P, π). Then thefollowing statements are equivalent [Lalonde and McDuff (2002)]:(i) Φ is isotopic to an element of Ham(P, π);(ii) Φ ∗ ({a}) = {a} for some Hamiltonian structure {a} on P ;(iii) Φ ∗ ({a}) = {a} for all Hamiltonian structures {a} on P .For any Hamiltonian bundle P → B, the group Aut(P, π, {a}) does notdepend on the choice of the Hamiltonian structure {a} put on P . Moreover,it contains Ham(P, π) and each element of Aut(P, π, {a}) is isotopic to anelement in Ham(P, π).The following characterization is now obvious:Let P be the product B ×M and {a} any Hamiltonian structure. Then:(i) Ham(P, π) consists of all maps from B to Ham(M, ω).(ii) Aut(P, π, {a}) consists of all maps Φ : B → Symp 0 (M, ω) for whichthe following composition is trivialπ 1 (B) −→ Φ∗π 1 (Symp 0 (M)) F −→ luxωH 1 (M, R).The basic reason why the above proposition holds is that Hamiltonianautomorphisms of (P, π) act trivially on the set of extensions of the fibersymplectic class. This need not be true for symplectic automorphisms. Forexample, ifπ : P = S 1 × M → S 1


630 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis a trivial bundle and Φ is given by a non–Hamiltonian loop φ inSymp 0 (M), then Φ is in Symp 0 (P, π) but it preserves no Hamiltonianstructure on P sinceΦ ∗ (a) = a + [dt] ⊗ F lux(φ).In general, if we choose a trivialization of P over B 1 , there are exactsequences{Id} → Aut(P, π, {a}) → Symp 0 (P, π) → Hom(π 1 (B), Γ ω ) → {id},{Id} → Ham(P, π, {a}) → Aut(P, π, {a}) → H 1 (M, R)/Γ ω → {0}.In particular, the subgroup of Aut(P, π, {a}) consisting of automorphismsthat belong to Ham(M b0 , ω b0 ) at the base point b 0 retracts toHam(P, π, {a}).4.12.1.3 Marked Hamiltonian StructuresAnother approach to characterizing a Hamiltonian structure is to define itin terms of a structure on the fiber that is preserved by elements of theHamiltonian group [Lalonde and McDuff (2002)].The so–called marked symplectic manifold (M, ω, [L]) is a pair consistingof a closed symplectic manifold (M, ω) together with a marking [L]. HereL is a collection {l 1 , . . . , l k } of loops l i : S 1 → M in M that projectsto a minimal generating set G L = {[l 1 ], . . . , [l k ]} for H 1 (M, Z)/torsion. Amarking [L] is an equivalence class of generating loops L, where L ∼ L ′ iffor each i there is an singular integral 2−chain c i whose boundary modulotorsion is l ′ i − l i such that ∫ c iω = 0.The symplectomorphism group acts on the space L of markings. Moreover,it is easy to check that if a symplectomorphism φ fixes one marking[L] it fixes them all. Hence the groupLHam(M, ω) = LHam(M, ω, [L]) = {φ ∈ Symp(M, ω) : φ ∗ [L] = [L]}independent of the choice of [L]. Its identity component is Ham(M, ω).There is a forgetful map [L] → G L from the space L of markings tothe space of minimal generating sets for the group H 1 (M, Z), and it is nothard to check that its fiber is (R/P) k , where is the image of the periodhomomorphismI [ω] : H 2 (M, Z) → R.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 631If P is not discrete, there is no nice topology one can put on L. However, ithas a pseudotopology, i.e., one can specify which maps of finite polyhedraX into L are continuous, namely: f : X → L is continuous iff every x ∈ Xhas a neighborhood U x such that f : U x → L lifts to a continuous map intothe space of generating loops L.Now, let us fix a marking [L] on (M, ω). A Hamiltonian structure ona symplectic bundle π : P → B is an isotopy class of symplectomorphisms(M, ω, [L]) → (M b0 , ω b0 , [L 0 ]) together with a continuous choice of marking[L b ] on each fiber (M b , ω b ) that is trivial over the 1−skeleton B 1 of B inthe sense that there is a symplectic trivializationΦ : π −1 (B 1 ) → B 1 × (M, ω, [L])that respects the markings on each fiber.Here is another way of thinking of a Hamiltonian structure due toPolterovich. He observed that there is an exact sequence0 → R/P → SH 1 (M, ω) → H 1 (M, Z) → 0,where SH 1 (M, ω) is the so–called strange homology group formed by quotientingthe space of integral 1−cycles by the image under d of the space ofintegral 2−chains with zero symplectic area. The group Symp(M, ω) actson SH 1 (M, ω). Moreover, if φ ∈ Symp 0 (M) and ã ∈ SH 1 (M) projects toa ∈ H 1 (M), then φ ∗ (ã) − ã ∈ R/P can be thought of as the value of theclass F lux(φ) ∈ H 1 (M, R)/Γ ω on a. It is easy to see that LHam(M, ω) isthe subgroup of Symp(M, ω) that acts trivially on SH 1 (M, Z). Further, amarking on (M, ω) is a pair consisting of a splitting of the above sequencetogether with a generating set L for H 1 (M, Z)/torsion.Given any symplectic bundle P −→ B there is an associated bundle ofabelian groups with fiber SH 1 (M, ω). A Hamiltonian structure on P −→ Bis a flat connection on this bundle that is trivial over the 1-skeleton B 1 ,under an appropriate equivalence relation.These ideas can obviously be generalized to bundles that are not trivialover the 1−skeleton. Equivalently, one can consider bundles with disconnectedstructural group. This group could be the whole of LHam(M, ω).One could also restrict to elements acting trivially on H ∗ (M) and/or tothose that act trivially on the groupsSH 2k−1 (M, ω) =integral (2k − 1)−cyclesd(2k−chains in the kernel of ω k ) .


632 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThese generalizations of SH 1 (M, ω) are closely connected to Reznikov’sFutaki type characters [Reznikov (1997)]. It is not yet clear what is themost natural disconnected extension of Ham(M, ω).4.12.1.4 StabilityAnother important property of Hamiltonian bundles is stability [Lalondeand McDuff (2002)]. A symplectic (resp. Hamiltonian) bundle π : P → Bwith fiber (M, ω) is said to be stable if π may be given a symplectic (resp.Hamiltonian) structure with respect to any symplectic form ω ′ on M thatis sufficiently close to (but not necessarily cohomologous to) ω, in such away that the structure depends continuously on ω ′ .Using Moser’s homotopy argument, it is easy to prove that any symplecticbundle is stable. The following characterization of Hamiltonian stabilityis an almost immediate consequence of above theorem:A Hamiltonian bundle π : P → B is stable iff the restriction mapH 2 (P, R) → H 2 (M, R) is surjective.The following result is less immediate: Every Hamiltonian bundle isstable.The proof uses the (difficult) stability property for Hamiltonian bundlesover S 2 that was established in [Lalonde et al. (1999); McDuff (2000)] aswell as the (easy) fact that the image of the evaluation map π 2 (Ham(M))−→ π 2 (M) lies in the kernel of [ω].4.12.1.5 Cohomological SplittingWe next extend the splitting results of [Lalonde et al. (1999); McDuff(2000)], which prove that the rational cohomology of every Hamiltonianbundle π : P → S 2 splits additively, i.e., there is an additive isomorphismH ∗ (P ) ∼ = H ∗ (S 2 ) ⊗ H ∗ (M).For short we will say in this situation that π is c−split. This is a deepresult, that requires the use of Gromov–Witten invariants for its proof. 16The results of the present subsection provide some answers to the following16 Recall that Gromov–Witten invariants are rational numbers that count pseudo–holomorphic curves meeting prescribed conditions in a given symplectic manifold. Theseinvariants may be packaged as a homology or cohomology class in an appropriate space,or as the deformed cup product of quantum cohomology. They have been used to distinguishsymplectic manifolds that were previously indistinguishable. They also play acrucial role in closed type IIA string theory.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 633question [Lalonde and McDuff (2002)]: Does any Hamiltonian fiber bundleover a compact CW–complex c−split?A special case is when the structural group of P → B can be reducedto a compact Lie subgroup G of Ham(M). Here c−splitting over any basefollows from the work of Atiyah–Bott [Atiyah and Bott (1984)]. In thiscontext, one usually discusses the universal Hamiltonian G−bundle withfiber MM −→ M G = EG × G M −→ BG.The cohomology of P = M G is known as the equivariant cohomologyHG ∗ (M) of M. Atiyah–Bott show that if G is a torus T that acts in aHamiltonian way on M then the bundle M T → BT is c−split. The resultfor a general compact Lie group G follows by standard arguments [Lalondeand McDuff (2002)].The following theorem describes conditions on the base B that implyc−splitting. Let (M, ω) be a closed symplectic manifold, and M ↩→ P → Ba bundle with structure group Ham(M) and with base a compact CW–complex B. Then the rational cohomology of P splits in each of the followingcases:(i) the base has the homotopy type of a coadjoint orbit or of a product ofspheres with at most three of dimension 1;(ii) the base has the homotopy type of a complex blow up of a product ofcomplex projective spaces;(iii) dim(B) ≤ 3.Case (ii) is a generalization of the foundational example B = S 2 and isproved by similar analytic methods. The idea is to show that the mapι : H ∗ (M) → H ∗ (P ) is injective by showing that the image ι(a) in Pof any class a ∈ H ∗ (M) can be detected by a nonzero Gromov–Witteninvariant of the form n P (ι(a), c 1 , . . . , c n ; σ), where c i ∈ H ∗ (P ) and σ ∈H 2 (P ) is a spherical class with nonzero image in H 2 (B). The proof shouldgeneralize to the case when all one assumes about the base is that there is anonzero invariant of the form n B (pt, pt, c 1 , . . . , c k ; A) [Lalonde et al. (1998);Lalonde et al. (1999)].The proofs of parts (i) and (iii) start from the fact of c−splitting overS 2 and proceed using purely topological methods. The following fact aboutcompositions of Hamiltonian bundles is especially useful. Let M ↩→ P →B be a Hamiltonian bundle over a simply connected base B and assume


634 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthat all Hamiltonian bundles over M as well as over B c−split. Thenany Hamiltonian bundle over P c−splits too. 17 This provides a powerfulrecursive argument which allows one to establish c−splitting over CP n byinduction on n, and is an essential tool in all arguments here [Lalonde andMcDuff (2002)].It is natural to wonder whether c−splitting is a purely homotopy–theoretic property. A c−symplectic manifold (M, a M ) is defined to be a2n−manifold together with a class a M ∈ H 2 (M) such that a n M > 0. In viewof above theorem one could define a c−Hamiltonian bundle over a simplyconnected base manifold B to be a bundle P → B with c−symplectic fiber(M, a M ) in which the symplectic class a M extends to a class a ∈ H 2 (P ).A variety of results about symplectic torus actions were discussed in [Allday(1998)], some of which do extend to the c−symplectic case and someof which do not. The next lemma shows that c−splitting in generalis a geometric rather than a homotopy–theoretic property: There is ac−Hamiltonian bundle over S 2 that is not c−split.It is also worth noting that it is essential to restrict to finite dimensionalspaces: c−splitting does not always hold for ‘Hamiltonian’ bundles withinfinite dimensional fiber [Lalonde and McDuff (2002)].4.12.1.6 Homological Action of Ham(M) on MThe action Ham(M) × M → M gives rise to mutually dual maps [Lalondeand McDuff (2002)]Tr φ : H k (Ham(M)) ⊗ H ∗ (M) −→ H k+∗ (M), (φ, Z) ↦→ Tr φ (Z), andTr ∗ φ : H k (Ham(M)) → Hom(H ∗ (M), H ∗−k (M)), (k ≥ 0).In this language, the flux of a loop φ ∈ π 1 (Ham(M)) is precisely the elementTr ∗ φ([ω]) ∈ H 1 (M). (Here we should use real rather than rational coefficientsso that [ω] ∈ H ∗ (M).) The following result is a consequence of abovetheorem: The maps Tr φ and Tr ∗ φ are zero for all φ ∈ H k (Ham(M)), k > 0.The argument goes as follows. Recall that the cohomology ring ofHam(M) is generated by elements dual to its homotopy. It therefore sufficesto consider the restriction of Tr k to the spherical elements φ. Butin this case it is not hard to see that the Tr k are precisely the connectinghomomorphisms in the Wang sequence of the bundle P φ → S k+1 withclutching function φ. These vanish because all Hamiltonian bundles over17 This fact is based on the characterization of Hamiltonian bundles in terms of closedextensions of the symplectic form.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 635spheres are c−split [Lalonde and McDuff (2002)].In particular, looking at the action on H 0 (M), we see that the pointevaluation mapev : Ham(M) → M :ψ ↦→ ψ(x)induces the trivial map on rational (co)homology. It also induces the trivialmap on π 1 . However, the map on π k , k > 1, need not be trivial. To see this,consider the action of Ham(M) on the symplectic frame bundle SF r(M)of M and the corresponding point evaluation maps. The obvious action ofSO(3) ≃ Ham(S 2 ) on SF r(S 2 ) ≃ RP 3 induces an isomorphismH 3 (SO(3)) ∼ = H 3 (SF r(S 2 )),showing that these evaluation maps are not homologically trivial. Moreover,its composite with the projection SF r(S 2 ) → S 2 gives rise to a nonzeromapπ 3 (SO(3)) = π 3 (Ham(S 2 )) → π 3 (S 2 ).Thus the corresponding Hamiltonian fibration over S 4 with fiber S 2 , thoughc−split, does not have a section.Note, however, that the extended evaluation mapπ 2l (X X ) ev → π 2l (X) → H 2l (X, Q), l > 0,is always zero, if X is a finite CW complex and X X is its space of self–maps. Indeed, because the cohomology ring H ∗ (X X , Q) is freely generatedby elements dual to π ∗ (X X ) ⊗ Q, there would otherwise be an elementa ∈ H 2l (X) that would pull back to an element of infinite order in thecohomology ring of the H−space X X . Hence a itself would have to haveinfinite order, which is impossible. A more delicate argument shows thatthe integral evaluation π 2l (X X ) → H 2l (X, Z) is zero [Gottlieb (1975)].A Hamiltonian automorphism of the product Hamiltonian bundle B ×M → B is simply a map B → B × Ham(M) of the form b ↦→ (b, φ b ). IfB is a closed manifold we will see that any Hamiltonian automorphism ofthe product bundle acts as the identity map on the rational cohomologyof B × M. The natural generalization of this result would claim that aHamiltonian automorphism of a bundle P acts as the identity map on therational cohomology of P . We do not know yet whether this is true ingeneral. However, we can show that it is closely related to the c−splitting


636 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionof Hamiltonian bundles. Thus we can establish it only under conditionssimilar to the conditions under which c−splitting holds.4.12.1.7 General Symplectic BundlesConsider the Wang sequence for a symplectic bundle π : P → S 2 withclutching map φ ∈ π 1 (Symp(M)):· · · −→ H k (M)∂−→ H k−1 u(M) −→ H k+1 (P ) restr−→ H k+1 (M) −→ · · ·Here the map u may be realized in de Rham cohomology by choosing anyextension of a given closed form α on M and then wedging it with thepullback of a normalized area form on the base. Further, as pointed outabove, the boundary map ∂ = ∂ φ is just Tr ∗ φ. Therefore, the bundle isHamiltonian iffTr ∗ φ([ω]) = ∂([ω]) = 0.In the Hamiltonian case the above splitting theorem implies that ∂ is identically0. In the general case, we know that the map ∂ : H ∗ (M) → H ∗−1 (M)is a derivation, i.e.,∂(ab) = ∂(a)b + (−1) deg(a) a∂(b).The following result is an easy consequence of the fact that the action ofπ 2 (Ham(M)) = π 2 (Symp(M)) on H ∗ (M)is trivial: The boundary map ∂ in the Wang rational cohomology sequenceof a symplectic bundle over S 2 has the basic property: ∂ ◦ ∂ = 0.The above result holds trivially when φ corresponds to a smooth (notnecessarily symplectic) S 1 −action since then ∂ is given in the de Rhamcohomology by contraction ι X by the generating vector field X. Moreover,the authors know of no smooth bundle over S 2 for which the above propositiondoes not hold, though it is likely that they exist. Such a bundle wouldhave no extension over CP 2 .One consequence is the following result about the boundary map ∂ = ∂ φin the case when the loop φ is far from being Hamiltonian. Recall (e.g.,from [Lalonde et al. (1999)]) that π 1 (Ham(M)) is included in (but notnecessarily equal to) the kernel of the evaluation map π 1 (Symp(M)) →π 1 (M). Any loop whose evaluation is homologically essential can therefore


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 637be thought of as ‘very non–Hamiltonian’. We have the following corollary:Ker ∂ = Im ∂iff the image of φ under the evaluation map π 1 (Symp(M)) → H 1 (M, Q) isnonzero. A similar result was obtained by [Allday (1998)] concerning S 1actions on c−symplectic manifolds. He was considering manifolds M thatsatisfy the weak Lefschetz condition, i.e., manifolds such that∧[ω] n−1 : H 1 (M, R) → H 2n−1 (M, R)is an isomorphism, in which case every non–Hamiltonian loop is ‘very non–Hamiltonian’.4.12.1.8 Existence of Hamiltonian Structures<strong>Geom</strong>etric proofs (such as those in [McDuff and Salamon (1998)]) applywhen P and B are smooth manifolds and π is a smooth surjection. However,as the following lemma makes clear, this is no restriction.Suppose that π : Q → W is a locally trivial bundle over a finite CWcomplex W with compact fiber (M, ω) and suppose that the structuralgroup G is equal either to Symp(M, ω) or to Ham(M, ω). Then thereis a smooth bundle π : P → B as above with structural group G and ahomeomorphism f of W onto a closed subset of B such that π : Q → W ishomeomorphic to the pullback bundle f ∗ (P ) → W .Let us embed W into some Euclidean space and let B be a suitable smallneighborhood of W . Then W is a retract of B so that the classifying mapW → B G extends to B. It remains to approximate this map B → B Gby a smooth map. To get a relation between the existence of the class aand the structural group it seems necessary to use the idea of a symplecticconnection. We begin with an easy lemma.Let P → B be a symplectic bundle with closed connection form τ. Thenthe holonomy of the corresponding connection ▽ τ round any contractibleloop in B is Hamiltonian.To prove it, suffices to consider the case when B = D 2 . Then thebundle π : P → D 2 is symplectically trivial and so may be identified withthe product D 2 × M in such a way that the symplectic form on each fiberis simply ω. Use this trivialization to identify the holonomy round theloop s ↦→ e 2πis ∈ ∂D 2 with a family of diffeomorphisms Φ s : M → M, s ∈[0, 1]. Since this holonomy is simply the flow along the null directions (orcharacteristics) of the closed form τ on the hypersurface ∂P , a standard


638 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncalculation shows that the Φ s are symplectomorphisms. Given a 1−cycleδ : S 1 → M in the fiber M over 1 ∈ ∂D 2 , consider the closed 2−cycle thatis the union of the following two cylinders:c 1 : [0, 1] × S 1 → ∂D 2 × M : (s, t) ↦→ (e 2πis , Φ s (δ(t))),c 2 : [0, 1] × S 1 → 1 × M : (s, t) ↦→ (1, Φ 1−s (δ(t))).This cycle is obviously contractible. Hence,τ(c 1 ) = −τ(c 2 ) = F lux({Φ s })(δ).But τ(c 1 ) = 0 since the characteristics of τ| ∂P are tangent to c 1 . Applyingthis to all δ, we see that the holonomy round ∂D 2 has zero flux and so isHamiltonian.If π 1 (B) = 0 then a symplectic bundle π : P → B is Hamiltonian iff theclass [ω b ] ∈ H 2 (M) extends to a ∈ H ∗ (P ).Suppose first that the class a exists. We can work in the smooth category.Then Thurston’s convexity argument allows us to construct a closedconnection form τ on P and hence a horizontal distribution Hor τ . Theprevious lemma shows that the holonomy around every contractible loopin B is Hamiltonian. Since B is simply connected, the holonomy round allloops is Hamiltonian. Using this, it is easy to reduce the structural group ofP → B to Ham(M). For more details, see [McDuff and Salamon (1998)].Next, suppose that the bundle is Hamiltonian. We need to show thatthe fiber symplectic class extends to P . The proof in [McDuff and Salamon(1998)] does this by the method of [Guillemin et. al. (1998)] and constructsa closed connection form τ, called the coupling form, starting from a connectionwith Hamiltonian holonomy. This construction uses the curvatureof the connection and is quite analytic. In contrast, we shall now use topologicalarguments to reduce to the cases B = S 2 and B = S 3 . These casesare then dealt with by elementary arguments.Consider the Leray–Serre cohomology spectral sequence for M → P →B. Its E 2 term is a product: E p,q2 = H p (B) ⊗ H q (M). 18 We need to showthat the class [ω] ∈ E 0,22 survives into the E ∞ term, which happens iff it isin the kernel of the two differentials d 0,22 , d0,2 3 . Nowd 0,22 : H 2 (M) → H 2 (B) ⊗ H 1 (M)is essentially the same as the flux homomorphism. More precisely, if c :S 2 → B represents some element (also called c) in H 2 (B), then the pullback18 Here H ∗ denotes cohomology over R.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 639of the bundle π : P → B by c is a bundle over S 2 that is determined bya loop φ c ∈ π 1 (Ham(M)) that is well defined up to conjugacy. Moreover,for each λ ∈ H 1 (M),d 0,22 ([ω])(c, λ) = Tr∗ φ c(λ),where Tr ∗ is as above. Hence d 0,22 ([ω]) = 0 because φ c is Hamiltonian[Lalonde and McDuff (2002)].To deal with d 3 observe first that because the inclusion of the 3−skeletonB 3 into B induces an injection H q (B) → H q (B 3 ) for q ≤ 3, d 0,23 ([ω])vanishes in the spectral sequence for P → B if it vanishes for the pullbackbundle over B 3 . Therefore we may suppose that B is a 3−dimensionalCW–complex whose 2−skeleton B 2 is a wedge of 2 spheres, as π 1 (B) = 0.Further, we can choose the cell decomposition so that the first k 3−cellsspan the kernel of the boundary map C 3 → C 2 in the cellular chain complexof B 3 . Because H 2 (B 2 ) = π 2 (B 2 ), the attaching maps of these first k−cellsare null homotopic. Hence there is a wedge B ′ of 2−spheres and 3−spheresand a map B ′ → B 3 that induces a surjection on H 3 . It therefore sufficesto show that d 0,23 ([ω]) vanishes in the pullback bundle over B′ . This willclearly be the case if it vanishes in every Hamiltonian bundle over S 3 .Now, a Hamiltonian fiber bundle over S 3 is determined by a mapI 2 /∂I 2 = S 2 −→ Ham(M) : (s, t) ↦→ φ s,t ,and it is easy to see that d 0,23 ([ω]) = 0 exactly when the the evaluation mapev x : Ham(M) −→ M : φ ↦→ φ(x)takes π 2 (Ham(M)) into the kernel of ω.Given a smooth map Ψ : (I 2 , ∂I 2 ) → (Ham(M), Id) and x ∈ M, letΨ x : (I 2 , ∂I 2 ) → M be the composite of Ψ with evaluation at x. Then wehave∫I 2 (Ψ x ) ∗ ω = 0, for all x ∈ M.For each s, t let X s,t (resp. Y s,t ) be the Hamiltonian vector field on Mthat is tangent to the flow of the isotopy s ↦→ Ψ x (s, t), (resp. t ↦→ Ψ x (s, t).)Then∫∫ ∫(Ψ x ) ∗ ω = ω(X s,t (Ψ x (s, t)), Y s,t (Ψ x (s, t))) dsdt.I 2


640 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe first observation is that this integral is a constant c that is independentof x, since the maps Ψ x : S 2 → M are all homotopic. Secondly, recall thatfor any Hamiltonian vector fields X, Y on M∫∫ω(X, Y )ω n = n ω(X, ·)ω(Y, ·)ω n−1 = 0,MMsince ω(X, ·), ω(Y, ·) are exact 1−forms. Taking X s,t = X s,t (Ψ x (s, t)) andsimilarly for Y , we have∫(∫)c ω n = ω(X s,t , Y s,t ) ω∫I n ds dt = 0.2MHence c = 0. This lemma can also be proved by purely topological methods[Lalonde and McDuff (2002)].Suppose that π B : P → B is Hamiltonian. It is classified by a mapB → B Ham(M). Because B Ham(M) is simply connected this factorsthrough a map C → B Ham(M), where C is obtained by collapsing the1−skeleton of B to a point. In particular condition (i) is satisfied. Toverify (ii), let π C : Q → C be the corresponding Hamiltonian bundle, sothat there is a commutative diagramP✲ Qπ Bπ C❄❄B ✲ C = B/B 1There is a class a C ∈ H 2 (Q) that restricts to [ω] on the fibers. Its pullbackto P is the desired class a.Conversely, suppose that conditions (i) and (ii) are satisfied. By (i),the classifying map B → B Symp(M) factors through a map f : C →B Symp(M), where C is as above. This map f depends on the choice of asymplectic trivialization of π over the 1−skeleton B 1 of B. We now showthat f can be chosen so that (ii) holds for the associated symplectic bundleQ f → C.We need to show that the differentials (d C ) 0,22 , (d C) 0,23 in the spectralsequence for Q f → C both vanish on [ω]. Let· · · → C k (B) ∂ → C k−1 (B) → · · ·be the cellular chain complex for B, and choose 2−cells e 1 , . . . , e k in Bwhose attaching maps α 1 , . . . , α k form a basis over Q for the image of ∂


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 641in C 1 (B). Then the obvious maps C k (B) → C k (C) (which are the identityfor k > 1) give rise to an isomorphismH 2 (B, Q) ⊕ ⊕ i Q[e i ] ∼ = H 2 (C, Q).By the naturality of spectral sequences, the vanishing of (d B ) 0,22 ([ω]) impliesthat (d C ) 0,22 ([ω]) vanishes on all cycles in H 2(C, Q) coming from H 2 (B, Q).Therefore we just need to check that it vanishes on the cycles e i . For this,we have to choose the trivialization over B 1 so that its pullback by each α igives rise to a Hamiltonian bundle over e i . For this it would suffice that itspullback by each α i is the “natural trivialization”, i.e the one that extendsover the 2-cell e i . To arrange this, choose any symplectic trivialization overB 1 = ∨ j γ j . Then comparing this with the natural trivialization gives riseto a homomorphismΦ : ⊕ i Ze i → π 1 Symp(M, Z) F lux−→ H 1 (M, R).Since the boundary map ⊕ i Ze i → C 1 (B) ⊗ Q is injective, we can nowchange the chosen trivializations over the 1−cells γ j in B 1 to make Φ = 0.This ensures that d 0,22 = 0 in the bundle over C. Since the mapH q (C) → H q (B) is an isomorphism when q ≥ 3, the vanishing of d 0,23for B implies that it vanishes for C. Therefore (ii) holds for Q → C. Bythe previous result, this implies that the structural group of Q → C reducesto Ham(M). Therefore, the same holds for P → B.So we have established the following result [Lalonde and McDuff (2002)]:Let C be the CW complex obtained by collapsing the 1−skeleton of B to apoint and f : B → C be the obvious map. Then any Hamiltonian bundleP → B is the pullback by f of some Hamiltonian bundle over C.The above theorem shows that there are two obstructions to the existenceof a Hamiltonian structure on a symplectic bundle. Firstly, thebundle must be symplectically trivial over the 1−skeleton B 1 , and secondlythe symplectic class on the fiber must extend. The first obstruction obviouslydepends on the 1−skeleton B 1 while the second, in principle, dependson its 3−skeleton (since we need d 2 and d 3 to vanish on [ω]). However, infact, it only depends on the 2−skeleton, as is shown in the following lemma:Every symplectic bundle over a 2−connected base B is Hamiltonian. Notethat we just have to show that d 0,23 ([ω]) = 0. Alternatively, let Symp 0 (resp.˜Ham) denote the universal cover of the group Symp 0 = Symp 0 (M, ω) (resp


642 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionHam(M)), and set π S = π 1 (Symp 0 ) so that there are fibrations˜Ham → Symp 0F lux→ H 1 (M, R), B(π S ) → B Symp 0 → B Symp 0 .The existence of the first fibration shows that ˜Ham is homotopy equivalentto Symp 0 so that B ˜Ham ≃ B Symp 0 , while the second implies that thereis a fibrationB Symp 0 → B Symp 0 → K(π S , 2),where K(π S , 2) is an Eilenberg–MacLane space. A symplectic bundle overB is equivalent to a homotopy class of maps B → B Symp 0 . If B is2−connected, the composite B → B Symp 0 → K(π S , 2) is null homotopic,so that the map B → B Symp 0 lifts to B Symp 0 and hence to the homotopicspace B ˜Ham. Composing this map B → B ˜Ham with the projectionB ˜Ham → B Ham we get a Hamiltonian structure on the given bundleover B.Equivalently, use the existence of the fibration ˜Ham → Symp 0 →H 1 (M, R) to deduce that the subgroup π 1 (Ham) of ˜Ham injectsinto π 1 (Symp 0 ). This implies that the relative homotopy groupsπ i (Symp 0 , Ham) vanish for i > 1, so thatπ i (B Symp 0 , B Ham) = π i−1 (Symp 0 , Ham) = 0, (i > 2).The desired conclusion now follows by obstruction theory. The second proofdoes not directly use the sequence0 −→ Ham −→ Symp 0 −→ H 1 /Γ ω −→ 0,since the flux group Γ ω may not be a discrete subgroup of H 1 .4.12.1.9 Classification of Hamiltonian StructuresThe previous subsection discussed the question of the existence of Hamiltonianstructures on a given bundle. We now look at the problem of describingand classifying them.Let π : P → B be a symplectic bundle satisfying the above conditionsand fix an identification of (M, ω) with (M b0 , ω b0 ). Let a be anyclosed extension of [ω], γ 1 , . . . , γ k be a set of generators of the first rationalhomology group of B, {c i } the dual basis of H 1 (B) and T 1 , . . . , T k symplectictrivializations round the γ i . Assume for the moment that each classf(T i , a) ∈ H 1 (M b0 ) = H 1 (M) has an extension ˜f(T i , a) to P . Subtracting


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 643from a the class ∑ ki=1 π∗ (c i ) ∪ ˜f(T i , a), we get a closed extension a ′ whosecorresponding classes f(T i , a ′ ) belong to Γ ω .There remains to prove that the extensions of the f(T i , a)’s exist inHamiltonian bundles. It is enough to prove that the fiber inclusion M → Pinduces an injection on the first homology group. One only needs to provethis over the 2−skeleton B 2 of B, and we can assume as well that B 2 is awedge of 2−spheres. Hence this is a consequence of the easy fact that theevaluation of a Hamiltonian loop on a point of M gives a 1−cycle of Mthat is trivial in rational homology, i.e., that the differential d 0,12 vanishes inthe cohomology spectral sequence for P → B (see [Lalonde et al. (1999)],where this is proved by elementary methods).The following result extends the above lemma: Let P → B be a symplecticbundle with a given symplectic trivialization of P over B 1 , and leta ∈ H 2 (P ) be a normalized extension of the fiber symplectic class. Thenthe restriction of a to π −1 (B 1 ) defines and is defined by a homomorphismΦ from π 1 (B) to Γ ω .As above, we can use the given trivialization to identify the holonomyround some loop s ↦→ γ(s) ∈ B 1 with a family of symplectomorphismsΦ γ s : M → M, s ∈ [0, 1]. Given a 1−cycle δ : S 1 → M in the fiber M over1 ∈ ∂D 2 , consider the closed 2−cycle C(γ, δ) = c 1 ∪ c 2 as before. Sinceτ(c 1 ) = 0,If we now setτ(C(γ, δ)) = τ(c 2 ) = −F lux({Φ γ s })(δ).Φ(γ) = −F lux({Φ γ s }),it is easy to check that Φ is a homomorphism [Lalonde and McDuff (2002)].Its values are in Γ ω by the definition of normalized extension classes.The next task is to prove the above theorem that characterizes Hamiltonianstructures. Thus we need to understand the homotopy classes of lifts ˜gof the classifying map g : B → B Symp(M, ω) of the underlying symplecticbundle to B Ham(M). We first consider the intermediate lift ĝ of g intoB Symp 0 (M, ω). In view of the fibration sequenceπ 0 (Symp) → B Symp 0 → B Symp → B(π 0 (Symp))in which each space is mapped to the homotopy fiber of the subsequent map,a map g : B → B Symp lifts to ĝ : B → B Symp 0 iff the symplectic bundlegiven by g can be trivialized over the 1−skeleton B 1 of B. Moreover such


644 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionlifts are in bijective correspondence with the elements of π 0 (Symp) and socorrespond to an identification (up to symplectic isotopy) of (M, ω) withthe fiber (M b0 , ω b0 ) at the base point b 0 (recall that B is always assumedto be connected).To understand the full lift ˜g, recall the above exact sequence{Id} −→ Ham(M, ω) −→ Symp 0 (M, ω) F lux−→ H 1 (M, R)/Γ ω −→ {0}.If Γ ω is discrete, then the space H 1 (M, R)/Γ ω is homotopy equivalent to atorus and we can investigate the liftings ˜g by homotopy theoretic argumentsabout the fibrationH 1 (M, R)/Γ ω → B Ham(M, ω) → B Symp 0 (M, ω).Now, suppose that a symplectic bundle π : P → B is given that satisfiesthe above conditions. Fix an identification of (M, ω) with (M b0 , ω b0 ).We have to show that lifts from B Symp 0 to B Ham are in bijective correspondencewith equivalence classes of normalized extensions a of the fibersymplectic class. There is a lift iff there is a normalized extension classa. Therefore, it remains to show that the equivalence relations correspond.The essential reason why this is true is that the induced mapπ i (Ham(M, ω)) → π i (Symp 0 (M, ω))is an injection for i = 1 and an isomorphism for i > 1. This, in turn, followsfrom the exactness of the sequence (∗).{Id} −→ Ham(M, ω) −→ Symp 0 (M, ω) F lux−→ H 1 (M, R)/Γ ω −→ {0}.Let us spell out a few more details, first when B is simply connected.Then the classifying map from the 2−skeleton B 2 to B Symp 0 has a lift toB Ham iff the image of the induced mapπ 2 (B 2 ) → π 2 (B Symp 0 (M)) = π 1 Symp 0 (M)lies in the kernel of the flux homomorphismF lux : π 1 (Symp 0 (M)) −→ Γ ω .Since π 1 (Ham(M, ω)) injects into π 1 (Symp 0 (M, ω)), there is only one suchlift up to homotopy. Standard arguments now show that this lift can beextended uniquely to the rest of B. Hence in this case there is a uniquelift. Correspondingly there is a unique equivalence class of extensions a[Lalonde and McDuff (2002)].


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 645Now let us consider the general case: We are given a map g : B →B Symp 0 and want to identify the different homotopy classes of liftings ofg to B Ham. Let C = B/B 1 as above. There is a symplectic trivializationT over B 1 that is compatible with the given identification of the base fiberand induces a map C → B Symp 0 which lifts to B Ham. Since this liftingg T,C of C → B Symp 0 is unique, each isotopy class T of such trivializationsover B 1 gives rise to a unique homotopy class g T of maps B → B Ham,namelyg T :B −→ C g T,C−→ B Ham.Note that g T is a lifting of f and that every lifting occurs this way.Standard arguments show that two such isotopy classes differ by a homomorphismπ 1 (B) −→ π 1 (Symp 0 ).Moreover, the corresponding maps g T and g T ′ ′are homotopic iff T and Tdiffer by a homomorphism with values in π 1 (Ham). Thus homotopy classesof liftings of g to B Ham are classified by homomorphisms π 1 (B) → Γ ω .These homomorphisms are precisely what defines the equivalence classes ofextensions a.4.12.2 Properties of General Hamiltonian BundlesThe key to extending results about Hamiltonian bundles over S 2 to higher–dimensional bases is their functorial properties, in particular their behaviorunder composition. Before discussing this, it is useful to establish the factthat this class of bundles is stable under small perturbations of the symplecticform on M [Lalonde and McDuff (2002)].4.12.2.1 StabilityMoser’s argument implies that for every symplectic structure ω on M thereis a Serre fibrationSymp(M, ω) −→ <strong>Diff</strong>(M) −→ S ω ,where S ω is the space of all symplectic structures on M that are diffeomorphicto ω. At the level of classifying spaces, this gives a homotopyfibrationS ω ↩→ BSymp(M, ω) −→ B<strong>Diff</strong>(M).


646 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAny smooth fiber bundle P → B with fiber M is classified by a map B →B <strong>Diff</strong>(M), and isomorphism classes of symplectic structures on it withfiber (M, ω) correspond to homotopy classes of sections of the associatedfibration W (ω) → B with fiber S ω . We will suppose that π is described bya finite set of local trivializations T i : π −1 (V i ) → V i ×M with the transitionfunctions φ ij : V i ∩ V j → <strong>Diff</strong>(M).Suppose that M → P → B is a smooth fibration constructed from acocycle (T , φ ij ) with the following property: there is a symplectic form ωon M such that for each x ∈ M the convex hull of the finite set of forms{φ ∗ ij(ω) : x ∈ V i ∩ V j } lies in the set S ω of symplectic forms diffeomorphicto ω. Then (M, ω) → P → B may be given the structure of a symplecticbundle [Lalonde and McDuff (2002)].It suffices to construct a section σ of W (ω) → B. The hypothesis impliesthat for each x the convex hull of the set of forms T i (x) ∗ (ω), x ∈ V i , lies inthe fiber of W (ω) at x. Hence we may take σ(x) = ρ i T ∗ i (ω), where ρ i is apartition of unity subordinate to the cover V i .Let P −→ B be a symplectic bundle with closed fiber (M, ω) and compactbase B. There is an open neighborhood of ω in the space (M) of allsymplectic forms on M such that, for all ω ′ ∈ U, the structural group ofπ : P −→ B may be reduced to Symp(M, ω ′ ).Trivialize P −→ B so that φ ∗ ij(ω) = ω for all i, j. Then the hypothesisof the lemma is satisfied for all ω ′ sufficiently close to ω by the openness ofthe symplectic condition.Thus the set π (M) of symplectic forms on M, with respect to which π issymplectic, is open. The aim of this section is to show that a correspondingstatement is true for Hamiltonian bundles. The following example showsthat the Hamiltonian property need not survive under large perturbationsof ω.Here is an example of a smooth family of symplectic bundles that isHamiltonian at all times 0 ≤ t < 1 but is non–Hamiltonian at time 1. Leth t (0 ≤ t ≤ 1) be a family of diffeomorphisms of M with h 0 = Id and defineQ = M × [0, 1] × [0, 1]/(x, 1, t) ≡ (h t (x), 0, t).Thus we can think of Q as a family of bundles π : P t → S 1 with monodromyh t at time t. It was shown in [Seidel (1997)] that there are smooth families ofsymplectic forms ω t and diffeomorphisms h t ∈ Symp(M, ω t ) for t ∈ [0, 1]such that h t is not in the identity component of Symp(M, ω t ) for t = 1but is in this component for t < 1. For such h t each bundle P t → S 1 issymplectic. Moreover, it is symplectically trivial and hence Hamiltonian


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 647for t < 1 but is non–Hamiltonian at t = 1.We have the following result [Lalonde and McDuff (2002)]: A Hamiltonianbundle π : P → B is stable iff the restriction map H 2 (P ) → H 2 (M)is surjective. If π : P → B is Hamiltonian with respect to ω ′ then [ω ′ ] is inthe image of H 2 (P ) → H 2 (M). If π is stable, then [ω ′ ] fills out a neighborhoodof [ω] which implies surjectivity. Conversely, suppose that we havesurjectivity. To check (i) let γ : S 1 → B be a loop in B and suppose thatγ ∗ (P ) is identified symplectically with the product bundle S 1 × (M, ω).Let ω t , 0 ≤ t ≤ ε, be a (short) smooth path with ω 0 = ω. Then, becauseP → B has the structure of an ω t −symplectic bundle for each t, each fiberM b has a corresponding smooth family of symplectic forms ω b,t of the formg ∗ b,t ψ∗ b(ω t ), where ψ b is a symplectomorphism (M b , ω b ) → (M, ω). Hence,for each t, γ ∗ (P ) can be symplectically identified with∪ s∈[0,1] ({s} × (M, g ∗ s,t(ω t ))), where g ∗ 1,t(ω t ) = ω tand the g s,t lie in an arbitrarily small neighborhood U of the identity in<strong>Diff</strong>(M). By Moser’s homotopy argument, we can choose U so small thateach g 1,t is isotopic to the identity in the group Symp(M, ω t ).Now, the pullback of a stable Hamiltonian bundle is stable [Lalonde andMcDuff (2002)]. Suppose that P −→ B is the pullback of P ′ −→ B ′ via B−→ B ′ so that there is a commutative diagramP ✲ P ′❄B ✲ ❄B ′By hypothesis, the restriction H 2 (P ′ ) −→ H 2 (M) is surjective. But thismap factors as H 2 (P ′ ) −→ H 2 (P ) −→ H 2 (M). Hence H 2 (P ) −→ H 2 (M) isalso surjective. Fe have the following lemma:(i) Every Hamiltonian bundle over S 2 is stable.(ii) Every symplectic bundle over a 2−connected base B is Hamiltonianstable.(i) holds because every Hamiltonian bundle over S 2 is c−split, in particularthe restriction map H 2 (P ) → H 2 (M) is surjective .The above theorem states that every Hamiltonian bundle is stable. Toprove this, first observe that we can restrict to the case when B is simplyconnected. For the map B → B Ham(M) classifying P factors through amap C → B Ham(M), where C = B/B 1 as before, and the stability of the


648 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioninduced bundle over C implies that for the original bundle.Next observe that a Hamiltonian bundle P → B is stable iff the differentialsd 0,2k: E 0,2k→ E k,3−kkin its Leray cohomology spectral sequencevanish on the whole of H 2 (M) for k = 2, 3. We can reduce the statementfor d 0,22 to the case B = S 2 . Thus d 0,22 = 0. Similarly, we can reduce thestatement for d 0,23 to the case B = S 3 .4.12.2.2 Functorial PropertiesWe begin with some trivial observations and then discuss composites ofHamiltonian bundles. The first lemma is true for any class of bundles withspecified structural group [Lalonde and McDuff (2002)].Suppose that π : P → B is Hamiltonian and that g : B ′ → B is acontinuous map. Then the induced bundle π ′ : g ∗ (P ) → B ′ is Hamiltonian.Recall that any extension τ of the forms on the fibers is called a connectionform.If P → B is a smooth Hamiltonian fiber bundle over a symplectic base(B, σ) and if P is compact then there is a connection form Ω κ on P that issymplectic.The bundle P carries a closed connection form τ. Since P is compact,the form Ω κ = τ + κπ ∗ (σ) is symplectic for large κ.Observe that the deformation type of the form Ω κ is unique for sufficientlylarge κ since given any two closed connection forms τ, τ ′ the linearisotopytτ + (1 − t)τ ′ + κπ ∗ (σ), 0 ≤ t ≤ 1,consists of symplectic forms for sufficiently large κ. However, it can happenthat there is a symplectic connection form τ such that τ + κπ ∗ (σ) is notsymplectic for small κ > 0, even though it is symplectic for large κ. (Forexample, suppose P = M × B and that τ is the sum ω + π ∗ (ω B ) whereω B + σ is not symplectic.)Let us now consider the behavior of Hamiltonian bundles under composition.If(M, ω) → P π P→ X, and (F, σ) → X π →XBare Hamiltonian fiber bundles, then the restrictionπ P :W = π −1P(F ) −→ F


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 649is a Hamiltonian fiber bundle. Since F is a manifold, we can assume withoutloss of generality that W → F is smooth. Moreover, since (F, σ) issymplectic, the manifold W carries a symplectic connection form Ω κ W , andit is natural to ask when the composite map π : P → B with fiber (W, Ω κ W )is itself Hamiltonian.Suppose in the above situation that B is simply connected and that Pis compact. Then π = π X ◦ π P : P → B is a Hamiltonian fiber bundle withfiber (W, Ω κ W ), whereΩκ W = τ W + κπ ∗ P (σ),τ W is any symplectic connection form on W , and κ is sufficiently large.By above lemma, we may assume that the base B as well as the fibrationsare smooth. We first show that there is some symplectic form onW for which π is Hamiltonian and then show that it is Hamiltonian withrespect to the given form Ω κ W .Let τ P (resp. τ X ) be a closed connection form with respect to thebundle π P , (resp. π X ), and let τ W be its restriction to W . Then Ω κ W isthe restriction to W of the closed formΩ κ P = τ P + κπ ∗ P (τ X ).By increasing κ if necessary we can ensure that Ω κ P restricts to a symplecticform on every fiber of π not just on the the chosen fiber W . Thisshows firstly that π : P → B is symplectic, because there is a well definedsymplectic form on each of its fibers, and secondly that it is Hamiltonianwith respect to this form Ω κ W on the fiber W . Hence H2 (P ) surjects ontoH 2 (W ).Now suppose that τ W is any closed connection form on π P : W → F .Because the restriction map H 2 (P ) → H 2 (W ) is surjective, the cohomologyclass [τ W ] is the restriction of a class on P and so, by Thurston’s construction,the form τ W can be extended to a closed connection form τ P for thebundle π P . Therefore the previous argument applies in this case too.We know that the restriction map H 2 (P ) → H 2 (M) is surjective, sothat the cohomology class [τ W ] of the connection form τ W of W → F isthe restriction of a class on P , and by Thurston’s construction the form τ Wcan be extended to a closed connection form τ P for the bundle π P . Choosealso a closed connection form τ X for the bundle π X . Then Ω κ W is therestriction to W of the closed form Ω κ P = τ P + κπ ∗ P (τ X). By increasingκ if necessary we can ensure that Ω κ P restricts to a symplectic form on


650 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionevery fiber of π not just on the the chosen fiber W . This shows firstly thatπ : P → B is symplectic, because there is a well defined symplectic form oneach of its fibers, and secondly that it is Hamiltonian with respect to thisform Ω κ W on the fiber W . Hence H2 (P ) surjects onto H 2 (W ).Now suppose that τ W is any closed connection form on π P : W → F .Because the restriction map H 2 (P ) → H 2 (W ) is surjective, the cohomologyclass [τ W ] is the restriction of a class on P and so, by Thurston’s construction,the form τ W can be extended to a closed connection form τ P for thebundle π P . Therefore the previous argument applies in this case too.Now let us consider the general situation, when π 1 (B) ≠ 0. The proofof the lemma above applies to show that the composite bundle π : P → Bis symplectic with respect to suitable Ω κ W and that it has a symplecticconnection form. However, even though π X : X → B is symplecticallytrivial over the 1-skeleton of B the same may not be true of the compositemap π : P → B. Moreover, in general it is not clear whether triviality withrespect to one form Ω κ W implies it for another. Therefore, we may concludethe following: If(M, ω) → P π P→ X and (F, σ) → X π →XBare Hamiltonian fiber bundles and P is compact, then the composite π =π X ◦ π P : P → B is a symplectic fiber bundle with respect to any form Ω κ Won its fiber W = π −1 (pt), provided that κ is sufficiently large. Moreoverif π is symplectically trivial over the 1−skeleton of B with respect to Ω κ Wthen π is Hamiltonian.In practice, we will apply these results in cases where π 1 (B) = 0. Wewill not specify the precise form on W , assuming that it is Ω κ W for a suitableκ.4.12.2.3 Splitting of Rational CohomologyWe write H ∗ (X), H ∗ (X) for the rational (co)homology of X. Recall that abundle π : P → B with fiber M is said to be c−split ifH ∗ (P ) ∼ = H ∗ (B) ⊗ H ∗ (M).This happens iff H ∗ (M) injects into H ∗ (P ). Dually, it happens iff therestriction map H ∗ (P ) → H ∗ (M) is onto. Note also that a bundle P → Bc−splits iff the E 2 term of its cohomology spectral sequence is a productand all the differentials d k , k ≥ 2, vanish [Lalonde and McDuff (2002)].


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 651The following surjection lemma is obvious but useful: Consider a commutativediagramP ′✲ PB ′❄❄✲ Bwhere P ′ is the induced bundle. Then:(i) If P → B is c−split so is P ′ → B ′ .(ii) If P ′ → B ′ is c−split and H ∗ (B ′ ) → H ∗ (B) is surjective, thenP → B is c−split.To prove (i), we can use the fact that P → B is c−split iff the mapH ∗ (M) → H ∗ (P ) is injective. To prove (ii), the induced map on theE 2 −term of the cohomology spectral sequences is injective. Therefore theexistence of a nonzero differential in the spectral sequence P → B impliesone for the pullback bundle P ′ → B ′ .As a corollary, suppose that P → W is a Hamiltonian fiber bundle overa symplectic manifold W and that its pullback to some blowup Ŵ of W isc−split. Then P → W is c−split.This follows immediately from (ii) above since the map H ∗ (Ŵ ) →H ∗ (W ) is surjective.If (M, ω) → π P → B is a compact Hamiltonian bundle over a simplyconnected CW–complex B and if every Hamiltonian fiber bundle over Mand B is c−split, then every Hamiltonian bundle over P is c−split.Let π E : E → P be a Hamiltonian bundle with fiber F and letF → W → M be its restriction over M. Then by assumption the latterbundle c−splits so that H ∗ (F ) injects into H ∗ (W ). The above compositionlemma implies that the composite bundle E → B is Hamiltonian with fiberW and therefore also c−splits. Hence H ∗ (W ) injects into H ∗ (E). ThusH ∗ (F ) injects into H ∗ (E), as required.If Σ is a closed orientable surface then any Hamiltonian bundle overS 2 × . . . × S 2 × Σ is c−split.Consider any degree one map f from Σ → S 2 . Because Ham(M, ω)is connected, B Ham(M, ω) is simply connected, and therefore any homotopyclass of maps from Σ → B Ham(M, ω) factors through f. Thus anyHamiltonian bundle over Σ is the pullback by f of a Hamiltonian bundleover S 2 . Because such bundles c−split over S 2 , the same is true over Σ.Any Hamiltonian bundle over S 2 × . . . × S 2 × S 1 is c−split. For each


652 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionk ≥ 1, every Hamiltonian bundle over S k c−splits.There is for each k a k−dimensional closed manifold X such that everyHamiltonian bundle over X c−splits. Given any Hamiltonian bundle P →S k consider its pullback to X by a map f : X → S k of degree 1. Since thepullback c−splits, the original bundle does too by the surjection lemma.As we shall see this result implies that the action of the homology groupsof Ham(M) on H ∗ (M) is always trivial. Here are some other examples ofsituations in which Hamiltonian bundles are c−split.Every Hamiltonian bundle over CP n1 × . . . × CP n kc−splits.Let us prove first that it splits over CP n . Use induction over n. Againit holds when n = 1. Assuming the result for n let us prove it for n + 1.Let B be the blowup of CP n+1 at one point. Then B fibers over CP n withfiber CP 1 . Thus every Hamiltonian bundle over B c−splits.Every Hamiltonian bundle whose structural group reduces to a subtorusT ⊂ Ham(M) c−splits [Lalonde and McDuff (2002)]. It suffices to considerthe universal modelM −→ ET × T M −→ BT,and hence to show that all Hamiltonian bundles over BT are c−split. Butthis is equal to CP ∞ × . . . × CP ∞ and the proof that the i th group ofhomology of the fiber injects in P → CP ∞ × . . . × CP ∞ may be reducedto the proof that it injects in the restriction of the bundle P over CP j ×. . . × CP j for a sufficiently large j.Note that the proof of the above corollary shows that every Hamiltonianbundle over CP ∞ ×. . .×CP ∞ c−splits. Since the structural group of sucha bundle can be larger than the torus T , the result presented here extendsthe Atiyah-Bott splitting theorem for Hamiltonian bundles with structuralgroup T .For completeness, we show how the above corollary leads to a proof ofthe splitting of G−equivariant cohomology where G is a Lie subgroup ofHam(M, ω).If G is a compact connected Lie group that acts in a Hamiltonian wayon M then any bundle P → B with fiber M and structural group G isc−split. In particular,H ∗ G(M) ∼ = H ∗ (M) ⊗ H ∗ (BG).We only need to prove the second statement, sinceM G = EG × G M −→ BG


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 653is the universal bundle. Every compact connected Lie group G is the imageof a homomorphism T × H → G, where the torus T maps onto the identitycomponent of the center of G and H is the semi–simple Lie group correspondingto the commutator subalgebra [Lie(G), Lie(G)] in the Lie algebraLie(G). It is easy to see that this homomorphism induces a surjection onrational homology BT × BH → BG. Therefore, we may suppose thatG = T × H. Let T max = (S 1 ) k be the maximal torus of the semi–simplegroup H. Then the induced map on cohomologyH ∗ (BH) → H ∗ (BT max ) = Q[a 1 , . . . , a k ]takes H ∗ (BH) bijectively onto the set of polynomials in H ∗ (BT max ) thatare invariant under the action of the Weyl group, by the Borel–Hirzebruchtheorem. Hence the mapsBT max → BH and BT × BT max → BGinduce a surjection on homology. Therefore the desired result follows fromthe surjection lemma and the Atiyah–Bott theorem.We have the following lemma [Lalonde and McDuff (2002)]: EveryHamiltonian bundle over a coadjoint orbit c−splits.This is an immediate consequence of the results by [Grossberg andKarshon (1994)] on Bott towers. Recall that a Bott tower is an iteratedfibration of Kähler manifoldsM k → M k−1 → · · · → M 1 = S 2where each map M i+1 → M i is a fibration with fiber S 2 . They show thatany coadjoint orbit X can be blown up to a manifold that is diffeomorphicto a Bott tower M k . Moreover the blow–down map M k → X inducesa surjection on rational homology. Every Hamiltonian bundle over M kc−splits. Hence the result follows from the surjection lemma.Every Hamiltonian bundle over a 3−complex X c−splits. As in theproof of stability given above, we can reduce this to the cases X = S 2 andX = S 3 . The only difference from the stability result is that we now requirethe differentials d 0,q2 , d0,q 3 to vanish for all q rather than just at q = 2.Every Hamiltonian bundle over a product of spheres c−splits, providedthat there are no more than 3 copies of S 1 .By hypothesis,B = ∏ S 2mi × ∏ S 2ni+1 × T k ,i∈Ij∈J


654 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere n i > 0 and 0 ≤ k ≤ 3. SetB ′ = ∏ i∈ICP mi × ∏ j∈JCP ni × T |J| × T l ,where l = k if k +|J| is even, and = k +1 otherwise. Since CP ni ×S 1 mapsonto S 2ni+1 by a map of degree 1, there is a homology surjection B ′ → Bthat maps the factor T l to T k . By the surjection lemma, it suffices to showthat the pullback bundle P ′ → B ′ is c−split.Consider the fibrationT |J| × T l → B ′ → ∏ i∈ICP mi × ∏ j∈JCP ni .Since |J|+l is even, we can think of this as a Hamiltonian bundle. Moreover,by construction, the restriction of the bundle P ′ → B ′ to T |J| × T l is thepullback of a bundle over T k , since the map T |J| → B is nullhomotopic.(Note that each S 1 factor in T |J| goes into a different sphere in B.) Becausek ≤ 3, the bundle over T k c−splits. Hence we can conclude that P ′ → B ′c−splits.Every Hamiltonian bundle whose fiber has cohomology generated by H 2is c−split. This is an immediate consequence of the stability theorem.Any Hamiltonian fibration c−splits if its base B is the image of a homologysurjection from a product of spheres and projective spaces, providedthat there are no more than three S 1 factors. One can also consider iteratedfibrations of projective spaces, rather than simply products. However, wehave not yet managed to deal with arbitrary products of spheres. In orderto do this, it would suffice to show that every Hamiltonian bundle over atorus T m c−splits. This question has not yet been resolved for m ≥ 4.4.12.2.4 Hamiltonian Bundles and Gromov–Witten InvariantsWe begin by sketching an alternate proof that every Hamiltonian bundleover B = CP n is c−split that generalizes the arguments in [McDuff (2000)].We will use the language of [McDuff (1999)], which is based on the Liu–Tian [Liu and Tian (1998)] approach to general Gromov–Witten invariants.Clearly, any treatment of general Gromov–Witten invariants could be usedinstead.We have the fundamental result: [Lalonde and McDuff (2002)] EveryHamiltonian bundle over CP n is c−split. To prove this, the basic idea is toshow that the inclusion ι : H ∗ (M) → H ∗ (P ) is injective by showing that for


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 655every nonzero a ∈ H ∗ (M) there is b ∈ H ∗ (M) and σ ∈ H 2 (P ; Z) for whichthe Gromov–Witten invariant n P (ι(a), ι(b); σ) is nonzero. Intuitively speaking,this invariant ‘counts the number of isolated J−holomorphic curves inP that represent the class σ and meet the classes ι(a), ι(b).’ More correctly,it is defined to be the intersection number of the image of the evaluationmapev : M ν 0,2(P, J, σ) −→ P × P,with the class ι(a) × ι(b), where M ν 0,2(P, J, σ) is a virtual moduli cyclemade from perturbed J−holomorphic curves with 2 marked points, and evis given by evaluating at these two points. As explained in [McDuff (1999);McDuff (2000)], M ν = M ν 0,2(P, J, σ) is a branched pseudo–manifold, i.e.,a kind of stratified space whose top dimensional strata are oriented andhave rational labels. Roughly speaking, one can think of it as a finitesimplicial complex, whose dimension d equals the ‘formal dimension’ ofthe moduli space, i.e., the index of the relevant operator. The elementsof M ν are stable maps [Σ, h, z 1 , z 2 ] where z 1 , z 2 are two marked points onthe nodal, genus 0, Riemann surface Σ, and the map h : Σ → P satisfiesa perturbed Cauchy–Riemann equation M J h = ν h . The perturbation νcan be arbitrarily small, and is chosen so that each stable map in M ν isa regular point for the appropriate Fredholm operator. Hence M ν is oftencalled a regularization of the unperturbed moduli space M = M 0,2 (P, J, σ)of all J−holomorphic stable maps.Given any Hamiltonian bundle P S → S 2 and any a ∈ H ∗ (M), it wasshown in [Lalonde et al. (1999); McDuff (2000)] that there is b ∈ H ∗ (M)and a lift σ S ∈ H 2 (P S ; Z) of the fundamental class of S 2 to P S such thatn PS (ι S (a), ι S (b); σ S ) ≠ 0,where ι S denotes the inclusion into P S . Therefore, if P S is identified withthe restriction of P to a complex line L 0 in B and if a, b and σ S are asabove, it suffices to prove thatn PS (ι S (a), ι S (b); σ S ) = n P (ι(a), ι(b); σ)where σ is the image of σ S in P . Note that a direct count shows that thedimensions of the appropriate virtual moduli spaces M ν 0,2(P S , J S , σ S ) andM ν 0,2(P, J, σ) differ by the codimension of P S × P S in P × P (which equalsthe codimension of CP 1 × CP 1 in CP n × CP n ) so that the both sides arewell–defined.


656 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAs was shown in [McDuff (2000)], one can construct the virtual modulicycle M ν (P S ) = M ν 0,2(P S , J S , σ S ) using an almost complex structure J Sand a perturbation ν that are compatible with the bundle. In particular,this implies that the projection P S → S 2 is (J S , j)−holomorphic (where jis the usual complex structure on S 2 ) and that every element of M ν (P S )projects to a j−holomorphic stable map in S 2 .Following [Lalonde and McDuff (2002)], we claim that this is also truefor the bundle P → B. In other words, we can choose J so that theprojection (P, J) → (B, j) is holomorphic, where j is the usual complexstructure on B = CP n , and we can choose ν so that every element inM ν (P ) projects to a j−holomorphic stable map in B. The proof is exactlyas before [McDuff (2000)]. The essential point is that every element of theunperturbed moduli space M 0,2 (CP n , j, [CP 1 ]) is regular. In fact, the topstratum in M 0,2 (CP n , j, [CP 1 ]) is the space L = M 0,2 (CP n , j, [CP 1 ])of all lines in CP n with 2 distinct marked points. The other stratumcompletes this space by adding in the lines with two coincident markedpoints, which are represented as stable maps by a line together with aghost bubble containing the two points.It follows that there is a projection mapproj : M ν 0,2(P, J, σ) −→ M 0,2 (CP n , j, [CP 1 ]).Moreover the inverse image of a line L ∈ can morally speaking be identifiedwith M ν 0,2(P S , J S , σ S ). The latter statement would be correct if we wereconsidering ordinary moduli spaces of stable maps, but the virtual modulispace is not usually built in such a way that the fibers (proj) −1 (L) have theneeded structure of a branched pseudo–manifold. However, we can chooseto construct M ν 0,2(P, J, σ) so that this is true for all lines near a fixed lineL 0 . A detailed recipe is given in [McDuff (2000)] for constructing M ν fromthe unperturbed moduli space M . The construction is based on the choiceof suitable covers {U i }, {V I } of M and of perturbations ν i over each U i .Because regularity is an open condition, one can make these choices firstfor all stable maps that project to the fixed line L 0 and then extend to theset of stable maps that project to nearby lines in such a way that M ν islocally a product near the fiber over L 0 (see the proof of [McDuff (2000)]for a very similar construction).Once this is done, the rest of the argument is easy. If we identify P S withπ −1 (L 0 ) and choose a representative α × β of ι S (a) × ι S (b) in P S × P S thatis transverse to the evaluation map from M ν 0,2(P S , J S , σ S ), its image in


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 657P × P will be transverse to the evaluation map from M ν 0,2(P, J, σ) becauseproj is a submersion at L 0 . Moreover, by [McDuff (2000)], we may supposethat α and β lie in distinct fibers of the projection P S → S 2 . Let x a , x bbe the corresponding points of CP n under the identification S 2 = L 0 .Then every stable map that contributes to n P (ι(a), ι(b); σ) projects to anelement of M 0,2 (CP n , j, [CP 1 ]) whose marked points map to the distinctpoints x a , x b . Since there is a unique line in CP n through two given points,in this case L 0 , every stable map that contributes to n P (ι(a), ι(b); σ) mustproject to L 0 and hence be contained in M ν 0,2(P S , J S , σ S ). One can thencheck thatn PS (ι S (a), ι S (b); σ S ) = n P (ι(a), ι(b); σ),as claimed. The only delicate point here is the verify that the sign of eachstable map on the left hand side is the same as the sign of the correspondingmap on the right hand side. But this is also a consequence of the localtriviality of the above projection map (see [Lalonde et al. (1998); Lalondeet al. (1999)] for more details).The above argument generalizes easily to the case when B is a complexblowup of CP n .Let B be a blowup of CP n along a disjoint union Q = ∐ Q i of complexsubmanifolds, each of complex codimension ≥ 2. Then every Hamiltonianbundle B is c−split.The above argument applies almost verbatim in the case when Q is afinite set of points. The top stratum of M 0,2 (B, j, [CP 1 ]) still consists oflines marked by two distinct points, and again all elements of this unperturbedmoduli space are regular.In the general case, there is a blow-down map ψ : B → CP n which isbijective over CP n − Q, and we can choose j on B so that the exceptionaldivisors ψ −1 (Q) are j−holomorphic, and so that j is pulled back from theusual structure on CP n outside a small neighborhood of ψ −1 (Q). Let L 0 bea complex line in CP n − Q. Then its pullback to B is still j−holomorphic.Hence, although the unperturbed moduli space M 0,2 (B, j, [CP 1 ]) may containnonregular and hence ‘bad’ elements, its top stratum does contain anopen set U L0 consisting of marked lines near L 0 that are regular. Moreover,if we fix two points x a , x b on L 0 , every element of M 0,2 (B, j, [CP 1 ]) whosemarked points map sufficiently close to x a , x b actually lies in this openset U L0 . We can then regularize M 0,2 (B, j, [CP 1 ]) to a virtual modulicycle that contains the open set U L0 as part of its top stratum. More-


658 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionover, because the construction of the regularization is local with respect toM 0,2 (B, j, [CP 1 ]), this regularization M ν 0,2(B, j, [CP 1 ]) will still have theproperty that each of its elements whose marked points map sufficientlyclose to x a , x b actually lies in this open set U L0 .We can now carry out the previous argument, choosing J on P to befibered, and constructing ν to be compatible with the fibration on thatpart of M 0,2 (P, J, σ) that projects to U L0[Lalonde and McDuff (2002)].Let X = #kCP 2 #lM 2 be the connected sum of k copies of CP 2 with lcopies of M 2 . If one of k, l is ≤ 1 then every bundle over X is c−split.By reversing the orientation of X we can suppose that k ≤ 1. The casek = 1 is covered in the previous proposition. When k = 0, pull the bundleback over the blowup of X at one point and then use the surjection lemma.The previous proof can easily be generalized to the case of a symplecticbase B that has a spherical 2-class A with Gromov–Witten invariantof the form n B (pt, pt, c 1 , . . . , c k ; A) absolutely equal to 1 19 . Herec 1 , . . . , c k are arbitrary homology classes of B and we assume that k ≥ 0.Again the idea is to construct the regularizations M ν 0,2+k(P, J, σ) andM ν 0,2+k(B, j, A) so that there is a projection from one to the other which isa fibration at least near the element of M ν 0,2+k(B, j, A) that contributes ton B (pt, pt, c 1 , . . . , c k ; A). Thus B could be the blowup of CP n along a symplecticsubmanifold Q that is disjoint from a complex line. One could alsotake similar blowups of products of projective spaces, or, more generally, ofiterated fibrations of projective spaces. For example, if B = CP m × CP nwith the standard complex structure then there is one complex line in thediagonal class [CP 1 ] + [CP 1 ] passing through any two points and a cycleH 1 × H 2 , where H i is the hyperplane class, and one could blow up alongany symplectic submanifold that did not meet one such line.It is also very likely that this argument can be extended to applywhen all we know about B is that some Gromov–Witten invariantn B (pt, pt, c 1 , . . . , c k ; A) is nonzero, for example, if B is a blowup of CP nalong arbitrary symplectic submanifolds. There are two new problems here:(a) we must control the construction of M ν 0,2+k(P, J, σ) in a neighborhoodof all the curves that contribute to n B (pt, pt, c 1 , . . . , c k ; A) and (b) we mustmake sure that the orientations are compatible so that curves in P projectingover different and noncancelling curves in B do not cancel eachother in the global count of the Gromov–Witten invariant in P . Note19 By this we mean that for some generic j on B the relevant moduli space containsexactly one element, which moreover parametrizes an embedded curve in B.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 659that the bundles given by restricting P to the different curves countedin n B (pt, pt, c 1 , . . . , c k ; A) are diffeomorphic, since, this being a homotopytheoretic question, we can always replace X by the simply connected spaceX/(X 1 ) in which these curves are homotopic [McDuff (2000)].4.12.2.5 Homotopy Reasons for SplittingIn this section we discuss c−splitting in a homotopy-theoretic context. Recallthat a c−Hamiltonian bundle is a smooth bundle P → B together witha class a ∈ H 2 (P ) whose restriction a M to the fiber M is c−symplectic, i.e.,(a M ) n ≠ 0 where 2n = dim(M). Further a closed manifold M is said to satisfythe hard Lefschetz condition with respect to the class a M ∈ H 2 (M, R)if the following maps are isomorphisms,∪(a M ) k : H n−k (M, R) → H n+k (M, R),(1 ≤ k ≤ n).In this case, elements in H n−k (M) that vanish when cupped with (a M ) k+1are called primitive, and the cohomology of M has an additive basis consistingof elements of the form b ∪ (a M ) l where b is primitive and l ≥ 0. 20Let M → P → B be a c−Hamiltonian bundle such that π 1 (B) actstrivially on H ∗ (M, R). If in addition M satisfies the hard Lefschetz conditionwith respect to the c−symplectic class a M , then the bundle c−splits[Blanchard (1956)].The proof is by contradiction. Consider the Leray spectral sequence incohomology and suppose that d p is the first non zero differential. Then,p ≥ 2 and the E p term in the spectral sequence is isomorphic to the E 2 termand so can be identified with the tensor product H ∗ (B) ⊗ H ∗ (M). Becauseof the product structure on the spectral sequence, one of the differentialsd 0,ip must be nonzero. So there is b ∈ Ep0,i ∼= H i (M) such that d 0,ip (b) ≠ 0.We may assume that b is primitive (since these elements together with a Mgenerate H ∗ (M).) Then b ∪ a n−iMWe can write≠ 0 but b ∪ an−i+1M= 0.d p (b) = ∑ je j ⊗f j , where e j ∈ H ∗ (B) and f j ∈ H l (M) (with l < i).Hence f j ∪a n−i+1M≠ 0 for all j by the Lefschetz property. Moreover, becausethe E p term is a tensor product(d p (b)) ∪ a n−i+1M= ∑ je j ⊗ (f j ∪ a n−i+1M) ≠ 0.20 These manifolds are sometimes called ‘cohomologically Kähler manifolds.’


660 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionBut this is impossible since this element is the image via d p of the trivialelement b ∪ a n−i+1M[Lalonde and McDuff (2002)].Here is a related argument due to [Kedra (2000)]. Every Hamiltonianbundle with 4-dimensional fiber c−splits.Consider the spectral sequence as above. We know that d 2 = 0 andd 3 = 0. Consider d 4 . We just have to check that d 0,34 = 0 since d 0,i4 = 0for i = 1, 2 for dimensional reasons, and = 0 for i = 4 since the top classsurvives.Suppose d 4 (b) ≠ 0 for some b ∈ H 3 (M). Let c ∈ H 1 (M) be suchthat b ∪ c ≠ 0. Then d 4 (c) = 0 and d 4 (b ∪ c) = d 4 (b) ∪ c ≠ 0 sinced 4 (b) ∈ H 4 (B) ⊗ H 0 (M). But we need d 4 (b ∪ c) = 0 since the top classsurvives. So d 4 = 0 and then d k = 0, k > 4 for reasons of dimension.Here is an example of a c−Hamiltonian bundle over S 2 that is notc−split. This shows that c−splitting is a geometric rather than a topological(or homotopy–theoretic) property.Observe that if S 1 acts on manifolds X, Y with fixed points p X , p Ythen we can extend the S 1 action to the connected sum X#Y opp at p X , p Ywhenever the S 1 actions on the tangent spaces at p X and p Y are the same. 21Now let S 1 act on X = S 2 × S 2 × S 2 by the diagonal action in the firsttwo spheres (and trivially on the third) and let the S 1 action on Y be theexample constructed in [McDuff (1988)] of a non–Hamiltonian S 1 actionthat has fixed points. The fixed points in Y form a disjoint union of 2−toriand the S 1 action in the normal directions has index ±1. In other words,there is a fixed point p Y in Y at which we can identify T pY Y with C ⊕C ⊕ C, where θ ∈ S 1 acts by multiplication by e iθ in the first factor, bymultiplication by e −iθ in the second and trivially in the third. Since thereis a fixed point on X with the same local structure, the connected sumZ = X#Y opp does support an S 1 − action. Moreover Z is a c−symplecticmanifold. There are many possible choices of c−symplectic class: underthe obvious identification of H 2 (Z) with H 2 (X) + H 2 (Y ) we will take thec−symplectic class on Z to be given by the class of the symplectic form onX.Let P X → S 2 , P Y → S 2 and P Z → S 2 be the corresponding bundles.Then P Z can be thought of as the connect sum of P X with P Y along thesections corresponding to the fixed points. By analyzing the correspondingMayer–Vietoris sequence, it is easy to check that the c−symplectic class onZ extends to P Z . Further, the fact that the symplectic class in Y does not21 Here Y opp denotes Y with the opposite orientation.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 661extend to P Y implies that it does not extend to P Z either. Hence P Z → S 2is not c−split.4.12.2.6 Action of the Homology of (M) on H ∗ (M)The action Ham(M) × M → M gives rise to mapsH k (Ham(M)) × H ∗ (M) → H ∗+k (M) :(φ, Z) ↦→ Tr φ (Z).We have the following result: These maps are trivial when k ≥ 1.To see this, let us first consider the action of a spherical elementφ : S n → Ham(M).It is not hard to check that the homomorphismsTr φ : H k (M) → H k+n (M)are precisely the connecting homomorphisms in the Wang sequence of thebundle P φ → S k+1 with clutching function φ, i.e., there is an exact sequence. . . H k (M) Tr φ→ H k+n (M) → H k+n (P ) ∩[M]→ H k−1 (M) → . . .Thus the fact that P φ → S k+1 is c−split immediately implies that the Tr φare trivial [Lalonde and McDuff (2002)].Next recall that in a H-space the rational cohomology ring is generatedby elements dual to the rational homotopy. It follows that there is a basisfor H ∗ (Ham(M)) that is represented by cycles of the formφ 1 × · · · × φ k : S 1 × . . . × S k → Ham(M),where the S j s are spheres and one defines the product of maps by usingthe product structure in Ham(M). Therefore it suffices to show that theseproduct elements act trivially. However, if a ∈ H ∗ (M) is represented bythe cycle α, then Tr Sk (α) is null-homologous, and so equals the boundary∂β of some chain β. Therefore,∂ ( Tr S1×...×S k−1(β) ) = ±Tr S1×...×S k−1(∂β)= ±Tr S1×...×S k−1(Tr Sk (α)) = Tr S1×...×S k(α).Hence Tr S1×...×S k(a) = 0.Let P → B be a trivial symplectic bundle. Then any Hamiltonianautomorphism Φ ∈ Ham(P, π) acts as the identity map on H ∗ (P ).


662 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAn element Φ ∈ Ham(P, π) is a map of the formΦ : B × M → B × M :(b, x) ↦→ (b, Φ b (x)),where Φ b ∈ Ham(M) for all b ∈ B. Let us denote the induced mapB ×M → M : (b, x) ↦→ Φ b (x) by α Φ . The previous proposition implies thatif B is a closed manifold of dimension > 0, or, more generally, if it carriesa fundamental cycle [B] of degree > 0,(α Φ ) ∗ ([B] ⊗ m) = Tr [B] (m) = 0, for all m ∈ H ∗ (M).We can also think of Φ : B × M → B × M as the compositeB × M diag B×Id M−→IdB × B × MB ×α−→ΦB × M.The diagonal class in B × B can be written as[B] ⊗ [pt] + ∑ i∈Ib i ⊗ b ′ i, where b i , b ′ i ∈ H ∗ (B) with dim(b ′ i) > 0.HenceΦ ∗ ([B] ⊗ m) = [B] ⊗ m + ∑ i∈Ib i ⊗ Tr b ′i(m) = [B] ⊗ m.More generally, given any class b ∈ H ∗ (B), represent it by the image of thefundamental class [X] of some polyhedron under a suitable map X → Band consider the pullback bundle P X → X. Since the class Φ ∗ ([X] ⊗ m)is represented by a cycle in X × M for any m ∈ H ∗ (M), we can work outwhat it is by looking at the pullback of Φ to X × M. The argument abovethen applies to show that Φ ∗ ([X] ⊗ m) = [X] ⊗ m whenever b has degree> 0. Thus Φ ∗ = Id on all cycles in H ∗>0 (B) ⊗ H ∗ (M). However, it clearlyacts as the identity on H 0 (B) ⊗ H ∗ (M) since the restriction of Φ to anyfiber is isotopic to the identity.We now show that there is a close relation between this question andthe problem of c−splitting of bundles. Given an automorphism Φ of asymplectic bundle M → P → B we define P Φ = (P × [0, 1])/Φ to bethe corresponding bundle over B × S 1 . If the original bundle and theautomorphism are Hamiltonian, so is P Φ → B × S 1 , though the associatedbundle P Φ → B × S 1 → S 1 over S 1 will not be, except in the trivial casewhen Φ is in the identity component of Ham(P, π).Assume that a given Hamiltonian bundle M → P → B c−splits. Thena Hamiltonian automorphism Φ ∈ Ham(P, π) acts trivially (i.e., as the


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 663identity) on H ∗ (P ) iff the corresponding Hamiltonian bundle P Φ → B × S 1c−splits.Clearly, the fibration P → B c−splits iff every basis of the Q−vectorspace H ∗ (M) can be extended to a set of classes in H ∗ (P ) that form a basisfor a complement to the kernel of the restriction map. We will call such aset of classes a Leray–Hirsch basis. It corresponds to a choice of splittingisomorphism H ∗ (P ) ∼ = H ∗ (B) ⊗ H ∗ (M). Now, the only obstruction toextending a Leray–Hirsch basis from P to P ′ is the nontriviality of theaction of Φ on H ∗ (P ). This shows the “only if ” part.Conversely, suppose that P Φ c−splits and let e j , j ∈ J, be a Leray–Hirsch basis for H ∗ (P Φ ). Then H ∗ (P Φ ) has a basis of the form e j ∪π ∗ (b i ), e j ∪ π ∗ (b i × [dt]) where b i runs through a basis for H ∗ (B) and [dt]generates H 1 (S 1 ). Identify P with P × {0} in P Φ and consider some cycleZ ∈ H ∗ (P ). Since the cycles Φ ∗ (Z) and Z are homologous in P Φ , theclasses e j ∪ π ∗ (b i ) have equal values on Φ ∗ (Z) and Z. But the restrictionof these classes to P forms a basis for H ∗ (P ). It follows that [Φ ∗ (Z)] = [Z]in H ∗ (P ).Let P → B be a Hamiltonian bundle. Then the group Ham(P, π) actstrivially on H ∗ (P ) if the base:(i) has dimension ≤ 2, or(ii) is a product of spheres and projective spaces with no more than twoS 1 factors, or(iii) is simply connected and has the property that all Hamiltonian bundlesover B are c−split.In all cases, the hypotheses imply that P → B c−splits. Thereforethe previous proposition applies and (i), (ii) follow immediately from thesplitting theorem. To prove (iii), suppose that B is a simply connectedcompact CW complex over which every Hamiltonian fiber bundle c−splits.Let M ↩→ P → B × S 1 be any Hamiltonian bundle – in particular oneof the form P Φ → B × S 1 . Consider its pull-back P ′ by the projectionmap B × T 2 → B × S 1 . This is still a Hamiltonian bundle. To showthat P c−splits, it is sufficient, to show that P ′ c−splits. B × T 2 may beconsidered as a smooth compact Hamiltonian fibration T 2 ↩→ (B×T 2 ) → Bwith simply connected base, so P ′ c−splits since any Hamiltonian bundleover B or over T 2 c−splits [Lalonde and McDuff (2002)].Finally, we prove the above statements about the automorphism groupsof Hamiltonian structures.We have to show that the following statements are equivalent for anyΦ ∈ Symp 0 (P, π):


664 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(i) Φ is isotopic to an element of Ham(P, π);(ii) Φ ∗ ({a}) = {a} for some Hamiltonian structure {a} on P ;(iii) Φ ∗ ({a}) = {a} for all Hamiltonian structures {a} on P .Recall that the relative homotopy groups π i (Symp(M), Ham(M)) allvanish for i > 1. Using this together with the fact that a ∈ H 2 (P ), we canreduce to the case when B is a closed oriented surface.Let us prove this first in the case where P → B is trivial, so that Φ is amap B → Symp 0 (M, ω). Suppose that Φ ∗ (a) = a for some extension classa. By isotoping Φ if necessary, we can suppose that Φ takes the base pointb 0 of B to the identity map. Then, the compositeπ 1 (B) −→ Φ∗π 1 (Symp 0 (M)) F −→ luxH 1 (M, R)must vanish. Thus the restriction of Φ : B → Symp 0 (M) to the1−skeleton of B homotops into Ham(M). Since the relative homotopygroups π i (Symp(M), Ham(M)) all vanish for i > 1, this implies that φhomotops to a map in Ham(M), as required [Lalonde and McDuff (2002)].Therefore, it remains to show that we can reduce the proof that (ii)implies (i) to the case when P → B is trivial. To this end, isotop Φ so thatit is the identity map on all fibers M b over some disc D ⊂ B. Since P → B istrivial over X = B − D, we can decompose P → B into the fiber connectedsum of a trivial bundle P B over B (where B is thought of as the spaceobtained from X by identifying its boundary to a point) and a nontrivialbundle Q over S 2 = D/∂D. Further, this decomposition is compatible withΦ, which can be thought of as the fiber sum of some automorphism Φ B ofP B together with the trivial automorphism of Q. Clearly, this reduces theproof that (ii) implies (i) to the case Φ B : P B → P B on trivial bundles, ifwe note that when Φ B is the identity over some disc D ⊂ B, the isotopybetween Φ and an element in Ham(P B ) can be constructed so that itremains equal to the identity over D.4.12.2.7 Cohomology of General Symplectic BundlesNow we discuss some consequences for general symplectic bundles of theabove results on Hamiltonian bundles. First, we prove the above proposition,which states that the boundary map ∂ in the rational homology Wangsequence of a symplectic bundle over S 2 has ∂ ◦ ∂ = 0.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 665The mapwhere∂ ◦ ∂ : H k (M) → H k+2 (M) is given by a ↦→ Ψ ∗ ([T 2 ] ⊗ a),Ψ : T 2 × M → M : (s, t, x) ↦→ φ s φ t (x).Let b(s, t) = φ −1s+tφ s φ t . The map (s, t) ↦→ b(s, t) factors throughT 2 −→ S 2 = T 2 /{s = 0} ∪ {t = 0}.Let Z → M represent a k−cycle. We have a mapT 2 × Z → A1S 1 × S 2 × Z → A2M,given by(s, t, z) ↦→ (φ s+t , b(s, t), z) ↦→ φ s+t b(s, t)z = φ s φ t (z),and want to calculate∫T 2 ×Z∫A ∗ 1A ∗ 2(α) =(A 1) ∗[T 2 ×Z]A ∗ 2(α),for some k + 2−form α on M. But (A 1 ) ∗ [T 2 × Z] ∈ H 2 (S 2 ) ⊗ H k (Z). 22Now observe that A ∗ 2(α) vanishes on H 2 (S 2 ) ⊗ H k (Z).The previous lemma is trivially true for any smooth (not necessarilysymplectic) bundle over S 2 that extends to CP 2 . For the differential d 2 inthe Leray cohomology spectral sequence can be written asd 2 (a) = ∂(a) ∪ u ∈ E 2,q−12 ,where a ∈ H q (M) ≡ E 0,q2 and u generates H 2 (CP 2 ) ≡ E 2,02 . Hence0 = d 2 (d 2 (a)) = d 2 (∂(a) ∪ u) = d 2 (∂(a)) ∪ u = ∂(∂(a)) ⊗ u 2 .If π : P → B is any symplectic bundle over a simply connected base,then d 3 ≡ 0.As above we can reduce to the case when B is a wedge of S 2 s and S 3 s.The differential d 3 is then given by restricting to the bundle over ∨S 3 . Sincethis is Hamiltonian, d 3 ≡ 0.The next lemma describes the Wang differential ∂ = ∂ φ in the case ofa symplectic loop φ with nontrivial image in H 1 (M): Suppose that φ is asymplectic loop such that [φ t (x)] ≠ 0 in H 1 (M). Then Ker∂ = Im∂, where∂ = ∂ φ : H k (M) → H k−1 (M) is the corresponding Wang differential.22 Note that there is no component in H 3 (S 1 × S 2 ) ⊗ H k−1 (Z) since A 1 = Id on theZ factor.


666 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet α ∈ H 1 (M) be such that α([φ t (x)]) = 1. So ∂α = 1. Then, for everyβ ∈ Ker∂, ∂(α∪β) = β. As ∂◦∂ = 0, this means that [Lalonde et al. (1998);Lalonde et al. (1999); Lalonde and McDuff (2002)]Moreover, the mapKer∂ ⊂ Im∂ =⇒ Ker∂ = Im∂.α∪ : H k (M) → H k+1 (M)is injective on Ker ∂ and H ∗ (M) decomposes as the direct sum Ker ∂ ⊕(α∪Ker ∂).The above corollary claims that for a symplectic loop φ,Ker ∂ = Im ∂ iff [φ t (x)] ≠ 0 in H 1 (M).Since 1 ∈ H 0 (M) is in Ker ∂ it must equal ∂(α) for some α ∈ H 1 (M). Thismeans that α([φ t (x)]) ≠ 0 so that [φ t (x)] ≠ 0. Note that the only placethat the symplectic condition enters in the above proof is in the claim that∂ ◦ ∂ = 0. Since this is always true when the loop comes from a circleaction, this lemma holds for all, not necessarily symplectic, circle actions.In this case, we can interpret the result topologically. For the hypothesis[φ t (x)] ≠ 0 in H 1 (X) implies that the action has no fixed points, so thatthe quotient M/S 1 is an orbifold with cohomology isomorphic to ker ∂.Thus, the argument shows that M has the same cohomology as the product(M/S 1 ) × S 1 .4.13 Clifford Algebras, Spinors and Penrose Twistors4.13.1 Clifford Algebras and ModulesIn this subsection, mainly following [Yang (1995)], we provide the generaltheory of Clifford algebra and subsequently consider its special 4D case.Let V be a vector space over R with a quadratic form Q on it. The Cliffordalgebra of (V, Q), denoted by C(V, Q), is the algebra over R generatedby V with the relationsv 1 · v 2 + v 2 · v 1 = −2Q(v 1 , v 2 ), (for all v 1 , v 2 ∈ V ).Since Q is symmetric, we have v 2 = −Q(v) for all v ∈ V .For fixed Q, we may abbreviate C(V, Q) and Q(v 1 , v 2 ) into C(V ) and(v 1 , v 2 ) respectively.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 667It is a basic fact in algebra that the Clifford algebra is the unique (upto isomorphism) solution to the following universal problem.If A is an algebra and c : V −→ A is a linear map satisfyingc(v 2 )c(v 1 ) + c(v 1 )c(v 2 ) = −2Q(v 1 , v 2 ), (for all v 1 , v 2 ∈ V ),then there is a unique algebra homomorphism from C(V, Q) to A extendingthe map c.The Clifford algebra may be realized as the quotient T (V )/I Q whereT (V ) =∞⊕T k (V )k=1is the tensor algebra of V with T k (V ) generated byand I Q is generated by{ v 1 ⊗ v 2 ⊗ · · · ⊗ v k | v 1 , v 2 , . . . , v k ∈ V }{v 1 ⊗ v 2 + v 2 ⊗ v 1 + 2Q(v 1 , v 2 ) | v 1 , v 2 ∈ V }The tensor algebra T (V ) has a Z 2 −grading obtained from the naturalN−grading after reduction mod 2:T (V ) = T + (V ) + T − (V ),whereT + (V ) = R ⊕ T 2 (V ) ⊕ T 4 (V ) ⊕ · · · ⊕ T 2k (V ) ⊕ . . . ,T − (V ) = V ⊕ T 3 (V ) ⊕ T 5 (V ) ⊕ · · · ⊕ T 2k+1 (V ) ⊕ . . .Therefore it forms a super–algebra.Similarly, for k = 0, 1, 2, 3, . . . , letC k (V ) = T k (V )/I Q ,and letC + (V ) = R ⊕ C 2 (V ) ⊕ C 4 (V ) ⊕ · · · ⊕ C 2k (V ) ⊕ . . . ,C − (V ) = V ⊕ C 3 (V ) ⊕ C 5 (V ) ⊕ · · · ⊕ C 2k+1 (V ) ⊕ . . .Since the ideal I Q is generated by elements from the evenly gradedsubalgebra T + (V ), C(V ) is itself a superalgebra and we have the gradingC(V ) = C + (V ) + C − (V ).Let E be a module over R or C which is Z 2 −graded,E = E + ⊕ E − .


668 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionE is called a Clifford module over a Clifford algebra C(V ) if there is aClifford action·cC(V ) × E −→ E( a , e ) ↦−→ a · ecor equivalently, an algebra homomorphismcC(V ) −→ End(E)a ↦−→ c(a)which is even with respect to this grading:with c(a) (e) = a ·c e,C + (V ) · E ± ⊂ E ± , C − (V ) · E ∓ ⊂ E ∓ .ccLet O(V, Q) be the group of linear transformations of V which preserveQ. That means for all φ ∈ O(V, Q) and v 1 , v 2 ∈ V,Q(φ v 1 , φv 2 ) = Q(v 1 , v 2 ).The action of O(V, Q) on generators of T (V ) is defined byφ(v 1 ⊗ v 2 ⊗ · · · ⊗ v k ) =k∑v 1 ⊗ · · · ⊗ φ(v i ) ⊗ · · · ⊗ v ki=1and extends to the whole T (V ) linearly.I Q is invariant under the action of O(V, Q). Hence C(V, Q) carries anatural action of O(V, Q).Let ∗ : a ↦→ a ∗ be the anti–automorphism of T (V ) induced by v ↦→ −von T , and satisfiesHence,Since(v 1 v 2 . . . v k ) ∗ =(a 1 a 2 ) ∗ = a ∗ 2 a ∗ 1.{(vk v k−1 . . . v 1 ), if k is even,−(v k v k−1 . . . v 1 ),v ∗ 1 ⊗ v ∗ 2 + v ∗ 2 ⊗ v ∗ 1 + 2Q(v ∗ 1, v ∗ 2)if k is odd.= (−v 1 ) ⊗ (−v 2 ) + (−v 2 ) ⊗ (−v 1 ) + 2Q(−v 1 , −v 2 )= v 1 ⊗ v 2 + v 2 ⊗ v 1 + 2Q(v 1 , v 2 ),


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 669I Q is invariant under ∗ . So it induces an anti–isomorphism a ↦→ a ∗ of C(V ).If Q is a positive–definite quadratic form, then a Clifford module E ofC(V ) with an inner product is said to be self–adjoint if c(a ∗ ) = c(a) ∗ ,where c(v) denote the action of v ∈ V on a Clifford module of C(V ) (whichmay be C(V ) itself).The inner product on E must be C(V ) − invariant:( c(a)e 1 , c(a)e 2 ) = ( e 1 , e 2 ), (for all a ∈ C(V ), e 1 , e 2 ∈ E).Hence for a self–dual module E,Especially, we have(c(a)e 1 , e 2 ) = (e 1 , c(a) ∗ e 2 ) = (e 1 , c(a ∗ )e 2 ).(c(v)e 1 , e 2 ) = (e 1 , c(v ∗ )e 2 ) = (e 1 , −c(v)e 2 ),(for all v ∈ V, e 1 , e 2 ∈ E).That means c(v) is skew–adjoint for all v ∈ V .Let E be a Z 2 −graded Clifford module over the Clifford algebra C(V ).We denote by End C(V ) (E) the algebra of homomorphisms of E supercommutingwith the action of C(V ).4.13.1.1 The Exterior AlgebraThe first interesting example of Clifford module is the exterior algebra.The exterior algebra ΛV of a vector space V is defined to beT (V )/I ∗ , where I ∗ is the ideal generated by elements of the form( v 1 ⊗ · · · ⊗ v i ⊗ v i+1 ⊗ · · · ⊗ v k ) + ( v 1 ⊗ · · · ⊗ v i+1 ⊗ v i ⊗ · · · ⊗ v k ),or equivalently,ΛV =∞⊕Λ k V, where Λ k V = T k (V )/I ∗ .k=1Letɛ : V −→ Hom( Λ k V , Λ k+1 V )be the action of V on ΛV by exterior product, i.e., for all v ∈ V ,ɛ(v) : Λ k V −→ Λ k+1 Vw ↦−→ v ∧ w .


670 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionExplicitly,Letɛ(v) (v 1 ∧ · · · ∧ v k ) = v ∧ v 1 ∧ · · · ∧ v k .ι : V −→ Hom( Λ k V , Λ k−1 V )be the action of V on ΛV by interior product (or contraction), i.e., for allv ∈ V ,Explicitly,ι(v) (v 1 ∧ · · · ∧ v k ) =ι(v) : Λ k V −→ Λ k−1 Vw ↦−→ Q(v, w) .k∑(−1) i−1 Q(v, v i ) v 1 ∧ · · · ∧ ̂v i ∧ · · · ∧ v k .i=1The Clifford action of v ∈ V on w ∈ ΛV is given byFor any v, w in V , we havev · w = c(v) w = ɛ(v) w − ι(v) w.cɛ(v) ι(w) + ι(w) ɛ(v) = Q(v , w).The action c : V −→ End(ΛV ) extends to an action of the Cliffordalgebra C(V ) on ΛV .The symbol map σ : C(V ) −→ ΛV is defined by σ(a) = c(a) 1 ΛV, where1 ΛV∈ Λ 0 V is the identity in the exterior algebra ΛV .If 1 C(V )denotes the identity in C(V ), then σ(1 C (V )) is the identity1 End(ΛV )in End(ΛV ).The Clifford algebra C(V ) is isomorphic to the tensor algebra ΛV andis therefore a 2 dim V dimensional vector space with generators{(c 1 ) n1 (c 2 ) n2 . . . (c dim V) n dim V | (n1 , n 2 , . . . , n dim V) ∈ {0, 1} dim V }.If we consider C(V ) and ΛV as Z 2 −graded O(V )−modules, then σ andpreserve the Z 2 −graded and the O(V ) action. Hence they are isomorphismsof Z 2 −graded O(V )−modules.There is a natural increasing filtration∞⋃C 0 (V ) ⊆ C 1 (V ) ⊆ . . . ⊆ C k (V ) ⊆ . . . ⊆ C i (V ) = C(V ),i=0


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 671whereC i (V ) = C 0 (V ) ⊕ C 1 (V ) ⊕ C 2 (v) ⊕ . . . ⊕ C i (V )and C 0 (V ) = R. It follows thatC i (V ) = span{ v 1 . . . v k | v j ∈ V ↩→ C(V ) for j = 1, . . . , k ≤ i }.The Clifford algebra C(V ) with this filtration is called the associatedgraded algebra of C(V ) and is denoted by gr C(V ). The ith grading ofgr C(V ) is denoted by gr i C(V ).The associated graded algebra gr C(V ) is naturally isomorphic tothe exterior algebra ΛV , where the isomorphism is given by sendinggr i (v 1 . . . v i ) ∈ gr i C(V ) to v 1 Λ . . . Λv i ∈ Λ i V . The symbol map σ extendsthe symbol mapσ i : C i (V ) −→ gr i C(V ) ∼ = Λ i V,in the sense that if a ∈ C i (V ), then σ(a) [i] = σ i (a). The filtration C i (V )may be written asC i (V ) =i∑q(Λ i V ).j=0Hence the Clifford algebra C(V ) may be identified with the exterior algebraΛV with a twisted, or quantized multiplication α ·Qβ.If v ∈ V ↩→ C(V ) and a ∈ C + (V ), then σ( [v, a] ) = −2 ι(v) σ(a).The space C 2 (V ) = q(Λ 2 V ) is a Lie subalgebra of C(V ), where theLie bracket is just the commutator in C(V ). It is isomorphic to the Liealgebra (V ), under the map τ : C 2 (V ) −→ so(V ), where any a ∈ C 2 (V ) ismapped into τ(a) which acts on any v ∈ V ∼ = C 1 (V ) by the adjoint action:τ(a) v = [ a , v ]. Here the bracket is the bracket of the Lie super–algebraC(V ), i.e.,[ a 1 , a 2 ] = a 1 a 2 −(−1) |a1| |a2| a 2 a 1 , (for a 1 ∈ C |a1| (V ), a 2 ∈ C |a2| (V )).It satisfies the following Axioms of Lie super–algebra:Supercommutativity [ a 1 , a 2 ] + (−1) |a1| |a2| [ a 2 , a 1 ] = 0, andJacobi’s identity[ a 1 , [a 2 , a 3 ] ] = [ [a 1 , a 2 ] , a 3 ] + (−1) |a1| |a2| [ a 2 , [a 1 , a 3 ] ].


672 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWe usually identify A ∈ so(V ) with∑( A e i , e j ) e i ∧e j ∈ Λ 2 V, so that we have q (A) = ∑ ( A e i , e j ) c i c j ,i 1, then the homomorphism τ : Spin(V ) −→ SO(V ) is adouble covering.4.13.1.3 4D CaseNow consider the most interesting case when V ∼ = R 4 . Fix a basis{ e 1 , e 2 , e 3 , e 4 } which is orthonormal with respect to the fixed quadratic


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 673form Q. Then any vector v ∈ V may be written as v = v k e k and theClifford algebra isC(R 4 ) = span{ c n11 cn2 2 cn3 3 cn4 4 | n i ∈ {0, 1} }, where c i = q( e i ).For convenience, we may denote c i by e i without ambiguity. Especially,C 2 (R 4 ) = span{ e 1 e 2 , e 1 e 3 , e 1 e 4 , e 2 e 3 , e 2 e 4 , e 3 e 4 }Now consider the isomorphism τ : C 2 (R 4 ) −→ so(4) = so(R 4 ). And thecorresponding τ : Spin(4) −→ so(4). We haveτ(e i e j ) · v = [ e i e j ,==4∑v k e k ] =k=14∑v k [ e i e j , e k ] =k=14∑v k ( e i e j e k − e k e i e j )k=14∑v k ( e i e j e k + e i e k e j − e i e k e j − e k e i e j )k=14∑v k ( −2 Q( e j , e k ) e i + 2 Q( e k , e i ) e j ) = −2 v j e i + 2 v i e j .k=1Hence τ ( e i e j ) corresponds to 2 · m(i, j) ∈ so(4) where m(i, j) =(m(i, j) αβ ) is a matrix with entriesExplicitly, we have⎧⎪⎨ 1, if α = j and β = i,m(i, j) αβ = −1, if α = i and β = j,⎪⎩ 0, otherwise.⎛ ⎞⎛ ⎞0 −1 0 00 0 −1 0τ( e 1 e 2 ) = 2 ⎜1 0 0 0⎟⎝0 0 0 0⎠ τ( e 1 e 3 ) = 2 ⎜0 0 0 0⎟⎝1 0 0 0⎠0 0 0 00 0 0 0⎛ ⎞⎛ ⎞0 0 0 −10 0 0 0τ( e 1 e 4 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 0 ⎠ τ( e 2 e 3 ) = 2 ⎜0 0 −1 0⎟⎝0 1 0 0⎠1 0 0 00 0 0 0


674 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction⎛ ⎞⎛ ⎞0 0 0 00 0 0 0τ( e 2 e 4 ) = 2 ⎜0 0 0 −1⎟⎝0 0 0 0 ⎠ τ( e 3 e 4 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 −1⎠ .0 1 0 00 0 1 0Notice that for t ∈ [0, 2π), τ( t 2 e ie j ) corresponds to the matrix t · m(i, j)and(t · m(i, j) αβ ) 2 = −t 2 · ∆(i, j),where ∆(i, j) is a matrix with(t · m(i, j) αβ ) 4 = t 4 · ∆(i, j),∆(i, j) αβ ={1, if α = β = i or α = β = j,0, otherwise.Therefore, we haveexp τ ( t 2 e ie j ) = (cos t − 1) · ∆(i, j) + 1 SO(4)+ (sin t) m(i, j).Since we have the commutative relation τ · exp = exp · τ, it followsthatτ exp ( t e i e j ) = exp τ ( te i e j ) = (cos 2t−1)·∆(i, j) + 1 SO(4)+ (sin 2t) m(i, j).Explicitly, we haveτ( exp( t e 1 e 2 ) ) =τ( exp( t e 1 e 3 ) ) =⎛⎞cos 2t − sin 2t 0 0⎜sin 2t cos 2t 0 0⎟⎝ 0 0 1 0⎠0 0 0 1⎛⎞cos 2t 0 − sin 2t 0⎜ 0 1 0 0⎟⎝sin 2t 0 cos 2t 0⎠0 0 0 1τ( exp( t e 1 e 4 ) ) =⎛⎞cos 2t 0 0 − sin 2t⎜ 0 1 0 0⎟⎝ 0 0 1 0 ⎠sin 2t 0 0 cos 2t


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 675τ( exp( t e 2 e 3 ) ) =τ( exp( t e 2 e 4 ) ) =τ( exp( t e 3 e 4 ) ) =⎛⎞1 0 0 0⎜0 cos 2t − sin 2t 0⎟⎝0 sin 2t cos 2t 0⎠0 0 0 1⎛⎞1 0 0 0⎜0 cos 2t 0 − sin 2t⎟⎝0 0 1 0 ⎠0 sin 2t 0 cos 2t⎛⎞1 0 0 0⎜0 1 0 0⎟⎝0 0 cos 2t − sin 2t⎠ .0 0 sin 2t cos 2t4.13.2 SpinorsIn this subsection, mainly following [Yang (1995)], we define spinors andthe spinor representation of the Clifford algebras.4.13.2.1 Basic PropertiesLet {e i } i=1,...,dim V be an oriented, orthonormal basis of V , that means thereis a preferred ordering of the basis elements modulo even permutations. Thechirality operator 23 isΓ = i p e 1 . . . e dim V∈ C(V ) ⊗ C, wherep =[ ]dim V + 12={dim V2if dim V is even,dim V +12if dim V is odd.The Chirality operator satisfies: Γ 2 = 1 C(V ), and it super–anticommutes with elements v ∈ V , i.e., Γ v + (−1) dim V v Γ = 0.For dim V odd, Γ is in the center of C(V ) ⊗ C.23 Also known as the volume element or the complex unit. Physicists also denote it asγ 5 for the four dimensional case.


676 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFor dim V even, every complex Clifford module E has a Z 2 −gradingdefined by the ±1 eigen–spaces of the chirality operator:E ± = {v ∈ E | Γv = ±v }.Especially, for dim V ≡ 0 mod 4, Γ ∈ C(V ), so in this case, the real Cliffordmodules are also Z 2 −graded.A polarization of a complex vector space V ⊗C is a subspace P ⊂ V ⊗Cwhich is isotropic, i.e.,and we have a splitingQ( v , v ) = 0, (for all v ∈ P ),V ⊗ C = P ⊕ P.Here the quadratic form Q extends from V to V ⊗ C complex linearly, i.e.,Q( a + ib , c + id ) = Q( a , c ) + i (Q( a , d ) + Q( b , c )) − Q( b , d )A polarization is called oriented, if there is an oriented orthonormalbasis {e i } of V , such that P is spanned by the vectors{ w i = ( e 2i−1 − ie 2i )√2| 1 ≤ i ≤ dim V2and therefore the complement P is spanned by the vectors{ w i = ( e 2i−1 + ie 2i )√2| 1 ≤ i ≤ dim V2The basis {w i } i=1,..., dim V and the corresponding conjugate2{w i } i=1,..., dim V satisfy the following equations: For 1 ≤ i ≤ dim V22,For 1 ≤ i ≠ j ≤ dim V2,w i w i = w i w i = 0, w i w i + w i w i = −2.w i w j = − w j w i , w i w j = − w j w i , w i w j = − w j w i .If V is an even–dimensional oriented Euclidean vector space, then thereis a unique Z 2 −graded Clifford module: S = S + ⊕ S − called the spinormodule, such that C(V )⊗C ∼ = End(S). Elements in S + and S − are calledpositive and negative spinors respectively.In particular, we havedim C(S) = 2 ( dim V2 ) , and dim C(S + ) = dim C(S − ) = 2 ( dim V2 −1) .},}.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 6774.13.2.2 4D CaseNow we look at the spinors in four dimensional vector space.Consider an oriented orthonormal basis {e 1 , e 2 , e 3 , e 4 } of the Euclideanvector space R 4 . Then the standard oriented polarization P of C 4 ∼ = R 4 ⊗Cis generated by{ w 1 = e 1 − i e 2√2, w 2 = e 3 − i e 4√2}, and we haveS + = span C 〈 1 ΛP, w 1 ∧ w 2 〉, S − = span C 〈 w 1 , w 2 〉.Together, we have the following standard basis of S = S + ⊕ S − :{ 1 ΛP, w 1 ∧ w 2 , w 1 , w 2 }.Under this basis, any spinor s ∈ S may be written as a column vectorwith componentss =⎛ ⎞s 1⎜s 2⎟⎝s 3⎠s 4and we have a splitings = s + ⊕ s − =⎛⎝s + 1⎞ ⎛⎠ ⊕ ⎝s − 1s + 2 s − 2⎞⎠ =⎛⎝⎞s 1⎠ ⊕s 2⎛⎝⎞s 3⎠ .s 4Since S ∼ = C 4 , we have a representation: C(V ) ⊗ C ↩→ End(C 4 ). Tofind out the exact correspondencs, let’s consider the Clifford action of w i


678 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand w i on S:⎧⎧√ 2 w11 ΛPc( w 1 ) :⎪⎨ w 1 ∧ w 2w 1↦→⎪⎨ 00⎪⎩w 2⎪⎩ √2 w1 ∧ w 2⎧1 ΛP⎧√ 2 w2c( w 2 ) :⎪⎨ w 1 ∧ w 2↦→⎪⎨ 0w 1− √ 2 w 1 ∧ w 2⎪⎩w 2⎪⎩0⎧1 ΛP⎧0c( w 1 ) :⎪⎨ w 1 ∧ w 2↦→⎪⎨ − √ 2 w 2w 1− √ 2 1 ΛP⎪⎩w 2⎪⎩0⎧1 ΛP⎧0c( w 2 ) :⎪⎨ w 1 ∧ w 2⎪⎩w 1w 2↦→√ ⎪⎨ 2 w1.0⎪⎩− √ 2 1 ΛP


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 679Therefore we have the following correspondence:w 1 = √ 2⎛ ⎞0 0 0 00 0 0 1⎜⎝ 1 0 0 0 ⎟⎠0 0 0 0w 2 = √ 2⎛ ⎞0 0 0 00 0 −1 0⎜⎝ 0 0 0 0 ⎟⎠1 0 0 0w 1 = √ 2⎛⎞0 0 −1 00 0 0 0⎜⎝ 0 0 0 0 ⎟⎠0 −1 0 0w 2 = √ 2⎛ ⎞0 0 0 −10 0 0 0.⎜⎝ 0 1 0 0 ⎟⎠0 0 0 0Now, by the relationse 2i−1 = ( w i + w i )√2, e 2i = ( −w i + w i )√2 i,


680 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwe can deduce the corresponding matrices:⎛⎞0 0 −1 0e 1 = ⎜ 0 0 0 1⎟⎝ 1 0 0 0 ⎠0 −1 0 0⎛ ⎞0 0 i 0e 2 = ⎜ 0 0 0 i⎟⎝ i 0 0 0 ⎠0 i 0 0⎛⎞0 0 0 −1e 3 = ⎜ 0 0 −1 0⎟⎝ 0 1 0 0 ⎠1 0 0 0⎛ ⎞0 0 0 ie 4 = ⎜ 0 0 −i 0⎟⎝ 0 −i 0 0⎠ .i 0 0 0As a result, we have the following Theorem: There are isomorphismsHom( S + , S − ) ∼ = Λ 1 C, Hom( S − , S + ) ∼ = Λ 1 C.The elements in C 2 (V ) correspond to the matrices⎛ ⎞−i 0 0 0⎛ ⎞−i 0 0 0e 1 e 2 = ⎜ 0 i 0 0⎟⎝ 0 0 i 0 ⎠ e 3 e 4 = ⎜ 0 i 0 0⎟⎝ 0 0 −i 0⎠0 0 0 −i0 0 0 i⎛0 −1 0⎞0⎛0 −1 0⎞0e 1 e 3 = ⎜ 1 0 0 0⎟⎝ 0 0 0 −1⎠ e 4 e 2 = ⎜ 1 0 0 0⎟⎝ 0 0 0 1 ⎠0 0 1 00 0 −1 0⎛ ⎞0 i 0 0⎛0 i 0⎞0⎜ i 0 0 0⎟⎜ i 0 0 0 ⎟e 1 e 4 = ⎜⎝0 0 0 i0 0 i 0There is a representation⎟⎠ e 2 e 3 = ⎜⎝0 0 0 −i0 0 −i 0⎟⎠ .Spin(4) −→ End(C 4 ), which splits into: SU(2) × SU(2).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 6814.13.2.3 (Anti) Self DualityThere is an operator on the exterior algebra which is similar to the chiralityoperator.Let the Hodge star operator⋆ : ΛV −→ ΛV,with respect to Q, be given by the following relation:e i1 ∧ · · · ∧ e ik ∧ ⋆(e i1 ∧ · · · ∧ e ik ) = ɛ i1...i ke 1 ∧ · · · ∧ e dim V,where {e i } i=1,...,dim Vwithɛ i1...i kis orthonormal with respect to Q and( )1 2 . . . . . . . . . . dim V= sgni 1 . . . i k i k+1 . . . i dim V(i k+1 , . . . , i dim V) = (1, 2, . . . , î 1 , . . . , î k , . . . , dim V ).Now we consider the case when dim V = 4.For a four dimensional vector space V , the restriction of the square ofthe Hodge star operator satisfies:and⋆ 2 | C ± (V )= ± 1 End( C ± (V ) ).For a 4D vector space V , Λ 2 V splits into ±1 eigen–spaces of ⋆:Λ 2 +V = {Λ 2 −V = {(1 + ⋆)2(1 − ⋆)2w | w ∈ ΛV }w | w ∈ ΛV },which are called the space of self–dual (SD) or anti–self–dual (ASD) 2–forms and are simplified as Λ + and Λ − respectively.The standard basis for Λ + and Λ − are the self–dual basis:{w + 1 = e 1 ∧e 2 +e 3 ∧e 4 , w + 2 = e 1 ∧e 3 +e 4 ∧e 2 , w + 3 = e 1 ∧e 4 +e 2 ∧e 3 },and the anti–self–dual basis:{w − 1 = e 1 ∧e 2 −e 3 ∧e 4 , w − 2 = e 1 ∧e 3 −e 4 ∧e 2 , w − 3 = e 1 ∧e 4 −e 2 ∧e 3 },respectively.


682 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNotice that in the above corollary, the notation w ± i refers to elementsof C(V ) ⊗ C. If we consider w ± i as elements of Λ C V = ΛV ⊗ C, then wehavew + 1 = i ( w 1 ∧ w 1 + w 2 ∧ w 2 )w + 2 = w 1 ∧ w 2 + w 1 ∧ w 2w + 3 = i ( w 1 ∧ w 2 − w 1 ∧ w 2 )w − 1 = i ( w 1 ∧ w 1 − w 2 ∧ w 2 )w − 2 = w 1 ∧ w 2 + w 1 ∧ w 2w − 3 = i ( w 1 ∧ w 2 − w 1 ∧ w 2 ).The difference in the two interpretations of w + 1arises from the fact thatw i w i = w i w i = −1 C(V )∈ C(V ) ⊗ C, butw i ∧ w i = w i ∧ w i = 0 ∈ Λ C V.By using the self-dual and anti-self dual basis, we can express our previousresults again in a ‘better’ way:The map τ : Spin(4) −→ SO(4) has the following images:⎛⎞cos 2t − sin 2t 0 0τ( exp( t w + 1 ) ) = ⎜sin 2t cos 2t 0 0⎟⎝ 0 0 cos 2t − sin 2t⎠0 0 sin 2t cos 2t⎛⎞cos 2t 0 − sin 2t 0τ( exp( t w + 2 ) ) = ⎜ 0 cos 2t 0 sin 2t⎟⎝sin 2t 0 cos 2t 0 ⎠0 − sin 2t 0 cos 2t⎛⎞cos 2t 0 0 − sin 2tτ( exp( t w + 3 ) ) = ⎜ 0 cos 2t − sin 2t 0⎟⎝ 0 sin 2t cos 2t 0 ⎠sin 2t 0 0 cos 2t


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 683⎛⎞cos 2t − sin 2t 0 0τ( exp( t w − 1 ) ) = ⎜sin 2t cos 2t 0 0⎟⎝ 0 0 cos 2t sin 2t⎠0 0 − sin 2t cos 2t⎛⎞cos 2t 0 − sin 2t 0τ( exp( t w − 2 ) ) = ⎜ 0 cos 2t 0 − sin 2t⎟⎝sin 2t 0 cos 2t 0 ⎠0 sin 2t 0 cos 2t⎛⎞cos 2t 0 0 − sin 2tτ( exp( t w − 3 ) ) = ⎜ 0 cos 2t sin 2t 0⎟⎝ 0 − sin 2t cos 2t 0 ⎠ .sin 2t 0 0 cos 2tThe map C + (R 4 ) −→ H ⊕ H has the following images1 ↦→ 1 ⊕ 1, Γ = −e 1 e 2 e 3 e 4 ↦→ 1 ⊕ −1w + 1 ↦→ −2 i ⊕ 0, w− 1 ↦→ 0 ⊕ 2 iw + 2 ↦→ −2 j ⊕ 0, w− 2w + 3 ↦→ 2 k ⊕ 0, w− 3↦→ 0 ⊕ −2 j↦→ 0 ⊕ 2 k,while the inverse H ⊕ H −→ C + (R 4 ) has the following images1 ⊕ 0 ↦→ 1 + Γ , 0 ⊕ 1 ↦→ 1 − Γ22i ⊕ 0 ↦→ − w+ 12j ⊕ 0 ↦→ − w+ 22, 0 ⊕ i ↦→ w− 12, 0 ⊕ j ↦→ − w− 22k ⊕ 0 ↦→ w+ 32 , 0 ⊕ k ↦→ w− 32 .So, the map Spin(4) −→ SU(2) × SU(2) hasexp( t Λ + ) −→ SU(2) × 1 SU(2),exp( t Λ − ) −→ 1 SU(2)× SU(2),


684 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongiven by( ) ( )exp(−i2t) 0 1 0exp( t w + 1 ) ↦→ ⊕0 exp(i2t) 0 1( ) ( )cos 2t − sin 2t 1 0exp( t w + 2 ) ↦→⊕sin 2t cos 2t 0 1( ) ( )cos 2t sin 2t i 1 0exp( t w + 3 ) ↦→⊕sin 2t i cos 2t 0 1( ) ( )1 0 exp(i2t) 0exp( t w − 1 ) ↦→ ⊕0 1 0 exp(−i2t)( ) ( )1 0 cos 2t − sin 2texp( t w − 2 ) ↦→ ⊕0 1 sin 2t cos 2t( ) ( )1 0 cos 2t sin 2t iexp( t w − 3 ) ↦→ ⊕.0 1 sin 2t i cos 2tThe action of w ± iin C(V ) ⊗ C on S ∼ = ΛP is given byw + 1w + 2w + 3⎛ ⎞⎛ ⎞−i 0 0 00 0 0 0= 2 ⎜ 0 i 0 0⎟⎝ 0 0 0 0 ⎠ w − 1 = 2 ⎜ 0 0 0 0⎟⎝ 0 0 i 0 ⎠0 0 0 00 0 0 −i⎛ ⎞⎛ ⎞0 −1 0 00 0 0 0= 2 ⎜ 1 0 0 0⎟⎝ 0 0 0 0 ⎠ w − 2 = 2 ⎜ 0 0 0 0⎟⎝ 0 0 0 −1⎠0 0 0 00 0 1 0⎛ ⎞⎛ ⎞0 i 0 00 0 0 0= 2 ⎜ i 0 0 0⎟⎝ 0 0 0 0 ⎠ w − 3 = 2 ⎜ 0 0 0 0⎟⎝ 0 0 0 i ⎠ .0 0 0 00 0 i 0Therefore, Λ + acts on S + while Λ − acts on S − .Now consider the complexified algebra of self–dual and anti–self–dualtwo formsΛ ±C = Λ ± ⊗ C.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 685Usually, we use the following basis for Λ +C :w 1 ∧ w 1 + w 2 ∧ w 2 = − i w + 1w 1 ∧ w 2 = w+ 2 − i w+ 32w 1 ∧ w 2 = w+ 2 + i w+ 32and the following basis for Λ −C :w 1 ∧ w 1 − w 2 ∧ w 2 = − i w − 1w 1 ∧ w 2 = w− 2 − i w− 32w 1 ∧ w 2 = w− 2 + i w− 32.


686 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionTheir quantization acts on S as⎛ ⎞−1 0 0 0q(w 1 ∧ w 1 + w 2 ∧ w 2 ) = 2 ⎜ 0 1 0 0⎟⎝ 0 0 0 0⎠0 0 0 0⎛ ⎞0 0 0 0q(w 1 ∧ w 2 ) = 2 ⎜1 0 0 0⎟⎝0 0 0 0⎠0 0 0 0⎛ ⎞0 −1 0 0q(w 1 ∧ w 2 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 0⎠0 0 0 0⎛ ⎞0 0 0 0q(w 1 ∧ w 1 − w 2 ∧ w 2 ) = 2 ⎜0 0 0 0⎟⎝0 0 1 0 ⎠0 0 0 −1⎛ ⎞0 0 0 0q(w 1 ∧ w 2 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 0⎠0 0 1 0⎛ ⎞0 0 0 0q(w 1 ∧ w 2 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 −1⎠ .0 0 0 0As a result, we have the following Theorem: there are isomorphismsEnd( S + ) ∼ = Λ 0 C ⊕ Λ +C , End( S − ) ∼ = Λ 0 C ⊕ Λ −C .4.13.2.4 Hermitian Structure on the SpinorsThere is a canonical Hermitian structure on the space of positive spinorsS + given by the Hermitian inner product 〈 · , · 〉, which takes the value〈 s + , t + 〉 = s + 1 t+ 1 + s+ 2 t+ 2


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 687on the spinorss + =⎛⎝s + 1⎞⎠ and t + =⎛⎝t + 1⎞⎠ ∈ S + .s + 2t + 2The above Hermitian form is Spin(4)−invariant.The dual vector space S +∗ consists of complex linear functionalsφ : S + −→ C.S +∗ is generated by the dual complex basis{ 1 ∗ C(V ) , (w 1 ∧ w 2 ) ∗ }, which satisfies1 ∗ C(V ) ( 1 C(V ) ) = 1, 1∗ C(V ) ( w 1 ∧ w 2 ) = 0,(w 1 ∧ w 2 ) ∗ ( 1 C(V )) = 0, (w 1 ∧ w 2 ) ∗ ( w 1 ∧ w 2 ) = 1.There is a Hermitian Riesz representationwith the following identification⎛ ⎞⎝s + 1S + ∼=−→ S +∗ ,⎛⎠ ↦→ 〈 ⎝s + 1⎞⎠ , · 〉.s + 2s + 2The Hermitian Riesz representation S + −→ S +∗ is given by⎛ ⎞ ⎛ ⎞∗⎝s + 1⎠ ↦→ ⎝s + 1⎠ ,which implies⎛⎝s + 1s + 2s + 2⎞⎠∗⎛= 〈 ⎝s + 2s + 1s + 2⎞⎠ , · 〉.By using the Hermitian Riesz representation, we haveS + ⊗ S + ∼ = End( S + ).Also, by using the Hermitian Riesz representation, we haveS + ⊗ S + ∼ = Λ0C ⊕ Λ +C .


688 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow consider the negative spinors S − . With respect to the standardbasis { w 1 , w 2 } we can define a similar Spin(4)− invariant Hermitian innerproduct, which takes the value〈 s − , t − 〉 = s − 1 t− 1 + s− 2 t− 2on the spinorss − =⎛⎝s − 1⎞⎠ and t − =⎛⎝t − 1⎞⎠ ∈ S − .s − 2t − 2We have a similar Hermitian Riesz representation on S − :with the following identification⎛ ⎞⎝s − 1S − ∼=−→ S −∗ ,⎛⎠ ↦→ 〈 ⎝s − 1⎞⎠ , · 〉,which, as in S + , satisfies⎛⎝s − 2s − 1s − 2⎞⎠∗⎛= 〈 ⎝s − 2s − 1s − 2⎞⎠ , · 〉.As in S + , by using the Hermitian Riesz representation, we have:S − ⊗ S − ∼ = End( S − ),S − ⊗ S − ∼ = Λ0C ⊕ Λ −C ,S + ⊗ S − ∼=−→ Hom( S + , S − ),S + ⊗ S − ∼ = Λ1C .Similarly, by interchanging S + and S − , and by using the HermitianRiesz representation, we have:S − ⊗ S + ∼ = Hom( S − , S + ),S − ⊗ S + ∼ = Λ1C .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 6894.13.2.5 Symplectic Structure on the SpinorsThere is a canonical symplectic structure on the space of positive spinorsS + given by the symplectic form { · , · }, which takes the value{s + , t + } = s + 1 t+ 2 − s+ 2 t+ 1on the spinorss + =⎛⎝s + 1⎞⎠ and t + =⎛⎝t + 1⎞⎠ ∈ S + .s + 2t + 2The above symplectic form is Spin(4)−invariant.There is a symplectic Riesz representationwith the following identification⎛ ⎞⎝s + 1S + ∼=−→ S +∗⎛⎠ ↦→ { ⎝s + 1⎞⎠ , ·}.s + 2s + 2The symplectic Riesz representationS + −→ S +∗is given by⎛⎝s + 1s + 2⎞⎠ ↦→⎛⎝−s + 2s + 1⎞∗⎠ .That means⎛⎝s + 1⎞⎠∗⎛= { ⎝s + 2⎞⎠ , ·}.s + 2−s + 1Just like the Hermitian case, we also have the following.By using the symplectic Riesz representation, we haveS + ⊗ S + ∼ = End( S + ),S + ⊗ S + ∼ = Λ0C ⊕ Λ +C .


690 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow consider the negative spinors S − . With respect to the standardbasis { w 1 , w 2 } we can define a similar Spin(4)−invariant symplecticform which takes the value{s − , t − } = s − 1 t− 2 − s− 2 t− 1on the spinorss − =⎛⎝s − 1⎞⎠ and t − =⎛⎝t − 1⎞⎠ ∈ S − .s − 2t − 2We have a similar symplectic Riesz representation on S − :with the following identification⎛ ⎞⎝s − 1S − ∼=−→ S −∗⎛⎠ ↦→ { ⎝s − 1⎞⎠ , ·},which, as in S + , satisfies⎛⎝s − 1s − 2s − 2⎞⎠∗⎛= { ⎝s − 2s − 2−s − 1⎞⎠ , ·}.As in S + , by using the symplectic Riesz representation, we have:S − ⊗ S − ∼ = End( S − ),S − ⊗ S − ∼ = Λ0C ⊕ Λ −C ,S + ⊗ S − ∼=−→ Hom( S + , S − ),S + ⊗ S − ∼ = Λ1C .Similarly, by interchanging S + and S − , and by using the symplecticRiesz representation, we haveS − ⊗ S + ∼ = Hom( S − , S + ),S − ⊗ S + ∼ = Λ1C .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 6914.13.3 Penrose Twistor CalculusRecall that the twistor theory, originally developed by Roger Penrose in1967, is the mathematical theory which maps the geometric objects of the4D Minkowski space–time into the geometric objects in the 4D complexspace with the metric signature (2, 2). The coordinates in such a space arecalled twistors.The twistor approach appears to be especially natural for solving theequations of motion of massless fields of arbitrary spin.Recently, Ed Witten used twistor theory to understand certain Yang–Mills amplitudes, by relating them to a certain string theory, the topologicalB model, embedded in twistor space. This field has come to be known astwistor string theory.4.13.3.1 Penrose Index FormalismExcept where otherwise indicated we use Penrose’s abstract index notation[Penrose and Rindler (1984)] which allows for easy explicit calculationswithout involving a choice of basis. Thus we may write, v A or v B for asection of the unprimed fundamental spinor bundle E A . Similarly w A ′ coulddenote a section of the primed fundamental spinor bundle E A ′. We write E Afor the dual bundle to E A and E A′ for the dual to E A ′. The tensor productsof these bundles yield the general spinor objects such as E AB := E A ⊗ E B ,EA ABC′ ′ B and so forth. The tensorial indices are also abstract indices. Recall′that E a = EA A is the tangent bundle, so E ′ a = EAA′ is the cotangent bundleand we may use the terms ‘spinor’ or ‘section of a spinor bundle’ to describetensor fields [Gover and Slovak (1999)].A spinor object on which some indices have been contracted will betermed a spinor contraction (of the underlying spinor). For example,vBC ABC′ ′ DE is a contraction of vABC′ DD ′ EF . In many cases the underlying spinorof interest is a tensor product of lower valence spinors. For example,v AB wB C′ u ACD is a contraction of v AB wC C′ u DEF . The same conventionsare used for the tensor indices and the twistor indices; the latter are tointroduced below. Standard notation is also used for the symmetrizationsand antisymmetrizations over some indices.Weights and scalesWe define line bundles of densities or weighted functions as follows [Goverand Slovak (1999)]. The weight -1 line bundle E[−1] over M is identified


692 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionp{ }} {with E|A ′ B ′ · · · C ′ | . Then, for integral w, the weight w line bundle E[w]is defined to be (E[−1]) −w . In the case of AG–geometries corresponding tothe real–split form SL(p + q, R) we can (locally) extend this definition toweights w ∈ R, by locally selecting a ray fibre subbundle of E[−1]. Callingthis say E + [−1], we can then define the ray bundles E + [w] := (E[−1]) −w .Finally these may be canonically extended to line bundles in the obviousway. In any case we write E A′ [w] for E A′ ⊗E[w] and so on, whenever defined.In view of the defining isomorphismh : ∧ q E A≃−→ ∧ p E A ′, we also have (4.165)q{ }} {E[−1] ∼ = E |AB · · · C|, E[1] ∼ = E||AB · · · C| ∼ = E|A ′ B ′ · · · C ′ |.} {{ }} {{ }qpWe write ɛ A′ B ′···C ′for the tautological section of E [A′ B ′···C ′] [1] giving the≃mapping E[−1] −→ ∧ p E A′ byf ↦→ fɛ A′ B ′···C ′ , (4.166)≃and ɛ D···E for similar object giving E[−1] −→ ∧ q E A . A scale for the AG–structure is a nowhere vanishing section ξ of E[1]. Note that such a choiceis equivalent to a choice of spinor ‘volume’ formɛ A′···C ′ξ := ξ −1 ɛ A′···C ′ , or to a choice of form, ɛ ξ D···E := ξ−1 ɛ D···E .Distinguished connectionsA connection ∇ a on M belongs to the given AG–structure (this really means∇ a comes from a principal connection on the bundle G 0 described below)iff it satisfies two conditions [Gover and Slovak (1999)]:• ∇ a is the tensor product of linear connections (both of which we shallalso denote ∇ a ) on the spinor bundles E A and E A ′, and• The defining isomorphism h in (4.165) is covariantly constant, i.e.,∇ a h = 0.Our conventions for the torsion T ab c and curvature R abcdof a connection∇ a on the tangent bundle T M are determined by the following equation,2∇ [a ∇ b] v c = T ab d ∇ d v c + R abcd v d .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 693Since T ab c is skew on its lower indices, T ab c = T [ab] c , it can be writtenas a sum of two termsT ab c = F ab c + ˜F ab c ,whereF ab c = : F A′ B ′ CABC ′ = F [A′ B ′ ]C(AB)C ′ , ˜Fab c =: ˜F A′ B ′ CABC ′ = F (A′ B ′ )C[AB]C ′ .The Cartan bundle G over the manifold M has the quotient G 0 , a principalfibre bundle with structure group G 0 . By the general theory, eachG 0 −equivariant section σ : G 0 → G of the quotient projection defines thedistinguished principal connection on G 0 , the pullback of the g 0 −part of ω.The whole class of these connections consists precisely of connections on G 0with the unique torsion taking values in the kernel of ∂ ∗ . A straightforwardcomputation shows that the latter condition is equivalent to the conditionthat both ˜F and F be completely trace–free. Each principal connectionon G 0 induces the induced connection on the bundle E[1] \ {0} which isassociated to G 0 and, moreover, the resulting correspondence between thesections σ and the latter connections is bijective. In particular, each sectionξ of the bundle E[1] \ {0} defines uniquely a reduction σ, such that thecorresponding distinguished connection leaves ξ horizontal.Therefore, given a scale ξ on an AG–structure there are unique connectionson E A and E A ′ such that F A′ B ′ CABC and ˜F ′A′ B ′ CABC are totally trace–free,′the induced covariant derivative preserves the isomorphism h of (4.165),and ∇ a ξ = 0. The torsion components F c ab and ˜F c ab of the induced connectionon T M are invariants of the so–called AG–structures [Bailey andEastwood (1991)].Note that in the special case of the four-dimensional conformal geometries,there is always a connection with vanishing torsion on G 0 and so bothF and ˜F are zero. The scales correspond to a choice of metric from theconformal class while the general distinguished connections (correspondingto the reduction parameter σ being not necessarily exact) are just the Weylgeometries.We may write ∇ ξ a to indicate a connection as determined by the Theorem,although mostly we omit the ξ. Thus we might write ∇ˆξa or simplyˆ∇ a to indicate the connection corresponding to a scale ˆξ and similar conventionswill be used for other operators and tensors that depend on ξ.In what follows, for the purpose of explicit calculations, we shall oftenchoose a scale and work with the corresponding connections. Objects arethen well defined, or invariant, (on the AG-structure) if they are independentof the choice of scale. Note that if we change the scale according to


694 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionξ ↦→ ˆξ = Ω −1 ξ, where Ω is a smooth non–vanishing function, then theconnection transforms as follows [Gover and Slovak (1999)]:E A :E A ′ :E B :E B′ :ˆ∇A′A u C = ∇ A′A u C + δ C AΥ A′B uBˆ∇A′A u C ′ = ∇ A′A u C ′ + δ A′C ′ΥB′ A u B ′ˆ∇A′A v B = ∇ A′A v B − Υ A′B v Aˆ∇A′A v B′ = ∇ A′A v B′ − Υ B′where Υ a := Ω −1 ∇ a Ω. ConsequentlyA vA′ (4.167)̂∇ a f = ∇ a f + wΥ a f if f ∈ E[w]. (4.168)CGiven a choice of scale ξ, we write R abD (or R (ξ)ab C D to emphasise thechoice of scale) for the curvature of ∇ a on E A Cand R ′ab D for the curvature′of ∇ a on E A ′, that is(2∇ [a ∇ b] − T ab e ∇ e )v C = R abCD v D , (2∇ [a ∇ b] − T ab e ∇ e )w D ′ = −R abC ′D ′w C ′.Then the curvature of the induced linear connection on T M isR abcd = R abC ′D ′δC D + R abCD δ C′D ′.Note that since ∇ a preserves the volume forms ɛ ξ A ′···C and ′ ɛξ D···Eit followsC Cthat R abD and R ′ab D are trace–free on the spinor indices displayed. Thus′the equationsC CR abD = U abD −δ C BP A′ B ′AD +δ C AP B′ A ′CBD , R ′ab D = U ′ ab C′D ′+δB′ D ′P A′ C ′AB −δ A′D ′P B′ C ′BAC Cdetermine the objects U abD , U ′ab D and the Rho–tensor, P ab, if we require′that U A′ B ′ CACD = 0 = U A′ D ′ C ′ABD . In this notation we have,′c cR abd = U abd + δ D′C ′ δC AP B′ A ′BD − δ D′C ′ δC BP A′ B ′AD − δ C Dδ A′C ′P B′ D ′BA + δ C Dδ B′C ′P A′ D ′AB (4.169) ,c Cwhere U abd = U abD δ D′C + U ′ ab D′C ′ δC D. (4.170)In the case of p = 2 = q this agrees with the usual decomposition ofthe curvature of the Levi–Civita connection into the conformally invariant(and trace–free) Weyl tensor part and the remaining part given by the Rho–tensor (see e.g., [Bailey et. al. (1994)]). Note that U’s are 2–forms valuedin g 0 coming from the curvature of the canonical Cartan connection andso they are in the kernel of ∂ ∗ . This is the source of the condition on thetrace, but they are not trace–free in general:U abCC = −U abC ′C ′ = 2P [ab]. (4.171)


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 695On the other hand, it follows from the Bianchi identity,R [abdc] + ∇ [a T bc] d + T [ab e T c]e d = 0,that2(p + q)P [ab] = −∇ c T ab c .The Rho–tensor P ab has the transformation equationˆP A′ B ′AB = P A′ B ′AB − ∇ A′A Υ B′B + Υ B′A Υ A′B . (4.172)We are most interested in the special case p = 2. Then the wholecomponent F c ab is irreducible and so it vanishes by our condition on thetrace, while the other component ˜F c ab of the torsion, together with thetrace–free part of U [A′ B ′ ]D(ABC)are the only local invariants of the structures.In all other cases 2 < p ≤ q, the two components of the torsion are the onlyinvariants.The totally symmetrized covariant derivatives of the Rho–tensors playa special role. We use the notationS a···b := ∇ (a ∇ b · · · ∇ d P ef)} {{ }sfor s = 2, 3 · · · .TwistorsVia the Cartan bundle G over M any P −module V gives rise to a naturalbundle (or induced bundle) V. Sections of V are identified with functionsf : G → V such that f(x.p) = ρ(p −1 )f(x), where x ↦→ x.p gives the actionof p ∈ P on x ∈ G while ρ is the action defining the P −module structure.Recall also that the Cartan bundle is equipped with a canonical connection,the so called normal Cartan connection ω. In view of this it isin our interests to work, where possible, with natural bundles V inducedfrom V where this is not merely a P −module but in fact a G−module.Then the Cartan connection induces an invariant linear connection on V.Let us write V α for the module corresponding to the standard representationof G on R p+q and write V α for the dual module. The index αis another Penrose-type abstract index and we write E α and E α for therespective bundles induced by these G-modules. All finite dimensionalG−modules are submodules in tensor products of the fundamental representationsV α and V α . Thus the bundles E α and E α play a special roleand we term these (local) twistor bundles [Bailey and Eastwood (1991);Penrose and Rindler (1986)]. In fact in line with the use of the word“tensor” we also describe any explicit subbundle of a tensor product of


696 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthese bundles as a twistor bundle and sections of such bundles as localtwistors. In particular observe that there is a canonical completely skewlocal–twistor (p + q)−form h αβ···γ on E α which is equivalent to the isomorphism(4.165). We write h αβ···γ for the dual completely skew twistorsatisfying h αβ···γ h αβ···γ = (p + q)!.All finite dimensional P −modules enjoy filtrations which split completelyas G 0 −modules. V α and V α , give the simplest cases and, asP −modules, admit filtrationsV α = V A + V A′ , V α = V A ′ + V A .(Our notational convention is that the ‘right ends’ in the formal sums aresubmodules while the ‘left ends’ are quotients.) These determine filtrationsof the twistor bundlesE α = E A + E A′ , E α = E A ′ + E A .We write XA α for the canonical section of E α ′ A ′ which gives the injectingmorphism E A′ → E α viav A′ ↦→ X α A ′vA′ . (4.173)Similarly Y A αdescribes the injection of E A into dual twistors,E A ∋ u A ↦→ Y A α u A ∈ E α . (4.174)It follows from standard representation theory that a choice of splittingof the exact sequence,0 → V A′ → V α → V A → 0is equivalent to the choice of subgroup of P which is isomorphic to G 0 . Itfollows immediately that a choice of splitting of the twistor bundle E α isequivalent to a reduction from G to G 0 . Such a splitting is a G 0 −equivarianthomomorphism ξ : E α → E A′ . We can regard ξ here as a section of E α ⊗E A′ = EαA′ and then in our index notation the homomorphism is determinedby v α ↦→ ξ A′α v α , for any section v α of E α . The composition of ξ with themonomorphism E A′ → E α must be the identity so we have,ξ A′β X β B = ′ δA′ B ′.A splitting ξ A′α of E α determines a dual splitting λ α A of E α , λ α A : E α → E A .Given such splittings we have E α = E A ⊕ E A′ and E α = E A ′ ⊕ E A , so we


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 697may write sections of these bundles as a “matrices” such as( ) u[u α A] ξ = ∈ [ΓE α ] ξ [v α ] ξ = (v A :: v A ′) ∈ [ΓE α ] ξ .u A′We will always work with splittings determined by a choice of scale ξ ∈ E[1],as discussed earlier. If u α and v α , as displayed, are expressed by such ascale then the change of scale ξ ↦→ ˆξ = Ω −1 ξ yields a transformation ofthese splittings. For example [u α ] ↦→ [u α ] bξ where[u α ] bξ =(ûA) (u A=û A′ u A′ − Υ A′B uB )With this understood we henceforth drop the notation [·] ξ and simply write,for example, v α ↦→ ˆv α whereˆv α = (ˆv A :: ˆv A ′) = (v A + Υ B′A v B ′ :: v A ′),for the corresponding transformation of v α . In particular, the objects ξ B′are not invariant andλ β A.α ,ˆξB ′α = ξ B′α − Y A α Υ B′A ,ˆλβ A = λ β A + Xβ B ′ΥB′ A .However, note that, in the splittings they determine, ξ B′α)(ξ B′α =(0 :: δ B′A , λ β δB ′ A = A0).and λ β Aare givenIn any such splitting the invariant objects XB α and Y A ′ βare given by(XB α = 0( ), Y A ′ β = δ A B :: 0 .)δ A′B ′The first four identities of the following display are immediate, while thefinal two items are useful definitions:Yβ AXβ A ′ = 0, ξA′ β λ β A = 0,Yβ Aλβ B = δA B, ξ A′β X β B = ′ δA′ B ′,(4.175)=: λγ β , ξA′ β X γ A =: ′ ξγ β .Y A β λγ AWe shall mostly deal with weighted twistors, i.e., tensor products of theform E α...β α...βγ...δ[w] = Eγ...δ ⊗ E[w]. All the above algebraic machinery works forthe weighted twistors. In fact we shall often omit the word ‘weighted’ eventhough, of course, these bundles do not come from G−modules for w ≠ 0.


698 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFinally, we observe that via this machinery any spinorial quantity maybe identified with a (weighted) twistor. For example, valence 1 spinors inE A′ [w 1 ] or E A [w 2 ] may be dealt with via (4.173) or (4.174) respectively.This determines an identification for tensor powers by treating each factorin this way. This does all cases since, via (4.166),p−1{ }} {E A ∼ = E|B · · · D|[1], E A ′∼ = E|B ′ · · · C ′ | [−1].} {{ }q−1Now, any irreducible representation of G 0 is given as a tensor product oftwo irreducible components in tensor products of the fundamental spinors(viewed as representations of the special linear groups, adjusted by aweight). Applying the corresponding Young symmetrizers [Penrose andRindler (1984); Fulton and Harris (1991)] to the tensor products of E α andE β , we get the explicit realization of each irreducible spinor bundle as thesubbundle of the (weighted) twistor bundle which is isomorphic to the injectingpart of the twistor bundle. Thus a section of a weighted irreduciblespinor bundle V may be identified with a twistor object which is zero in allits composition factors except the first. So, in fact, this non–zero factor isalso the projecting part of the twistor. We write Ṽ for this twistor (sub–)bundle satisfying V ∼ = Ṽ. Altogether, we have established the followingresult [Gover and Slovak (1999)]: Any irreducible spinor object v can beidentified with the twistor ṽ which has the spinor as its projecting part.This identification is provided in a canonical algebraic way.4.13.3.2 Twistor CalculusGiven a choice of scale ξ, a twistor connection ∇ a on E α and E α is givenby [Penrose and MacCallum (1972); Dighton (1974); Bailey and Eastwood(1991)]and∇ P ′A( ) (vB=v B′∇ P ′A v B + δ B Av P ′∇ P ′A v B′ − P P ′ B ′AB vB )(4.176)∇ P ′A (u B :: u B ′) = (∇ P ′A u B + P P ′ B ′AB u B ′ ::: ∇ P ′A u B ′ − δ P ′B ′u A). (4.177)Notice that whereas on the left hand side ∇ indicates the twistor connection,on the right hand side the symbol ∇ indicates the usual spinor connection


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 699determined by the choice of scale. Although we have fixed a choice ofscale to present explicit formulae for these connections, it is easily verifieddirectly using the formulae (4.167) that the twistor connections are in factindependent of the choice of scale.A straightforward calculation gives( ) v([∇ a , ∇ b ] − T d Cab ∇ d ) =v C′(=U abCD v D − T abCD ′v D′−2∇ [a P b]C ′D vD + T abEE ′P E′ C ′ED vD + U abC ′D ′vD′ )Thus the curvature of the twistor connection is given, in this scale, by(C Cδ = UabD −T abD ′C, where Q−2Q ′abc := ∇ [a P b]c − 1ab 2 T ab e P ec .(4.178)Note that since the twistor connection is invariant it follows that this twistorγcurvature W ab δis invariant. In fact, viewed as a g−valued 2–form on theCartan bundle G, this is just the curvature of the normal Cartan connection.In particular, we know that the structures are torsion–free (in the sense ofthe Cartan connection) if and only if the torsion part T c ab vanishes andγthey are locally flat if and only if the whole W ab δ vanishes.W abγThe D−operators)D U ab C′D ′If f ∈ E[w] then it follows easily from (4.168) that the spinor–twistor objectD A′β f := (∇ A′B f wδ A′B ′f)is invariant. We may regard this as an injecting part of the invariant twistorobject Dβ αf := Xα A ′DA′ β f. By regarding, in this formula for Dα β, ∇ to bethe coupled twistor–spinor connection it is easily verified that the operatorDβ α is well defined and invariant on sections of the weighted twistor bundlesEα···γ[w]. ρ···µ The invariant operators Dβ α : E ρ···µαρ···µδ···γ[w] → Eβδ···γ [w] are called thetwistor–D operators.For many calculations, where a choice of scale is made, it is useful toallow Dβ α to operate on spinors and their tensor products, although in thiscase the result is not independent of the scale. For example, if v C ∈ E C [w]thenD A′β v C := (∇ A′B v C wδ A′B ′v C)..


700 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSince the operator Dβα and its concatenations will have an importantrole in the following discussions we develop notation for their target spaces.First let F ρ be defined as follows,Then we writeF ρ := Ker(Y Aρ :: E ρ → E A ).F ρ···σα···β := F ρ ⊗ · · · ⊗ F σ ⊗ E α ⊗ · · · ⊗ E β ,and F ρ···σρ···σα···β[w] = Fα···β ⊗ E[w]. Finally letS ρ···σ[w] := (⊙α · · · βk Fα) ρ ⊗ E[w].} {{ }kNote that sections of Fρα (= Sρ α ) are not generally trace–free, but that Fραis in a complement to the trace–part of Eρ α .Now if f ∈ E[w] then Dαf ρ ∈ Fα[w]. ρ Similarly observe that if v σ ∈ F σthenis in Fαρσ . ThusD ρ αv σ − δ σ αv ρgives an invariant operatorD ρσαβ := 1 2 (Dρ αD σ β + D σ βD ρ α − δ σ αD ρ β − δρ β Dσ α)D ρσSimilarly we define D ρσµαβγ byαβ : E µ···νγ···δ[w] → Sρσαβ ⊗ E µ···νγ···δ [w].D ρσµαβγ := 1 3 ((Dρ αD σµβγ + Dσ βD ρµαγ + D µ γ D ρσαβ − δρ αD σµβγ − δσ βD ρµαγ − δ µ γD ρσαβ )and so on for Dρ···ν α···δ . Notice that the construction of these is designed in sucha way that the resulting operators are annihilated if composed (contracted)with YνB on any index.The Splitting MachineryIn terms of the algebraic projectors and embeddings introduced in the last


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 701section, the twistor–D operator is given byD ρ αf = X ρ R ′Y A α ∇ R′A f + wξ ρ αf, (4.179)where f is any weighted twistor-spinor object. Using this and the expressions(4.176), (4.177) for the twistor connection, the following identities areeasily established:DαX ρ β C = ′ Xρ C ′λβ K Y α K DαY ρ β C = −Yα C X ρ K ′ξK′ βD ρ αξ S′β= P ρS′αβD ρ αλ σ B = −P ρσαBX α B ′Dγ αf = wX γ B ′fY Bγ D γ αf = 0ξ B′γ D γ αf = D B′α f λ α BD B′α f = ∇ B′B f,where, again, f is any weighted twistor-spinor and we writeP ρσαβ:= P R′ S ′AB Xρ R ′Xσ S ′Y α A Yβ B, P ρS′αβ:= P R′ S ′AB Xρ R ′Y α A Yβ B, P ρσP R′ S ′ABXρ R ′Xσ S ′Y A α , etc.(4.180)αB :=Notice also that the objects ξ B′α and λ β Adescribing the splitting of thetwistors can be viewed as the projecting parts of ξ β α := ξ −1 Dαξ β andδ β α − ξ β α, respectively.D−CurvatureFor f ∈ E[w] the projecting part of Dαf ρ is 1 p Xα ′P ′DP α f = wf. Althoughthis is 0th order in f, this part of Dαf ρ behaves like a first order operatorbecause of the weight factor, w. In particular 1 p Xα ′P ′DP α satisfies a Leibnizrule and so therefore so does Dα. ρ It follows immediately that, acting onE µ [w], [Dα, ρ Dβ σ ] decomposes into a 0th order curvature part and a 1st ordertorsion part. In fact it is easy using the identities (4.171) and (4.180) toverify that[D ρ α, D σ β]v µ = W ρσµαβγ vγ − W ρσναβγ Dγ ν v µ + δ σ αD ρ β vµ − δ ρ β Dσ αv µ ,whereW ρσµαβγ = Xρ A ′Xσ B ′Y A α Y B β W A′ B ′ µABγ .4.13.4 Application: Rovelli’s Loop Quantum Gravity4.13.4.1 Introduction to Loop Quantum GravityRecall (from subsection 3.10.4 above) that Carlo Rovelli developed (in thelast decade of the 20th Century) the so–called loop approach to quantum


702 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongravity (see [Rovelli (1998)] and references therein). The first announcementof this approach was given in [Rovelli and Smolin (1987)]. Togetherwith string theory, this approach provides another serious candidate theoryof quantum gravity. It provides a physical picture of Planck scale quantumgeometry, calculation techniques, definite quantitative predictions, and atool for discussing classical problems such as black hole thermodynamics.String theory and loop quantum gravity differ not only because theyexplore distinct physical hypotheses, but also because they are expressionsof two separate communities of scientists, which have sharply distinct prejudices,and view the problem of quantum gravity in surprisingly differentmanners. As Rovelli says: “I heard the following criticism to loop quantumgravity: ‘Loop quantum gravity is certainly physically wrong, because:(1) it is not supersymmetric, and(2) is formulated in four dimensions’.But experimentally, the world still insists on looking four–dimensional andnot supersymmetric. In my opinion, people should be careful of not beingblinded by their own speculation, and mistaken interesting hypotheses(such as supersymmetry and high–dimensions) for established truth. Butstring theory may claim extremely remarkable theoretical successes and istoday the leading and most widely investigated candidate theory of quantumgravity” [Rovelli (1998)].High energy physics has obtained spectacular successes during this Century,culminated with the (far from linear) establishment of quantum fieldtheory as the general form of dynamics and with the comprehensive successof the SU(3) × SU(2) × U(1) Standard Model. Thanks to this success,now a few decades old, physics is in a condition in which it has been veryrarely: there are no experimental results that clearly challenge, or clearlyescape, the present fundamental theory of the world. The theory we haveencompasses virtually everything – except gravitational phenomena. Fromthe point of view of a particle physicist, gravity is then simply the last andweakest of the interactions. It is natural to try to understand its quantumproperties using the strategy that has been so successful for the restof microphysics, or variants of this strategy. The search for a conventionalquantum field theory capable of embracing gravity has spanned severaldecades and, through an adventurous sequence of twists, moments of excitementand disappointments, has lead to string theory. The foundationsof string theory are not yet well understood; and it is not yet entirely clearhow a supersymmetric theory in 10 or 11 dimensions can be concretely used


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 703for deriving comprehensive univocal predictions about our world.In string theory, gravity is just one of the excitations of a string (orother extended object) living over some background metric space. Theexistence of such background metric space, over which the theory is defined,is needed for the formulation and for the interpretation of the theory, notonly in perturbative string theory, but in the recent attempts of a nonperturbativedefinition of the theory, such as M theory, as well, in myunderstanding. Thus, for a physicist with a high energy background, theproblem of quantum gravity is now reduced to an aspect of the problemof understanding what is the mysterious non–perturbative theory that hasperturbative string theory as its perturbation expansion, and how to extractinformation on Planck scale physics from it.For a relativist, on the other hand, the idea of a fundamental descriptionof gravity in terms of physical excitations over a background metric spacesounds physically very wrong. The key lesson learned from general relativityis that there is no background metric over which physics happens.The world is more complicated than that. Indeed, for a relativist, generalrelativity is much more than the field theory of a particular force. Rather,it is the discovery that certain classical notions about space and time areinadequate at the fundamental level; they require modifications which arepossibly as basics as the ones that quantum mechanics introduced. Oneof such inadequate notions is precisely the notion of a background metricspace (flat or curved), over which physics happens. This profound conceptualshift has led to the understanding of relativistic gravity, to the discoveryof black holes, to relativistic astrophysics and to modern cosmology.From Newton to the beginning of this Century, physics has had a solidfoundation in a small number of key notions such as space, time, causalityand matter. In spite of substantial evolution, these notions remained ratherstable and self-consistent. In the first quarter of this Century, quantum theoryand general relativity have modified this foundation in depth. The twotheories have obtained solid success and vast experimental corroboration,and can be now considered as established knowledge. Each of the two theoriesmodifies the conceptual foundation of classical physics in a (more orless) internally consistent manner, but we do not have a novel conceptualfoundation capable of supporting both theories. This is why we do not yethave a theory capable of predicting what happens in the physical regimein which both theories are relevant, the regime of Planck scale phenomena,10 −33 cm.General relativity has taught us not only that space and time share the


704 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionproperty of being dynamical with the rest of the physical entities, but also(more crucially) that space–time location is relational only. Quantum mechanicshas taught us that any dynamical entity is subject to Heisenberg’suncertainty at small scale. Thus, we need a relational notion of a quantumspace–time, in order to understand Planck scale physics. Thus, fora relativist, the problem of quantum gravity is the problem of bringing avast conceptual revolution, started with quantum mechanics and with generalrelativity, to a conclusion and to a new synthesis (see [Rovelli (1997);Smolin (1997)].) In this synthesis, the notions of space and time need to bedeeply reshaped, in order to keep into account what we have learned withboth our present ‘fundamental’ theories.Unlike perturbative or non–perturbative string theory, loop quantumgravity is formulated without a background space–time, and is thus a genuineattempt to grasp what is quantum space–time at the fundamentallevel. Accordingly, the notion of space–time that emerges from the theoryis profoundly different from the one on which conventional quantum fieldtheory or string theory are based.According to Rovelli, the main merit of string theory is that it provides asuperbly elegant unification of known fundamental physics, and that it hasa well defined perturbation expansion, finite order by order. Its main incompletenessis that its non–perturbative regime is poorly understood, andthat we do not have a background–independent formulation of the stringtheory. In a sense, we do not really know what is the theory we are talkingabout. Because of this poor understanding of the non perturbative regimeof the theory, Planck scale physics and genuine quantum gravitational phenomenaare not easily controlled: except for a few computations, there isnot much Planck scale physics derived from string theory so far. Thereare, however, two sets of remarkable physical results. The first is givenby some very high energy scattering amplitudes that have been computed.An intriguing aspect of these results is that they indirectly suggest thatgeometry below the Planck scale cannot be probed –and thus in a sensedoes not exist– in string theory. The second physical achievement of stringtheory (which followed the D–branes revolution) is the derivation of theBekenstein–Hawking black hole entropy formula for certain kinds of blackholes.On the other hand, the main merit of loop quantum gravity is thatit provides a well–defined and mathematically rigorous formulation of abackground–independent non–perturbative generally covariant quantumfield theory. The theory provides a physical picture and quantitative pre-


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 705dictions on the world at the Planck scale. The main incompleteness of thetheory regards the dynamics, formulated in several variants. The theoryhas lead to two main sets of physical results. The first is the derivation ofthe (Planck scale) eigenvalues of geometrical quantities such as areas andvolumes. The second is the derivation of black hole entropy for ‘normal’black holes (but only up to the precise numerical factor).The main physical hypotheses on which loop quantum gravity reliesare only general relativity and quantum mechanics. In other words, loopquantum gravity is a rather conservative ‘quantization’ of general relativity,with its traditional matter couplings. In this sense, it is very different fromstring theory, which is based on a strong physical hypothesis with no directexperimental support ‘that the world is made by strings’.Finally, strings and loop gravity, may not necessarily be competing theories:there might be a sort of complementarity, at least methodological,between the two. This is due to the fact that the open problems of stringtheory regard its background–independent formulation, and loop quantumgravity is precisely a set of techniques for dealing non–perturbatively withbackground independent theories. Perhaps the two approaches might even,to some extent, converge. Undoubtedly, there are similarities between thetwo theories: first of all the obvious fact that both theories start with theidea that the relevant excitations at the Planck scale are one dimensionalobjects – call them loops or strings. [Smolin (1997)] also explored the possiblerelations between string theory and loop quantum gravity.Loop quantum gravity is a quantum field theory on a differentiable4–manifold. We have learned with general relativity that the space–timemetric and the gravitational field are the same physical entity. Thus, aquantum theory of the gravitational field is a quantum theory of the space–time metric as well. It follows that quantum gravity cannot be formulatedas a quantum field theory over a metric manifold, because there is no(classical) metric manifold whatsoever in a regime in which gravity (andtherefore the metric) is a quantum variable [Rovelli (1998)].One could conventionally split the space–time metric into two terms:one to be consider as a background, which gives a metric structure to space–time; the other to be treated as a fluctuating quantum field. This, indeed,is the procedure on which old perturbative quantum gravity, perturbativestrings, as well as current non-perturbative string theories (M–theory), arebased. In following this path, one assumes, for instance, that the causalstructure of space–time is determined by the underlying background metricalone, and not by the full metric. Contrary to this, in loop quantum


706 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongravity we assume that the identification between the gravitational fieldand the metric–causal structure of space–time holds, and must be takeninto account, in the quantum regime as well. Thus, no split of the metricis made, and there is no background metric on space–time.We can still describe space–time as a (differentiable) manifold (a spacewithout metric structure), over which quantum fields are defined. A classicalmetric structure will then be defined by expectation values of thegravitational field operator. Thus, the problem of quantum gravity is theproblem of understanding what is a quantum field theory on a manifold,as opposed to quantum field theory on a metric space. This is what givesquantum gravity its distinctive flavor, so different than ordinary quantumfield theory. In all versions of ordinary quantum field theory, the metricof space–time plays an essential role in the construction of the basic theoreticaltools (creation and annihilation operators, canonical commutationrelations, gaussian measures, propagators ); these tools cannot be used inquantum field over a manifold.Technically, the difficulty due to the absence of a background metric iscircumvented in loop quantum gravity by defining the quantum theory asa representation of a Poisson algebra of classical observables, which can bedefined without using a background metric. The idea that the quantumalgebra at the basis of quantum gravity is not the canonical commutationrelation algebra, but the Poisson algebra of a different set of observableshas long been advocated by [Isham (1984)], whose ideas have been veryinfluential in the birth of loop quantum gravity. The algebra on which loopgravity is the loop algebra [Rovelli and Smolin (1990)].In choosing the loop algebra as the basis for the quantization, we areessentially assuming that Wilson loop operators are well defined in theHilbert space of the theory. In other words, that certain states concentratedon one dimensional structures (loops and graphs) have finite norm.This is a subtle non trivial assumptions entering the theory. It is the keyassumption that characterizes loop gravity. If the approach turned out tobe wrong, it will likely be because this assumption is wrong. The Hilbertspace resulting from adopting this assumption is not a Fock space. Physically,the assumption corresponds to the idea that quantum states can bedecomposed on a basis of Faraday lines–excitations (as Minkowski QFTstates can be decomposed on a particle basis).Furthermore, this is an assumption that fails in conventional quantumfield theory, because in that context well defined operators and finite normstates need to be smeared in at least three dimensions, and 1D objects are


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 707too singular. The fact that at the basis of loop gravity there is a mathematicalassumption that fails for conventional Yang–Mills quantum field theoryis probably at the origin of some of the resistance that loop quantum gravityencounters among some high energy theorists. What distinguishes gravityfrom Yang–Mills (YM) theories, however, and makes this assumption viablein gravity even if it fails for YM theory is diffeomorphism invariance.The loop states are singular states that span a ‘huge’ non–separable statespace. Non–perturbative diffeomorphism invariance plays two roles. First,it wipes away the infinite redundancy. Second, it ‘smears’ a loop state intoa knot state, so that the physical states are not really concentrated in onedimension, but are, in a sense, smeared all over the entire manifold by thenon–perturbative diffeomorphisms [Rovelli (1998)].Conventional field theories are not invariant under a diffeomorphismacting on the dynamical fields. Every field theory, suitably formulated, istrivially invariant under a diffeomorphism acting on everything. Generalrelativity, on the contrary is invariant under such transformations. Moreprecisely, every general relativistic theory has this property. Thus, diffeomorphisminvariance is not a feature of just the gravitational field: it isa feature of physics, once the existence of relativistic gravity is taken intoaccount. Thus, one can say that the gravitational field is not particularly‘special’ in this regard, but that diffeomorphism invariance is a propertyof the physical world that can be disregarded only in the approximation inwhich the dynamics of gravity is neglected.Now, diffeomorphism invariance is the technical implementation of aphysical idea, due to Einstein. The idea is a deep modification of the pre–general–relativistic (pre–GR) notions of space and time. In pre–GR physics,we assume that physical objects can be localized in space and time withrespect to a fixed non–dynamical background structure. Operationally,this background space–time can be defined by means of physical reference–system objects, but these objects are considered as dynamically decoupledfrom the physical system that one studies. This conceptual structure failsin a relativistic gravitational regime. In general relativistic physics, thephysical objects are localized in space and time only with respect to eachother. Therefore if we ‘displace’ all dynamical objects in space–time at once,we are not generating a different state, but an equivalent mathematicaldescription of the same physical state. Hence, diffeomorphism invariance.Accordingly, a physical state in GR is not ‘located’ somewhere. Pictorially,GR is not physics over a stage, it is the dynamical theory of (orincluding) the stage itself. Loop quantum gravity is an attempt to imple-


708 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionment this subtle relational notion of space–time localization in quantumfield theory. In particular, the basic quantum field theoretical excitationscannot be localized somewhere as, say, photons are. They are quantumexcitations of the ‘stage’ itself, not excitations over a stage. Intuitively,one can understand from this discussion how knot theory plays a role inthe theory. First, we define quantum states that correspond to loop–likeexcitations of the gravitational field, but then, when factoring away diffeomorphisminvariance, the location of the loop becomes irrelevant. The onlyremaining information contained in the loop is then its knotting (a knot isa loop up to its location). Thus, diffeomorphism invariant physical statesare labelled by knots. A knot represent an elementary quantum excitationof space. It is not here or there, since it is the space with respect to whichhere and there can be defined. A knot state is an elementary quantum ofspace. In this manner, loop quantum gravity ties the new notion of spaceand time introduced by general relativity with quantum mechanics.4.13.4.2 Formalism of Loop Quantum GravityThe starting point is classical general relativity formulated in terms of theAshtekar phase–space formalism (see [Ashtekar (1991)]). Recall that classicalgeneral relativity can be formulated in the phase–space form as follows.We fix a 3D manifold M (compact and without boundaries) and considera smooth real SU(2)−connection A i a(x) and a vector density Ẽa i (x)(transforming in the vector representation of SU(2)) on M. We usea, b, . . . = 1, 2, 3 for spatial indices and i, j, . . . = 1, 2, 3 for internal indices.The internal indices can be viewed as labelling a basis in the Liealgebra of SU(2) or the three axis of a local triad. We indicate coordinateson M with x. The relation between these fields and conventional metricgravitational variables is as follows: Ẽi a (x) is the (densitized) inverse triad,related to the 3D metric g ab (x) of constant–time surfaces bywhere g is the determinant of g ab ; andg g ab = Ẽa i Ẽb i , (4.181)A i a(x) = Γ i a(x) + γk i a(x); (4.182)Γ i a(x) is the spin connection associated to the triad,(defined by ∂ [a e i b] = Γi [a e b]j, where e i a is the triad).k i a(x) is the extrinsic curvature of the constant time three surface.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 709In (4.182), γ is a constant, denoted the Immirzi parameter, that canbe chosen arbitrarily (it will enter the Hamiltonian constraint). <strong>Diff</strong>erentchoices for γ yield different versions of the formalism, all equivalentin the classical domain. If we choose γ to be equal to the imaginary unit,γ = √ −1, then A is the standard Ashtekar connection, which can be shownto be the projection of the self–dual part of the 4D spin connection on theconstant time surface. If we choose γ = 1, we get the real Barbero connection.The Hamiltonian constraint of Lorentzian general relativity hasa particularly simple form in the γ = √ −1 formalism, while the Hamiltonianconstraint of Euclidean general relativity has a simple form whenexpressed in terms of the γ = 1 real connection. Other choices of γ areviable as well. In particular, it has been argued that the quantum theorybased on different choices of γ are genuinely physical inequivalent, becausethey yield ‘geometrical quanta’ of different magnitude [Rovelli (1998)]. Apparently,there is a unique choice of γ yielding the correct 1/4 coefficient inthe Bekenstein–Hawking formula.The spinorial version of the Ashtekar variables is given in terms of thePauli matrices σ i , i = 1, 2, 3, or the su(2) generators τ i = − i 2 σ i, byẼ a (x) = −i Ẽ a i (x) σ i = 2Ẽa i (x) τ i , (4.183)A a (x) = − i 2 Ai a(x) σ i = A i a(x) τ i . (4.184)Thus, A a (x) and Ẽa (x) are 2 × 2 anti–Hermitian complex matrices.The theory is invariant under local SU(2) gauge transformations, threedimensionaldiffeomorphisms of the manifold on which the fields are defined,as well as under (coordinate) time translations generated by the Hamiltonianconstraint. The full dynamical content of general relativity is capturedby the three constraints that generate these gauge invariances (see [Ashtekar(1991)]).4.13.4.3 Loop AlgebraCertain classical quantities play a very important role in the quantum theory.These are: the trace of the holonomy of the connection, which islabelled by loops on the three manifold; and the higher order loop variables,obtained by inserting the E field (in n distinct points, or ‘hands’)into the holonomy trace. More precisely, given a loop α in M and the points


710 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductions 1 , s 2 , . . . , s n ∈ α we define:and, in generalT [α] = −Tr[U α ], (4.185)T a [α](s) = −Tr[U α (s, s)Ẽa (s)] (4.186)T a1a2 [α](s 1 , s 2 ) = −Tr[U α (s 1 , s 2 )Ẽa2 (s 2 )U α (s 2 , s 1 )Ẽa1 (s 1 )], (4.187)T a1...a N[α](s 1 . . . s N ) = −Tr[U α (s 1 , s N )Ẽa N(s N )U α (s N , s N−1 ) . . . Ẽa1 (s 1 )]where U α (s 1 , s 2 ) ∼ P exp{ ∫ s 2s 1A a (α(s))ds} is the parallel propagator ofA a along α, defined bydds U α(1, s) = dα a(s)dsA a (α(s)) U α (1, s). (4.188)These are the loop observables, previously introduced in YM theories.The loop observables coordinate the phase space and have a closedPoisson algebra, denoted the loop algebra. This algebra has a remarkablegeometrical flavor. For instance, the Poisson bracket between T [α]and T a [β](s) is non vanishing only if β(s) lies over α; if it does, the resultis proportional to the holonomy of the Wilson loops obtained by joiningα and β at their intersection (by rerouting the 4 legs at the intersection).More preciselyHere{T [α], T a [β](s)} = ∆ a [α, β(s)] [ T [α#β] − T [α#β −1 ] ] . (4.189)∫∆ a [α, x] =ds dαa (s)dsδ 3 (α(s), x) (4.190)is a vector distribution with support on α and α#β is the loop obtainedstarting at the intersection between α and β, and following first α and thenβ. β −1 is β with reversed orientation.A (non–SU(2) gauge invariant) quantity that plays a role in certainaspects of the theory, particularly in the regularization of certain operators,is obtained by integrating the E field over a two dimensional surface S∫E[S, f] = dS a Ẽi a f i , (4.191)where f is a function on the surface S, taking values in the Lie algebra ofSU(2). In alternative to the full loop observables (4.185,4.186,4.187), onealso can take the holonomies and E[S, f] as elementary variables.S


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 7114.13.4.4 Loop Quantum GravityThe kinematic of a quantum theory is defined by an algebra of ‘elementary’operators (such as x and id/dx, or creation and annihilation operators)on a Hilbert space H. The physical interpretation of the theory is based onthe connection between these operators and classical variables, and on theinterpretation of H as the space of the quantum states. The dynamics isgoverned by a Hamiltonian, or, as in general relativity, by a set of quantumconstraints, constructed in terms of the elementary operators. To assurethat the quantum Heisenberg equations have the correct classical limit,the algebra of the elementary operator has to be isomorphic to the Poissonalgebra of the elementary observables. This yields the heuristic quantizationrule: ‘promote Poisson brackets to commutators’. In other words, definethe quantum theory as a linear representation of the Poisson algebra formedby the elementary observables. The kinematics of the quantum theory isdefined by a unitary representation of the loop algebra.We can start à la Schrödinger, by expressing quantum states by meansof the amplitude of the connection, namely by means of functionals Ψ(A) ofthe (smooth) connection. These functionals form a linear space, which wepromote to a Hilbert space by defining a inner product. To define the innerproduct, we choose a particular set of states, which we denote ‘cylindricalstates’ and begin by defining the scalar product between these.Pick a graph Γ, say with n links, denoted γ 1 . . . γ n , immersed in themanifold M. For technical reasons, we require the links to be analytic.Let U i (A) = U γi , i = 1, . . . , n be the parallel transport operator of theconnection A along γ i . U i (A) is an element of SU(2). Pick a functionf(g 1 . . . g n ) on [SU(2)] n . The graph Γ and the function f determine afunctional of the connection as followsψ Γ,f (A) = f(U 1 (A), . . . , U n (A)), (4.192)(these states are called cylindrical states because they were previously introducedas cylindrical functions for the definition of a cylindrical measure).Notice that we can always ‘enlarge the graph’, in the sense that if Γ is asubgraph of Γ ′ we can writeψ Γ,f (A) = ψ Γ′ ,f ′(A), (4.193)by simply choosing f ′ independent from the U i ’s of the links which are inΓ ′ but not in Γ. Thus, given any two cylindrical functions, we can alwaysview them as having the same graph (formed by the union of the two


712 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongraphs). Given this observation, we define the scalar product between anytwo cylindrical functions, by∫(ψ Γ,f , ψ Γ,h ) = dg 1 . . . dg n f(g 1 . . . g n ) h(g 1 . . . g n ). (4.194)SU(2) nwhere dg is the Haar measure on SU(2). This scalar product extends bylinearity to finite linear combinations of cylindrical functions. It is notdifficult to show that (4.194) defines a well defined scalar product on thespace of these linear combinations. Completing the space of these linearcombinations in the Hilbert norm, we get a Hilbert space H. This is the(unconstrained) quantum state space of loop gravity. 24 H carries a naturalunitary representation of the diffeomorphism group and of the group of thelocal SU(2) transformations, obtained transforming the argument of thefunctionals. An important property of the scalar product (4.194) is that itis invariant under both these transformations.H is non-separable. At first sight, this may seem as a serious obstaclefor its physical interpretation. But we will see below that after factoringaway diffeomorphism invariance we may get a separable Hilbert space.Also, standard spectral theory holds on H, and it turns out that using spinnetworks (discussed below) one can express H as a direct sum over finitedimensional subspaces which have the structure of Hilbert spaces of spinsystems; this makes practical calculations very manageable.Finally, we will use a Dirac notation and writeΨ(A) = 〈A|Ψ〉, (4.195)in the same manner in which one may write ψ(x) = 〈x|Ψ〉 in ordinaryquantum mechanics. As in that case, however, we should remember that|A〉 is not a normalizable state.4.13.4.5 Loop States and Spin Network StatesA subspace H 0 of H is formed by states invariant under SU(2) gaugetransformations. We now define an orthonormal basis in H 0 . This ba-24 This construction of H as the closure of the space of the cylindrical functions ofsmooth connections in the scalar product (4.194) shows that H can be defined without theneed of recurring to C ∗ algebraic techniques, distributional connections or the Ashtekar-Lewandowski measure. The casual reader, however, should be warned that the resultingH topology is different than the natural topology on the space of connections: if asequence Γ n of graphs converges point–wise to a graph Γ, the corresponding cylindricalfunctions ψ do not converge to ψ Γn,f Γ,f in the H Hilbert space topology.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 713sis represents a very important tool for using the theory. It was introducedin [Rovelli and Smolin (1995)] and developed in [Baez (1996a);Baez (1996b)]; it is denoted spin network basis.First, given a loop α in M, there is a normalized state ψ α (A) in H,which is obtained by taking Γ = α and f(g) = −Tr(g). Namelyψ α (A) = −Tr(U α (A)). (4.196)We introduce a Dirac notation for the abstract states, and denote this stateas |α〉. These sates are called loop states. Using Dirac notation, we canwriteψ α (A) = 〈A|α〉, (4.197)It is easy to show that loop states are normalizable. Products of loop statesare normalizable as well. Following tradition, we denote with α also a multi–loop, namely a collection of (possibly overlapping) loops {α 1 , . . . , α n , }, andwe callψ α (A) = ψ α1(A) × . . . × ψ αn(A) (4.198)– a multi–loop state. Multi–loop states represented the main tool for loopquantum gravity before the discovery of the spin network basis. Linearcombinations of multi–loop states over–span H, and therefore a genericstate ψ(A) is fully characterized by its projections on the multi–loop states,namely byψ(α) = (ψ α , ψ). (4.199)The ‘old’ loop representation was based on representing quantum statesin this manner, namely by means of the functionals ψ(α) over loop spacedefined in(4.199).Next, consider a graph Γ. A ‘coloring’ of Γ is given by the following.(1) Associate an irreducible representation of SU(2) to each link of Γ.Equivalently, we may associate to each link γ i a half integer numbers i , the spin of the irreducible, or, equivalently, an integer number p i ,the ‘color’ p i = 2s i .(2) Associate an invariant tensor v in the tensor product of the representationss 1 . . . s n , to each node of Γ in which links with spins s 1 . . . s n meet.An invariant tensor is an object with n indices in the representationss 1 . . . s n that transform covariantly. If n = 3, there is only one invarianttensor (up to a multiplicative factor), given by the Clebsh–Gordon


714 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncoefficient. An invariant tensor is also called an intertwining tensor.All invariant tensors are given by the standard Clebsch–Gordon theory.More precisely, for fixed s 1 . . . s n , the invariant tensors form a finite dimensionallinear space. Pick a basis v j is this space, and associate oneof these basis elements to the node. Notice that invariant tensors existonly if the tensor product of the representations s 1 . . . s n contains thetrivial representation. This yields a condition on the coloring of thelinks. For n = 3, this is given by the well known Clebsh–Gordan condition:each color is not larger than the sum of the other two, and thesum of the three colors is even.We indicate a colored graph by {Γ, ⃗s, ⃗v}, or simply S = {Γ, ⃗s, ⃗v}, anddenote it a ‘spin network’. (It was R. Penrose who first had the intuitionthat this mathematics could be relevant for describing the quantum propertiesof the geometry, and who gave the first version of spin network theory[Penrose (1971a); Penrose (1971b)].)Given a spin network S, we can construct a state Ψ S (A) as follows. Wetake the propagator of the connection along each link of the graph, in therepresentation associated to that link, and then, at each node, we contractthe matrices of the representation with the invariant tensor. We get a stateΨ S (A), which we also write asOne can then show the following.ψ S (A) = 〈A|S〉. (4.200)• The spin network states are normalizable. The normalization factor iscomputed in [DePietri and Rovelli (1996)].• They are SU(2) gauge invariant.• Each spin network state can be decomposed into a finite linear combinationof products of loop states.• The (normalized) spin network states form an orthonormal basis forthe gauge SU(2) invariant states in H (choosing the basis of invarianttensors appropriately).• The scalar product between two spin network states can be easily computedgraphically and algebraically.The spin network states provide a very convenient basis for the quantumtheory.The spin network states defined above are SU(2) gauge invariant. Thereexists also an extension of the spin network basis to the full Hilbert space.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 7154.13.4.6 Diagrammatic Representation of the StatesA diagrammatic representation for the states in H is very useful in concretecalculations. First, associate to a loop state |α〉 a diagram in M, formedby the loop α itself. Next, notice that we can multiply two loop states, obtaininga normalizable state. We represent the product of n loop states bythe diagram formed by the set of the n (possibly overlapping) correspondingloops (we denote this set ‘multi–loop’). Thus, linear combinations ofmulti–loops diagrams represent states in H. Representing states as linearcombinations of multi–loops diagrams makes computation in H easy.Now, the spin network state defined by the graph with no nodes α,with color 1, is clearly, by definition, the loop state |α〉, and we representit by the diagram α. The spin network state |α, n〉 determined by thegraph without nodes α, with color n can be obtained as follows. Draw nparallel lines along the loop α; cut all lines at an arbitrary point of α, andconsider the n! diagrams obtained by joining the legs after a permutation.The linear combination of these n! diagrams, taken with alternate signs(namely with the sign determined by the parity of the permutation) isprecisely the state |α, n〉. The reason of this key result can be found inthe fact that an irreducible representation of SU(2) can be obtained as thetotally symmetric tensor product of the fundamental representation withitself (for details, see [DePietri and Rovelli (1996)]).Next, consider a graph Γ with nodes. Draw n i parallel lines alongeach link γ i . Join pairwise the end points of these lines at each node(in an arbitrary manner), in such a way that each line is joined with aline from a different link. In this manner, one get a multi–loop diagram.Now antisymmetrize the n i parallel lines along each link, obtaining a linearcombination of diagrams representing a state in H. One can show that thisstate is a spin network state, where n i is the color of the links and thecolor of the nodes is determined by the pairwise joining of the legs chosen[DePietri and Rovelli (1996)]. Again, simple SU(2) representation theoryis behind this result.More in detail, if a node is trivalent (has 3 adjacent links), then wecan join legs pairwise only if the total number of the legs is even, and ifthe number of the legs in each link is smaller or equal than the sum ofthe number of the other two. This can be immediately recognized as theClebsch–Gordan condition. If these conditions are satisfied, there is only asingle way of joining legs. This corresponds to the fact that there is onlyone invariant tensor in the product of three irreducible of SU(2). Higher


716 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionvalence nodes can be decomposed in trivalent ‘virtual’ nodes, joined by‘virtual’ links. Orthogonal independent invariant tensors are obtained byvarying over all allowed colorings of these virtual links (compatible with theClebsch–Gordan conditions at the virtual nodes). <strong>Diff</strong>erent decompositionsof the node give different orthogonal bases. Thus the total (links and nodes)coloring of a spin network can be represented by means of the coloring ofthe real and the virtual links (see Figure 4.10).Fig. 4.10 Construction of ‘virtual’ nodes links over an n−valent node in a graph Γ.Viceversa, multi–loop states can be decomposed in spin network statesby simply symmetrizing along (real and virtual) nodes. This can be doneparticularly easily diagrammatically, as illustrated by the graphical formulaein [Rovelli and Smolin (1995); DePietri and Rovelli (1996)]. These arestandard formulae. In fact, it is well known that the tensor algebra of theSU(2) irreducible representations admits a completely graphical notation.This graphical notation has been widely used for instance in nuclear andatomic physics. The application of this diagrammatic calculus to quantumgravity is described in detail in [DePietri and Rovelli (1996)].4.13.4.7 Quantum OperatorsNow, we define the quantum operators, corresponding to the T −variables,as linear operators on H. These form a representation of the loop variablesPoisson algebra. The operator T [α] acts diagonallyT [α]Ψ(A) = −Tr U α (A) Ψ(A),(recall that products of loop states and spin–network states are normalizablestates). In diagrammatic notation, the operator simply adds a loop to


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 717a (linear combination of) multi–loopsT [α] |Ψ〉 = |α〉|Ψ〉.Higher order loop operators are expressed in terms of the elementary‘grasp’ operation. Consider first the operator T a (s)[α], with one hand inthe point α(s). The operator annihilates all loop states that do not crossthe point α(s). Acting on a loop state |β〉, it givesT a (s)[α] |β〉 = l 2 0 ∆ a [β, α(s)] [ |α#β〉 − |α#β −1 〉 ] , (4.201)where we have introduced the elementary length l 0 byl 2 0 = G = 16πG Newtonc 3 = 16π l 2 P lanck (4.202)and ∆ a and # were defined above. This action extends by linearity, continuityand by the Leibniz rule to products and linear combinations of loopstates, and to the full H. In particular, it is not difficult to compute itsaction on a spin network state [DePietri and Rovelli (1996)]. Higher orderloop operators act similarly. It is easy to verify that these operators providea representation of the classical Poisson loop algebra.All the operators in the theory are then constructed in terms of thesebasics loop operators, in the same way in which in conventional QFT oneconstructs all operators, including the Hamiltonian, in terms of creationand annihilation operators. The construction of the composite operatorsrequires the development of regularization techniques that can be used inthe absence of a background metric.4.13.4.8 Loop v.s. Connection RepresentationImagine we want to quantize the one dimensional harmonic oscillator. Wecan consider the Hilbert space of square integrable functions ψ(x) on thereal line, and express the momentum and the Hamiltonian as differentialoperators. Denote the eigenstates of the Hamiltonian as ψ n (x) = 〈x|n〉. Itis well known that the theory can be expressed entirely in algebraic form interms of the states |n〉. In doing so, all elementary operators are algebraic:ˆx|n〉 = √ 12(|n − 1〉 + (n + 1)|n + 1〉), ˆp|n〉 = √ −i2(|n − 1〉 − (n + 1)|n + 1〉).Similarly, in quantum gravity we can directly construct the quantum theoryin the spin–network (or loop) basis, without ever mentioning functionals ofthe connections. This representation of the theory is denoted the looprepresentation.


718 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionA section of the first paper on loop quantum gravity by [Rovelli andSmolin (1990)] was devoted to a detailed study of ‘transformation theory’(in the sense of Dirac) on the state space of quantum gravity, and in particularon the relations between the loop statesψ(α) = 〈α|ψ〉 (4.203)and the states ψ(A) giving the amplitude for a connection field configurationA, and defined byψ(A) = 〈A|ψ〉. (4.204)Here |A〉 are ‘eigenstates of the connection operator’, or, more precisely(since the operator corresponding to the connection is ill defined in thetheory) the generalized states that satisfyT [α] |A〉 = −Tr[Pe R α A ] |A〉. (4.205)However, at the time of [Rovelli and Smolin (1990)] the lack of a scalarproduct made transformation theory quite involved.On the other hand, the introduction of the scalar product (4.194) givesa rigorous meaning to the loop transform. In fact, we can write, for everyspin network S, and every state ψ(A)ψ(S) = 〈S|ψ〉 = (ψ S , ψ). (4.206)This equation defines a unitary mapping between the two presentationsof H: the ‘loop representation’, in which one works in terms of the basis|S〉; and the ‘connection representation’, in which one uses wave functionalsψ(A).The development of the connection representation followed a windingpath through C ∗ −algebraic and measure theoretical methods. The work of[DePietri (1997)] has proven the unitary equivalence of the two formalisms.4.14 Application: Seiberg–Witten Monopole Field TheorySome of the most important physical problems of contemporary theoreticalphysics concern the behavior of gauge theories and string theory at strongcoupling. For gauge theories, these include the problems of confinementof color, of dynamical chiral symmetry breaking, of the strong couplingbehavior of chiral gauge theories, of the dynamical breaking of supersymmetry.In each of these areas, major advances have been achieved over the


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 719past few years, and a useful resolution of some of these difficult problemsappears to be within sight. For string theory, these include the problemsof dynamical compactification of the 10D theory to string vacua with 4Dand of supersymmetry breaking at low energies. Already, it has becomeclear that, at strong coupling, the string spectrum is radically altered andeffectively derives from the unique 11D M–theory.This rapid progress was driven in large part by the Seiberg–Witten (SW)solution of N = 2 supersymmetric Yang–Mills (YM) theory for SU(2) gaugegroup [Seiberg and Witten (1994a); Seiberg and Witten (1994b)] and bythe discovery of D−branes in string theory. Some of the key ingredientsunderlying these developments are [D’Hoker and Phong (1998a); D’Hokerand Phong (1997)]:• Restriction to solving for the low energy behavior of the non–perturbative dynamics, summarized by the low energy effective actionof the theory.• High degrees of supersymmetry. This has the effect of imposing certainholomorphicity constraints on parts of the low energy effective action,and thus of restricting its form considerably. For gauge theories in 4D,the following degrees of supersymmetry can be distinguished:(1) N = 1 supersymmetry supports chiral fermions and is the startingpoint for the Minimal Supersymmetric Standard Model, the simplestextension of the Standard Model to include supersymmetricpartners.(2) N = 2 supersymmetry only supports non–chiral fermions and isthus less realistic as a particle physics model, but appears better‘solvable’. This is where the SW–solution was constructed.(3) N = 4 is the maximal amount of supersymmetry, and a specialcase of N = 2 supersymmetry with only non–chiral fermions andvanishing renormalization group β−function. Dynamically, thelatter theory is the simplest amongst 4D gauge theories, and offersthe best hopes for admitting an exact solution.• Electric–magnetic and Montonen–Olive duality [Montonen and Olive(1977)]. Recall that the free Maxwell equations are invariant underelectric–magnetic duality when E ⃗ → B ⃗ and B ⃗ → −E. ⃗ In the presenceof matter, duality will require the presence of both electric charge eand magnetic monopole charge g whose magnitude is related by Diracquantization: e · g ∼ . Thus, weak electric coupling is related tolarge magnetic coupling. Conversely, problems of large electric coupling


720 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(such as confinement of the color electric charge of quarks) are mappedby duality into problems of weak magnetic charge. It was conjecturedby [Montonen and Olive (1977)] that the N = 4 supersymmetric YMtheory for any gauge algebra g is mapped under the interchange ofelectric and magnetic charges, i.e., under e ↔ 1/e into the theory withdual gauge algebra g ∨ . When combined with the shift–invariance ofthe instanton angle θ this symmetry is augmented to the duality groupSL(2, Z), or a subgroup thereof.Of central interest to many of these exciting developments is the 4Dsupersymmetric YM theory with maximal supersymmetry, N = 4, andwith arbitrary gauge algebra g. As an N = 2 supersymmetric theory,the theory has a g−gauge multiplet, and a hypermultiplet in the adjointrepresentation of g with mass m.This generalized theory enjoys many of the same properties as theN = 4 theory: it has the same field contents; it is ultra–violet finite; it hasvanishing renormalization group β−function, and it is expected to haveMontonen–Olive duality symmetry. For vanishing hypermultiplet massm = 0, the N = 4 theory is recovered. For m −→ ∞, it is possible tochoose dependences of the gauge coupling and of the gauge scalar expectationvalues so that the limiting theory is one of many interesting N = 2supersymmetric YM theories. Amongst these possibilities for g = SU(N)for example, are the theories with any number of hypermultiplets in thefundamental representation of SU(N), or with product gauge algebrasSU(N 1 ) × SU(N 2 ) × · · · × SU(N p ), and hypermultiplets in fundamentaland bi-fundamental representations of these product algebras.An outstanding problem in non–Abelian gauge theory has been to makereliable predictions about the (non–perturbative) strong coupling region.N. Seiberg and E. Witten in [Seiberg and Witten (1994a); Seiberg andWitten (1994b)] studied N = 2 supersymmetric gauge theories in 4D withmatter multiplets. For all such models for which the gauge group is SU(2),they derived the exact metric on the moduli space of quantum vacua andthe exact spectrum of the stable massive states. Seiberg and Witten haveshown that the local part of the effective action is governed by a singleanalytic function F of a complex variable; they made an Ansatz for the Fthat satisfies all the physical criteria and embodies electromagnetic duality,thus directly connecting the weak to the strong coupling regions. A numberof new physical phenomena occurred, such as chiral symmetry breakingthat was driven by the condensation of magnetic monopoles that carried


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 721global quantum numbers. For those cases in which conformal invariancewas broken only by mass terms, their formalism automatically gave resultsthat were invariant under electric–magnetic duality.4.14.1 SUSY Formalism4.14.1.1 N = 2 SupersymmetryFirst, recall the essentials of the N = 2 supersymmetry algebra (for detailssee, e.g., [West (1990)]). The algebra is given by [Flume et. al. (1996)]{Q i α, ¯Q k β} = δ ik σ µ αβ P µ, {Q i α, Q k β} = ɛ ik ɛ αβ Z,plus the Hermitian conjugate of the second relation, where i, k = 1, 2 andZ is a central charge. This algebra is realized on the simplest possiblenon–trivial super–multiplet: Ψ ⊃ {φ, ψ, A µ ; F, D}, where φ is a complexscalar field, ψ is a Dirac spinor, A µ is a gauge–field, while F and D arecomplex and real dummy–fields respectively. This N = 2 superfield actuallyconsists of two N = 1 superfields: Φ ⊃ {φ, q, F} and V ⊃{A µ , f, D} or W α ⊃ {F µν , f, D}, where φ and V/W α are chiral and vectormultiplets respectively, the q and f fields being Weyl spinors of oppositechirality. Since the gauge–field A µ belongs to the adjoint representation ofthe gauge group G and all the fields belong to the same multiplet, theymust all belong to the adjoint representation of G.4.14.1.2 N = 2 Super–ActionThe super–action for the N = 2 superfield just described is given by [Flumeet. al. (1996)]∫(Ψ) 2.A = Im Tr d 4 xd 2 θ α d 2¯θβOn expanding this action in terms of the N = 1 superfields, it becomes∫) ∫ (A = Im d 4 xd 2 θ α d(Āe 2¯θβ −2g oV A + τ o d 4 xd 2 θ α W α W α), (4.207)whereτ o = θ o2π + 4πi ,g 2 othe parameter g o is the usual gauge–coupling constant and θ o is theQCD−vacuum–angle. The exponential in the first term is just the supersymmetricgeneralization of the covariant derivative. Expanding this


722 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionaction further in terms of conventional fields, we get [Flume et. al. (1996)]∫A = Tr∫+ Tr{ 1d 4 x2 (φ† D 2 φ) + ¯ψDψ + g o (φ[¯ψ, γ 5 ψ]) + go[φ 2 † , φ] 2}{ 1d 4 x4go2 F µν F µν + θ o32 ˜F}µν F µν . (4.208)The action (4.208) can be recognized as the standard action for a Quark–Gluon–Higgs system in which all the fields are in the adjoint representationand the coupling constants are reduced to g and θ by the supersymmetry.Thus it is not very exotic. Indeed it could be the QCD action except forthe fact that the quarks are in the adjoint and presence of the scalar field.The action (4.208) embodies all the properties of quantum gauge theorythat have surfaced over the past thirty years and could even be used as amodel to teach quantum gauge theory. It might be worthwhile to list theseproperties [Flume et. al. (1996)]:1. It contains a gauge–field coupled to matter2. It is asymptotically free3. It is scale–invariant, but with a scale–anomaly4. It has spontaneous symmetry breaking5. It has central charges (Z and ¯Z)6. It admits both instantons (see [Belavin et. al. (1975)]) and monopolesBecause of the supersymmetry it has some further special properties, whosesignificance will become clear later, namely,7. It generalizes the Montonen–Olive mass formula [Montonen and Olive(1977)] for gauge–fields and monopoles:from(M = |v| N e + 1 )g 2 n mwhere Z = (an e + a d n m ),toM = |Z|,where n e and n m denote the gauge–field and monopole charges respectively,while the so–called prepotential coefficients a and a d will be explained later.8. It is symmetric with respect to a Z 4 symmetry, which is the relic ofthe R−symmetry: (θ α → e iɛ θ α ), which itself survives the axial anomalybreakdown.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 7239. It has a holomorphic structure10. It has a duality that connects the weak and strong coupling regimes12. The duality generalizes to an SL(2, Z) symmetry.4.14.1.3 Spontaneous Symmetry–BreakingFor SU(2) this concept is very simple. From the form of the Higgs potentialin (4.208) we see that there is a Higgs vacuum for φ = vσ where v isany complex number and σ is any fixed generator of SU(2). Furthermore,for v ≠ 0 this breaks the gauge-symmetry from SU(2) to U(1). For othergauge-groups G the corresponding statement is that v must lie in the Cartansubalgebra of G. On the other hand there is no spontaneous breakdownof supersymmetry. Thus the full breakdown isSU(2) → U(1) : N = 2 supersymmetry unbroken.Indeed it is the fact that the supersymmetry is unbroken that gives themodel its nice properties, since otherwise the classical properties would notbe preserved after quantization.After the spontaneous breakdown the restriction of the N = 1 form of theclassical action (4.207) to the massless U(1) fields takes the form∫) ∫ (A = Im d 4 xd 2 θ α d(ĀA 2¯θβ + τ o Im d 4 xd 2 θ α W α W α).Since the adjoint representation of U(1) is trivial this action is a free-fieldone. However, in the quantum theory this does not mean that the effectiveLagrangian is also free because, through the quantum fluctuations, themassive fields induce interaction term for the massless ones. The first greatvirtue of the SW model is that these interactions have a very specific form.In fact, they have shown that, due to the N = 2 supersymmetry the localpart of the effective Lagrangian can only be of the formA = 1 ∫) ∫)d 4 xd 2 θd 2¯θ(ĀAd −2ĀdA + Im d 4 xd 2 θ(τ(A))(W α W α ,(4.209)where A d = F ′ (A) and τ(A) = F ′′ (A),for some function F(A). Thus the effective Lagrangian is completely governedby the single function F(A). Note that (4.209) is very similar to theclassical action (4.207) which is the special case for which F(A) = 1 2 τ oA 2 .


724 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe SW solution is actually a special Ansatz for the functional form ofF(A) [Flume et. al. (1996)].4.14.1.4 Holomorphy and DualityIt is now easy to quantify what is meant by holomorphy and duality. Holomorphyis simply the statement that F(A) depends only on A and not on Ā.Duality means that the physics described by the effective action (4.209) isinvariant with respect to the duality transformation [Flume et. al. (1996)]:( ) A→A d( 0 1−1 0) ( AA d), D α W α → D ˙α W ˙α , τ(A) → (τ(A)) −1 .(4.210)Note that the duality transformation is closely linked to the Legendre transformof F(A) with respect to A. By noting that in the free classical theorywith θ o = 0 the transformation (4.210) reduces to⃗E → ⃗ B and g → 1 g ,we see that it is the generalization of the well–known Maxwell–Dirac duality.Thus the action (4.209) not only generalizes Maxwell–Dirac duality, butputs it into a genuine dynamical model. Furthermore, the duality (4.210)generalizes to( AA d)→( ) ( ) p q Ar s A dandτ(A) → pτ(A) + qrτ(A) + s ,where the matrix with entries (p, q, r, s) is in SL(2, Z). The integer–valuedness of the transformation follows from the requirement that, in theperturbation theory at least, it should change the θ angle only by multiplesof 2π and leaves the mass–formula form–invariant.4.14.1.5 The SW PrepotentialThe formulation of the SW solution [Seiberg and Witten (1994a); Seibergand Witten (1994b)] itself is relatively simple, as follows (see, e.g., [Marshakov(1997)]). Supersymmetry requires the metric on moduli space ofmassless complex scalars from N = 2 vector supermultiplets to be of ‘specialKähler form’, i.e., the Kähler potentialK(a, ā) = Im ∑ iā i∂F∂a i


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 725needs to be expressed through a certain holomorphic function F = F(a),called a prepotential. Then the Coulomb–branch, low–energy, effective actionfor the 4D N = 2 SUSY YM vector multiplets can be described interms of an auxiliary Riemann surface (i.e., a complex curve) Σ, equippedwith a meromorphic 1–form dS. The required Riemann surface Σ has somepeculiar properties:• The number of ‘live’ moduli (of complex structure) of Σ is stronglyrestricted (roughly ‘3 times’ less than for a generic Riemann surface).The genus of Σ for the SU(N) gauge theories is exactly equal 25 to therank of gauge group – i.e., to the number of independent moduli.• The variation of generating 1–form dS over these moduli gives holomorphicdifferentials.• The periods of the generating 1–form∮∮a = dS and a D = dSAgive the set of ‘dual’ masses, the W −bosons and the monopoles, whilethe period matrix T ij (Σ) represents the set of couplings in the low–energy effective theory. The prepotential F = F(a) is a function of halfof the variables (a, a D ), so that we havea i D = ∂F∂a iand T ij = ∂ai D∂a j= ∂2 F∂a i ∂a j.These data mean that the effective SW theory is formulated in terms ofa classical finite–gap integrable system (see, e.g., [Dubrovin et. al. (1985)])and their Whitham deformations [Marshakov (1997)].B4.14.2 Clifford Actions, Dirac Operators and Spinor BundlesRecall that the SW monopole equations were written in terms of a sectionof a Spinor bundle and a U(1) connection on a line bundle L. The first25 For generic gauge groups one should speak instead of genus (i.e., dimension of Jacobianof a spectral curve) – about the dimension of Prym–variety. Practically it meansthat for other than A N −type gauge theories one should consider the spectral curves withinvolution and only the invariant under the involution cycles possess physical meaning.We consider in detail only the A N theories, the generalization to the other gauge groups isstraightforward: for example, instead of periodic Toda chains [Toda (1981)], correspondingto A N theories, one has to consider the ‘generalized’ Toda chains, first introducedfor different Lie–algebraic series (B, C, D, E, F and G) in [Bogoyavlensky (1981)].


726 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionequation just says that the spinor section ψ has to be in the kernel of theDirac operator. The second equation describes a relation between the self–dual part of the curvature associated to the connection A and the sectionψ in terms of the Clifford action.The mathematical setting for Witten’s gauge theory is considerably simplerthan Donaldson’s analogue: first of all it deals with U(1)−principalbundles (Hermitian line bundles) rather than with SU(2)−bundles, andthe Abelian structure group allows simpler calculations; moreover the equation,which plays a role somehow analogous to the previous anti–self–dualequation for SU(2)−instantons (see [Donaldson and Kronheimer (1990)]),involves Dirac operators and Spin c −structures, which are well known andlong developed mathematical tools (see [Roe (1988)] or [Libermann andMarle (1987)]).The main differences between the two theories arise when it comes tothe properties of the moduli space of solutions of the monopole equation upto gauge transformations. The SW invariant, which depends on the Chernclass of the line bundle L, is given by the number of points, counted withorientation, in a zero–dimensional moduli space.In Witten’s paper [Witten (1994)] the monopole equation is introduced,and the main properties of the moduli space of solutions are deduced.The dimension of the moduli space is computed by an index theory technique,following an analogous proof for Donaldson’s theory, as in [Atiyahet. al. (1978)]; and the circumstances under which the SW invariants providea topological invariant of the four–manifold are illustrated in a similarway to the analogous result regarding the Donaldson polynomials.The tool that is of primary importance in proving the results about themoduli space of Abelian instantons is the Weitzenböck formula for the Diracoperator on the Spin c −bundle S + ⊗ L: such a formula is a well known (see[Roe (1988)]) decomposition of the square of the Dirac operator on a spinbundle twisted with a line bundle L.A first property which follows from the Weitzenböck formula is a boundon the number of solutions: the moduli space is empty for all but finitelymany choices of the line bundle L.Moreover, as shown in [Kronheimer and Mrowka (1994a)], the modulispace is always compact: a fact that avoids the complicated analytic techniquesthat were needed for the compactification of the moduli space ofSU(2)−instantons (see [Donaldson and Kronheimer (1990)]).Another advantage of this theory is that the singularities of the modulispace (again this is shown in [Kronheimer and Mrowka (1994a)]) only ap-


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 727pear at the trivial section ψ ≡ 0, since elsewhere the action of the gaugegroup is free. Hence, by perturbing the equation, it is possible to get asmooth moduli space.The analogue of the vanishing Theorem for Donaldson polynomials ona manifold that splits as a connected sum can be proven, reinforcing theintuitive feeling that the two sets of invariants ought to be the same.Moreover, explicit computations can be done in the case of Kähler manifolds,by looking at the SW invariants associated to the canonical linebundle.The latter result has a generalization due to [Taubes (1994)], where itis shown that the value ±1 of the SW invariants is achieved on symplecticfour–manifolds, with respect to the canonical line bundle, by a techniquethat involves estimates of solutions of a parametrized family of perturbedmonopole equations.4.14.2.1 Clifford Algebras and Dirac OperatorsRecall that the Clifford algebra C(V ) of a (real or complex) vector space Vwith a symmetric bilinear form (, ) is the algebra generated by the elements[Marcolli (1995)] {e ɛ11 · · · eɛn n , }, where ɛ i = 0, or 1 and {e i } is an orthogonalbasis of V , subject to the relations e·e ′ +e ′·e = −2(e, e ′ ). The multiplicationof elements of V in the Clifford algebra is called Clifford multiplication.In particular given a differentiable manifold X we shall consider theClifford algebra associated to the tangent space at each point. The Cliffordalgebra of the tangent bundle of X is the bundle that has fibre over eachpoint x ∈ X the Clifford algebra C(T x X). We shall denote this bundleC(T X).If dim V = 2m, there is a unique irreducible representation of the Cliffordalgebra C(V ). This representation has dimension 2 m .A spinor bundle over a Riemannian manifold X is the vector bundleassociated to C(T X) via this irreducible representation, endowed with aHermitian structure such that the Clifford multiplication is skew–symmetricand compatible with the Levi–Civita connection on X (see [Roe (1988)]).Not all manifolds admit a spinor bundle; it has been proved in [Roe(1988)] that the existence of such a bundle is equivalent to the existence ofa Spin c –structure on the manifold X: we shall discuss Spin c −-structuresin the next paragraph. If such a bundle exists, it splits as a direct sumof two vector bundles, S = S + ⊕ S − , where the splitting is given by theinternal grading of the Clifford algebra.


728 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet X be a manifold that admits a spinor bundle S. Let {e i } be a localorthonormal basis of sections of the tangent bundle T X and 〈 , 〉 be theHermitian structure as in the above definition of spinors. Then, for anysection ψ ∈ Γ(X, S), the expression 〈e i e j ψ, ψ〉 is purely imaginary at eachpoint x ∈ X. As a proof, by skew–adjointness of Clifford multiplicationand the fact that the basis is orthonormal,〈e i e j ψ, ψ〉 = − 〈e j ψ, e i ψ〉 = 〈ψ, e j e i ψ〉 − 〈ψ, e i e j ψ〉 = −〈e i e j ψ, ψ〉.Given a spinor bundle S over X, the Dirac operator on S is a first orderdifferential operator on the smooth sections D : Γ(X, S + ) → Γ(X, S − ),defined as the compositionD : Γ(X, S + ) ∇ → Γ(X, S + ) ⊗ T ∗ X g → Γ(X, S + ) ⊗ T X • → Γ(X, S − ),where the first map is the covariant derivative, with the Spin–connectioninduced by the Levi–Civita connection on X (see [Roe (1988)]), the secondis the Legendre transform given by the Riemannian metric, and the thirdis Clifford multiplication.It is easy to check (for details see [Roe (1988)]) that this corresponds tothe following expression in coordinates: Ds = e k · ∇ k s.An essential tool in Spin geometry, which is very useful in SW gaugetheory (see e.g. [Jost et. al. (1995); Kronheimer and Mrowka (1994a);Taubes (1994); Witten (1994)]), is the Weitzenböck formula.Given a smooth vector bundle E over a Spin c −-manifold X and aconnection A on E, the twisted Dirac operator D A : Γ(X, S + ⊗ E) →Γ(X, S − ⊗E) is the operator acting on a section s⊗e as the Dirac operatoron s and the composite of the covariant derivative ˜∇ A and the Cliffordmultiplication on e:D A : Γ(X, S + ⊗ E) ∇⊗1+1⊗ →˜∇ AΓ(X, S + ⊗ E ⊗ T ∗ X) →gg→ Γ(X, S + ⊗ E ⊗ T X) → • Γ(X, S − ⊗ E).The twisted Dirac operator D A satisfies the Weitzenböck formula:D 2 As = (∇ ∗ A∇ A + κ 4 + −i4 F A)s,where ∇ ∗ A is the formal adjoint of the covariant derivative with respectto the Spin−-connection on the Spinor bundle, and with respect to theconnection A on E, ∇ A = ∇ ⊗ 1 + 1 ⊗ ˜∇ A ; κ is the scalar curvature on


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 729X, F A is the curvature of the connection A, and s ∈ Γ(X, S + ⊗ E). As aproof, in local coordinates (i ≤ j)D 2 As = e i ∇ Ai (e j ∇ Aj s) = e i e j ∇ Ai ∇ Aj s = −∇ A2i s+e i e j (∇ Ai ∇ Aj −∇ Aj ∇ Ai )s,where ∇ A = ∇ ⊗ 1 + 1 ⊗ ˜∇ A . The first summand is ∇ ∗ A∇ A [Roe (1988)],and the second splits into a term which corresponds to the scalar curvatureon X [Roe (1988)] and the curvature -iF A of the connection A (note thathere we identify the Lie algebra of U(1) with iR).4.14.2.2 Spin and Spin c StructuresThe group Spin(n) is the universal covering of SO(n).The group Spin c (n) is defined via the following extension:1 → Z 2 → Spin c (n) → SO(n) × U(1) → 1, (4.211)i.e., Spin c (n) = (Spin(n) × U(1))/Z 2 .The extension (4.211) determines the exact sheaf–cohomology sequence:· · · → H 1 (X; Spin c (n)) → H 1 (X; SO(n)) ⊕ H 1 (X; U(1)) δ → H 2 (X; Z 2 ).(4.212)By the standard fact that H 1 (X; G) represents the equivalence classesof principal G−-bundles over X, we see that the connecting homomorphismof the sequence (4.212) is given byδ : (P SO(n) , P U(1) ) ↦→ w 2 (P SO(n) ) + ¯c 1 (P U(1) ),where ¯c 1 (P U(1) ) is the reduction mod 2 of the first Chern class of the linebundle associated to the principal bundle P U(1) by the standard representationand w 2 is the second Stiefel–Whitney class.A manifold X has a Spin c −structure if the frame bundle lifts to aprincipal Spin c (n) bundle. It has a Spin−structure if it lifts to a Spin(n)principal bundle.From the above considerations on the cohomology sequence (4.212), andanalogous considerations on the group Spin(n), it follows that a manifold Xadmits a Spin c −-structure iff w 2 (X) is the reduction mod 2 of an integralclass. It has a Spin−-structure iff w 2 (X) = 0. <strong>Diff</strong>erent Spin c −-structureson X are parametrized by 2H 2 (X; Z) ⊕ H 1 (X; Z 2 ). This follows directlyform (4.212): see [Libermann and Marle (1987)].


730 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWith this characterization of Spin c −structures we have the followingTheorem, which has been proved in [Hirzebruch and Hopf (1958)]: Everyoriented 4–manifold admits a Spin c −structure.4.14.2.3 Spinor BundlesBoth Spin(n) and Spin c (n) can be thought of as lying inside the Cliffordalgebra C(R n ), [Roe (1988); Libermann and Marle (1987)]. Therefore to aprincipal Spin(n) or Spin c (n) bundle we can associate a vector bundle viathe unique irreducible representation of the Clifford algebra. This will bethe bundle of Spinors over X associated to the Spin c or Spin structure, asdefined in the definition of spinors above.This can be given an explicit description in terms of transition functions(see [Libermann and Marle (1987)]). In fact let g αβ be the transitionfunctions of the frame bundle over X, which take values in SO(n). Thenlocally they can be lifted to functions ˜g αβ which take values in Spin(n),since on a differentiable manifold it is always possible to choose open setswith contractible intersections that trivialise the bundle.However, if we have a Spin c −-manifold that is not Spin, the ˜g αβ willnot form a cocycle, as ˜g αβ˜g βγ ˜g γα = 1 means exactly that the second Stiefel–Whitney class vanishes.Because of the Spin c structure we know that w 2 is the reduction of anintegral class c ∈ H 2 (X, Z), which represents a complex line bundle, saywith transition functions λ αβ with values in U(1). Such functions will havea square root λ 1/2αβlocally; however, the line bundle will not have a squareroot globally (which is to say that the λ 1/2αβwon’t form a cocycle), since byconstruction the first Chern class is not divisible by 2.However, the relation w 2 (X)+c = 0 mod 2 that comes from (4.212) saysthat the product ˜g αβ λ 1/2αβis a cocycle. These are the transition functionsof S ⊗ L, where S would be the Spinor bundle of a Spin structure and Lwould be the square root of a line bundle: neither of these objects is definedglobally, but the tensor product is. This is the description of the SpinorBundle of a Spin c structure that we shall use in the following.Also, note that symplectic geometry plays a prominent role in theSW gauge theory. Many computations are possible for the case of symplectic4–manifolds ([Taubes (1994)], [Taubes (1995a)], [Kotschik et. al.(1995)]); moreover, although the invariants are defined in terms of U(1)−connections,and sections of Spinor bundles, as we shall illustrate below,they turn out to be strictly related to invariants of symplectic man-


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 731ifolds, known as Gromov invariants (see [Gromov and Lawson (1980);Taubes (1995b)]). The fact that the SW invariants have a ‘more basic’structure also led to a conjecture, suggested by Taubes, that symplecticmanifolds may be among the most basic building blocks of the whole geometryof 4–manifolds.4.14.2.4 The Gauge Group and Its EquationsRecall that the gauge group of a G−-bundle is defined as the group of selfequivalences of the bundle, namely the group of smooth mapsλ α : U α → G, λ β = g βα λ α g αβ ,where the bundle is trivial over U α and has transition functions g αβ . It isoften useful to consider this space of maps endowed with Sobolev norms,and, by completing with respect to the norm, to consider gauge groups of L 2 kfunctions, as in [Donaldson and Kronheimer (1990); Freed and Uhlenbeck(1984)].The gauge group is an infinite dimensional manifold; if the structuregroup is Abelian then it has a simpler description as G = M(X, G), thespace of maps from X to G.In the case G = U(1), the set of connected components of the gaugegroup G is H 1 (X, Z).The equations of the gauge theory are given in terms of a pair (A, ψ) ofindeterminates, of which A is a connection on L and ψ is a smooth sectionof S + ⊗ L.The equations areD A ψ = 0, (F + A ) ij = 1 4 〈e ie j ψ, ψ〉 e i ∧ e j ,where D A is the Dirac operator twisted by the connection A, and F + Ais theself–dual part of the curvature associated to A. Here {e i } is a local basisof T X that acts on ψ by Clifford multiplication, {e i } is the dual basis ofT ∗ X, and is the inner product on the fibres of S + ⊗ L.The gauge group of L is well defined although L is not globally definedas a line bundle, since the definition of the gauge group is given just interms of the transition functions. In particular, as in the case of a linebundle, G = M(X, U(1)).There’s an action of the gauge group on the space of pairs (A, ψ), where


732 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionA is a connection on L and ψ a section of S + ⊗ L, given byλ : (A, ψ) ↦→ (A − 2iλ −1 dλ, iλψ). (4.213)The action defined in (4.213) induces an action of G on the space ofsolutions to the SW equations. As a proof, it is enough to check thatD A−2iλ −1 dλ(λψ) = iλD A ψ + idλ · ψ − 2idλ · ψ + idλ · ψ,and in the second equationF + A−2iλ −1 dλ = F + A − 2D+ (iλ −1 dλ) = F + A ,and 〈e i e j λψ, λψ〉 =| λ | 2 〈e i e j ψ, ψ〉 = 〈e i e j ψ, ψ〉.It is clear from (4.213) that the action of G on the space of solutions isfree iff ψ is not identically zero; while for ψ ≡ 0 the stabiliser of the actionis U(1), the group of constant gauge transformations.4.14.3 Original SW Low Energy Effective Field ActionNow, let us examine the original SW theory [Seiberg and Witten (1994a);Seiberg and Witten (1994b)] in more detail. Recall that the classical potentialof the pure N = 2 theory (without hypermultiplets) isV (φ) = 1 g 2 Tr[φ, φ† ] 2 . (4.214)For this to vanish, it is not necessary that φ should vanish; it is enoughthat φ and φ † commute. The classical theory therefore has a family ofvacuum states. For instance, if the gauge group is SU(2), then up to gaugetransformation we can take φ = 1 2 aσ3 , with σ 3 = diag(1, −1) and a acomplex parameter labelling the vacua. The Weyl group of SU(2) actsby a ↔ −a, so the gauge invariant quantity parameterizing the space ofvacua is u = 1 2 a2 = Tr φ 2 . For non–zero a the gauge symmetry is brokento U(1) and the global Z 8 symmetry is broken to Z 4 . The residual Z 4acts trivially on the u plane since the U(1) R charge of u is 4. The globalsymmetry group acts on the u plane as a spontaneously broken Z 2 , actingby u ↔ −u. Classically, there is a singularity at u = 0, where the fullSU(2) gauge symmetry is restored and more fields become massless.The next thing studied in [Seiberg and Witten (1994a); Seiberg andWitten (1994b)] was the low energy effective action of the light fields on themoduli space. For generic 〈φ〉 the low energy effective Lagrangian containsa single N = 2 vector multiplet, A. The terms with at most two derivatives


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 733and not more than four fermions are constrained by N = 2 supersymmetry.They are expressed in terms of a single holomorphic function F(A). InN = 1 superspace, the Lagrangian is[∫14π Imd 4 θ ∂F(A) ∫∂A Ā +d 2 θ 1 ∂ 2 ]F(A)2 ∂A 2 W α W α , (4.215)where A is the N = 1 chiral multiplet in the N = 2 vector multiplet Awhose scalar component is a. Here, Seiberg and Witten made the followingcomments:1. For large a, asymptotic freedom takes over and the theory is weaklycoupled. Moreover, since it is impossible to add an N = 2 invariant superpotentialto (4.215), the vacuum degeneracy cannot be removed quantummechanically. Therefore, the quantum theory has a non–trivial modulispace which is in fact a complex 1D Kähler manifold. The Kähler potentialcan be written in terms of the effective low energy F function as( ) ∂F(A)K = Im∂A Ā .The metric is thus concretely( ∂(ds) 2 2 F(a)= Im∂a 2)da dā . (4.216)In the classical theory, F can be read off from the tree level Lagrangianof the SU(2) gauge theory and is F(A) = 1 2 τ clA 2 with τ cl =θ2π + i 4πg. 2Asymptotic freedom means that this formula is valid for large a if g 2 isreplaced by a suitable effective coupling. The small a behavior will howeverturn out to be completely different. Classically, the θ parameter has noconsequences. Quantum mechanically, the physics is θ dependent, but sincethere is an anomalous symmetry, it can be absorbed in a redefinition of thefields. Therefore, we can set θ = 0.2. The formula for the Kähler potential does not look covariant – theKähler potential can be written in this way only in a distinguished classof coordinate systems. In fact, A is related by N = 2 supersymmetry tothe ‘photon’ A µ , which has a natural linear structure; this gives a naturalcoordinate system (or what will turn out to be a natural class of coordinatesystems) for A.3. The low energy values of the gauge coupling constant and thetaparameter can be read off from the Lagrangian. If we combine them in


734 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe form τ = θ2π + i 4πg, and denote the effective couplings in the vacuum2parametrized by a as τ(a), then τ(a) = ∂2 F∂a. 24. The generalization to an arbitrary compact gauge group G of rankr is as follows. The potential is always given by (4.214), so the classicalvacua are labelled by a complex adjoint–valued matrix φ with [φ, φ † ] = 0.The unbroken gauge symmetry at the generic point on the moduli space isthe Cartan subalgebra and therefore the complex dimension of the modulispace is r. The low energy theory is described in terms of r Abelian chiralmultiplets A i , and the generalization of (4.215) is[∫14π Imd 4 θ ∂F(A)∂A i Ā i +∫d 2 θ 1 ∂ 2 ]F(A)2 ∂A i ∂A j W αW i αj . (4.217)Here i labels the generators in the Cartan subalgebra and locally F is anarbitrary holomorphic function of r complex variables.5. The SU(2) theory, studied on the flat direction with u ≠ 0, has inaddition to the massless chiral or vector multiplet A, additional chargedmassive vector multiplets. One can easily write a gauge invariant effectiveaction for the triplet of chiral multiplets A a , a = 1 . . . 3, which reduces atlow energies to (4.215) for the massless fields and incorporates the massiveones. Using the same function F as above, we set F( √ A · A) = H(A · A)and write[∫12π Imd 4 θH ′ A a ( e V ) ∫ab Āb +(H ′ δ ab + 2H ′′ A a A b) ]WαW a αb ,d 2 θ 1 2(4.218)where the SU(2)−invariant metric δ ab has been used to raise and lowerindices. (4.218) has N = 2 supersymmetry and manifest gauge invariance,and reduces at low energies to (4.218).6. The Lagrangian (4.218) is unchanged if we add to F terms linear inA. This has the effect of shifting ∂F/∂A by a constant.As already mentioned, classically the F function isThe one–loop contributions add up toF 0 = 1 2 τ clA 2 . (4.219)F one−loop = i 12π A2 ln A2Λ 2 , (4.220)where Λ is the dynamically generated scale. This logarithm is related to theone–loop beta function and also ensures the anomalous transformation laws


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 735under U(1) R . Higher order perturbative corrections are absent. Instantonslead to new terms. The anomaly and the instanton action suggest thatF = i 12π A2 ln A2∞ Λ 2 + ∑( 4k ΛF k AA) 2 ,where the k’th term arises as a contribution of k instantons. A detailedcalculation of the k = 1 term indicates that F 1 ≠ 0. Also, corrections tothe classical formula 4.219 are related to the beta function, and for N =4 supersymmetric YM theory, whose beta function vanishes, the formula(4.219) is exact.k=14.14.4 QED With MatterFollowing the original SW approach [Seiberg and Witten (1994a); Seibergand Witten (1994b)], we first consider Abelian gauge theories with N = 2supersymmetry and charged matter hypermultiplets – that is, the N = 2analog of ordinary QED.The ‘photon’, A µ is accompanied by its N = 2 superpartners – twoneutral Weyl spinors λ and ψ that are often called ‘photinos’, and a complexneutral scalar a. They form an irreducible N = 2 representation that canbe decomposed as a sum of two N = 1 representations: a and ψ are ina chiral representation, A, while A µ and λ are in a vector representation,W α .We take the charged fields, the ‘electrons’, to consist of k hypermultipletsof electric charge one. Each hypermultiplet, for i = 1 . . . k, consists oftwo N = 1 chiral multiplets M i and ˜M i with opposite electric charge; suchan N = 1 chiral multiplet contains a Weyl fermion and a complex scalar.The renormalizable N = 2 invariant Lagrangian is described in an N =1 language by canonical kinetic terms and minimal gauge couplings for allthe fields as well as a superpotentialW = √ 2AM i ˜Mi + ∑ im i M i ˜Mi . (4.221)The first term in related by N = 2 supersymmetry to the gauge couplingand the second one leads to N = 2 invariant mass terms.The classical moduli space of the N = 2, SU(2) gauge theory isparametrized by u = 〈Tr( φ 2 )〉 where φ is a complex scalar field in theadjoint representation of the gauge group. For u ≠ 0 the gauge symmetryis broken to U(1). At u = 0 the space is singular and the gauge symmetry


736 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis unbroken. Our main goal is to determine – as quantitatively as possible– how this picture is modified quantum mechanically.The quantum moduli space is described by the global supersymmetryversion of special geometry. The Kähler potential, K =Im(a D (u)ā(ū)), determines the metric (or, equivalently the kinetic terms).The pair (a D , a) is a holomorphic section of an SL(2, Z) bundle over thepunctured complex u plane. They are related by N = 2 supersymmetry toa U(1) gauge multiplet. a is related by N = 2 to the semiclassical ‘photon’while a D is related to its dual – ‘the magnetic photon’. The gauge kineticenergy is proportional to∫d 2 θ ∂a D∂a W 2 α. (4.222)In this N = 2 theory, the one–loop approximation to K is exact (there areno higher order perturbative corrections and there are no U(1) instantonson R 4 ) leading toa D = − ik2π a log(a/Λ).The lack of asymptotic freedom appears here as a breakdown of the theoryat |a| = Λ/e, where the metric on the moduli space Im( ∂a D∂a) vanishes andthe effective gauge coupling is singular. This is the famous Landau pole.For large |u| the theory is semiclassical anda ∼ = √ 2u, a D∼2 = i a log a. (4.223)πThese expressions are modified by instanton corrections. The exact expressionswere determined as the periods on a torusy 2 = (x 2 − Λ 4 )(x − u) (4.224)of the meromorphic 1–form λ = √ 2 dx (x−u)2π y. In (4.224), Λ is the dynamicallygenerated mass scale of the theory.The spectrum contains dyons labelled by various magnetic and electriccharges. Stable states with magnetic and electric charges (n m , n e ) havemasses given by the BPS formulaM 2 = 2|Z| 2 = 2|n e a(u) + n m a D (u)| 2 . (4.225)There are two singular points on the quantum moduli space at u = ±Λ 2 ;they are points at which a magnetic monopole becomes massless. When an


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 737N = 2 breaking but N = 1 preserving mass term is added to the theory,these monopoles condense, leading to confinement (see below).When the masses in (4.221) are not zero, the moduli space changes.The singularities on the Coulomb branch can move. Whenever a = − √ 12m ione of the electrons becomes massless. Thereforea D = − i2π∑(a + m i / √ (2) logia + m i / √ 2ΛIf some of the masses are equal, the corresponding singularities on theCoulomb branch coincide and there are more massless particles there. Inthis case a Higgs branch with non–zero expectation values for these electronstouches the Coulomb branch at the singularity. When there is onlyone massless electron hypermultiplet, the |D| 2 term in the potential preventsa Higgs branch from developing.).4.14.5 QCD With MatterFollowing ([Seiberg and Witten (1994a); Seiberg and Witten (1994b)]), wenow turn to QCD with an SU(2) gauge group. The gluons are accompaniedby Dirac fermions and complex scalars φ in the adjoint representation of thegauge group. We also add N f hypermultiplets of quarks in the fundamentalrepresentation. As in the previous subsection, each hypermultiplet containsa Dirac fermion and four real scalars. In terms of N = 1 superfields thehypermultiplets contain two chiral superfields Q i a and ˜Q ia (i = 1, ..., N fis the flavor index and a = 1, 2 the color index) and the N = 2 gaugemultiplets include N = 1 gauge multiplets and chiral multiplets Φ. Thesuperpotential for these chiral superfields isW = √ 2 ˜Q i ΦQ i + ∑ im i ˜Qi Q i .We now begin the analysis of the quantum moduli space. The firstbasic fact is that for large fields, the theory is weakly coupled and thequantum moduli space is well approximated by the classical moduli space.We parameterize the Coulomb branch by the gauge invariant coordinateu = 〈Tr φ 2 〉.For N f = 0, the metric and the dyon masses are determined by aholomorphic section of an SL(2, Z) bundle:a = 1 2√2u + . . . ,aD = i 4 − N f ua(u) log2π Λ 2 + . . . ,N f


738 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere the ellipses represent instanton corrections and Λ Nf is the dynamicallygenerated scale of the theory with N f flavors. The metric isds 2 = Im(a ′ Dā′ )du dū and the dyon masses M 2 = 2|Z 2 | are expressed interms of Z = n e a + n m a D , where n m , n e are the magnetic and electriccharges, respectively.4.14.6 DualityNext, Seiberg and Witten performed SL(2, Z) duality transformation onthe low energy fields. Although they are non–local on the photon field A µ ,they act simply on (a D , a). Several new issues appeared when matter fieldswere present.First, consider the situation of one massive quark with mass m Nf andexamine what happens when a approaches m Nf / √ 2 where one of the elementaryquarks becomes massless. Loop diagrams in which this quarkpropagates make a logarithmic contribution to a D . The behavior neara = m Nf / √ 2 is thusa ≈ a 0 , a D ≈ c − i2π (a − a 0) ln(a − a 0 ),with a 0 = m Nf / √ 2 and c a constant. The monodromy 26 around a = a 0 is26 Recall that monodromy is the study of how geometrical objects behave as they ‘runaround’ a singularity. It is closely associated with covering maps and their degenerationinto ramification; the aspect giving rise to monodromy phenomena is that certain functionswe may wish to define fail to be single–valued as we ‘run around’ a path encircling asingularity. The failure of monodromy is best measured by defining a monodromy group:a group of transformations acting on the data that codes what does happen as we ‘runaround’.In the case of a covering map, we look at monodromy as a special case of a fibration,and use the homotopy lifting property to ‘follow’ paths on the base space X (we assumeit is path–connected, for simplicity) as they are lifted up into the cover C. If we followround a loop based at a point x ∈ X, which we lift to start at c above x, we end at somec ∗ again above x; it is quite possible that c ≠ c ∗ , and to code this, one considers theaction of the fundamental group π 1 (X, x) as a permutation group on the set of all c, asa monodromy group in this context.An analogous geometrical role is played by parallel transport. In a principal bundle Bover a smooth manifold M, a connection allows ‘horizontal’ movement from fibers abovea point m ∈ M to adjacent ones. The effect when applied to loops based at m is todefine a holonomy group of translations of the fiber at m; if the structure group of Bis G, it is a subgroup of G that measures the deviation of B from the product bundleM × G.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 739thusa → a,a D → a D + a − a 0 = a D + a − m N√2f. (4.226)Thus, under monodromy, the pair (a D , a) is not simply transformed bySL(2, Z); they also pick up additive constants. This possibility is not realizedfor the pure N = 2 gauge theory. The above simple consideration of amassless quark shows that this possibility does enter for N f > 0.If one arranges a D , a, and the bare mass m as a 3D column vector(m/ √ 2, a D , a), then the monodromy in (4.226) can be written in the generalform⎛ ⎞1 0 0M = ⎝r k l⎠ , (4.227)q n pwith det M = kp − nl = 1. This is the most general form permitted bythe low energy analysis. The specific form of the first row in (4.227) meansthat m is monodromy–invariant; intuitively this reflects the fact that m isa ‘constant’, not a ‘field’. Now, if one arranges the charges as a row vectorW = (S, n m , n e ), then W transforms by W → W M −1 . Explicitly,⎛⎞1 0 0M −1 = ⎝ lq − pr p −l⎠nr − kq −n kThus, the electric and magnetic charges n e and n m mix among themselvesbut do not get contributions proportional to the global symmetry charge S.On the other hand, the S charge can get contributions proportional to gaugecharges n e or n m . Equivalently, the global symmetry can be transformedto a linear combination of itself and a gauge symmetry but not the otherway around. Notice that the monodromy matrix mixing the charges in thisway survives even if the bare mass m vanishes.Now, it was noted above that locally, by virtue of N = 2 supersymmetry,the metric on the moduli space is of the form(ds) 2 = Im(τ(a) da dā), (4.228)with τ(a) the holomorphic function τ = ∂ 2 F/∂a 2 . The one–loop formula(4.220) shows that for large |a|, τ(a) ≈ i ( ln(a 2 /Λ 2 ) + 3 ) /π is a multivaluedfunction whose imaginary part is single–valued and positive. However, ifIm[τ(a)] is globally defined it cannot be positive definite as the harmonic


740 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfunction Im(τ) cannot have a minimum. This indicates that the abovedescription of the metric must be valid only locally.To what extent is it possible to change variables from a to some otherlocal parameter, while leaving the metric in the form (4.228)? The answerto this question is at the heart of the physics. We define a D = ∂F/∂a. Themetric can then be written(ds) 2 = Im(da D dā) = − i 2 (da Ddā − da dā D ) . (4.229)This formula is completely symmetric in a and a D , so if we use a D as thelocal parameter, the metric will be in the same general form as (4.228),with a different harmonic function replacing Im(τ). This transformationcorresponds to electric–magnetic duality.4.14.6.1 Witten’s FormalismNow, following [Seiberg and Witten (1994a); Seiberg and Witten (1994b)],to be able to treat the formalism in a way that is completely symmetricbetween a and a D , we introduce an arbitrary local holomorphic coordinateu, and treat a and a D as functions of u, which is a local coordinate on acomplex manifold M – the moduli space of vacua of the theory. Eventuallywe pick u to be the expectation value of Tr( φ 2 ) – a good physical parameter– but for now u is arbitrary.Introduce a 2D complex space X ∼ = C 2 with coordinates (a D , a). EndowX with the symplectic form ω = Im(da D ∧dā). The functions (a D (u), a(u))give a map f from M to X. The metric on M is(ds) 2 = Im( daDdudādū)dudū = − i 2( daDdudādū − dadu)dā Ddudū. (4.230)dūThis formula is valid for an arbitrary local parameter u on M. If one picksu = a, one gets back the original formula (4.216) for the metric. 27 Noticethat ω had no particular positivity property and thus, if a(u) and a D (u)are completely arbitrary local holomorphic functions, the metric (4.230) isnot positive. We will eventually construct a(u) and a D (u) in a particularway that will ensure positivity.It is easy to see what sort–of transformations preserve the general structureof the metric. If we set a α = (a D , a), α = 1, 2, and let ɛ αβ be the27 This formula can be described in a coordinate–free way by saying that the Kählerform of the induced metric on M is f ∗ (ω).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 741antisymmetric tensor with ɛ 12 = 1, then(ds) 2 = − i 2 ɛ da α dā βαβ dudū. (4.231)du dūThis is manifestly invariant under linear transformations that preserve ɛ andcommute with complex conjugation (the latter condition ensures that a αand ā α transform the same way). These transformations make the groupSL(2, R) (or equivalently Sp(2, R)). Also, (4.231) is obviously invariantunder adding a constant to a D or a. So if we arrange (a D , a) as a columnvector v, the symmetries that preserve the general structure are: v →Mv + c, where M is a 2 × 2 matrix in SL(2, R), and c is a constant vector.In general, this group of transformations „ « can be thought of as the group of1 03 × 3 matrices of the form , acting on the three objects (1, ac MD , a).Generalization to Dimension Greater than OneNow, let us briefly discuss the generalization to other gauge groups. If thegauge group G has rank r, then M has complex dimension r. Locally, from(4.217), it follows that the metric is( ∂(ds) 2 2 )F= Im∂a i ∂a j da i dā j ,with distinguished local coordinates a i and a holomorphic function F. Weagain reformulate this by introducingThen we can writea D,j = ∂F∂a j . (4.232)(ds) 2 = Im ∑ ida D,i dā i .To formulate this invariantly, we introduce a complex space X ∼ = C 2r withcoordinates a i , a D,j . We endow X with the symplectic formω = i 2∑ (da i ∧ dā D,i − da D,i ∧ dā i)of type (1, 1) and also with the holomorphic 2–formiω h = ∑ ida i ∧ da D,i .


742 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThen we introduce arbitrary local coordinates u s , (s = 1, . . . , r), on themoduli space M, and describe a map f : M → X by functions a i (u),a D,j (u). We require f to be such that f ∗ (ω h ) = 0; this precisely ensuresthat locally, if we pick u i = a i , then a D,j must be of the form in (4.232)with some holomorphic function F. Then we take the metric on M to bethe one whose Kähler form is f ∗ (ω), i.e.,(ds) 2 = Im ∑ s,t,i∂a D,i∂u s∂ā i∂ū t dus dū t .If again we arrange a, a D as a 2r−component column vector v, then theformalism is invariant under transformations v → Mv + c, with M amatrix in Sp(2r, R) and c a constant vector. Again, considerations involvingthe charges will eventually require that M be in Sp(2r, Z) and imposerestrictions on c.Physical Interpretation via DualitySo far we have seen that the spin zero component of the N = 2 multiplethas a Kähler metric of a very special sort, constructed using a distinguishedset of coordinate systems. This rigid structure is related by N = 2 supersymmetryto the natural linear structure of the gauge field. We have foundthat, for the spin zero component, the distinguished parametrization is notcompletely unique; there is a natural family of parameterizations relatedby SL(2, R). How does this SL(2, R) (which will actually be reduced toSL(2, Z)) act on the gauge fields?SL(2, R) is generated by the transformationsT b =( 1 b0 1), and S =( 0 1−1 0with real b. The former acts as a D → a D + ba, a → a; this acts triviallyon the distinguished coordinate a, and can be taken to act trivially on thegauge field. By inspection of (4.218), the effect of a D → a D + ba on thegauge kinetic energy is just to shift the θ angle by 2πb; in the Abeliantheory, this has no effect until magnetic monopoles (or at least non–trivialU(1) bundles) are considered. Once that is done, the allowed shifts in theθ angle are by integer multiples of 2π; that is why b must be integral andgives essentially our first derivation of the reduction to SL(2, Z).The remaining challenge is to understand what S means in terms of thegauge fields. We will see that it corresponds to electric-magnetic duality. To),


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 743see this, let us see how duality works in Lagrangians of the sort introducedabove.We work in Minkowski space and consider first the purely bosonicterms involving only the gauge fields. We use conventions such thatF 2 µν = −( ∗ F) 2 µν and ∗ ( ∗ F) = −F where ∗ F denotes the dual of F. Therelevant terms are∫132π Imτ(a) · (F + i ∗ F) 2 = 116π Im ∫τ(a) · (F 2 + i ∗ FF). (4.233)Duality is carried out as follows. The constraint dF = 0 (which in theoriginal description follows from F = dA) is implemented by adding aLagrange multiplier vector field V D . Then F is treated as an independentfield and integrated over. The normalization is set as follows. The U(1) ⊂SU(2) is normalized such that all SO(3) fields have integer charges (mattermultiplets in the fundamental representation of SU(2) therefore have halfinteger charges). Then, a magnetic monopole corresponds to ɛ 0µνρ ∂ µ F νρ =8πδ (3) (x). For V D to couple to it with charge one, we add to (4.233)∫1V Dµ ɛ µνρσ ∂ ν F ρσ = 1 ∫∗ F D F = 1 ∫8π8π16π Re ( ∗ F D − iF D )(F + i ∗ F),where F Dµν = ∂ µ V Dν − ∂ ν V Dµ is the the field strength of V D . We can nowperform the Gaussian functional integral over F and find an equivalentLagrangian for V D ,(1 −132π Im τ)(F D + i ∗ F D ) 2 = 1 ( −116π Im τ)(F 2 D + i ∗ F D F D ).We now repeat these steps in N = 1 superspace. We treat W α in∫18π Im d 2 θτ(A)W 2as an independent chiral field. The superspace version of the Bianchi identitydF = 0 isIm(DW ) = 0,where D is the supercovariant derivative. It can be implemented by a realvector superfield V D Lagrange multiplier. We add to the action∫14π Imd 4 xd 4 θV D DW = 14π Re ∫d 4 xd 4 θiDV D W = − 14π Im ∫d 4 xd 2 θW D W.


744 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionPerforming the Gaussian integral over W we find an equivalent Lagrangian∫18π Im d 2 θ −1τ(A) W D. 2 (4.234)To proceed further, we need to transform the N = 1 chiral multiplet Ato A D . The kinetic term∫Im d 4 θh(A)Ā is transformed byA D = h(A) to Im∫d 4 θh D (A D )ĀD,where h D (h(A)) = −A is minus the inverse function. Then using h ′ (A) =τ(A) the coefficient of the gauge kinetic term (4.234) becomes− 1τ(A) = − 1h ′ (A) = h′ D(A D ) = τ D (A D ).Note that a shift of h by a constant does not affect the Lagrangian. Therefore,the duality transformation has a freedom to shift A D by a constant.The relations A D = h(A) and h D = −A mean that the duality transformationprecisely implements the missing SL(2, Z) generator S. Thefunction τ = h ′ is mapped by τ D (A D ) = − 1 Remembering thatτ(a) = θ(a)2πτ(A) .+ i 4πg(a) 2 , we see that the duality transformation inverts τ ratherthan the low energy gauge coupling g(a).It is important to stress that unlike τ → τ +1, the duality transformationis not a symmetry of the theory. It maps one description of the theory toanother description of the same theory.For other gauge groups G the low energy Lagrangian has several Abelianfields, A i , in the Cartan subalgebra.Then (A D ) i = h i (A i ) = ∂ i F(A i ),which leads to h i D(h j (A k )) = −A i , and the ‘metrics’τ ij (A) = ∂ i ∂ j F(A) = ∂ j h i (A), τ ij D (A D) = ∂ i ∂ j F D (A D ) = ∂ j h i D(A D )satisfyf ij f jkD = −δk i .The above transformation together with the more obvious shifts: A Di →A Di + M ij A j – generate Sp(2r, Z).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 745Coupling to GravityNow, we would like to compare the structure we have found to the ‘specialgeometry’ that appears if the chiral multiplet is coupled to N = 2supergravity. In N = 2 supergravity, the general Kähler metric for a systemof r chiral superfields is described locally by a holomorphic functionG 0 (a 1 , . . . , a r ) of r complex variables a i . The Kähler potential is(K grav = − ln2i(G 0 − Ḡ0) + i 2∑(iā i ∂G ) )0 ∂Ḡ0− ai∂ai ∂ā i . (4.235)In global supersymmetry we had a local holomorphic function F withK = −i ∑(ā i ∂F )∂ ¯F− ai2 ∂ai ∂ā i . (4.236)iOne would expect that there is some limit in which gravitational effects aresmall and (4.235) would reduce to (4.236). How does this occur?It suffices to setG 0 = −i M P l 2+ F, (4.237)4with M P l the Planck mass. Then if M P l is much larger than all relevantparameters, we getK grav = − ln M P l 2 +KM P l2 + O(M P l −4 ).The constant term -ln M 2 P l does not contribute to the Kähler metric, soup to a normalization factor of 1/MP 2 l, the Kähler metric with supergravityreduces to that of global N = 2 supersymmetry as M P l → ∞ keepingeverything else fixed.More fundamentally, we would like to compare the allowed monodromygroups. In supergravity, the global structure is exhibited as follows. Oneintroduces an additional variable a 0 and sets G = (a 0 ) 2 G 0 . One also introducesa D,j = ∂G/∂a j for j = 0, . . . , r. Then one finds that the specialKähler structure of (4.235) allows Sp(2r + 2, R) transformations acting on(a Di , a j ). 28 Now, in decoupling gravity, we consider G to be of the specialform in (4.237). In that case, a D,0 = −i M P l2 . The other ai , a D,j are independentof M P l . To preserve this situation in which M P l appears only in28 In the gauge fields this is reduced to Sp(2r + 2, Z). The symplectic form preservedby Sp(2r + 2, R) becomes the usual one P i dai ∧ da D,i .


746 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona D,0 , we must consider only those Sp(2r + 2, R) transformations in whichthe transformations of all fields are independent of a D,0 . These transformationsall leave a 0 invariant. There is no essential loss then in scaling the a’sso that a 0 = 1. Arrange the a D,i , a j with i, j = 1 . . . r as a column vectorv. The Sp(2r + 2, R) transformations that leave invariant a 0 = 1 act on vby v → Mv + c where M ∈ Sp(2r, R) and c is a constant. This is preciselythe duality group that we found in the global N = 2 theory.4.14.7 Structure of the Moduli Space4.14.7.1 Singularity at InfinityIt is actually quite easy to see explicitly the appearance of non–trivialmonodromies. In fact, asymptotic freedom implies a non–trivial monodromyat infinity. The renormalization group corrected classical formulaF one loop = iA 2 ln(A 2 /Λ 2 )/2π gives for large aa D = ∂F∂a ≈ 2iaπ ln(a/Λ) + ia π . (4.238)It follows that a D is not a single-valued function of a for large a. If werecall that the physical parameter is really u = 1 2 a2 (at least for large u anda), then the monodromy can be determined as follows. Under a circuit ofthe u plane at large u, one has ln u → ln u + 2πi, and hence ln a → ln a + πi.So the transformation is a D → −a D + 2a, a → −a. Thus, there is anon–trivial monodromy at infinity in the u plane,(( )) −1 2M ∞ = P T −2 =, (4.239)0 −1„ « 1 1where P is the element -1 of SL(2, Z), and as usual T = .0 1The factor of P in the monodromy exists already at the classical level.As we said above, a and -a are related by a gauge transformation (theWeyl subgroup of the SU(2) gauge group) and therefore we work on theu plane rather than its double cover, the a plane. In the anomaly freeZ 8 subgroup of the R symmetry group U(1) R , there is an operation thatacts on a by a → −a; when combined with a Weyl transformation, this isthe unbroken symmetry that we call P . Up to a gauge transformation itacts on the bosons by φ → −φ, so it reverses the sign of the low energyelectromagnetic field which in terms of SU(2) variables is proportional toTr (φF). Hence it reverses the signs of all electric and magnetic charges and


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 747acts as -1 ∈ SL(2, Z). The P monodromy could be removed by (perhapsartificially) working on the a plane instead of the u plane.The main new point here is the factor of T −2 which arises at the quantumlevel. This factor of T −2 has a simple physical explanation in terms ofthe electric charge of a magnetic monopole. Magnetic monopoles labelledby (n m , n e ) have anomalous electric charge n e + θ eff2π n m. The appropriateeffective theta parameter is the low energy oneθ eff = 2π Re[τ(a)] = 2π Re( daDda)= 2π Re( ) daD /du.da/duFor large |a|, we have θ eff ≈ −4 arg(a), which can be understood from theanomaly in the U(1) R symmetry. The monodromy at infinity transformsthe row vector (n m , n e ) to (−n m , −n e − 2n m ), which implies that (a D , a)transforms to (−a D +2a, −a). The electric charge of the magnetic monopolecan in fact be seen in the formula for Z, which if we take a D from (4.238)and set a = a 0 e −iθ eff /4 (with a 0 > 0) isZ ≈ a 0 e −iθ eff /4{(n e + θ eff n m2π)+ in m( 2 ln a0 /Λ + 1π)}.The monodromy under θ eff → θ eff + 4π is easily seen from this formula totransform (n m , n e ) in the expected fashion. Obviously, this simple formuladepended on the semiclassical expression (4.238) for a D ; with the exactexpressions we presently propose, the results are much more complicated,in part because the effective theta angle is no longer simply the argumentof a.4.14.7.2 Singularities at Strong CouplingThe monodromy at infinity means that there must be an additional singularitysomewhere in the u−plane. If M ′ is the moduli space of vacua withall singularities deleted, then the monodromies must give a representationof the fundamental group of M ′ in SL(2, Z). Can this representation beAbelian? If the monodromies all commute with P T −2 , then a 2 is a goodglobal complex coordinate, and the metric is globally of the form (4.228)with a global harmonic function Im τ(a). As we have already noted, sucha metric could not be positive.The alternative is to assume a non–Abelian representation of the fundamentalgroup. This requires at least two more punctures of the u plane(in addition to infinity). Since there is a symmetry u ↔ −u acting on


748 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe u plane, the minimal assumption is that there are precisely two morepunctures exchanged by this symmetry.The most natural physical interpretation of singularities in the u plane isthat some additional massless particles are appearing at a particular valueof u. For instance, in the classical theory, at u = a = 0, the SU(2) gaugesymmetry is restored; all the gluons become massless. In fact classicallya D = 4πia/g 2 also vanishes at this point, and the monopoles and dyonsbecome massless as well. One might be tempted to believe that the missingsingularity comes from an analogous point in the quantum theory at whichthe gauge boson masses vanish. Though this behavior might seem unusualin asymptotically free theories in general, there are good indications thatsome N = 1 theories have an infrared fixed point with massless non–Abeliangluons.4.14.7.3 Effects of a Massless MonopoleNow, following [Seiberg and Witten (1994a); Seiberg and Witten (1994b)],let us analyze the behavior of the effective Lagrangian near a point u 0 on themoduli space where magnetic monopoles become massless, that is, wherea D (u 0 ) = 0.Since monopoles couple in a non-local way to the original photon, we cannotuse that photon in our effective Lagrangian. Instead, we should perform aduality transformation and write the effective Lagrangian in terms of thedual vector multiplet A D . The low energy theory is therefore an Abeliangauge theory with matter (an N = 2 version of QED). The unusual factthat the light matter fields are magnetically charged rather than electricallycharged does not make any difference to the low energy physics. The onlyreason we call these particles monopoles rather than electrons is that thislanguage is appropriate at large |u| where the theory is semiclassical.The dominant effect on the low energy gauge coupling constant is dueto loops of light fields. In our case, these are the light monopoles. The lowenergy theory is not asymptotically free and therefore its gauge couplingconstant becomes smaller as the mass of the monopoles becomes smaller.Since the mass is proportional to a D , the low energy coupling goes to zero asu → u 0 . The electric coupling constant which is the inverse of the magneticone diverges at that point.More quantitatively, using the one–loop beta function, near the pointwhere a D = 0, the magnetic coupling is τ D ≈ − i π ln a D. Since a D is a


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 749good coordinate near that point, a D ≈ c 0 (u − u 0 ), with some constant c 0 .Using τ D = dh D /da D , we learn thata(u) = −h D (u) ≈ a 0 + i π a D ln a D ≈ a 0 + i π c 0(u − u 0 ) ln(u − u 0 ),for some constant a 0 = a(u = u 0 ). This constant a 0 cannot be zero becauseif it had been zero, all the electrically charged particles would have beenmassless at u = u 0 and the computation using light monopoles only wouldnot be valid.Now we can read off the monodromy. When u circles around u 0 ,soln(u − u 0 ) → ln(u − u 0 ) + 2πi,then one has a D → a D , a → a − 2a D . (4.240)This effect is a sort–of dual of the monodromy at infinity. Near infinity,the monopole gains electric charge, and near u = u 0 , the electron gainsmagnetic charge. (4.240) can be represented by the 2 × 2 monodromymatrix( ) 1 0M 1 = ST 2 S −1 = . (4.241)−2 14.14.7.4 The Third SingularityWith our assumption that there are only three singularities (counting u =∞) and with two of the three monodromies determined in (4.239) and(4.241), we can now determine the third monodromy, which we call M −1 .With all of the monodromies taken in the counter clockwise direction, themonodromies must obey M 1 M −1 = M ∞ , and from this we get( ) −1 2M −1 = (T S)T 2 (T S) −1 = .−2 3The matrix M −1 is conjugate to M 1 . Actually,( ) −1 1If A = T M 1 = , (4.242)−2 1Then M −1 = AM 1 A −1 . (4.243)Hence, M −1 can arise from a massless particle, just like M 1 . (4.243) wouldalso hold if Ais replaced by AM r 1 for any integer r.What kind of particle should become massless to generate this singularity?If one arranges the charges as a row vector q = (n m , n e ), then


750 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe massless particle that produces a monodromy M has qM = q. Forinstance, monodromy M 1 arises from a massless monopole of charge vectorq 1 = (1, 0), and using the known form of M 1 , one has q 1 M 1 = q 1 . Dualitysymmetry implies that this must be so not just for the particular monodromyM 1 but for any monodromy coming from a massless particle. Uponsetting q −1 = (1, −1), we get q −1 M −1 = q −1 , and hence the monodromyM −1 arises from vanishing mass of a dyon of charges (1, −1).It seems that we are seeing massless particles of charges (1, 0) or (1, −1).However, there is in fact a complete democracy among dyons. The BPSsaturateddyons that exist semiclassically have charges (1, n) (or (−1, −n))for arbitrary integer n. The monodromy at infinity brings about a shift(1, n) → (1, n − 2). If one carries out this shift ntimes before proceeding tothe singularity at u = 1 or u = −1, the massless particles producing thosesingularities would have charges (1, −2n) and (1, −1 − 2n), respectively.This amounts to conjugating the representation of the fundamental groupby M n ∞ .The particular matrix A in (4.242) obeys A 2 = −1, which is equivalentto the identity as an automorphism of SL(2, Z). Conjugation byAimplements the underlying Z 2 symmetry of the quantum moduli space,which according to our assumptions, exchanges the two singularities. TheZ 2 maps M 1 → M 1 ′ = M −1 , M −1 → M −1 ′ = M 1 and M ∞ → M ∞ ′ =M 1M ′ −1 ′ = M −1 M 1 . Note that M ∞ ′ is not just obtained from M ∞ by conjugation,but the relation M ∞ = M 1 M −1 is preserved. The reason for thatis that (as in any situation in which one is considering a representation ofthe fundamental group of a manifold in a non–Abelian group), the definitionof the monodromies requires a choice of base point. The operationu → −u acts on the base point, and this has to be taken into account indetermining how M ∞ transforms under Z 2 .One can go farther and show that if one assumes the existence of a Z 2symmetry between M 1 and M −1 , then they must be conjugate to T 2 , andnot some other power of T . In our derivation of the monodromy (4.241),the 2 came from something entirely independent of the assumption of a Z 2symmetry, namely, from the charges and multiplicities of the monopolesthat exist semiclassically.4.14.7.5 Monopole Condensation and ConfinementWe recall that the underlying N = 2 chiral multiplet A decomposes underN = 1 supersymmetry as a vector multiplet W α and a chiral multiplet


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 751Φ. Breaking N = 2 down to N = 1, one can add a superpotential W =mTr(Φ 2 ) for the chiral multiplet. This gives a bare mass to Φ, reducing thetheory at low energies to a pure N = 1 gauge theory. The low energy theoryhas a Z 4 chiral symmetry. This theory is strongly believed to generate amass gap, with confinement of charge and spontaneous breaking of Z 4 toZ 2 . Furthermore, there is no vacuum degeneracy except what is producedby this symmetry breaking, so that there are precisely two vacuum states.How can this be mimicked in the low energy effective N = 2 theory?That theory has a moduli space M of quantum vacua. The massless spectrumat least semiclassically consists solely of the Abelian chiral multipletA of the unbroken U(1) subgroup of SU(2). If those are indeed the onlymassless particles, the effect in the low energy theory of turning on m canbe analyzed as follows. The operator Tr( Φ 2 ) is represented in the low energytheory by a chiral superfield U. Its first component is the scalar fieldu whose expectation value is〈u〉 = 〈Tr( φ 2 )〉,where φ is the θ = 0 component of the superfield Φ. This is a holomorphicfunction on the moduli space. At least for small m we should add to ourlow energy Lagrangian an effective superpotential W eff = mU.Turning on the superpotential mU would perhaps eliminate almost all ofthe vacua and in the surviving vacua give a mass to the scalar componentsof A. But if there are no extra degrees of freedom in the discussion, thegauge field in Awould remain massless. To get a mass for the gauge field,as is needed since the microscopic theory has a mass gap for m ≠ 0, oneneeds either(i) extra light gauge fields, giving a non–Abelian gauge theory and possiblestrong coupling effects, or(ii) light charged fields, making possible a Higgs mechanism.Thus we learn, as we did in discussing the monodromies, that somewhereon Mextra massless states must appear. The option (i) does not seemattractive, for reasons that we have already discussed. Instead we consideroption (ii), with the further proviso, from our earlier discussion, that thelight charged fields in question are monopoles and dyons.Near the point at which there are massless monopoles, the monopolescan be represented in an N = 1language by ordinary (local) chiral superfieldsMand ˜M, as long as we describe the gauge field by the dual to the


752 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionoriginal photon, A D . The superpotential isŴ = √ 2A D M ˜M + mU(A D ), (4.244)where the first term is required by N = 2 invariance of the m = 0 theory,and the second term is the effective contribution to the superpotentialinduced by the microscopic perturbation m Tr(Φ 2 ).The presence of a term mTr( Φ 2 ) in the microscopic superpotentialshows that the parameter m carries charge two. The low energy superpotentialis holomorphic in its variables Ŵ (m, M ˜M, A D ) and should havecharge two under U(1) J . Imposing that it is regular at m = M ˜M = 0, wefind that it is of the form Ŵ = mf 1(A D ) + M ˜Mf 2 (A D ). The functions f 1and f 2 are independent of m and can be determined by examining the limitof small m, leading to (4.244).The low energy vacuum structure is easy to analyze. Vacuum statescorrespond to solutions of dŴ = 0 (up to gauge transformation), whichobey the additional condition |M| = |˜M| (we denote by M and ˜M boththe superfields and their first components). The latter condition comesfrom vanishing of the D terms. Implementing these conditions, one finds ifm = 0 that vacuum states correspond to M = ˜M = 0 with arbitrary a D ;this is simply the familiar moduli space M. If m ≠ 0 the result is quitedifferent. We get√2M ˜M + mduda D= 0, a D M = a D ˜M = 0.Assuming that du ≠ 0, the first equation requires M, ˜M ≠ 0, while thesecond equation requires a D = 0.Expanding around this vacuum, it is easy to see that there is a massgap. For instance, the gauge field gets a mass by the Higgs mechanism,since M, ˜M ≠ 0. The Higgs mechanism in question is a magnetic Higgsmechanism, since the fields with expectation values are monopoles! Condensationof monopoles will induce confinement of electric charge. Thus,we get an explanation in terms of the low energy effective action of why themicroscopic theory becomes confining when the mTr( Φ 2 ) superpotential isadded.4.14.8 Masses and PeriodsThe particle masses and the low energy metric and couplings were in[Seiberg and Witten (1994a); Seiberg and Witten (1994b)] determined by


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 753equating a and a D with periods of a certain meromorphic 1–form λ on thecurve E. λ has two characteristics:(i) λ may have poles but (as long as the monodromies are in SL(2, Z))its residues vanish; and(ii) To achieve positivity of the metric on the quantum moduli space,its derivative with respect to u is proportional to dx y .Condition (i) means that the definition of a and a D by contour integrals∮∮a = λ, a D = λ,γ 1 γ 2(with γ 1 and γ 2 some contours on E) is invariant under deformation of theγ i , even across poles of λ. This ensures that only the homology classes ofthe γ i matter and reduces the monodromies to a group SL(2, Z) that actson H 1 (E, Z). In the presence of bare masses, this is too strong a conditionsince when the bare masses are non–zero the monodromies are not quite inSL(2, Z).As for condition (ii), the differential form dx yhas no poles and representsa cohomology class on E of type (1, 0). Having dλ/du = f(u) dx/y leadsto positivity of the metric. The function f(u) is determined by requiringthe right behavior at the singularities, for instance a ≈ 2√ 1 2u for large u,while f is a constant. The proper relation is in fact√dλ 2du = 8πdxy . (4.245)Up to an inessential sign, this is 1/2 the value in the ‘old’ conventions.By integration with respect to u, (4.245) determines λ (once the curve isknown) for all values of N f . This relation is only supposed to hold up to atotal differential in x; λ is supposed to be meromorphic in x.The massless N f = 3 curve is given byy 2 = x 2 (x − u) − (x − u) 2 .The polynomial on the right hand side has zeroes at x 0 = u and atx ± = 1 2(1 ±√1 − 4u).In particular, at u = 1/4, x + = x − , giving the singularity that we haveattributed to a massless state of (n m , n e ) = (2, 1). To show that this stateis semiclassical, we interpolate on the positive u axis from the semiclassical


754 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionregime of u −→ ∞ to the singularity at u = 1/4. For u > 1/4, x + and x −are complex conjugates.We have∮dadu =γ 1∮dλdu =γ 1ω,da Ddu = ∮γ 2∮dλdu =γ 2ω,where ω = ( √ 2/8π)dx/y, γ 1 is a circle in the x plane that loops around x +and x − but not x 0 , and γ 2 is a contour that loops round x 0 and x + butnot x − . Complex conjugation leaves x 0 alone and exchanges x + with x − ;hence γ 1 is invariant under complex conjugation, but complex conjugationturns γ 2 into a contour γ 3 that loops around x 0 and x − while avoiding x + .So a is real but the complex conjugate of a D is given by∫da Ddu = ω. (4.246)γ 3γ 3 , however, is homotopic to the sum of -γ 1 and −γ 2 (the minus sign comesfrom keeping track of the orientations of the contours). Hence, (4.246) givesIn other words, 29a D = −a − a D .a D = − a 2 + imaginary.4.14.9 ResiduesSince the jumps in a or a D are integral linear combinations of m i / √ 2 (withm i the bare masses) and are 2πi times the residues of λ, the residues of λshould be of the formRes λ = ∑ in i m i2πi √ 2 , with n i ∈ Z.The N f = 4 theory is controlled by a curve y 2 = F(x, u, m i , τ) and a29 This equation has the following interpretation. The curve is real for real u, thatis, the coefficients in the equation are real. There are two types of real elliptic curve:τ can have real part zero or 1/2. (Thus τ is either invariant or transformed by theSL(2, Z) transformation τ → τ − 1 under the complex conjugation operation τ → −¯τ.)The two possibilities correspond in a suitable basis to a real, a D imaginary, or a real,a D = −a/2 + imaginary. For u > 1/4 we have the second possibility.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 755differential form λ obeyingdλdu= ω + exact form in x, (4.247)√2 dx8π y .with ω =For N = 4 the structure is the same, except that 8π is replaced by 4π. Fshould be such that the residues of λ are linear in the quark bare masses.This is a severe restriction on F; we see that it determines F uniquely (upto the usual changes of variables) independently of most of the argumentsthat we have used up to this point.Let us write (4.247) in a more symmetrical form. If λ = dx a(x, u), then(4.247) means√2 dx ∂a= dx8π y ∂u + dx ∂ f(x, u); (4.248)∂xthe arbitrary total x−derivative dx ∂f/∂x is allowed here because it doesnot contribute to the periods. (4.248) can be understood much better ifwritten symmetrically in x and u. Henceforth, instead of using a 1–formω = ( √ 2/8π) · dx/y, we use a 2–formω =√28πdx du.ySimilarly, we combine the functions a, f appearing in (4.248) into a 1–formλ = −a(x, u)dx + f(x, u)du. The change in notation for ω and λ shouldcause no confusion. Then equation (4.248) can be more elegantly writtenasω = dλ. (4.249)The meaning of the problem of finding λ can now be stated. Let X bethe (noncompact) complex surface defined by the equation y 2 = F(x, u)(we suppress the parameters m i and τ). Being closed, ω defines an element[ω] ∈ H 2 (X, C). A smooth differential λ obeying (4.249) exists if and onlyif [ω] = 0. Moreover, by standard theorems, in the absence of restrictionson the growth of λ at infinity, if λ exists it can be chosen to be holomorphicand of type (1, 0).If on the other hand [ω] ≠ 0, then (4.249) has no smooth, much lessholomorphic, solution. However, X has the property that if one throwsaway a sufficient number of complex curves C a , then X ′ = X − ∪ a C a


756 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionhas H 2 (X ′ , C) = 0. (The necessary C a are explicitly described later.) Soif we restrict to X ′ , the cohomology class of ω vanishes and λ exists. λmay however have poles on the C a , perhaps with residues, which we callRes Ca (λ). 30 If λ does have residues, then dλ contains delta functions, andif one works on X instead of X ′ , one really has not (4.249) butω = dλ − 2πi ∑ aRes Ca (λ) · [C a ] (4.250)where [C a ] (which represents the cohomology class known as the Poincarédual of C a ) is a delta function supported on C a .In cohomology, (4.250) simply means[ω] = −2πi ∑ aRes Ca (λ) · [C a ]. (4.251)Thus, if we pick the C a so that the [C a ] are a basis of H 2 (X, C), thenthe residues Res Ca (λ) are simply the coefficients of the expansion of [ω]in terms of the [C a ]. To find the residues we need not actually find λ; itsuffices to understand the cohomology class of ω by any method that maybe available.For instance, if X were compact, we could proceed as follows. Firstcompute the intersection matrixM ab = #(C a · C b )(that is, the number of intersection points of C a and C b , after perhapsperturbing the C a so that they intersect generically). This is an invertiblematrix. Second, calculate the periods∫c a = ω.C aThen[ω] = ∑ a,bc a M −1ab [C b].Comparing to (4.246), we get0.Res Ca (λ) = − 12πi∑bM −1ab c b.30 The residues of λ along C a are constants, since dRes Ca (λ) = Res Ca dλ = Res Ca ω =


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 7574.14.10 SW Monopole Equations and Donaldson TheoryDevelopments in the understanding of N = 2 supersymmetric YM theory in4D suggest a new point of view about Donaldson theory [Donaldson (1990);Donaldson (1986); Donaldson (1987); Donaldson and Kronheimer (1990)] offour manifolds: instead of defining 4–manifold invariants by counting SU(2)instantons, one can define equivalent 4–manifold invariants by countingsolutions of a nonlinear equation with an Abelian gauge group. This is a‘dual’ equation in which the gauge group is the dual of the maximal torusof SU(2). This new viewpoint, proposed by Witten in [Witten (1994)],suggests many new results about the Donaldson invariants.Let X be an oriented, closed 4–manifold on which we pick a Riemannianstructure with metric tensor g. Λ p T ∗ X, or simply Λ p , will denote thebundle of real–valued p−forms, and Λ 2,± will be the sub–bundle of Λ 2consisting of self–dual or anti–self–dual forms.The monopole equations relevant to SU(2) or SO(3) Donaldson theorycan be described as follows. If w 2 (X) = 0, then X is a spin manifold andone can pick positive and negative spin bundles S + and S − , of rank two. 31In that case, introduce a complex line bundle L; the data in the monopoleequation will be a connection A on L and a section M of S + ⊗ L. Thecurvature 2–form of A will be called F or F(A); its self–dual and anti–self–dual projections will be called F + and F − .If X is not spin, the S ± do not exist, but their projectivizations P S ±do exist (as bundles with fibers isomorphic to CP 1 ). A Spin c structure(which exists on any oriented four-manifold can be described as a choiceof a rank two complex vector bundle, which we write as S + ⊗ L, whoseprojectivization is isomorphic to P S + . In this situation, L does not exist asa line bundle, but L 2 does; 32 the motivation for writing the Spin c bundle asS + ⊗L is that the tensor powers of this bundle obey isomorphisms suggestedby the notation. For instance, (S + ⊗ L) ⊗2 ∼ = L 2 ⊗ (Λ 0 ⊕ Λ 2,+ ). The data ofthe monopole equation are now a section M of S + ⊗ L and a connection onS + ⊗ L that projects to the Riemannian connection on P S + . The symbolF(A) will now denote 1/2 the trace of the curvature form of S + ⊗ L.Since L 2 is an ordinary line bundle, one has an integral cohomologyclass x = −c 1 (L 2 ) ∈ H 2 (X, Z). Note that x reduces modulo two to w 2 (X);31 If there is more than one spin structure, the choice of a spin structure will not matteras we ultimately sum over twistings by line bundles.32 One might be tempted to call this bundle L and write the Spin c bundle as S + ⊗L 1/2 ;that amounts to assigning magnetic charge 1/2 to the monopole and seems unnaturalphysically.


758 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionin particular, if w 2 (X) = 0, then L exists as a line bundle and x = −2c 1 (L).To write the monopole equations, recall that S + is symplectic or pseudoreal,so that if M is a section of S + ⊗ L, then the complex conjugate ¯Mis a section of S + ⊗ L −1 . The product M ⊗ ¯M would naturally lie in(S + ⊗ L) ⊗ (S + ⊗ L −1 ) ∼ = Λ 0 ⊕ Λ 2,+ . F + also takes values in Λ 2,+ makingit possible to write the following equations. Introduce Clifford matrices Γ i(with anticommutators {Γ i , Γ j } = 2g ij ), and set Γ ij = 1 2 [Γ i, Γ j ]. Then theequations are 33 F + ij = − i 2 ¯MΓ ij M,∑Γ i D i M = 0.iIn the second equation, ∑ i Γi D i is the Dirac operator D that maps sectionsof S + ⊗ L to sections of S − ⊗ L. We will sometimes abbreviate the first asF + = (M ¯M) + . Alternatively, if positive spinor indices are written A, B, C,and negative spinor indices as A ′ , B ′ , C ′ , 34 the equations can be written asF AB = i 2(MA ¯MB + M B ¯MA), DAA ′M A = 0.As a first step in understanding these equations, let us work out thevirtual dimension of the moduli space M of solutions of the equations upto gauge transformation. The linearization of the monopole equations fitsinto an elliptic complex0 → Λ 0 s→ Λ 1 ⊕ (S + ⊗ L) t → Λ 2,+ ⊕ (S − ⊗ L) → 0.Here t is the linearization of the monopole equations, and s is the map fromzero forms to deformations in A, M given by the infinitesimal action of thegauge group. Since we wish to work with real operators and determine thereal dimension of M, we temporarily think of S ± ⊗L as real vector bundles(of rank four). Then an elliptic operatorT : Λ 1 ⊕ (S + ⊗ L) −→ Λ 0 ⊕ Λ 2,+ ⊕ (S − ⊗ L)is defined by T = s ∗ ⊕ t. The virtual dimension of the moduli space isgiven by the index of T . By dropping terms in T of order zero, T can be33 To physicists the connection form A on a unitary line bundle is real; the covariantderivative is d A = d + iA and the curvature is F = dA or in components F ij = ∂ i A j −∂ j A i .34 Spinor indices are raised and lowered using the invariant tensor in Λ 2 S + . In components,if M A = (M 1 , M 2 ), then M A = (−M 2 , M 1 ). One uses the same operation ininterpreting ¯M as a section of S + ⊗ L, so ¯M A = ( ¯M 2 , − ¯M 1 ). Also F AB = 1 4 F ijΓ ij AB .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 759deformed to the direct sum of the operator d + d ∗ 35 from Λ 1 to Λ 0 ⊕ Λ 2,+and the Dirac operator from S + ⊗L to S − ⊗L. The index of T is the indexof d + d ∗ plus twice what is usually called the index of the Dirac operator;the factor of two comes from looking at S ± ⊗ L as real bundles of twicethe dimension. Let χ and σ be the Euler characteristic and signature ofX. Then the index of d + d ∗ is -(χ + σ)/2, while twice the Dirac indexis -σ/4 + c 1 (L) 2 . The virtual dimension of the moduli space is the sum ofthese or2χ + 3σW = − + c 1 (L) 2 .4When this number is negative, there are generically no solutions of themonopole equations. When W = 0, that is, when x = −c 1 (L 2 ) = −2c 1 (L)obeysx 2 = 2χ + 3σ, (4.252)then the virtual dimension is zero and the moduli space generically consistsof a finite set of points P i,x , i = 1 . . . t x . With each such point, one canassociate a sign ɛ i,x = ±1 – the sign of the determinant of T as we discussmomentarily. Once this is done, define for each x obeying (4.252) an integern x byn x = ∑ iɛ i,x .We will see later that n x = 0 – indeed, the moduli space is empty – for allbut finitely many x. Under certain conditions that we discuss in a moment,the n x are topological invariants.Note that W = 0 iff the index of the Dirac operator is∆ = χ + σ .4In particular, ∆ must be an integer to have non–trivial n x . Similarly, ∆must be integral for the Donaldson invariants to be non–trivial (otherwiseSU(2) instanton moduli space is odd–dimensional).For the sign of the determinant of T to make sense one must trivializethe determinant line of T . This can be done by deforming T as above tothe direct sum of d + d ∗ and the Dirac operator. If the Dirac operator,which naturally has a non–trivial complex determinant line, is regarded asa real operator, then its determinant line is naturally trivial – as a complex35 What is meant here is a projection of the d + d ∗ operator to self–dual forms.


760 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionline has a natural orientation. The d + d ∗ operator is independent of A andM (as the gauge group is Abelian), and its determinant line is trivializedonce and for all by picking an orientation of H 1 (X, R) ⊕ H 2,+ (X, R). Notethat this is the same data needed by Donaldson [Donaldson (1987)] to orientinstanton moduli spaces for SU(2); this is an aspect of the relation betweenthe two theories.If one replaces L by L −1 , A by -A, and M by ¯M, the monopole equationsare invariant, but the trivialization of the determinant line is multiplied by(−1) δ with δ the Dirac index. Hence the invariants for L and L −1 arerelated byn −x = (−1) ∆ n x .For W < 0, the moduli space is generically empty. For W > 0 one cantry, as in Donaldson theory, to define topological invariants that involveintegration over the moduli space. Donaldson theory does not detect thoseinvariants at least in known situations. We will see below that even whenW > 0, the moduli space is empty for almost all x.4.14.10.1 Topological InvarianceIn general, the number of solutions of a system of equations weighted bythe sign of the determinant of the operator analogous to T is always atopological invariant if a suitable compactness holds. If as in the caseat hand one has a gauge invariant system of equations, and one wishesto count gauge orbits of solutions up to gauge transformations, then onerequires (i) compactness; and (ii) free action of the gauge group on thespace of solutions.Compactness fails if a field or its derivatives can go to infinity. Toexplain the contrast with Donaldson theory, note that for SU(2) instantonscompactness fails precisely because an instanton can shrink to zero size.This is possible because(i) the equations are conformally invariant,(ii) they have non–trivial solutions on a flat R 4 , and(iii) embedding such a solution, scaled to very small size, on any fourmanifoldgives a highly localized approximate solution of the instantonequations (which can sometimes be perturbed to an exact solution). Themonopole equations by contrast are scale invariant but they have no nonconstantL 2 solutions on flat R 4 (or after dimensional reduction on flat R nwith 1 ≤ n ≤ 3). So there is no analog for the monopole equations of the


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 761phenomenon where an instanton shrinks to zero size.On the other hand, an obstruction does arise, just as in Donaldsontheory from the question of whether the gauge group acts freely on thespace of solutions of the monopole equations. The only way for the gaugegroup to fail to act freely is that there might be a solution with M = 0, inwhich case a constant gauge transformation acts trivially. A solution withM = 0 necessarily has F + = 0, that is, it is an Abelian instanton.Since F/2π represents the first Chern class of the line bundle L, it isintegral; in particular if F + = 0 then F/2π lies in the intersection of theintegral lattice in H 2 (X, R) with the anti-self–dual subspace H 2,− (X, R).As long as b + 2 ≥ 1, so that the self–dual part of H2 (X, R) is non-empty, theintersection of the anti-self–dual part and the integral lattice genericallyconsists only of the zero vector. In this case, for a generic metric on X,there are no Abelian instantons (except for x = 0, which we momentarilyexclude) and n x is well–defined.To show that the n x are topological invariants, one must further showthat any two generic metrics on X can be joined by a path along whichthere is never an Abelian instanton. As in Donaldson theory, this can fail ifb + 2 = 1. In that case, the self–dual part of H2 (X, R) is 1D, and in a genericone–parameter family of metrics on X, one may meet a metric for whichthere is an Abelian instanton. When this occurs, the n x can jump. Letus analyze how this happens, assuming for simplicity that b 1 = 0. Givenb 1 = 0 and b + 2 = 1, one has W = 0 precisely if the index of the Diracequation is 1. Therefore, there is generically a single solution M 0 of theDirac equation DM = 0.The equation F + (A) = 0 cannot be obeyed for a generic metric on X,but we want to look at the behavior near a special metric for which it doeshave a solution. Consider a one–parameter family of metrics parametrizedby a real parameter ɛ, such that at ɛ = 0 the self–dual subspace in H 2 (X, R)crosses a ‘wall’ and a solution A 0 of F + (A) = 0 appears. Hence for ɛ = 0,we can solve the monopole equations with A = A 0 , M = 0. Let us seewhat happens to this solution when ɛ is very small but non-zero. We setM = mM 0 , with m a complex number, to obey DM = 0, and we writeA = A 0 + ɛδA. The equation F + (A) − (M ¯M) + = 0 becomesF + (A 0 ) + (dδA) + − |m| 2 (M 0 ¯M0 ) + = 0. (4.253)As the cokernel of A −→ F + (A) is 1D, δA can be chosen to project theleft hand side of equation (4.253) into a 1D subspace (as b 1 = 0, this canbe done in a unique way up to a gauge transformation). The remaining


762 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionequation looks near ɛ = 0 like cɛ − m ¯m = 0, with c a constant. The ɛterm on the left comes from the fact that F + (A 0 ) is proportional to ɛ.Now we can see what happens for ɛ ≠ 0 to the solution that at ɛ = 0 hasA = A 0 , M = M 0 . Depending on the sign of c, there is a solution for m,uniquely determined up to gauge transformation, for ɛ > 0 and no solutionfor ɛ < 0, or vice–versa. Therefore n x jumps by ±1, depending on the signof c, in passing through ɛ = 0.The trivial Abelian instanton with x = 0 is an exception to the abovediscussion, since it cannot be removed by perturbing the metric. To definen 0 , perturb the equation F AB = i 2 (M A ¯M B + M B ¯MA ) toF AB = i 2 (M A ¯M B + M B ¯MA ) − p AB , (4.254)with p a self–dual harmonic 2–form; with this perturbation, the gauge groupacts freely on the solution space. Then define n 0 as the number of gaugeorbits of solutions of the perturbed equations weighted by sign in the usualway. This is invariant under continuous deformations of p for p ≠ 0; aslong as b + 2 > 1, so that the space of possible p’s is connected, the integern 0 defined this way is a topological invariant.The perturbation just pointed out will be used later in the case that p isthe real part of a holomorphic 2–form to compute the invariants of Kählermanifolds with b + 2 > 1. It probably has other applications; for instance,the case that p is a symplectic form is of interest.4.14.10.2 Vanishing TheoremsSome of the main properties of the monopole equations can be understoodby means of vanishing theorems. The general strategy in deriving suchvanishing theorems is quite standard, but some unusual cancellations (requiredby the Lorentz invariance of the underlying untwisted theory) leadto unusually strong results.If we set s = F + − M ¯M, k = DM, then a small calculation gives∫d 4 x √ ( )1gX 2 |s|2 + |k| 2 = (4.255)∫d 4 x √ ( 1g2 |F + | 2 + g ij D i M A D j ¯MA + 1 2 |M|4 + 1 )4 R|M|2 .XHere g is the metric of X, R the scalar curvature, and d 4 x √ g the Riemannianmeasure. A salient feature here is that a term F AB M A ¯M B , which


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 763appears in either |s| 2 or |k| 2 , cancels in the sum. This sharpens the implicationsof the formula, as we see. One can also consider the effect hereof the perturbation in (4.254); the sole effect of this is to replace 1 2 |M|4 in(4.255) by∫X⎛⎞d 4 x √ g ⎝F + ∧ p + ∑ ∣ 1∣∣∣∣2 (M ¯M2A B + M B ¯MA ) − p AB⎠ . (4.256)A,BThe second term is non-negative, and the first is simply the intersectionpairing2πc 1 (L) · [p]. (4.257)An obvious inference from (4.255) is that if X admits a metric whosescalar curvature is positive, then for such a metric any solution of themonopole equations must have M = 0 and F + = 0. But if b + 2 > 0, thenafter a generic small perturbation of the metric (which will preserve thefact that the scalar curvature is positive), there are no Abelian solutions ofF + = 0 except flat connections. Therefore, for such manifolds and metrics,a solution of the monopole equations is a flat connection with M = 0.These too can be eliminated using the perturbation in (4.254). 36 Hence afour-manifold for which b + 2 > 0 and n x ≠ 0 for some x does not admit ametric of positive scalar curvature.We can extend this to determine the possible four-manifolds X withb + 2 > 0, some n x ≠ 0, and a metric of non–negative scalar curvature. 37 IfX obeys those conditions, then for any metric of R ≥ 0, any basic class xis in H 2,− modulo torsion (so that L admits a connection with F + = 0,enabling (4.255) to vanish); in particular if x is not torsion then x 2 < 0.Now consider the effect of the perturbation (4.254). As x ∈ H 2,− , (4.257)vanishes; hence if R ≥ 0, R must be zero, M must be covariantly constantand (M ¯M) + = p. For M covariantly constant, (M ¯M) + = p implies that pis covariantly constant also; but for all p ∈ H 2,+ to be covariantly constantimplies that X is Kähler with b + 2 = 1 or is hyper-Kähler. Hyper-Kählermetrics certainly have R = 0, and there are examples of metrics with R = 0on Kähler manifolds with b + 2 = 1 [LeBrun (1991)].36 Flat connections can only arise if c 1 (L) is torsion; in that case, c 1 (L) · [p] = 0.37 If b + 2 = 1, the nx are not all topological invariants, and we interpret the hypothesisto mean that with at least one sign of the perturbation in (4.254), the n x are not allzero.


764 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAs an example, for a Kähler manifold with b + 2 ≥ 3, the canonical divisorK always arises as a basic class, so except in the hyper-Kähler case, suchmanifolds do not admit a metric of non-negative scalar curvature.Even if the scalar curvature is not positive, we can get an explicit boundfrom (4.255) showing that there are only finitely many basic classes. Since∫d 4 x √ ( 1g2 |M|4 + 1 )4 R|M|2 ≥ − 1 ∫d 4 x √ gR 2 , (4.258)32Xit follows from (4.255), even if we throw away the term |D i M| 2 , that∫d 4 x √ g|F + | 2 ≤ 1 ∫d 4 x √ gR 2 .16XOn the other hand, basic classes correspond to line bundles L with c 1 (L) 2 =(2χ + 3σ)/4, or∫1(2π) 2XXd 4 x √ g ( |F + | 2 − |F − | 2) =2χ + 3σ. (4.259)4Therefore, for a basic class both I + = ∫ d 4 x √ g|F + | 2 and I − =∫d 4 x √ g|F − | 2 are bounded. For a given metric, there are only finitelymany isomorphism classes of line bundles admitting connections with givenbounds on both I + and I − , so the set of basic classes is finite.The basic classes correspond to line bundles on which the moduli spaceof solutions of the monopole equations is of zero virtual dimension. Wecan analyze in a similar way components of the moduli space of positivedimension. Line bundles L such that c 1 (L) 2 < (2χ + 3σ)/4 are not of muchinterest in that connection, since for such line bundles the moduli spacehas negative virtual dimension and is generically empty. But if c 1 (L) 2 >(2χ + 3σ)/4, then (4.259) is simply replaced by the stronger bound∫1(2π) 2d 4 x √ g ( |F + | 2 − |F − | 2) >2χ + 3σ.4The set of isomorphism classes of line bundles admitting a connection obeyingthis inequality as well as (4.258) is once again finite. So we concludethat for any given metric on X, the set of isomorphism classes of line bundlesfor which the moduli space is non–empty and of non–negative virtualdimension is finite; for a generic metric on X, there are only finitely manynon–empty components of the moduli space.For further consequences of (4.255), we turn to a basic case in the studyof four-manifolds: the case that X is Kähler.


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 7654.14.10.3 Computation on Kähler ManifoldsIf X is Kähler and spin, then S + ⊗ L has a decomposition S + ⊗ L ∼ =(K 1/2 ⊗ L) ⊕ (K −1/2 ⊗ L), where K is the canonical bundle and K 1/2 is asquare root. If X is Kähler but not spin, then S + ⊗L, defined as before, hasa similar decomposition where now K 1/2 and L are not defined separatelyand K 1/2 ⊗ L is characterized as a square root of the line bundle K ⊗ L 2 .We denote the components of M in K 1/2 ⊗L and in K −1/2 ⊗L as α and-i¯β, respectively. The equation F + (A) = M ¯M can now be decomposed asF 2,0 = αβ, Fω 1,1 = − ω (|α| 2 − |β| 2) , F 0,2 = ᾱ¯β. (4.260)2Here ω is the Kähler form and Fω1,1 is the (1, 1) part of F + . (4.255) can berewritten∫d 4 x √ ( ) ∫1gX 2 |s|2 + |k| 2 = d 4 x √ ( 1gX 2 |F + | 2 + g ij D i ᾱD j α + g ij D i¯βDj β+ 1 2 (|α|2 + |β| 2 ) 2 + 1 )4 R(|α|2 + |β| 2 ) . (4.261)The right hand side of (4.261) is not manifestly non-negative (unlessR ≥ 0), but the fact that it is equal to the left hand side shows that itis non-negative and zero precisely for solutions of the monopole equations.Consider the operationA −→ A, α −→ α, β −→ −β. (4.262)This is not a symmetry of the monopole equations. But it is a symmetryof the right hand side of (4.261). Therefore, given a zero of the righthand side of (4.261) – that is, a solution of the monopole equations – theoperation (4.262) gives another zero of the right hand side of (4.261) – thatis, another solution of the monopole equations. So, though not a symmetryof the monopole equations, the transformation (4.262) maps solutions ofthose equations to other solutions.Vanishing of αβ means that α = 0 or β = 0. If α = 0, then the Diracequation for M reduces to¯∂ A β = 0,where ¯∂ A is the ¯∂ operator on L. Similarly, if β = 0, then the Dirac equationgives¯∂ A α = 0.


766 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionKnowing that either α or β is zero, we can deduce which it is. Integratingthe (1, 1) part of (4.260) gives∫1ω ∧ F = − 1 ∫ω ∧ ω ( |α| 2 − |β| 2) . (4.263)2π4πXThe left hand side of (4.263) is a topological invariant which can be interpretedasXJ = [ω] · c 1 (L).The condition that there are no non–trivial Abelian instantons is that J isnon-zero; we only wish to consider metrics for which this is so. If J < 0,we must have α ≠ 0, β = 0, and if J > 0, we must have α = 0, β ≠ 0.The equation that we have not considered so far is the (1, 1) part of(4.260). This equation can be interpreted as follows. Suppose for examplethat we are in the situation with β = 0. The space of connections A andsections α of K 1/2 ⊗ L can be interpreted as a symplectic manifold, thesymplectic structure being defined by∫∫〈δ 1 A, δ 2 A〉 = ω∧δ 1 A∧δ 2 A, 〈δ 1 α, δ 2 α〉 = −i ω∧ω (δ 1 αδ 2 α − δ 2 ᾱδ 1 α) .XOn this symplectic manifold acts the group of U(1) gauge transformations.The moment map µ for this action is the quantity that appears in the (1, 1)equation that we have not yet exploited, that isµω = F 1,1ω + ω|α| 2 .By analogy with many similar problems, setting to zero the moment mapand dividing by the group of U(1) gauge transformations should be equivalentto dividing by the complexification of the group of gauge transformations.In the present case, the complexification of the group of gaugetransformations acts by α −→ tα, ¯∂ A −→ t ¯∂ A t −1 , where t is a map from Xto C ∗ .Conjugation by t has the effect of identifying any two A’s that definethe same complex structure on L. This can be done almost uniquely: theambiguity is that conjugation by a constant t does not change A. Obviously,a gauge transformation by a constant t rescales α by a constant. The resulttherefore, for J < 0, is that the moduli space of solutions of the monopoleequations is the moduli space of pairs consisting of a complex structure onL and a non-zero holomorphic section, defined up to scaling, of K 1/2 ⊗ L.X


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 767For J > 0, it is instead β that is non-zero, and K 1/2 ⊗ L is replaced byK 1/2 ⊗ L −1 .This result can be stated particularly nicely if X has b 1 = 0. Then thecomplex structure on L, assuming that it exists, is unique. The modulispace of solutions of the monopole equations is therefore simply a complexprojective space, P H 0 (X, K 1/2 ⊗ L) or P H 0 (X, K 1/2 ⊗ L −1 ), dependingon the sign of J.Identifying The Basic ClassesWe would now like to identify the basic classes. The above description ofthe moduli space gives considerable information: basic classes are of theform x = −2c 1 (L), where L is such that J < 0 and H 0 (X, K 1/2 ⊗ L) isnon-empty, or J > 0 and H 0 (X, K 1/2 ⊗ L −1 ) is non-empty. This, however,is not a sharp result.That is closely related to the fact that the moduli spaces P H 0 (X, K 1/2 ⊗L ±1 ) found above very frequently have a dimension bigger than the ‘generic’dimension of the moduli space as predicted by the index Theorem. In fact,Kähler metrics are far from being generic. In case the expected dimensionis zero, one would have always n x > 0 if the moduli spaces behaved ‘generically’(given the complex orientation, an isolated point on the moduli spacewould always contribute +1 to n x ; this is a special case of a discussion below).Since the n x are frequently negative, moduli spaces of non–genericdimension must appear.When the moduli space has greater than the generically expected dimension,one can proceed by integrating over the bosonic and fermioniccollective coordinates in the path integral. This gives a result that can bedescribed topologically: letting T be the operator that arises in linearizingthe monopole equations, the cokernel of T is a vector bundle V (the‘bundle of antighost zero modes’) over the moduli space M; its Euler classintegrated over M is the desired n x .Alternatively, one can perturb the equations to more generic ones. Weuse the same perturbation as before. For a Kähler manifold X, the conditionb + 2 > 1 is equivalent to H2,0 (X) ≠ 0, so we can pick a non-zeroholomorphic 2–form η. We perturb the monopole equations (4.260) toF 2,0 = αβ − η, F 1,1ω = −ω ( |α| 2 − |β| 2) , F 0,2 = ᾱ¯β − ¯η, (4.264)leaving unchanged the Dirac equation for M.


768 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIt suffices to consider the case that the first Chern class of L is of type(1, 1), since the unperturbed moduli space vanishes otherwise. That beingso, we have∫∫0 = F 2,0 ∧ ¯η = F 0,2 ∧ η.XUsing this, one finds that (4.261) generalizes to∫d 4 x √ ( ) ∫ ( 1 1gX 2 |s|2 + |k| 2 = d 4 xX 2 |F + | 2 + g ij D i ᾱD j α + g ij D i¯βDj β+ 1 2 (|α|2 − |β| 2 ) 2 + 2|αβ − η| 2 + R )4 (|α|2 + |β| 2 ) . (4.265)Witten now makes an argument of a sort we have already seen: thetransformationXA −→ A, α −→ α, β −→ −β, η −→ −η, (4.266)though not a symmetry of (4.264), is a symmetry of the right hand side of(4.265). As solutions of (4.264) are the same as zeroes of the right handside of (4.265), we deduce that the solutions of (4.264) with a 2–form η aretransformed by (4.266) to the solutions with -η. The terms in (4.264) evenor odd under the transformation must therefore separately vanish, so anysolution of (4.264) has0 = F 0,2 = F 2,0 = αβ − η.The condition F 0,2 = 0 means that the connection still defines a holomorphicstructure on L.The condition αβ = η gives our final criterion for determining thebasic classes: they are of the form x = −2c 1 (L) where, for any choice ofη ∈ H 0 (X, K), one has a factorization η = αβ with holomorphic sectionsα and β of K 1/2 ⊗ L ±1 , and x 2 = c 1 (K) 2 .To make this completely explicit, suppose the divisor of η is a union ofirreducible components C i of multiplicity r i . Thus the canonical divisor isc 1 (K) = ∑ ir i [C i ],where [C i ] denotes the cohomology class that is Poincaré dual to the curveC i . The existence of the factorization η = αβ means that the divisor of


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 769K 1/2 ⊗ L isc 1 (K 1/2 ⊗ L) = ∑ is i [C i ],where s i are integers with 0 ≤ s i ≤ r i . The first Chern class of L is thereforec 1 (L) = ∑ i(s i − 1 2 r i)[C i ].And the basic classes are of the form x = −2c 1 (L) orx = − ∑ i(2s i − r i )[C i ].An x of this form is is of type (1, 1) and congruent to c 1 (K) modulo two,but may not obey x 2 = c 1 (K) 2 . It is actually possible to prove using theHodge index Theorem that for x as above, x 2 ≤ c 1 (K) 2 . This is clear fromthe monopole equations: perturbed to η ≠ 0, these equations have at mostisolated solutions (from the isolated factorization η = αβ) and not a modulispace of solutions of positive dimension. So, for Kähler manifolds, the non–empty perturbed moduli spaces are at most of dimension zero; invariantsassociated with monopole moduli spaces of higher dimension vanish.4.14.11 SW Theory and Integrable SystemsRemarkably, the SW theory for N = 2 supersymmetric YM theory forarbitrary gauge algebra g appears to be intimately related with the existenceof certain classical mechanics integrable systems. This relation wasfirst suspected on the basis of the similarity between the SW curves andthe spectral curves of certain integrable models [Gorskii et. al. (1995);Nakatsu and Takasaki (1996)]. Then, arguments were developed that SWtheory naturally produces integrable structures [Donagi and Witten (1996)].For the N = 2 supersymmetric YM theory with massive hypermultiplet,the relevant integrable system appears to be the elliptic Calogero–Moser system [Calogero (1975); Moser (1975)]. For SU(N) gauge group,Donagi and Witten [Donagi and Witten (1996)] proposed that the spectralcurves of the SU(N) Hitchin system should play the role of theSW curves. [Nekrasov (1996)] recognized that the SU(N) Hitchin systemspectral curves are identical to those of the SU(N) elliptic Calogero–Moser (CM) integrable system. That the SU(N) elliptic CM–curves (andassociated SW differential) do indeed provide the SW solution for the


770 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionN = 2 theory with one massive hypermultiplet was fully established bythe authors in [D’Hoker and Phong (1998a); D’Hoker and Phong (1998b);D’Hoker and Phong (1998c)], where it was shown that:(i) The resulting effective prepotential F (and thus the low energy effectiveaction) reproduces correctly the logarithmic singularities predictedby perturbation theory.(ii) F satisfies a renormalization group type equation which determinesexplicitly and efficiently instanton contributions to any order.(iii) The prepotential in the limit of large hypermultiplet mass m (as wellas large gauge scalar expectation value and small gauge coupling) correctlyreproduces the prepotentials for N = 2 super–YM theory with any numberof hypermultiplets in the fundamental representation of the gauge group.The N = 2 theory for arbitrary gauge algebra g and with one massivehypermultiplet in the adjoint representation was one such outstandingcase when g ≠ SU(N). Actually, as discussed previously, upon takingsuitable limits, this theory contains a very large number of modelswith smaller hypermultiplet representations R, and in this sense has auniversal aspect. It appeared difficult to generalize directly the Donagi–Witten construction of Hitchin systems to arbitrary g, and it was thusnatural to seek this generalization directly amongst the elliptic CM integrablesystems. It has been known now for a long time, thanks to thework of Olshanetsky and Perelomov [Olshanetsky and Perelomov (1976);Olshanetsky and Perelomov (1981)], that CM–systems can be defined forany simple Lie algebra. Olshanetsky and Perelomov also showed that theCM–systems for classical Lie algebras were integrable, although the existenceof a spectral curve (or, a Lax pair with a spectral parameter) aswell as the case of exceptional Lie algebras remained open. Thus severalimmediate questions are:(i) Does the elliptic CM–system for general Lie algebra g admit a Laxpair with spectral parameter?(ii) Does it correspond to the N = 2 supersymmetric gauge theory withgauge algebra g and a hypermultiplet in the adjoint representation?(ii) Can this correspondence be verified in the limiting cases when themass m tends to 0 with the theory acquiring N = 4 supersymmetry andwhen m → ∞, with the hypermultiplet decoupling in part to smaller representationsof g?According to [D’Hoker and Phong (1998a); D’Hoker and Phong (1998b);D’Hoker and Phong (1998c)], the answers to these questions can be statedsuccinctly as follows:


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 771(i) The elliptic CM–systems defined by an arbitrary simple Lie algebrag do admit Lax pairs with spectral parameters.(ii) The correspondence between elliptic g CM–systems and N = 2supersymmetric g gauge theories with matter in the adjoint representationholds directly when the Lie algebra g is simply–laced. When g is not simply–laced, the correspondence is with new integrable models, the twisted ellipticCM–systems.(iii) The new twisted elliptic CM–systems also admit a Lax pair withspectral parameter.(iv) In the scaling limit m = Mq − 1 2 δ → ∞, (with M fixed), the twisted(respectively untwisted) elliptic g CM–systems tend to the Toda system for(g (1) ) ∨ (respectively g (1) ) for δ = 1h(respectively δ = 1∨gh g). Here h g andh ∨ g are the Coxeter and the dual Coxeter numbers of g.4.14.11.1 SU(N) Elliptic CM SystemThe original elliptic CM–system is the system defined by the HamiltonianH(x, p) = 1 N∑p 2 i − 1 ∑ 2 2 m2 ℘(x i − x j ). (4.267)i=1 i≠jHere m is a mass parameter, and ℘(z) is the Weierstrass ℘−function, definedon a torus C/(2ω 1 Z + 2ω 2 Z). As usual, we denote by τ = ω 2 /ω 1 themoduli of the torus, and set q = e 2πiτ . The well–known trigonometric andrational limits with respective potentials− 1 2 m2 ∑ i≠j14 sh 2 ( xi−xj2)and− 1 2 m2 ∑ i≠j1(x i − x j ) 2 ,arise in the limits ω 1 = −iπ, ω 2 → ∞ and ω 1 , ω 2 → ∞. All these systemshave been shown to be completely integrable in the sense of Liouville, i.e.,they all admit a complete set of integrals of motion which are in involution.However, we require a notion of integrability which is in some sense morestringent, namely the existence of a Lax pair L(z), M(z) with spectralparameter z. The Hamiltonian system (4.267) is equivalent to the Laxequation˙L(z) = [L(z), M(z)], (4.268)


772 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwith L(z) and M(z) given by the following N × N matricesL ij (z) = p i δ ij − m(1 − δ ij )Φ(x i − x j , z), (4.269)∑M ij (z) = mδ ij ℘(x i − x k ) − m(1 − δ ij )Φ ′ (x i − x j , z).k≠iThe function Φ(x, z) is defined byΦ(x, z) =σ(z − x)σ(z)σ(x) exζ(z) , (4.270)where σ(z), ζ(z) are the usual Weierstrass σ and ζ functions on the torusC/(2ω 1 Z+2ω 2 Z). The function Φ(x, z) satisfies the key functional equationΦ(x, z)Φ ′ (y, z) − Φ(y, z)Φ ′ (x, z) = (℘(x) − ℘(y))Φ(x + y, z). (4.271)It is well–known that functional equations of this form are required forthe Hamilton equations of motion to be equivalent to the Lax equation(4.268) with a Lax pair of the form (4.269). Often, solutions had beenobtained under additional parity assumptions in x (and y), which preventthe existence of a spectral parameter. The solution Φ(x, z) with spectralparameter z is obtained by dropping such parity assumptions for generalz. Conversely, general functional equations of the form (4.271) essentiallydetermine Φ(x, z).4.14.11.2 CM Systems Defined by Lie AlgebrasThe Hamiltonian system (4.267) is only one example of a whole series ofHamiltonian systems associated with each simple Lie algebra. More precisely,given any simple Lie algebra g, we have the system with Hamiltonian[Olshanetsky and Perelomov (1976); Olshanetsky and Perelomov (1976)]H(x, p) = 1 2r∑p 2 i − 1 2i=1∑α∈R(g)m 2 |α| ℘(α · x), (4.272)where r is the rank of g, R(g) denotes the set of roots of g, and the m |α|are mass parameters. To preserve the invariance of the Hamiltonian (4.272)under the Weyl group, the parameters m |α| depend only on the orbit |α| ofthe root α, and not on the root α itself. In the case of A N−1 = SU(N), itis common practice to use N pairs of dynamical variables (x i , p i ), since theroots of A N−1 lie conveniently on a hyperplane in C N . The dynamics of thesystem are unaffected if we shift all x i by a constant, and the number of degreesof freedom is effectively N −1 = r. Now the roots of SU(N) are given


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 773by α = e i −e j , 1 ≤ i, j ≤ N, i ≠ j. Thus we recognize the original ellipticCM–system as the special case of (4.272) corresponding to A N−1 . As in theoriginal case, the elliptic systems (4.272) admit rational and trigonometriclimits. Olshanetsky and Perelomov [Olshanetsky and Perelomov (1976);Olshanetsky and Perelomov (1976)] succeeded in constructing a Lax pairfor all these systems in the case of classical Lie algebras, albeit withoutspectral parameter.4.14.11.3 Twisted CM–Systems Defined by Lie AlgebrasIt turns out that the Hamiltonian systems (4.272) are not the only naturalextensions of the basic elliptic CM–system. A subtlety arises for simple Liealgebras g which are not simply–laced, i.e., algebras which admit roots ofuneven length. This is the case for the algebras B n , C n , G 2 , and F 4 in Cartan’sclassification. For these algebras, the following twisted elliptic CM–systems were introduced by the authors in [D’Hoker and Phong (1998a);D’Hoker and Phong (1998b); D’Hoker and Phong (1998c)]H twistedg = 1 2r∑p 2 i − 1 2i=1∑α∈R(g)m 2 |α| ℘ ν(α)(α · x). (4.273)Here the function ν(α) depends only on the length of the root α. If g issimply–laced, we set ν(α) = 1 identically. Otherwise, for g non simply–laced, we set ν(α) = 1 when α is a long root, ν(α) = 2 when α is a shortroot and g is one of the algebras B n , C n , or F 4 , and ν(α) = 3 when α is ashort root and g = G 2 . The twisted Weierstrass function ℘ ν (z) is definedbyν−1∑ σ℘ ν (z) = ℘(z + 2ω a ), (4.274)νσ=0where ω a is any of the half–periods ω 1 , ω 2 , or ω 1 +ω 2 . Thus the twisted anduntwisted CM–systems coincide for g simply laced. The original motivationfor twisted CM–systems was based on their scaling limits.


774 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction4.14.11.4 Scaling Limits of CM–SystemsFor the standard elliptic CM–systems corresponding to A N−1 , in the scalinglimit we have [Inozemtsev (1989a); Inozemtsev (1989b)]m = Mq − 12N , q → 0, (4.275)ix i = X i − 2ω 2 ,N1 ≤ i ≤ N, (4.276)where M is kept fixed, the elliptic A N−1 CM Hamiltonian tends to thefollowing Hamiltonian(H Toda = 1 N∑p 2 i − 1 N−1)∑e Xi+1−Xi + e X1−X N. (4.277)2 2i=1i=1The roots e i − e i+1 , 1 ≤ i ≤ N − 1, and e N − e 1 can be recognized as thesimple roots of the affine algebra A (1)N−1. Thus (4.277) can be recognized asthe Hamiltonian of the Toda system defined by A (1)N−1 .Scaling Limits based on the Coxeter NumberThe key feature of the above scaling limit is the collapse of the sum overthe entire root lattice of A N−1 in the CM Hamiltonian to the sum over onlysimple roots in the Toda Hamiltonian for the Kac–Moody algebra A (1)N−1 .Our task is to extend this mechanism to general Lie algebras. For this, weconsider the following generalization of the preceding scaling limitm = Mq − 1 2 δ , x = X − 2ω 2 δρ ∨ , (4.278)Here x = (x i ), X = (X i ) and ρ ∨ are rD vectors. The vector x is thedynamical variable of the CM–system. The parameters δ and ρ ∨ dependon the algebra g and are yet to be chosen. As for M and X, they have thesame interpretation as earlier, namely as respectively the mass parameterand the dynamical variables of the limiting system. Setting ω 1 = −iπ, thecontribution of each root α to the CM potential can be expressed asm 2 ℘(α · x) = 1 2 M 2∞ ∑n=−∞e 2δω2ch(α · x − 2nω 2 ) − 1 . (4.279)It suffices to consider positive roots α. We shall also assume that 0 ≤δ α · ρ ∨ ≤ 1. The contributions of the n = 0 and n = −1 summands in


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 775(4.279) are proportional to e 2ω2(δ−δ α·ρ∨) and e 2ω2(δ−1+δ α·ρ∨) , respectively.Thus the existence of a finite scaling limit requires thatδ ≤ δ α · ρ ∨ ≤ 1 − δ. (4.280)Let α i , 1 ≤ i ≤ r be a basis of simple roots for g. If we want all simpleroots α i to survive in the limit, we must require thatα i · ρ ∨ = 1, 1 ≤ i ≤ r.This condition characterizes the vector ρ ∨ as the level vector. Next, thesecond condition in (4.274) can be rewritten as δ{1 + max α (α · ρ ∨ )} ≤ 1.Buth g = 1 + max α (α · ρ ∨ ) (4.281)is precisely the Coxeter number of g, and we must have δ ≤ 1h g. Thus whenδ < 1h g, the contributions of all the roots except for the simple roots of gtend to 0. On the other hand, when δ = 1h g, the highest root α 0 realizingthe maximum over α in (4.281) survives. Since -α 0 is the additional simpleroot for the affine Lie algebra g (1) , we arrive in this way at the followingTheorem:Under the limit (4.278), with δ = 1h g, and ρ ∨ given by the level vector,the Hamiltonian of the elliptic CM–system for the simple Lie algebra gtends to the Hamiltonian of the Toda system for the affine Lie algebra g (1) .Scaling Limit based on the Dual Coxeter NumberIf the SW–spectral curve of the N = 2 supersymmetric gauge theory witha hypermultiplet in the adjoint representation is to be realized as the spectralcurve for a CM–system, the parameter m in the CM–system shouldcorrespond to the mass of the hypermultiplet. In the gauge theory, thedependence of the coupling constant on the mass m is given byτ =i2π h∨ g ln m2M 2 ⇐⇒ m = Mq − 12h ∨ g , (4.282)where h ∨ g is the quadratic Casimir function of the Lie algebra g. This showsthat the correct physical limit, expressing the decoupling of the hypermultipletas it becomes infinitely massive, is given by (4.271), but with δ = 1h. ∨gTo establish a closer parallel with our preceding discussion, we recall that


776 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe quadratic Casimir h ∨ g coincides with the dual Coxeter number of g,defined byh ∨ g = 1 + max α (α ∨ · ρ), (4.283)∑α>0α is thewhere α ∨ = 2αα 2 is the coroot associated to α, and ρ = 1 2well–known Weyl vector.For simply laced Lie algebras g (ADE algebras), we have h g = h ∨ g , andthe preceding scaling limits apply. However, for non simply–laced algebras(B n , C n , G 2 , F 4 ), we have h g > h ∨ g , and our earlier considerations show thatthe untwisted elliptic CM Hamiltonians do not tend to a finite limit, q → 0,M is kept fixed. This is why the twisted Hamiltonian systems (4.273) haveto be introduced. The twisting produces precisely to an improvement inthe asymptotic behavior of the potential which allows a finite, non–triviallimit. More precisely, we can writem 2 ℘ ν (x) = ν22We have the following Theorem:Under the limit∞∑n=−∞m 2ch ν(x − 2nω 2 ) − 1 . (4.284)1x = X + 2ω 2h ∨ ρ, m = Mq − 12h ∨ g ,gwith ρ the Weyl vector and q → 0, the Hamiltonian of the twisted ellipticCM–system for the simple Lie algebra g tends to the Hamiltonian of theToda system for the affine Lie algebra (g (1) ) ∨ .So far we have discussed only the scaling limits of the Hamiltonians.However, similar arguments show that the Lax pairs constructed belowalso have finite, non–trivial scaling limits whenever this is the case for theHamiltonians. The spectral parameter z should scale as e z = Zq 1 2 , with Zfixed. The parameter Z can be identified with the loop group parameterfor the resulting affine Toda system.4.14.11.5 Lax Pairs for CM–SystemsLet the rank of g be n, and d be its dimension. Let Λ be a representation ofg of dimension N, of weights λ I , 1 ≤ I ≤ N. Let u I ∈ C N be the weightsof the fundamental representation of GL(N, C). Project orthogonally theu I ’s onto the λ I ’s as su I = λ I + u I and λ I ⊥ v J . It is easily verifiedthat s 2 is the second Dynkin index. Then α IJ = λ I − λ J is a weight of


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 777Λ ⊗ Λ ∗ associated to the root u I − u J of GL(N, C). The Lax pairs for bothuntwisted and twisted CM–systems will be of the formL = P + X, M = D + X,where the matrices P, X, D, and Y are given byX = ∑ I≠JC IJ Φ IJ (α IJ , z)E IJ , Y = ∑ I≠jC IJ Φ ′ IJ(α IJ , z)E IJand by P = p · h, D = d · (h ⊕ ˜h) + ∆.Here h is in a Cartan subalgebra H g for g, ˜h is in the Cartan–Killingorthogonal complement of H g inside a Cartan subalgebra H for GL(N, C),and ∆ is in the centralizer of H g in GL(N, C). The functions Φ IJ (x, z)and the coefficients C IJ are yet to be determined. We begin by statingthe necessary and sufficient conditions for the pair L(z), M(z) of (4.275)to be a Lax pair for the (twisted or untwisted) CM–systems. For this, it isconvenient to introduce the following notationΦ IJ = Φ IJ (α IJ · x),℘ ′ IJ = Φ IJ (α IJ · x, z)Φ ′ JI(−α IJ · x, z) − Φ IJ (−α IJ · x, z)Φ ′ JI(α IJ · x, z).Then the Lax equation ˙L(z) = [L(z), M(z)] implies the CM–system ifand only if the following three identities are satisfied (K ≠ I ≠ J)∑C IJ C JI ℘ ′ IJα IJ = s 2 m 2 |α| ℘ ν(α)(α · x), C IJ C JI ℘ ′ IJ(v I − v J ) = 0,α∈R(g)C IK C KJ (Φ IK Φ ′ KJ − Φ ′ IKΦ KJ ) =sC IJ Φ IJ d · (v I − v J ) + ∆ IJ C KJ Φ KJ − C IK Φ IK ∆ KJWe have the following Theorem [D’Hoker and Phong (1998b)]:A representation Λ, functions Φ IJ , and coefficients C IJ with a spectralparameter z satisfying (4.280–4.281) can be found for all twisted and untwistedelliptic CM–systems associated with a simple Lie algebra g, exceptpossibly in the case of twisted G 2 . In the case of E 8 , we have to assumethe existence of a ±1 cocycle.Lax Pairs for Untwisted CM SystemsHere are some important features of the Lax pairs obtained in this manner[D’Hoker and Phong (1998a); D’Hoker and Phong (1998b); D’Hoker andPhong (1998c)]:


778 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn the case of the untwisted CM–systems, we can choose Φ IJ (x, z) =Φ(x, z), ℘ IJ (x) = ℘(x) for all g.∆ = 0 for all g, except for E 8 .For A n , the Lax pair (4.269–4.270) corresponds to the choice of thefundamental representation for Λ. A different Lax pair can be found bytaking Λ to be the antisymmetric representation.For the BC n system, the Lax pair is obtained by imbedding B n inGL(N, C) with N = 2n + 1. When z = ω a (half-period), the Lax pairobtained this way reduces to the Lax pair obtained in [Olshanetsky andPerelomov (1976); Olshanetsky and Perelomov (1976)].For the B n and D n systems, additional Lax pairs with spectral parametercan be found by taking Λ to be the spinor representation.For G 2 , a first Lax pair with spectral parameter can be obtained by theabove construction with Λ chosen to be the 7 of G 2 . A second Lax pairwith spectral parameter can be obtained by restricting the 8 of B 3 to the7 ⊕ 1 of G 2 .For F 4 , a Lax pair can be obtained by taking Λ to be the 26 ⊕ 1 of F 4 ,viewed as the restriction of the 27 of E 6 to its F 4 subalgebra.For E 6 , Λ is the 27 representation.Lax Pairs for Twisted CM SystemsRecall that the twisted and untwisted CM–systems differ only for nonsimplylaced Lie algebras, namely B n , C n , G 2 and F 4 . These are the onlyalgebras we discuss in this paragraph. The construction (4.277–4.281) givesthen Lax pairs for all of them, with the possible exception of twisted G 2 .Unlike the case of untwisted Lie algebras however, the functions Φ IJ haveto be chosen with care, and differ for each algebra.For B n , the Lax pair is of dimension N = 2n, admits two independentcouplings m 1 and m 2 , and{ Φ(x, z), if I − J ≠ 0, ±n;Φ IJ (x, z) =Φ 2 ( 1 2x, z),Here a new function Φ 2 (x, z) is defined byif I − J = ±n.Φ 2 ( 1 2 x, z) = Φ( 1 2 x, z)Φ( 1 2 x + ω 1, z).Φ(ω 1 , z)For C n , the Lax pair is of dimension N = 2n+2, admits one independent


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 779coupling m 2 , andwhere ω IJ are given byΦ IJ (x, z) = Φ 2 (x + ω IJ , z),⎧⎨ 0, if I ≠ J = 1, ..., 2n + 1;ω IJ = ω⎩ 2 , if 1 ≤ I ≤ 2n, J = 2n + 2;−ω 2 , if 1 ≤ J ≤ 2n, I = 2n + 2.For F 4 , the Lax pair is of dimension N = 24, two independent couplingsm 1 and m 2 ,⎧⎨ Φ(x, z), if λ · µ = 0;Φ λµ (x, z) = Φ⎩ 1 ( 1 2 x, z), if λ · µ = 1 2 ;Φ 2 ( 1 2x, z), if λ · µ = −1.where the function Φ 1 (x, z) is defined byΦ 1 (x, z) = Φ(x, z) − e πiζ(z)+η 1 z Φ(x + ω 1 , z).Here it is more convenient to label the entries of the Lax pair directly bythe weights λ = λ I and µ = λ J instead of I and J.4.14.11.6 CM and SW Theory for SU(N)The correspondence between SW theory for N = 2 super–YM theorywith one hypermultiplet in the adjoint representation of the gauge algebra,and the elliptic CM–systems was first established in [D’Hoker andPhong (1998a)], for the gauge algebra g = SU(N). We describe it herein some detail (see also [D’Hoker and Phong (1998b); D’Hoker and Phong(1998c)]).All that we shall need here of the elliptic CM–system is its Lax operatorL(z), whose N × N matrix elements are given byL ij (z) = p i δ ij − m(1 − δ ij )Φ(x i − x j , z). (4.285)Notice that the Hamiltonian is simply given in terms of L by H(x, p) =12 TrL(z)2 + C℘(z) with C = − 1 2 m2 N(N − 1).The correspondence between the data of the elliptic CM–system andthose of the SW theory is as follows:(i) The parameter m in (4.285) is the hypermultiplet mass;


780 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(ii) The gauge coupling g and the θ−angle are related to the modulusof the torus Σ = C/(2ω 1 Z + 2ω 2 Z) byτ = ω 2ω 1= θ2π + 4πig 2 ;(iii) The SW curve Γ is the spectral curve of the elliptic CM model,defined byΓ = {(k, z) ∈ C × Σ, det(kI − L(z)) = 0}and the SW 1–orm is dλ = k dz. Γ is invariant under the Weyl group ofSU(N).(iv) Using the Lax equation ˙L = [L, M], it is clear that the spectral curveis independent of time, and can be dependent only upon the constants ofmotion of the CM–system, of which there are only N. These integrals ofmotion may be viewed as parametrized by the quantum moduli of the SWsystem.(v) Finally, dλ = kdz is meromorphic, with a simple pole on each of theN sheets above the point z = 0 on the base torus. The residue at each ofthese poles is proportional to m, as required by the general set–up of SWtheory.Four Fundamental Theorems1. The spectral curve equation det(kI − L(z)) = 0 is equivalent to( 1ϑ 1 (z − m ∂ )2ω 1 ∂k )|τ H(k) = 0,where H(k) is a monic polynomial in k of degree N, whose zeros (or equivalentlywhose coefficients) correspond to the moduli of the gauge theory. IfH(k) = ∏ Ni=1 (k − k i), then∮limq→012πiA ikdz = k i − 1 2 m.Here, ϑ 1 is the Jacobi ϑ−function, which admits a simple series expansionin powers of the instanton factor q = e 2πiτ , so that the curve equation mayalso be rewritten as a series expansion∑(−) n q 1 2 n(n−1) e nz H(k − n · m) = 0, (4.286)n∈Z


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 781where we have set ω 1 = −iπ without loss of generality. The series expansion(4.286) is superconvergent and sparse in the sense that it receivescontributions only at integers that grow like n 2 .2. The prepotential of the SW theory obeys a renormalization group–type equation that simply relates F to the CM Hamiltonian, expressed interms of the quantum order parameters a ja j = 1 ∮dλ,2πi A j∂F∂τ | a j= H(x, p) = 1 2 TrL(z)2 + C℘(z). (4.287)Furthermore, in an expansion in powers of the instanton factor q = e 2πiτ ,the quantum order parameters a j may be computed by residue methods interms of the zeros of H(k). The proof of (4.287) requires Riemann surfacedeformation theory. The fact that the quantum order parameters may beevaluated by residue methods arises from the fact that A j −cycles may bechosen on the spectral curve Γ in such a way that they will shrink to zeroas q → 0. As a result, contour integrals around full-fledged branch cutsA j reduce to contour integrals around poles at single points, which may becalculated by residue methods only. Knowing the quantum order parametersin terms of the zeros k j of H(k) = 0 is a relation that may be invertedand used in (4.287) to get a differential relation for all order instanton corrections.It is now only necessary to evaluate explicitly the τ−independentcontribution to F, which in field theory arises from perturbation theory.This may be done easily by retaining only the n = 0 and n = 1 terms inthe expansion of the curve (4.286), so that z = ln H(k) − ln H(k − m). Theresults of the calculations to two instanton order may be summarized inthe following Theorem:3. The prepotential, to 2 instanton order is given by F = F pert +F (1) +F (2) . The perturbative contribution is given byF pert = τ 2∑ia 2 i − 18πi∑ [(ai − a j ) 2 ln(a i − a j ) 2 − (a i − a j − m) 2 ln(a i − a j − m) 2i,j(4.288)while all instanton corrections are expressed in terms of a single functionS i (a) =∏ Nj=1 [(a i − a j ) 2 − m 2 ]∏j≠i (a − a j) 2 , as follows:


782 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionF (1) =q2πiF (2) = q28πi∑S i (a i ), (4.289)⎡i⎣ ∑ iS i (a i )∂ 2 i S i (a i ) + 4 ∑ i≠j⎤S i (a i )S j (a j )(a i − a j ) 2 − S i(a i )S j (a j )⎦(a i − a j − m) 2 .The perturbative corrections to the prepotential of (4.288) indeed preciselyagree with the predictions of asymptotic freedom. The formulas (4.289) forthe instanton corrections F (1) and F (2) are new, as they have not yet beencomputed by direct field theory methods. The moduli k i , 1 ≤ i ≤ N, of thegauge theory are evidently integrals of motion of the system. To identifythese integrals of motion, denote by S be any subset of {1, · · · , N}, andlet S ∗ = {1, · · · , N} \ S, ℘(S) = ℘(x i − x j ) when S = {i, j}. Let also p Sdenote the subset of momenta p i with i ∈ S.4. For any K, 0 ≤ K ≤ N, let σ K (k 1 , · · · , k N ) = σ K (k) be the K−thsymmetric polynomial of (k 1 , · · · , k N ), defined byH(u) =N∑(−) K σ K (k)u N−K .K=0[K/2]∑σ K (k) = σ K (p) +l=1m 2l∑|S i ∩S j |=2δ ij1≤i,j≤lThen4.14.11.7 CM and SW Theory for General Lie Algebral∏σ K−2l (p (∪ li=1 S ∗) i) [℘(S i ) + η 1].ω 1Now, we consider the N = 2 supersymmetric gauge theory for a generalsimple gauge algebra g and a hypermultiplet of mass m in the adjointrepresentation. Then we have the following results [D’Hoker and Phong(1998d)].The SW curve of the theory is given by the spectral curveΓ = {(k, z) ∈ C × Σ; det(kI − L(z)) = 0}of the twisted elliptic CM–system associated to the Lie algebra g. The SWdifferential dλ is given by dλ = kdz.The function R(k, z) = det(kI − L(z)) is polynomial in k and meromorphicin z. The spectral curve Γ is invariant under the Weyl group of g.It depends on n complex moduli, which can be thought of as independentintegrals of motion of the CM–system.The differential dλ = kdz is meromorphic on Γ, with simple poles.i=1


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 783The position and residues of the poles are independent of the moduli. Theresidues are linear in the hypermultiplet mass m (unlike the case of SU(N),their exact values are difficult to determine for general g.)In the m → 0 limit, the CM–system reduces to a free system, thespectral curve Γ is just the producr of several unglued copies of the basetorus Σ, indexed by the constant eigenvalues of L(z) = p · h. Let k i ,1 ≤ i ≤ n, be n independent eigenvalues, and A i , B i be the A and B cycleslifted to the corresponding sheets. For each i, we readily geta i = 1 ∮dλ = k ∮i2πi A i2πia Di = 1 ∮dλ = k ∮i2πi B i2πiABdz = 2ω 12πi k i,dz = 2ω 12πi τk i.Thus the prepotential F is given by F = τ ∑ n2 i=1 a2 i . This is the classicalprepotential and hence the correct answer, since in the m → 0 limit,the theory acquires an N = 4 supersymmetry, and receives no quantumcorrections.The m → ∞ limit is the crucial consistency check, which motivatedthe introduction of the twisted CM–systems in the first place. In the limitm → ∞, q → 0, with1x = X + 2ω 2h ∨ ρ, m = Mq − 12h ∨ g ,gwith X and M kept fixed, the Hamiltonian and spectral curve for thetwisted elliptic CM–system with Lie algebra g reduce to the Hamiltonianand spectral curve for the Toda system for the affine Lie algebra (g (1) ) ∨ .This is the correct answer. Indeed, in this limit, the gauge theory withadjoint hypermultiplet reduces to the pure YM theory, and the SW spectralcurves for pure YM with gauge algebra g have been shown to be the spectralcurves of the Toda system for (g (1) ) ∨ . The effective prepotential can beevaluated explicitly in the case of g = D n for n ≤ 5. Its logarithmicsingularity does reproduce the logarithmic singularities expected from fieldtheory considerations.As in the known correspondences between SW theory and integrablemodels, we expect the following equation to hold [D’Hoker and Phong(1998d)]∂F∂τ = Htwisted g (x, p).


784 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNote that the left hand side can be interpreted in the gauge theory as arenormalization group equation.For simple laced g, the curves R(k, z) = 0 are modular invariant. Physically,the gauge theories for these Lie algebras are self–dual. For nonsimply–laced g, the modular group is broken to the congruence subgroupΓ 0 (2) for g = B n , C n , F 4 , and to Γ 0 (3) for G 2 . The Hamiltonians of thetwisted CM–systems for non–simply laced g are also transformed underLanden transformations into the Hamiltonians of the twisted CM–systemfor the dual algebra g ∨ . It would be interesting to determine whether suchtransformations exist for the spectral curves or the corresponding gaugetheories themselves.4.14.12 SW Theory and WDVV EquationsAs presented above, N. Seiberg and E. Witten proposed in [Seiberg andWitten (1994a); Seiberg and Witten (1994b)] a new way to deal with thelow–energy effective actions of N = 2 4D supersymmetric gauge theories,both pure gauge theories (i.e., containing only vector super–multiplet) andthose with matter hypermultiplets. Among other things, they have shownthat the low–energy effective actions (the end–points of the renormalizationgroup flows) fit into universality classes depending on the vacuum of thetheory. If the moduli space of these vacua is a finite–dimensional variety,the effective actions can be essentially described in terms of system withfinite–dimensional phase space (number of degrees of freedom is equal tothe rank of the gauge group), although the original theory lives in a many–dimensional space–time.4.14.12.1 WDVV EquationsNow, it turns out that the prepotential of SW effective theory satisfies thefollowing set of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations:F i F −1kF j = F j F −1k F i (i, j, k = 1, . . . , n), (4.290)where F i are the matrices on a moduli space M of third order derivatives∂ 3 F(F i ) jk =∂a i ∂a j ∂a kof a function F(a 1 , . . . , a n ), with the prepotential variables a i .


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 785Although generally there is a lot of solutions to the matrix equations(4.290), it is extremely non–trivial task to express all the matrix elementsthrough the only function F. In fact, there have been only known the twodifferent classes of the non–trivial solutions to the WDVV equations, bothbeing intimately related to the 2D topological theories of type A (quantumcohomologies) and of type B (N = 2 SUSY Landau-Ginzburg (LG)theories). Thus, the existence of a new class of solutions connected withthe 4D theories looks quite striking. It is worth noting that both the 2Dtopological theories and the SW theories reveal the integrability structuresrelated to the WDVV equations.This system of nonlinear equations is satisfied by the SW prepotentialdefining the low-energy effective action. Moreover the leading perturbativeapproximation to this exact SW prepotential should satisfy this set ofequations by itself. For instance, for the gauge group SU(n) the expression[Mironov (1998)]F pert = 1 4∑i≤i


786 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand the matrices F i (the third derivatives of the prepotential) are⎛ ⎞0 0 1F 1 = ⎝ 0 1 0 ⎠ ,⎛ ⎞0 1 0F 2 = ⎝ 1 0 0 ⎠ ,⎛ ⎞1 0 0F 3 = ⎝ 0 0 q ⎠ .1 0 00 0 q0 q 0The second example is the quantum cohomologies of CP 2 . In this case,the prepotential is given by the formulaF = 1 2 a 1a 2 2 + 1 2 a2 1a 3 +∞∑k=1N k a 3k−13(3k − 1)! eka2 , (4.292)where the coefficients N k (describing the rational Gromov–Witten classes)count the number of the rational curves in CP 2 and are to be calculated.Since the matrices F i have the form⎛ ⎞ ⎛⎞ ⎛⎞0 0 10 1 01 0 0F 1 = ⎝ 0 1 0 ⎠ , F 2 = ⎝ 1 F 222 F 223⎠ , F 3 = ⎝ 0 F 223 F 233⎠ ,1 0 00 F 223 F 233 0 F 233 F 333the WDVV equations are equivalent to the identity [Mironov (1998)]F 333 = F 2 223 − F 222 F 233 ,which, in turn, results into the recurrent relation defining the coefficientsN k :N k(3k − 4)! = ∑a+b=ka 2 b(3b − 1)b(2a − b)N a N b .(3a − 1)!(3b − 1)!The crucial feature of the presented examples is that, in both cases, thereexists a constant matrix F 1 . One can consider it as a flat metric on themoduli space. In fact, in its original version, the WDVV equations havebeen written in a slightly different form, that is, as the associativity conditionof some algebra. Having distinguished the (constant) metric η ≡ F 1 ,one can naturally rewrite (4.290) as the equationsC i C j = C j C i (4.293)for the matrices (C i ) jk≡ η −1 F i , i.e., C j ik = ηjl F ilk . Formula (4.293)is equivalent to (4.290) with j = 1. Moreover, this particular relationis already sufficient to reproduce the whole set of the WDVV equations(4.290).


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 787Let us also note that, although the WDVV equations can be fulfilledonly for some specific choices of the coordinates a i on the moduli space,they still admit any linear transformation. This defines the flat structureson the moduli space, and we often call a i flat coordinates.4.14.12.2 Perturbative SW PrepotentialsBefore going into the discussion of the WDVV equations for the completeSW prepotentials, let us note that the leading perturbative part of themshould satisfy the equations (4.290) by itself (since the classical quadraticpiece does not contribute into the third derivatives). On the other hand,if the WDVV equations are fulfilled for perturbative prepotential, it is anecessary condition for them to hold for complete prepotential.To determine the one–loop perturbative prepotential from the field theorycalculation, let us note that there are two origins of masses in N = 2SUSY YM models: first, they can be generated by vacuum values of thescalar φ from the gauge super–multiplet. For a super–multiplet in representationR of the gauge group G this contribution to the prepotential isgiven by the analog of the Coleman-Weinberg formula (from now on, weomit the classical part of the prepotential from all expressions):F R = ± 1 4 Tr R φ 2 log φ, (4.294)and the sign is ‘+’ for vector super–multiplets (normally they are in theadjoint representation) and ‘−’ for matter hypermultiplets. Second, thereare bare masses m R which should be added to φ in (4.294). As a result,the general expression for the perturbative prepotential isF = 1 4− 1 4∑vectormplets∑hypermpletsTr A (φ + M n I A ) 2 log(φ + M n I A ) −Tr R (φ + m R I R ) 2 log(φ + m R I R ) + f(m), (4.295)where the term f(m), depending only on masses, is not fixed by the (perturbative)field theory, but can be read off from the non-perturbative description,and I R denotes the unit matrix in the representation R.As an example, consider the SU(n) gauge group. Then, e.g., perturba-


788 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiontive prepotential for the pure gauge theory acquires the formF pertV= 1 4∑(a i − a j ) 2 log (a i − a j ) .ijThis formula states that when eigenvalues of the scalar fields in the gaugesuper–multiplet are non-vanishing (perturbatively a r are eigenvalues of thevacuum expectation matrix 〈φ〉), the fields in the gauge multiplet acquiremasses m rr ′ = a r − a r ′ (the pair of indices (r, r ′ ) label a field in the adjointrepresentation of G). In the SU(n) case, the eigenvalues are subjectto the condition ∑ i a i = 0. Analogous formulas for the adjoint mattercontribution to the prepotential isF pertA= − 1 4∑(a i − a j + M) 2 log (a i − a j + M) ,ijwhile the contribution of the fundamental matter readsF pertF= − 1 ∑(a i + m) 2 log (a i + m) .4iThe perturbative prepotentials have the following characteristics[Mironov (1998)]:(i) The WDVV equations always hold for the pure gauge theories:F pert = F pertV.(ii) If one considers the gauge super–multiplets interacting with the matterhypermultiplets in the first fundamental representation with masses m αF pert = F pertV+ rF pertF+ Kf F (m) (where r and K are some undeterminedcoefficients), the WDVV equations do not hold unlessK = r 2 /4, f F (m) = 1 ∑(m α − m β ) 2 log (m α − m β ) ,4α,βthe masses being regarded as moduli (i.e., the equations (4.290) contain thederivatives with respect to masses).(iii) If in the theory the adjoint matter hypermultiplets are presented,i.e., F pert = F pertV+ F pertA+ f A (m), the WDVV equations never hold.From the investigation of the WDVV equations for the perturbativeprepotentials, one can learn the following lessons:• Masses are to be regarded as moduli.• As an empiric rule, one may say that the WDVV equations are satisfiedby perturbative prepotentials which depend only on the pairwise sums


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 789of the type (a i ± b j ), where moduli a i and b j are either periods ormasses 38 . This is the case for the models that contain either massivematter hypermultiplets in the first fundamental representation (or itsdual), or massless matter in the square product of those. Troubles arisein all other situations because of the terms with a i ± b j ± c k ± . . ..• At value r = 2, like a i ’s lying in irrep of G, masses m α ’s can be regardedas lying in irrep of some ˜G so that if G = A n , C n , D n , ˜G = An , D n ,C n accordingly. This correspondence ‘explains’ the form of the massterm in the prepotential f(m).4.14.12.3 Associativity ConditionsIn the context of the 2D LG topological theories, the WDVV equationsarose as associativity condition of some polynomial algebra. Mironov hasproved in [Mironov (1998)] that the equations in the SW theories have thesame origin.In this case, one deals with the chiral ring formed by a set of polynomials{Φ i (λ)} and two co-prime (i.e., without common zeroes) fixed polynomialsQ(λ) and P (λ). The polynomials Φ form the associative algebra with thestructure constants Cij k given with respect to the product defined by moduloP ′ :the associativity condition beingΦ i Φ j = C k ijΦ k Q ′ + (∗)P ′ −→ C k ijΦ k Q ′ , (4.296)(Φ i Φ j ) Φ k = Φ i (Φ j Φ k ) , (4.297)i.e., C i C j = C j C i , (C i ) j k = Cj ik . (4.298)Now, in order to get from these conditions the WDVV equations, one needsto choose properly the flat moduli:a i = −n ( )i(n − i) Res P i/n dQ , n = ord(P ).Then, there exists the prepotential whose third derivatives are given by theresidue formulaF ijk = 12πi Res Φ i Φ j Φ kP ′ =0 P ′ . (4.299)38 This general rule can be easily interpreted in D–brane terms, since the interactionof branes is caused by strings between them. The pairwise structure (a i ± b j ) exactlyreflects this fact, a i and b j should be identified with the ends of string.


790 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOn the other hand, from the associativity condition (4.298) and residueformula (4.299), one obtains thatF ijk = (C i ) l j F Q ′ lk, i.e., C i = F i F −1Q ′ . (4.300)Substituting this formula for C i into (4.298), one finally reaches the equationsof the WDVV type. The choice Q ′ = Φ l gives the standard equations(4.290). In 2D topological theories, there is always the unity operator thatcorresponds to Q ′ = 1 and leads to the constant metric F Q ′.Thus, from this short study of the WDVV equations in the LG theories,we can get three main ingredients necessary for these equations to hold.These are:• associative algebra• flat moduli (coordinates)• residue formulaIn the SW theory only the first ingredient requires a non–trivial check.4.14.12.4 SW Theories and Integrable SystemsNow we turn to the WDVV equations in the SW construction [Seibergand Witten (1994a); Seiberg and Witten (1994b)] and show how they arerelated to an integrable system underlying the corresponding SW theory.The most important result of [Seiberg and Witten (1994a); Seiberg andWitten (1994b)], from this point of view, is that the moduli space of vacuaand low energy effective action in SYM theories are completely given bythe following input data:• Riemann surface C• moduli space M (of the curves C)• meromorphic 1–form dS on CThis input can be naturally described in the framework of some underlyingintegrable system. Let us consider a concrete example: the SU(n) puregauge SYM theory that can be described by the periodic Toda chain withn sites. This integrable system is entirely given by the Lax operator⎛p 1 e q1−q2 wL(w) =e q1−q2 p 2 .⎜⎝. ⎟. . . .. . ⎠ . (4.301)1w. . . p n⎞


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 791The Riemann surface C of the SW data is nothing but the spectral curveof the integrable system, which is given by the equation: det (L(w) − λ) =0. Taking into account (4.301), one can get from this formula the equationw + 1 n w = P (λ) = ∏(λ − λ i ) ,i=1∑λ i = 0, (4.302)iwhere the ramification points λ i are Hamiltonians (i.e., integrals of motion)parametrizing the moduli space M of the spectral curves. The substitutionY ≡ w − 1/w transforms the curve (4.302) to the standard hyper–ellipticform Y 2 = P 2 − 4, the genus of the curve being n − 1.As to the meromorphic 1–formdS = λ dw w = λdP Y ,it is just the shorten action ‘pdq’ along the non–contractible contours on theHamiltonian tori. Its defining property is that the derivatives of dS withrespect to the moduli (ramification points) are holomorphic differentials onthe spectral curve.Now, following [Mironov (1998)], let us describe the general integrableframework for the SW construction and start with the theories withoutmatter hypermultiplets. First, matter hypermultiplets. First, we introducebare spectral curve E that is torus: y 2 = x 3 + g 2 x 2 + g 3 for the UV finiteSYM theories with the associated holomorphic 1–form: dω = dx/y. Thisbare spectral curve degenerates into the double–punctured sphere (annulus)for the asymptotically free theories:x −→ w + 1/w, y −→ w − 1/w, dω = dw/w. On this bare curve, there isgiven a matrix–valued Lax operator L(x, y). The dressed spectral curve isdefined from the formula det(L − λ) = 0. This spectral curve is a ramifiedcovering of E given by the equationP(λ; x, y) = 0 (4.303)In the case of the gauge group G = SU(n), the function P is a polynomialof degree n in λ.Thus, the moduli space M of the spectral curve is given just by coefficientsof P. The generating 1-form dS ∼ = λdω is meromorphic on C(hereafter the equality modulo total derivatives is denoted by ‘ ∼ =’).


792 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe prepotential and other ‘physical’ quantities are defined in terms ofthe cohomology class of dS:∮a i = dS, a D i ≡ ∂F ∮= dS, A I ◦ B J = δ IJ . (4.304)A i∂a i B iThe first identity defines here the appropriate flat moduli, while the secondone – the prepotential. The derivatives of the generating differential dSgive holomorphic 1–differentials:∂dS∂a i= dω i (4.305)and, therefore, the second derivative of the prepotential is the period matrixof the curve C:∂ 2 F∂a i ∂a j= T ij .The latter formula allows one to identify prepotential with logarithm of theτ−function of Whitham hierarchy: F = log τ.So far we reckoned without massive hypermultiplets. In order to includethem, one just needs to consider the surface C with punctures. Then, themasses are proportional to residues of dS at the punctures, and the modulispace has to be extended to include these mass moduli. The correspondencebetween SYM theories and integrable systems is built through the SWconstruction in most of known cases that are collected in the followingtable [Mironov (1998)].SUSY 4D pure gauge 4D SYM with 4D SYM with 5d pure gaugephysical SYM theory, fundamental adjoint matter SYM theorytheory gauge group G matterUnderlying Toda chain Rational Calogero–Moser Relativisticintegrable for the dual spin chain system Toda chainsystem affine Ĝ∨ of XXX typeBarespectral sphere sphere torus spherecurveDressedspectral hyper–elliptic hyper–elliptic non-hyper–elliptic hyper–ellipticcurveGeneratingmeromorphic1–form dSλ dw wλ dw wλ dx ylog λ dw wCorrespondence: SUSY gauge theories ⇐⇒ integrable systems


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 793To complete this table, we describe the dressed spectral curves in each casein more explicit terms. Let us note that in all but adjoint matter cases thecurves are hyper–elliptic and can be described by the general formulaP(λ, w) = 2P (λ) − w − Q(λ)w . (4.306)Here P (λ) is characteristic polynomial of the algebra G itself, i.e.,P (λ) = det(G − λI) = ∏ i(λ − λ i ),where determinant is taken in the first fundamental representation and λ i ’sare the eigenvalues of the algebraic element G. For the pure gauge theorieswith the classical groups, Q(λ) = λ 2s andA n−1 : P (λ) =B n : P (λ) = λC n : P (λ) =D n : P (λ) =n∏(λ − λ i ), s = 0; (4.307)i=1n∏(λ 2 − λ 2 i ), s = 2; (4.308)i=1n∏(λ 2 − λ 2 i ), s = −2; (4.309)i=1n∏(λ 2 − λ 2 i ), s = 2 (4.310)i=1For exceptional groups, the curves arising from the characteristic polynomialsof the dual affine algebras do not acquire the hyper–elliptic form.Therefore, in this case, the line ‘dressed spectral curve’ in the table has tobe corrected.In order to include n F massive hypermultiplets in the first fundamentalrepresentation, one can just change λ 2s for Q(λ) = λ 2s ∏ n Fι=1 (λ − m ι) ifG = A n and for Q(λ) = λ 2s ∏ n Fι=1 (λ2 − m 2 ι ) if G = B n , C n , D n .At last, the 5D theory is just described by Q(λ) = λ n/2 .In the Calogero–Moser case, the spectral curve is non–hyper–elliptic,since the bare curve is elliptic. Therefore, it can be described as somecovering of the hyper–elliptic curve.4.14.12.5 WDVV Equations in SW TheoriesAs we already discussed, in order to derive the WDVV equations along theline used in the context of the LG theories, we need three crucial ingredients:


794 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionflat moduli, residue formula and associative algebra. However, the firsttwo of these are always contained in the SW construction provided theunderlying integrable system is known. Indeed, one can derive (see [4,5])the following residue formuladω i dω j dω kF ijk = Res, (4.311)dω=0 dωdλwhere the proper flat moduli a i ’s are given by formula (4.304). Thus, theonly point is to be checked is the existence of the associative algebra. Theresidue formula (4.311) hints that this algebra is to be the algebra Ω 1 ofthe holomorphic differentials dω i . In the forthcoming discussion we restrictourselves to the case of pure gauge theory, the general case being treatedin complete analogy.Let us consider the algebra Ω 1 and fix three differentials dQ, dω, dλ ∈Ω 1 . The product in this algebra is given by the expansiondω i dω j = C k ijdω k dQ + (∗)dω + (∗)dλ (4.312)that should be factorized over the ideal spanned by the differentials dω anddλ. This product belongs to the space of quadratic holomorphic differentials:Ω 1 · Ω 1 ∈ Ω 2 ∼ = Ω1 · (dQ ⊕ dω ⊕ dλ) . (4.313)Since the dimension of the space of quadratic holomorphic differentials isequal to 3g − 3, the l.h.s. of (6.6) with arbitrary dω i ’s is the vector spaceof dimension 3g − 3. At the same time, at the r.h.s. of (6.6) there are garbitrary coefficients Cij k in the first term (since there are exactly so manyholomorphic 1–forms that span the arbitrary holomorphic 1–form Cij k dω k),g −1 arbitrary holomorphic differentials in the second term (one differentialshould be subtracted to avoid the double counting) and g − 2 holomorphic1–forms in the third one. Thus, totally we get that the r.h.s. of (6.6) isspanned also by the basis of dimension g + (g − 1) + (g − 2) = 3g − 3.This means that the algebra exists in the general case of the SW construction.However, this algebra generally is not associative. This is because,unlike the LG case, when it was the algebra of polynomials and,therefore, the product of the two belonged to the same space (of polynomials),product in the algebra of holomorphic 1–differentials no longer belongsto the same space but to the space of quadratic holomorphic differentials.Indeed, to check associativity, one needs to consider the triple product of


<strong>Applied</strong> Bundle <strong>Geom</strong>etry 795Ω 1 :Ω 1 · Ω 1 · Ω 1 ∈ Ω 3 = Ω 1· (dQ) 2 ⊕ Ω 2· dω ⊕ Ω 2· dλ (4.314)Now let us repeat our calculation: the dimension of the l.h.s. of thisexpression is 5g − 5 that is the dimension of the space of holomorphic3–differentials. The dimension of the first space in expansion of ther.h.s. is g, the second one is 3g − 4 and the third one is 2g − 4. Sinceg + (3g − 4) + (2g − 4) = 6g − 8 is greater than 5g − 5 (unless g ≤ 3),formula (4.314) does not define the unique expansion of the triple productof Ω 1 and, therefore, the associativity spoils.The situation can be improved if one considers the curves with additionalinvolutions. As an example, let us consider the family of hyper–elliptic curves: y 2 = P ol 2g+2 (λ). In this case, there is the involution,σ : y −→ −y and Ω 1 is spanned by the σ−odd holomorphic 1–forms xi−1 dxy,i = 1, ..., g. Let us also note that both dQ and dω are σ−odd, while dλ isσ−even. This latter fact means that dλ can be only meromorphic unlessthere are punctures on the surface (which is, indeed, the case in the presenceof the mass hypermultiplets). Thus, formula (6.6) can be replaced bythat without dλΩ 2 + = Ω 1 − · dQ ⊕ Ω 1 − · dω, (4.315)where we have expanded the space of holomorphic 2–forms into the partswith definite σ−parity: Ω 2 = Ω 2 + ⊕ Ω 2 −, which are manifestly given bythe differentials xi−1 (dx) 2y, i = 1, ..., 2g − 1 and xi−1 (dx) 22 y, i = 1, ..., g − 2respectively. Now it is easy to understand that the dimensions of the l.h.s.and r.h.s. of (4.315) coincide and are equal to 2g − 1.Analogously, in this case, one can check the associativity. It is given bythe expansionΩ 3 − = Ω 1 − · (dQ) 2 ⊕ Ω 2 + · dω,where both the l.h.s. and r.h.s. have the same dimension: 3g − 2 = g +(2g − 2). Thus, the algebra of holomorphic 1–forms on hyper–elliptic curveis really associative [Mironov (1998)].


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Chapter 5<strong>Applied</strong> Jet <strong>Geom</strong>etryModern formulation of generalized Lagrangian and Hamiltonian dynamicson fibre bundles is developed in the language of jet spaces, or jetmanifolds (see [Kolar et al. (1993); Saunders (1989); Griffiths (1983);Bryant et. al. (1991); Bryant et al. (2003); Giachetta et. al. (1997);Mangiarotti et. al. (1999); Mangiarotti and Sardanashvily (2000a); Saunders(1989); Sardanashvily (1993); Sardanashvily (1995); Sardanashvily(2002a)]).Roughly speaking, given two smooth manifolds M and N, the twosmooth maps f, g : M → N between them are said to determine the samek−jet at a point x ∈ M, if they have the kth order contact (or, the kth ordertangency) at x [Kolar et al. (1993); Arnold (1988a)]. A set of all k−jetsfrom M to N is a jet space J k (M, N). It is a generalization of a tangentbundle that makes a new smooth fiber bundle out of a given smooth fiberbundle – following the recursive n−categorical process. It makes it possibleto write differential equations on sections of a fiber bundle in an invariantform. Historically, jet spaces are attributed to C. Ehresmann, and were anadvance on the method of prolongation of E. Cartan, of dealing geometricallywith higher derivatives, by imposing differential form conditions onnewly–introduced formal variables.5.1 Intuition Behind a Jet SpaceThe concept of jet space is based on the idea of higher–order tangency, orhigher–order contact, at some designated point on a smooth manifold (see[Arnold (1988a); Kolar et al. (1993)]). Namely, a pair of smooth manifold797


798 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmaps (see Figure 5.1),f 1 , f 2 : M → Nare said to be k−tangent (or tangent of order k, or have a kth order contact)at a point x on a domain manifold M, denoted by f 1 ∼ f 2 , ifff 1 (x) = f 2 (x) called 0 − tangent,∂ x f 1 (x) = ∂ x f 2 (x), called 1 − tangent,∂ xx f 1 (x) = ∂ xx f 2 (x), called 2 − tangent,... etc. to the order k.Fig. 5.1 An intuitive geometrical picture behind the k−jet concept, based on the ideaof higher–order tangency or contact (see text for explanation).In this way defined k−tangency is an equivalence relation, i.e.,f 1 ∼ f 2 ⇒ f 2 ∼ f 1 , f 1 ∼ f 2 ∼ f 3 ⇒ f 1 ∼ f 3 , f 1 ∼ f 1.Now a k−jet (or, a jet of order k), denoted by j k xf, of a smooth mapf : M → N at a point x ∈ M (see Figure 5.1), is defined as an equivalenceclass of k−tangent maps at x,j k xf = {f ′ : f ′ is k − tangent to f at x}.The point x is called the source and the point f(x) is the target of thek−jet j k xf.We choose local coordinates on M and N in the neighborhood of thepoints x and f(x), respectively. Then the k−jet j k xf of any map close tof, at any point close to x, can be given by its Taylor–series expansion at


<strong>Applied</strong> Jet <strong>Geom</strong>etry 799x, with coefficients up to degree k. Therefore, in a fixed coordinate chart,a k−jet can be identified with the collection of Taylor coefficients up todegree k.The set of all k−jets of smooth maps from M to N is called the k−jetspace and denoted by J k (M, N). It has a natural smooth manifold structure.Also, a map from a k−jet space J k (M, N) to a smooth manifold Mor N is called a jet bundle (we will make this notion more precise later).For example, consider a simple function f : X → Y, x ↦→ y = f(x),mapping the X−axis into the Y −axis. In this case, M = X is a domainand N = Y is a codomain. A 0−jet at a point x ∈ X is given by its graph(x, f(x)). A 1−jet is given by a triple (x, f(x), f ′ (x)), a 2−jet is givenby a quadruple (x, f(x), f ′ (x), f ′′ (x)), and so on up to the order k (wheref ′ (x) = df(x)dx, etc.). The set of all k−jets from X to Y is called the k−jetspace J k (X, Y ).Fig. 5.2 Common spaces associated with a function f on a smooth manifold M (seetext for explanation).In case of a function of two variables, f(x, y), the common spaces relatedto f, including its 1–jet j 1 f, are depicted in Figure 5.2 (see [Omohundro(1986)]). Recall that a hypersurface is a codimension–1 submanifold. Givena sample function f(x, y) = x 2 + y 2 in M = R 2 , then: (a) shows its graphas a hypersurface in R × M; (b) shows its level sets in M; (c) shows its


800 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondifferential form df = 2xdx + 2ydy in the cotangent bundle T ∗ M; and (d)shows a tangent hyperplane at a point (x 0 , y y ) ∈ M to its graph in R × M,which is a 1–jet j(x 1 f to f at (x 0,y 0) 0, y y ). Note that j(x 1 0,y 0)f is parallel todf, which means that its 1–jet space J 1 (R, M) is an (n + 1)D extension ofthe cotangent bundle T ∗ M.In mechanics we will consider a pair of maps f 1 , f 2 : R → M from thereal line R, representing the time t−axis, into a smooth nD (configuration)manifold M. We say that the maps f 1 = f 1 (t) and f 2 = f 2 (t) have thesame k−jet jxf k at a specified time instant t 0 ∈ R, iff:(1) f 1 (t) = f 2 (t) at t 0 ∈ R, and also(2) the first k terms of their Taylor–series expansion around t 0 ∈ R areequal.The k−jet space J k (R, M) is the set of all k−jets j k xf from R to M.Now, the fundamental geometrical construct in time–dependent mechanicsis its configuration fibre bundle (see section 5.6 below). Given a configurationfibre bundle M → R over the time axis R, we say that the 1−jetspace J 1 (R, M) is the set of equivalence classes j 1 t s of sections s i : R → Mof the bundle M → R, which are identified by their values s i (t), and by thevalues of their partial derivatives ∂ t s i = ∂ t s i (t) at time points t ∈ R. The1–jet space J 1 (R, M) is coordinated by (t, x i , ẋ i ), so the 1–jets are localcoordinate mapsj 1 t s : t ↦→ (t, x i , ẋ i ).Similarly, the 2−jet space J 2 (R, M) is the set of equivalence classes j 2 t sof sections s i : R → M of the bundle M → R, which are identified bytheir values s i (t), as well as the values of their first and second partialderivatives, ∂ t s i and ∂ tt s i , at time points t ∈ R. The 2–jet space J 2 (R, M)is coordinated by (t, x i , ẋ i , ẍ i ), so the 2–jets are local coordinate mapsj 2 t s : t ↦→ (t, x i , ẋ i , ẍ i ).Generalization to the k−jet space J k (R, M) is obvious. This mechanicaljet formalism will be developed in section 5.6 below.More generally, in a physical field context, instead of the mechanicalconfiguration bundle over the time axis R, we have some general physicalfibre bundle Y → X over some smooth manifold (base) X. In this generalcontext, the k−jet space J k (X, Y ) of a bundle Y → X is the set of equivalenceclasses j k xs of sections s i : X → Y , which are identified by their values


<strong>Applied</strong> Jet <strong>Geom</strong>etry 801s i (x), as well as the first k terms of their Taylor–series expansion at pointsx ∈ X. This has two important physical consequences:(1) The k−jet space of sections s i : X → Y of a fibre bundle Y → X isitself an nD smooth manifold, and(2) A kth–order differential operator on sections s i (x) of a fibre bundleY → X can be described as a map of J k (X, Y ) to a vector bundle overthe base X.A map from a k−jet space J k (X, Y ) to a smooth manifold Y or X is calleda jet bundle.As a consequence, the dynamics of mechanical and physical field systemsis played out on nD configuration and phase manifolds. Moreover, thisdynamics can be phrased in geometrical terms due to the 1–1 correspondencebetween sections of the jet bundle J 1 (X, Y ) → Y and connectionson the fibre bundle Y → X.In the framework of the standard first–order Lagrangian formalism, thenD configuration space of sections s i : X → Y of a fibre bundle Y → Xis the 1–jet space J 1 (X, Y ), coordinated by (x α , y i , y i α), where (x α , y i ) arefibre coordinates of Y , while y i α are the so–called ‘derivative coordinates’ or‘velocities’. A first–order Lagrangian density 1 on the configuration manifoldJ 1 (X, Y ) is given by an exterior one–form (the so–called horizontal density)L = L(x α , y i , y i α)ω, with ω = dx 1 ∧ ... ∧ dx n .This physical jet formalism will be developed below.5.2 Definition of a 1–Jet SpaceAs introduced above, a 1–jet is defined as an equivalence class of functionshaving the same value and the same first derivatives at some designatedpoint of the domain manifold (see Figure 5.1). Recall that in mechanical1 Recall that in classical field theory, a distinction is made between the Lagrangian L,of which the action is the time integral S[x i ] = R L[x i , ẋ i ]dt and the Lagrangian densityL, which one integrates over all space–time to get the action S[ϕ k ] = R L[ϕ k [x i ]]d 4 x.The Lagrangian is then the spatial integral of the Lagrangian density. However, L is alsofrequently simply called the Lagrangian, especially in modern use; it is far more useful inrelativistic theories since it is a locally defined, Lorentz scalar field. Both definitions ofthe Lagrangian can be seen as special cases of the general form, depending on whetherthe spatial variable x i is incorporated into the index i or the parameters s in ϕ k [x i ].Quantum field theories are usually described in terms of L, and the terms in this formof the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams.


802 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsettings, the 1–jets are local coordinate mapsj 1 t s : t ↦→ (t, x i , ẋ i ).More generally, given a fibre bundle Y → X with bundle coordinates(x α , y i ), consider the equivalence classes j 1 xs of its sections s i : X → Y ,which are identified by their values s i (x) and the values of their first–orderderivatives ∂ α s i = ∂ α s i (x) at a point x on the domain (base) manifold X.They are called the 1–jets of sections s i at x ∈ X. One can justify thatthe definition of jets is coordinate–independent by observing that the setJ 1 (X, Y ) of 1–jets j 1 xs is a smooth manifold with respect to the adaptedcoordinates (x α , y i , y i α), such that [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)]y i α(j 1 xs) = ∂ α s i (x), y ′iα = ∂xµ∂x ′ α (∂ µ + y j µ∂ j )y ′i .J 1 (X, Y ) is called the 1−jet space of the fibre bundle Y → X.In other words, the 1–jets j 1 xs : x α ↦→ (x α , y i , y i α), which are first–orderequivalence classes of sections of the fibre bundle Y → X, can be identifiedwith their codomain set of adapted coordinates on J 1 (X, Y ),j 1 xs ≡ (x α , y i , y i α).Note that in a section 5.7 below, the mechanical 1–jet space J 1 (R, M) ≡R × T M will be regarded as a fibre bundle over the base product–manifoldR × M (see [Neagu and Udrişte (2000a); Udriste (2000); Neagu (2002);Neagu and Udrişte (2000b); Neagu (2000)] for technical details).The jet space J 1 (X, Y ) admits the natural fibrationsπ 1 : J 1 (X, Y ) ∋ j 1 xs ↦→ x ∈ X, and (5.1)π 1 0 : J 1 (X, Y ) ∋ j 1 xs ↦→ s(x) ∈ Y, (5.2)which form the commutative triangle:π 1 0J 1 (X, Y )✲ Y❅π 1 ❅❅❅❘ π✠X


<strong>Applied</strong> Jet <strong>Geom</strong>etry 803It is convenient to call π 1 (5.1) the jet bundle, while π 1 0 (5.2) is called theaffine jet bundle. Note that, if Y → X is a vector or an affine bundle, italso holds for the jet bundle π 1 (5.1) [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)].There exist several equivalent ways in order to give the 1–jet spaceJ 1 (X, Y ) with the smooth manifold structure. Let Y → X be a fibrebundle with fibred coordinate atlases (4.4). The 1–jet space J 1 (X, Y ) ofthe bundle Y → X admits the adapted coordinate atlases(x α , y i , y i α), (x α , y i , y i α)(j 1 xs) = (x α , s i (x), ∂ α s i ), (5.3)y ′iα = ( ∂y′ i∂y j yj µ + ∂y′ i∂x µ ) ∂xµ∂x ′ α , (5.4)and thus satisfies the conditions which are required of a manifold. Thesurjection (5.1) is a bundle. The surjection (5.2) is a bundle. If Y → X isa bundle, so is the surjection (5.1).The transformation law (5.4) shows that the jet bundle J 1 (X, Y ) → Yis an affine bundle. It is modelled on the vector bundle T ∗ X ⊗ V Y → Y.In particular, if Y is the trivial bundleπ 2 : V × R m −→ R m ,the corresponding jet bundle J 1 (X, Y ) −→ R m (5.1) is a trivial bundle.There exist the following two canonical bundle monomorphisms of thejet bundle J 1 (X, Y ) −→ Y [Sardanashvily (1993); Sardanashvily (1995);Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a); Sardanashvily(2002a)]:• the contact mapλ : J 1 (X, Y ) ↩→ T ∗ X ⊗ T Y, λ = dx α ⊗ ̂∂ α = dx α ⊗ (∂ α + y i α∂ i ),(5.5)• the complementary mapθ : J 1 (X, Y ) ↩→ T ∗ Y ⊗V Y, θ = ̂dy i ⊗∂ i = (dy i −y i αdx α )⊗∂ i . (5.6)These canonical maps enable us to express the jet–space machinery in termsof tangent–valued differential forms (see section 4.10 above).The operatorŝ∂ α = ∂ α + y i α∂ i


804 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionare usually called the total derivatives, or the formal derivatives, while theformŝdy i = dy i − y i αdx αare conventionally called the contact forms.Identifying the 1–jet space J 1 (X, Y ) to its images under the canonicalmaps (5.5) and (5.6), one can represent 1–jets j 1 xs ≡ (x α , y i , y i α) by tangent–valued formsdx α ⊗ (∂ α + y i α∂ i ), and (dy i − y i αdx α ) ⊗ ∂ i . (5.7)There exists a jet functor J : Bun → Jet, from the category Bunof fibre bundles to the category Jet of jet spaces. It implies the naturalprolongation of maps of bundles to maps of jet spaces.Every bundle map Φ : Y −→ Y ′ over a diffeomorphism f of X has the1–jet prolongation to the bundle map j 1 Φ : J 1 (X, Y ) −→ J 1 (X, Y ) ′ , givenbyj 1 Φ : j 1 xs ↦→ j 1 f(x) (Φ ◦ s ◦ f −1 ), (5.8)y ′ iα ◦ j 1 Φ = ∂ α (Φ i ◦ f −1 ) + ∂ j (Φ i y j α ◦ f −1 ).It is both an affine bundle map over Φ and a fibred map over the diffeomorphismf. The 1–jet prolongations (5.8) of fibred maps satisfy the chainrulesj 1 (Φ ◦ Φ ′ ) = j 1 Φ ◦ j 1 Φ ′ , j 1 (Id Y ) = Id J 1 (X,Y ) .If Φ is a surjection (resp. an injection), so is j 1 Φ.In particular, every section s of a bundle Y → X admits the 1–jetprolongation to the section j 1 xs of the jet bundle J 1 (X, Y ) → X, given byWe have(y i , y i α) ◦ j 1 xs = (s i (x), ∂ α s i ).λ ◦ j 1 xs = T s,where λ is the contact map (5.5).Every projectable vector–field u on a fibre bundle Y → X,u = u α (x)∂ α + u i (y)∂ i


<strong>Applied</strong> Jet <strong>Geom</strong>etry 805has the 1−jet lift to the projectable vector–field j 1 u on the 1–jet spaceJ 1 (X, Y ), given byj 1 u ≡ u = r 1 ◦ j 1 u : J 1 (X, Y ) → T J 1 (X, Y ),j 1 u ≡ u = u α ∂ α + u i ∂ i + (d α u i − y i µ∂ α u µ )∂ α i . (5.9)<strong>Geom</strong>etrical applications of jet spaces are based on the canonical mapover J 1 (X, Y ),J 1 (X, Y ) × T X → J 1 (X, Y ) × T Y,which means the canonical horizontal splitting of the tangent bundle T Ydetermined over J 1 (X, Y ) as follows [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)].The canonical maps (5.5) and (5.6) induce the bundle monomorphismŝλ : J 1 (X, Y ) × T X → J 1 (X, Y ) × T Y, ∂ α ↦→ ̂∂ α = ∂ α ⌋λ (5.10)̂θ : J 1 (X, Y ) × V ∗ Y → J 1 (X, Y ) × T ∗ Y, dy i ↦→ ̂dy i = θ⌋dy i (5.11)The map (5.10) determines the canonical horizontal splitting of the pull–backJ 1 (X, Y ) × T Y = ̂λ(T X) ⊕ V Y, (5.12)ẋ α ∂ α + ẏ i ∂ i = ẋ α (∂ α + y i α∂ i ) + (ẏ i − ẋ α y i α)∂ i .Similarly, the map (5.11) induces the dual canonical horizontal splitting ofthe pull–backJ 1 (X, Y ) × T ∗ Y = T ∗ X ⊕ ̂θ(V ∗ Y ), (5.13)ẋ α dx α + ẏ i dy i = (ẋ α + ẏ i y i α)dx α + ẏ i (dy i − y i αdx α ).Building on the canonical splittings (5.12) and (5.13), one gets the followingcanonical horizontal splittings of• a projectable vector–field on a fibre bundle Y → X,u = u α ∂ α + u i ∂ i = u H + u V = u α (∂ α + y i α∂ i ) + (u i − u α y i α)∂ i , (5.14)• an exterior 1–formσ = σ α dx α + σ i dy i = (σ α + y i ασ i )dx α + σ i (dy i − y i αdx α ),


806 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction• a tangent–valued projectable horizontal formφ = dx α1 ∧ · · · ∧ dx αr ⊗ (φ µ α 1...α r∂ µ + φ i α 1...α r∂ i )= dx α1 ∧ · · · ∧ dx αr ⊗ [φ µ α 1...α r(∂ µ + y i µ∂ i ) + (φ i α 1...α r− φ µ α 1...α ry i µ)∂ i ]and, e.g., the canonical 1–formθ Y = dx α ⊗ ∂ α + dy i ⊗ ∂ i = α + θ = dx α ⊗ ̂∂ α + ̂dy i ⊗ ∂ i= dx α ⊗ (∂ α + y i α∂ i ) + (dy i − y i αdx α ) ⊗ ∂ i . (5.15)The splitting (5.15) implies the canonical horizontal splitting of theexterior differentiald = d θY = d H + d V = d α + d θ . (5.16)Its components d H and d Vact on the pull–backsφ α1...α r(y)dx α1 ∧ · · · ∧ dx αrof horizontal exterior forms on a bundle Y → X onto J 1 (X, Y ) by π 01 . Inthis case, d H makes the sense of the total differentiald H φ α1...α r(y)dx α1 ∧· · ·∧dx αr = (∂ µ +y i µ∂ i )φ α1...α r(y)dx µ ∧dx α1 ∧· · ·∧dx αr ,whereas d Vis the vertical differentiald V φ α1...α r(y)dx α1 ∧· · ·∧dx αr = ∂ i φ α1...α r(y)(dy i −y i µdx µ )∧dx α1 ∧· · ·∧dx αr .If φ = ˜φω is an exterior horizontal density on Y → X, we havedφ = d V φ = ∂ i˜φdy i ∧ ω.5.3 Connections as Jet FieldsRecall that one can introduce the notion of connections on a general fibrebundle Y −→ X in several equivalent ways. In this section, following [Giachettaet. al. (1997); Kolar et al. (1993); Mangiarotti and Sardanashvily(2000a); Saunders (1989)], we start from the traditional definition of a connectionas a horizontal splitting of the tangent space to Y at every pointy ∈ Y .A connection on a fibre bundle Y → X is usually defined as a linearbundle monomorphismΓ : Y × T X → T Y, Γ : ẋ α ∂ α ↦→ ẋ α (∂ α + Γ i α(y)∂ i ), (5.17)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 807which splits the exact sequence (4.13), i.e.,π T ◦ Γ = Id Y ×T X .The image HY of Y × T X by a connection Γ is called the horizontal distribution.It splits the tangent bundle T Y asT Y = HY ⊕ V Y, giving (5.18)ẋ α ∂ α + ẏ i ∂ i = ẋ α (∂ α + Γ i α∂ i ) + (ẏ i − ẋ α Γ i α)∂ i .Similarly, horizontal splitting of the cotangent bundle,T ∗ Y = T ∗ X ⊕ Γ(V ∗ X),givesẋ α dx α + ẏ i dy i = (ẋ α + Γ i αẏ i )dx α + ẏ i (dy i − Γ i αdx α ).Alternatively, a connection on a fibre bundle Y −→ X can be definedas a jet field, i.e., a section of the affine jet bundle J 1 (X, Y ) −→ Y . Thisconnection is called the Ehresmann connection, and historically it was aprimary reason for C. Ehresmann to develop the concept of jet spaces.Due to the Theorem that says [Hirzebruch (1966)]: Every exact sequenceof vector bundles (4.10) is split, a jet–field connection on a fibre bundlealways exists.A connection on a fibre bundle Y −→ X is defined to be a tangent–valued projectable horizontal one–form Γ on Y such that Γ⌋φ = φ for allexterior horizontal 1–forms φ on Y . It is given by the coordinate expressionΓ = dx α ⊗ (∂ α + Γ i α(y)∂ i ),Γ ′ iα = ( ∂y′ i∂y j Γj µ + ∂y′ i∂x µ ) ∂xµ∂x ′ α , (5.19)such that Γ(∂ α ) = ∂ α ⌋Γ. Conversely, every horizontal tangent–valued 1–form on a fibre bundle Y −→ X which projects onto the canonical tangent–valued form θ X (4.137) on X defines a connection on Y −→ X.In an equivalent way, the horizontal splitting (5.18) is given by thevertical–valued formwhich determines the epimorphismΓ = (dy i − Γ i αdx α ) ⊗ ∂ i , (5.20)Γ : T Y → V Y, ẋ α ∂ α + ẏ i ∂ i ↦→ (ẋ α ∂ α + ẏ i ∂ i )⌋Γ = (ẏ i − ẋ α Γ i α)∂ i .Let Y → X be a vector bundle. A linear connection on Y readsΓ = dx α ⊗ [∂ α − Γ i jα(x)y j ∂ i ]. (5.21)


808 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet Y → X be an affine bundle modelled on a vector bundle Y → X. Anaffine connection on Y readsΓ = dx α ⊗ [∂ α + (−Γ i jα(x)y j + Γ i α(x))∂ i ],whereΓ = dx α ⊗ [∂ α − Γ i jα(x)y j ∂ i ] is a linear connection on Y .Since the affine jet bundle J 1 (X, Y ) −→ Y is modelled on the vectorbundle Y −→ X, Ehresmann connections on Y −→ X constitute an affinespace modelled on the linear space of soldering forms on Y . If Γ is aconnection and σ is a soldering form (4.138) on Y , its sumΓ + σ = dx α ⊗ [∂ α + (Γ i α + σ i α)∂ i ]is a connection on Y . Conversely, if Γ and Γ ′ are connections on Y , thenΓ − Γ ′ = (Γ i α − Γ ′ iα)dx α ⊗ ∂ iis a soldering form.Given a connection Γ, a vector–field u on a fibre bundle Y −→ X iscalled horizontal if it lives in the horizontal distribution HY , i.e., takes theformu = u α (y)(∂ α + Γ i α(y)∂ i ). (5.22)Any vector–field τ on the base X of a fibre bundle Y −→ X admits thehorizontal liftΓτ = τ⌋Γ = τ α (∂ α + Γ i α∂ i ) (5.23)onto Y by means of a connection Γ on Y −→ X.Given the splitting (5.17), the dual splitting of the exact sequence (4.14)isΓ : V ∗ Y → T ∗ Y, dy i ↦→ Γ⌋dy i = dy i − Γ i αdx α , (5.24)where Γ is the vertical-valued form (5.20).There is 1–1 correspondence between the connections on a fibre bundleY → X and the jet fields, i.e., global sections of the affine jet bundleJ 1 (X, Y ) → Y . Indeed, given a global section Γ of J 1 (X, Y ) → Y , thetangent–valued formλ ◦ Γ = dx α ⊗ (∂ α + Γ i α∂ i )


<strong>Applied</strong> Jet <strong>Geom</strong>etry 809gives the horizontal splitting (5.18) of T Y . Therefore, the vertical–valuedformθ ◦ Γ = (dy i − Γ i αdx α )⊗∂ ileads to the dual splitting (5.24).It follows immediately from this definition that connections on a fibrebundle Y → X constitute an affine space modelled over the vector spaceof soldering forms σ (4.138). They obey the coordinate transformation law[Giachetta et. al. (1997); Saunders (1989)]Γ ′i α = ∂xµ∂x ′ α (∂ µ + Γ j µ∂ j )y ′i .In particular, a linear connection K on the tangent bundle T X of amanifold X and the dual connection K ∗ to K on the cotangent bundleT ∗ X are given by the coordinate expressionsK α β = −K α νβ(x)ẋ ν , K ∗ αβ = K ν αβ(x)ẋ ν . (5.25)Also, given a connection Γ on Y → X, the vertical tangent map V Γ :V Y → J 1 (R, V )Y induces the vertical connectionV Γ = dx α ⊗ (∂ α + Γ i α∂ y i + ∂ V Γ i α∂ẏi), ∂ V Γ i α = ẏ j ∂ j Γ i α, (5.26)on the bundle V Y → X. The connection V Γ is projectable to the connectionΓ on Y , and it is a linear bundle map over Γ. The dual coverticalconnection on the bundle V ∗ Y → X readsV ∗ Γ = dx α ⊗ (∂ α + Γ i α∂ y i − ∂ j Γ i αẏ i ∂ẏi). (5.27)Connections on a bundle Y → X constitute the affine space modelledon the linear space of soldering 1–forms on Y . It means that, if Γ is aconnection and σ is a soldering form on a bundle Y , its sumΓ + σ = dx α ⊗ [∂ α + (Γ i α + σ i α)∂ i ]is a connection on Y . Conversely, if Γ and Γ ′ are connections on a bundleY , then their differenceis a soldering form on Y .Γ − Γ ′ = (Γ i α − Γ ′i α)dx α ⊗ ∂ i


810 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionGiven a fibre bundle Y → X, let f : X ′ → X be a manifold map andf ∗ Y the pull–back of Y over X ′ . Any connection Γ (5.20) on Y → Xinduces the pull–back connectionf ∗ Γ = (dy i − Γ i α(f µ (x ′ ν), y j ) ∂f α∂x ′ µ dx′µ ) ⊗ ∂ i (5.28)on the pull–back fibre bundle f ∗ Y → X ′ .Since the affine jet bundle J 1 (X, Y ) −→ Y is modelled on the vectorbundle Y −→ X, connections on a fibre bundle Y constitute the affine spacemodelled on the linear space of soldering forms on Y . It follows that, if Γis a connection andσ = σ i αdx α ⊗ ∂ iis a soldering form on a fibre bundle Y , its sumΓ + σ = dx α ⊗ [∂ α + (Γ i α + σ i α)∂ i ]is a connection on Y . Conversely, if Γ and Γ ′ are connections on a fibrebundle Y , thenΓ − Γ ′ = (Γ i α − Γ ′ iα)dx α ⊗ ∂ iis a soldering form on Y .The key point for physical applications lies in the fact that every connectionΓ on a fibre bundle Y −→ X induces the first–order differentialoperatorD Γ : J 1 (X, Y ) → T ∗ X ⊗ V Y, D Γ = λ − Γ ◦ π 1 0 = (y i α − Γ i α)dx α ⊗ ∂ i ,(5.29)called the covariant differential relative to the connection Γ. If s : X → Yis a section, one defines its covariant differentialand its covariant derivative∇ Γ s = D Γ ◦ j 1 s = (∂ α s i − Γ i α ◦ s)dx α ⊗ ∂ i (5.30)∇ Γ τ s = τ⌋∇ Γ s (5.31)along a vector–field τ on X. A (local) section s of Y → X is said to be anintegral section of a connection Γ (or parallel with respect to Γ) if s obeysthe equivalent conditions∇ Γ s = 0 or j 1 s = Γ ◦ s. (5.32)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 811Furthermore, if s : X → Y is a global section, there exists a connectionΓ such that s is an integral section of Γ. This connection is defined asan extension of the local section s(x) ↦→ j 1 s(x) of the affine jet bundleJ 1 (X, Y ) → Y over the closed imbedded submanifold s(X) ⊂ Y .Note that every connection Γ on the bundle Y −→ X defines a system offirst–order differential equations on Y (in the spirit of [Bryant et. al. (1991);Krasil’shchik et. al. (1985); Pommaret (1978)]) which is an imbedded subbundleΓ(Y ) = Ker D Γ of the jet bundle J 1 (X, Y ) −→ Y . It is given by thecoordinate relationsy i α = Γ i (y). (5.33)Integral sections for Γ are local solutions of (5.33), and vice versa.We can introduce the following basic forms involving a connection Γand a soldering form σ:• the curvature of a connection Γ is given by the horizontal verticalvaluedtwo–form:R = 1 2 d ΓΓ = 1 2 Ri αµdx α ∧ dx µ ⊗ ∂ i ,R i αµ = ∂ α Γ i µ − ∂ µ Γ i α + Γ j α∂ j Γ i µ − Γ j µ∂ j Γ i α; (5.34)• the torsion of a connection Γ with respect to σ:Ω = d σ Γ = d Γ σ = 1 2 Ωi αµdx α ∧ dx µ ⊗ ∂ i• the soldering curvature of σ:= (∂ α σ i µ + Γ j α∂ j σ i µ − ∂ j Γ i ασ j µ)dx α ∧ dx µ ⊗ ∂ i ; (5.35)ε = 1 2 d σσ = 1 2 εi αµdx α ∧ dx µ ⊗ ∂ i= 1 2 (σj α∂ j σ i µ − σ j µ∂ j σ i l)dx α ∧dx µ ⊗ ∂ i . (5.36)They satisfy the following relations:Γ ′ = Γ + σ, R ′ = R + ε + Ω, Ω ′ = Ω + 2ε.In particular, the curvature (5.34) of the linear connection (5.21) readsRαµ(y) i = −R i jαµ(x)y j ,R i jαµ = ∂ α Γ i jµ − ∂ µ Γ i jα + Γ k jµΓ i kα − Γ k jαΓ i kµ.


812 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet Y and Y ′ be vector bundles over X. Given linear connections Γ andΓ ′ on Y and Y ′ respectively, there is the unique linear connection Γ ⊗ Γ ′ onthe tensor product Y ⊗Y ′ → X, such that the following diagram commutes:J 1 (X, Y ) × J 1 (X, Y ) ′ J 1 ⊗ ✲ J 1 (Y ⊗ Y ′ )Γ × Γ ′❄Y × Y ′ ✲ ❄⊗Y ⊗ Y ′Γ ⊗ Γ ′It is called the tensor–product connection and has the coordinate expression(Γ ⊗ Γ ′ ) ikα = Γ i jαy jk + Γ ′ kjαy ij .Every connection Γ on Y → X, by definition, induces the horizontaldistribution on Y ,Γ : T X ↩→ T Y, locally given by ∂ α ↦→ ∂ α + Γ i α(y)∂ i .It is generated by horizontal liftsτ Γ = τ α (∂ α + Γ i α∂ i )onto Y of vector–fields τ = τ α ∂ α on X. The associated Pfaffian system islocally generated by the forms (dy i − Γ i αdx α ).The horizontal distribution Γ(T X) is involutive iff Γ is a curvature–freeconnection. As a proof, straightforward calculations show that [τ Γ , τ ′ Γ] =([τ, τ ′ ]) Γ iff the curvature R (5.34) of Γ vanishes everywhere.Not every bundle admits a curvature–free connection. If a principal bundleover a simply–connected base (i.e., its first homotopy group is trivial)admits a curvature–free connection, this bundle is trivializable [Kobayashiand Nomizu (1963/9)].The horizontal distribution defined by a curvature–free connection iscompletely integrable. The corresponding foliation on Y is transversal tothe foliation defined by the fibration π : Y −→ X. It is called the horizontalfoliation. Its leaf through a point y ∈ Y is defined locally by the integralsection s y of the connection Γ through y. Conversely, let Y admits a horizontalfoliation such that, for each point y ∈ Y , the leaf of this foliationthrough y is locally defined by some section s y of Y −→ X through y. Then,the following map is well definedΓ : Y −→ J 1 (X, Y ), Γ(y) = j 1 s s y , π(y) = x.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 813This is a curvature–free connection on Y . There is the 1–1 correspondencebetween the curvature–free connections and the horizontal foliations on abundle Y −→ X.Given a horizontal foliation on Y −→ X, there exists the associated atlasof bundle coordinates (x α , y i ) of Y such that (i) every leaf of this foliation islocal generated by the equations y i = const, and (ii) the transition functionsy i −→ y ′i (y j ) are independent on the coordinates x α of the base X [Kamberand Tondeur (1975)]. It is called the atlas of constant local trivializations.Two such atlases are said to be equivalent if their union also is an atlas ofconstant local trivializations. They are associated with the same horizontalfoliation.There is the 1–1 correspondence between the curvature–free connectionsΓ on a bundle Y −→ X and the equivalence classes of atlases Ψ c of constantlocal trivializations of Y such that Γ i α = 0 relative to the coordinates of thecorresponding atlas Ψ c [Canarutto (1986)].Connections on a bundle over a 1D base X 1 are curvature–free connections.In particular, let Y −→ X 1 be such a bundle (X 1 = R or X 1 = S 1 ). Itis coordinated by (t, y i ), where t is either the canonical parameter of R orthe standard local coordinate of S 1 together with the transition functionst ′ = t+const. Relative to this coordinate, the base X 1 admits the standardvector–field ∂ t and the standard one–form dt. Let Γ be a connection on Y−→ X 1 . Such a connection defines a horizontal foliation on Y −→ X 1 . Itsleaves are the integral curves of the horizontal liftτ Γ = ∂ t + Γ i ∂ i (5.37)of ∂ t by Γ. The corresponding Pfaffian system is locally generated by theforms (dy i − Γ i dt). There exists an atlas of constant local trivializations(t, y i ) such that Γ i = 0 and τ Γ = ∂ t relative to these coordinates.A connection Γ on Y → X 1 is called complete if the horizontal vector–field (5.37) is complete. Every trivialization of Y → R defines a completeconnection. Conversely, every complete connection on Y → R defines atrivialization Y ≃ R×M. The vector–field (5.37) becomes the vector–field ∂ t on R×M. As a proof, every trivialization of Y → R defines a 1–parameter group of isomorphisms of Y → R over Id R , and hence a completeconnection. Conversely, let Γ be a complete connection on Y → R. Thevector–field τ Γ (5.37) is the generator of a 1–parameter group G Γ which actsfreely on Y . The orbits of this action are the integral sections of τ Γ . Hencewe get a projection Y → M = Y/G Γ which, together with the projection


814 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionY → R, defines a trivialization Y ≃ R×M.Let us consider a bundle π : Y → X which admits a composite fibrationY → Σ → X, (5.38)where Y → Σ and Σ → X are bundles. It is equipped with the bundlecoordinates (x α , σ m , y i ) together with the transition functionsx α → x ′ α (x µ ), σ m → σ ′ m (x µ , σ n ), y i → y ′ i (x µ , σ n , y j ),where (x µ , σ m ) are bundle coordinates of Σ → X. For example, we havethe composite bundlesLetT Y → Y → X, V Y → Y → X, J 1 (X, Y ) → Y → X.A = dx α ⊗ (∂ α + A i α∂ i ) + dσ m ⊗ (∂ m + A i m∂ i ) (5.39)be a connection on the bundle Y −→ Σ andΓ = dx α ⊗ (∂ α + Γ m α ∂ m )a connection on the bundle Σ −→ X. Given a vector–field τ on X, let usconsider its horizontal lift τ Γ onto Σ by Γ and then the horizontal lift (τ Γ ) Aof τ Γ onto Y by the connection (5.39).There exists the connectionγ = dx α ⊗ [∂ α + Γ m α ∂ m + (A i mΓ m α + A i α)∂ i ]. (5.40)on Y → X such that the horizontal lift τ γ onto Y of any vector–field τ onX consists with the above lift (τ Γ ) A [Sardanashvily (1993); Sardanashvily(1995)]. It is called the composite connection.Given a composite bundle Y (5.38), the exact sequence0 → V Y Σ ↩→ V Y → Y × V Σ → 0over Y take place, where V Y Σ is the vertical tangent bundle of Y → Σ.Every connection (5.39) on the bundle Y → Σ induces the splittingV Y = V Y Σ ⊕ (Y × V Σ),given byẏ i ∂ i + ˙σ m ∂ m = (ẏ i − A i m ˙σ m )∂ i + ˙σ m (∂ m + A i m∂ i ).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 815Due to this splitting, one can construct the first–order differential operator˜D = π 1 ◦ D γ : J 1 (X, Y ) → T ∗ X ⊗ V Y → T ∗ X ⊗ V Y Σ ,˜D = dx α ⊗ (y i α − A i α − A i mσ m α )∂ i , (5.41)on the composite manifold Y , where D γ is the covariant differential (5.29)relative to the composite connection (5.40). We call ˜D the vertical covariantdifferential.5.3.1 Principal ConnectionsThe above general approach to connections as jet fields is suitable to formulatethe classical concept of principal connections. In this section, astructure group G of a principal bundle is assumed to be a real finite–dimensional Lie group (of positive dimension dim G > 0).A principal connection A on a principal bundle P → Q is defined to bea G−equivariant global jet field on P such thatj 1 R g ◦ A = A ◦ R gfor each canonical map (4.31). We haveA ◦ R g = j 1 R g ◦ A, (g ∈ G), (5.42)A = dq α ⊗ (∂ α + A m α (p)e m ), (p ∈ P ),A m α (qg) = A m α (p)adg −1 (e m ).A principal connection A determines splitting T Q ↩→ T G P of the exactsequence (4.38). We will refer toA = A − θ Q = A m α dq α ⊗ e m (5.43)as a local connection form.Let J 1 (Q, P ) be the 1–jet space of a principal bundle P → Q with astructure Lie group G. The jet prolongationJ 1 (Q, P ) × J 1 (Q × G) → J 1 (Q, P )of the canonical action (4.31) brings the jet bundle J 1 (Q, P ) → Q into ageneral affine bundle modelled on the group bundleJ 1 (Q × G) = G × (T ∗ Q ⊗ g l ) (5.44)over Q. However, the jet bundle J 1 (Q, P ) → Q fails to be a principal bundlesince the group bundle (5.44) is not a trivial bundle over Q in general. At


816 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe same time, J 1 (Q, P ) is the G principal bundle C × P → C over thequotientC = J 1 (Q, P )/G (5.45)of the jet bundle J 1 (Q, P ) → P by the 1–jet prolongations of the canonicalmaps (4.31).Let J 1 (Q, P ) be the 1–jet space of a principal G−bundle P → Q. Itsquotient (5.45) by the jet prolongation of the canonical action R G (4.31) isa fibre bundle over Q.Given a bundle atlas of P and the associated bundle atlas of V G P , theaffine bundle C admits affine bundle coordinates (t, q i , a q α), and its elementsare represented by T G P −valued 1–formsa = dq α ⊗ (∂ α + a q αe q ) (5.46)on Q. One calls C (5.45) the connection bundle because its sections arenaturally identified with principal connections on the principal bundle P →Q as follows.There is the 1–1 correspondence between the principal connections ona principal bundle P → Q and the global sections of the quotient bundleC = J 1 (Q, P )/G −→ Q.We shall call C the principal connection bundle.modelled on the vector bundleIt is an affine bundleand there is the canonical vertical splittingC = T ∗ Q ⊗ V G P, (5.47)V C = C × C.Given a bundle atlas Ψ P of P , the principal connection bundle C admitsthe fibre coordinates (q µ , k m µ ) so that(k m µ ◦ A)(q) = A m µ (q)are coefficients of the local connection one–form (5.43). The 1–jet spaceJ 1 (Q, C) of C is with the adapted coordinates(q µ , k m µ , k m µλ). (5.48)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 817The affine jet bundle J 1 (Q, C) → C is modelled on the vector bundleThere exists the canonical splittingover C whereT ∗ Q ⊗ (C × T ∗ Q ⊗ V G P ).J 1 (Q, C) = (J 2 P/G) ⊗ (∧ 2 T ∗ Q ⊗ V G P ) (5.49)C − = C × ∧ 2 T ∗ Q ⊗ V G Pand C + → C is the affine bundle modelled on the vector bundleC + = ∧ 2 T ∗ Q ⊗ V G P.In the coordinates (5.48), the splitting (5.49) readsk m µλ = 1 2 (km µλ + k m λµ + c m nlk n αk l µ) + 1 2 (km µλ − k m λµ − c m nlk n αk l µ)where c k mn are structure constants of the Lie algebra g r with respect to itsbasis {I m }.There are the corresponding canonical projections given byS = π 1 : J 1 (Q, C) → C + , S m λµ = k m µλ + k m λµ + c m nlk n αk l µ,and F = π 2 : J 1 (Q, C) → C − , withF = 1 2 F m λµdq α ∧ dq µ ⊗ I m , F m λµ = k m µλ − k m λµ − c m nlk n αk l µ.For every principal connection A, we observe thatF◦j 1 A = F, F = 1 2 F m λµdq α ∧dq µ ⊗I m , F m λµ = ∂ α A m µ −∂ µ A m α −c m nkA n αA k µ,is the strength of A.Given a symmetric linear connection K ∗ on the cotangent bundle T ∗ Qof Q, every principal connection A on a principal bundle P induces theconnectionS A : C → C + , S A ◦ A = S ◦ j 1 A,on the principal connection bundle C. In the coordinates (5.48), the connectionS A readsS Amµλ = 1 2 [cm nlk n αk l µ +∂ µ A m α +∂ α A m µ −c m nl(k n µA l α +k n αA l µ)]−K β µλ(A m β −k m β ).(5.50)


818 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe P −associated bundle Y admits atlases Ψ = {U ξ , ψ ξ } associatedwith atlases Ψ P = {U ξ , z ξ } of the principal bundle P as follows:ψ −1ξ (q × V ) = [z ξ(q)] V (V ), (q ∈ U ξ ),where by [p] V is denoted the restriction of the canonical map P × V → Yto the subset p × V .Every principal connection A on a principal bundle P induces the associatedconnection Γ on a P −associated bundle Y such that the followingdiagram commutes:J 1 (Q, P ) × V ✲ J 1 (X, Y )A × Id V❄P × VΓ❄✲ YWe call it the associated principal connection. With respect to the associatedatlases Ψ of Y and Ψ P of P , this connection is writtenΓ = dq α ⊗ [∂ α + A m µ (q)I mij y j ∂ i ] (5.51)where A m µ (q) are coefficients of the local connection one–form (5.43) andI m are generators of the structure group G on the standard fibre V of thebundle Y . The curvature of the connection (5.51) readsR i λµ = F m λµI mij y i .5.4 Definition of a 2–Jet SpaceAs introduced above, a 2−jet is defined as a second–order equivalence classof functions having the same value and the same first derivatives at somedesignated point of the domain manifold. Recall that in mechanical settings,the 2–jets are local coordinate mapsj 2 t s : t ↦→ (t, x i , ẋ i , ẍ i ).In general, if we recursively apply the jet functor J : Bun → Jet to thejet bundles, we come to the higher order jet spaces (see [Kolar et al. (1993);Sardanashvily (1993); Sardanashvily (1995); Giachetta et. al. (1997);Mangiarotti and Sardanashvily (2000a); Sardanashvily (2002a)]).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 819In particular, taking the 1–jet space of the 1–jet bundle J 1 (X, Y ) −→ X,we get the repeated jet space J 1 (X, J 1 (X, Y )), which admits the adaptedcoordinateswith transition functions(x α , y i , y i α, ŷ i µ, y i µα)ŷ ′ iα = ∂xα∂x ′ α d αy ′ i , y′iµα = ∂xα∂x ′ µ d αy ′ iα, d α = ∂ α + ŷ j α∂ j + y j να∂ ν j .The 2−jet space J 2 (X, Y ) of a fibre bundle Y → X is coordinatedby (x α , y i , yα, i yαµ), i with the local symmetry condition yαµ i = yµα.i Themanifold J 2 (X, Y ) is defined as the set of equivalence classes jxs 2 of sectionss i : X → Y of the bundle Y → X, which are identified by their values s i (x)and the values of their first and second–order partial derivatives at pointsx ∈ X, respectively,y i α(j 2 xs) = ∂ α s i (x),y i αµ(j 2 xs) = ∂ α ∂ µ s i (x).In other words, the 2–jets j 2 xs : x α ↦→ (x α , y i , y i α, y i αµ), which are second–order equivalence classes of sections of the fibre bundle Y → X, can beidentified with their codomain set of adapted coordinates on J 2 (X, Y ),j 2 xs ≡ (x α , y i , y i α, y i αµ).Let s be a section of a fibre bundle Y → X, and let j 1 s be its 1–jetprolongation to a section of the jet bundle J 1 (X, Y ) → X. The latterinduces the section j 1 j 1 s of the repeated jet bundle J 1 (X, J 1 (X, Y )) → X.This section takes its values into the 2–jet space J 2 (X, Y ). It is called the2–jet prolongation of the section s, and is denoted by j 2 s.We have the following affine bundle monomorphismsJ 2 (X, Y ) ↩→ Ĵ 2 (X, Y )(X, Y ) ↩→ J 1 (X, J 1 (X, Y ))over J 1 (X, Y ) and the canonical splittingĴ 2 (X, Y )(X, Y ) = J 2 (X, Y ) ⊕ (∧ 2 T ∗ X ⊗ V Y ),given locally byy i αµ = 1 2 (yi αµ + y i µα) + ( 1 2 (yi αµ − y i µα).In particular, the repeated jet prolongation j 1 j 1 s of a section s : X → Yof the fibre bundle Y → X is a section of the jet bundle J 1 (X, J 1 (X, Y )) →


820 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionX. It takes its values into J 2 (X, Y ) and consists with the 2–jet prolongationj 2 s of s:j 1 j 1 s(x) = j 2 s(x) = j 2 xs.Given a 2–jet space J 2 (X, Y ) of the fibre bundle Y → X, we have(i) the fibred map r 2 : J 2 (Y, T Y ) → T J 2 (X, Y ), given locally by(ẏ i α, ẏ i αµ) ◦ r 2 = ((ẏ i ) α − y i µẋ µ α, (ẏ i ) αµ − y i µẋ µ αµ − y i αµẋ µ α),where J 2 (Y, T Y ) is the 2–jet space of the tangent bundle T Y, and(ii) the canonical isomorphism V J 2 (X, Y ) = J 2 (X, V Y ), where V J 2 (X, Y )is the vertical tangent bundle of the fibre bundle J 2 (X, Y ) → X, andJ 2 (X, V Y ) is the 2–jet space of the fibre bundle V Y → X.As a consequence, every vector–field u on a fibre bundle Y → X admitsthe 2−jet lift to the projectable vector–fieldj 2 u = r 2 ◦ j 2 u : J 2 (X, Y ) → T J 2 (X, Y ).In particular, if2–jet lift readsu = u α ∂ α + u i ∂ i is a projectable vector–field on Y , itsj 2 u = u α ∂ α + u i ∂ i + (∂ α u i + y j α∂ j u i − y i µ∂ α u µ )∂ α i (5.52)+ [(∂ α + y j α∂ j + y j βα ∂β j )(∂ α + y k α∂ k )u i − y i µẋ µ αβ − yi µβẋ µ α]∂ αβi .Generalizations of the contact and complementary maps (5.5–5.6) tothe 2–jet space J 2 (X, Y ) readλ : J 2 (X, Y ) → T ∗ X ⊗ T J 1 (X, Y )is locally given byλ = dx α ⊗ ̂∂ α = dx α ⊗ (∂ α + yα∂ i i + yµα∂ i µ i ), while (5.53)θ : J 2 (X, Y ) → T ∗ J 1 (X, Y ) ⊗ V J 1 (X, Y ) is locally given byθ = (dy i − y i αdx α ) ⊗ ∂ i + (dy i µ − y i µαdx α ) ⊗ ∂ µ i . (5.54)The contact map (5.53) defines the canonical horizontal splitting of theexact sequence0 → V J 1 (X, Y ) ↩→ T J 1 (X, Y ) → J 1 (X, Y ) × T X → 0.Hence, we get the canonical horizontal splitting of a projectable vector–fieldj 1 u ≡ u on J 1 (X, Y ) over J 2 (X, Y ):j 1 u = u H + u V = u α [∂ α + y i α + y i µα] + [(u i − y i αu α )∂ i + (u i µ − y i µαu α )∂ µ i ].


<strong>Applied</strong> Jet <strong>Geom</strong>etry 821Building on the maps (5.53) and (5.54), one can get the horizontalsplittings of the canonical tangent–valued 1–form on J 1 (X, Y ),θ J 1 (X,Y ) = dx α ⊗ ∂ α + dy i ⊗ ∂ i + dy i µ ⊗ ∂ µ i= α + θand the associated exterior differentiald = d θJ 1 (X,Y )= d α + d θ = d H + d V . (5.55)They are similar to the horizontal splittings (5.15) and (5.16).A 2–jet field (resp. a 2–connection) Γ on a fibre bundle Y → X is definedto be a 1–jet field (resp. a 1–connection) on the jet bundle J 1 (X, Y ) → X,i.e., Γ is a section (resp. a global section) of the bundle J 1 (X, J 1 (X, Y )) →J 1 (X, Y ).In the coordinates (y i α, y i (µ) , yi αµ) of the repeated jet spaceJ 1 (X, J 1 (X, Y )), a 2–jet field Γ is given by the expression(y i α, y i (µ) , yi αµ) ◦ Γ = (y i α, Γ i (µ), Γ i αµ).Using the contact map (5.53), one can represent it by the tangent–valuedhorizontal 1–form on the jet bundle J 1 (X, Y ) → X,Γ = dx µ ⊗ (∂ µ + Γ i (µ)∂ i + Γ i αµ∂ α i ). (5.56)A 2–jet field Γ on a fibre bundle Y → X is called a sesquiholonomic(resp. holonomic) 2–jet field if it takes its values into the subbundleĴ 2 (X, Y ) (resp. J 2 (X, Y )) of J 1 (X, J 1 (X, Y )). We have the coordinateequality Γ i (µ) = yµ i for a sesquiholonomic 2–jet field and the additionalequality Γ i αµ = Γ i µα for a holonomic 2–jet field.Given a symmetric connection K on the cotangent bundle T ∗ X, everyconnection Γ on a fibre bundle Y → X induces the connectionjΓ = dx µ ⊗ [∂ µ + Γ i µ∂ i + (∂ α Γ i µ + ∂ j Γ i µy j α − K α αµ(y i α − Γ i α))∂ α i ]on the jet bundle J 1 (X, Y ) → X. Note that the curvature R of a connectionΓ on a fibre bundle Y → X induces the soldering form σ R on J 1 (X, Y ) →X,σ R = R i αµdx µ ⊗ ∂ α i .


822 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.5 Higher–Order Jet SpacesThe notion of 1– and 2–jet spaces is naturally extended to higher–order jetspaces. The k−jet space J k (X, Y ) of a fibre bundle Y → X is defined asthe disjoint union of the equivalence classes j k xs of sections s i : X → Y ofthe fibre bundle Y → X, identified by their values and the values of the firstk terms of their Taylor–series expansion at points x i in the domain (base)manifold X. J k (X, Y ) is a smooth manifold with the adapted coordinates(x α , y i α k ...α 1), wherey i α k···α 1(j k xs) = ∂ αk · · · ∂ α1 s i (x),(0 ≤ k ≤ k).The transformation law of these coordinates readsy ′ iα+α k ...α 1= ∂xµ∂ ′ x α d µy ′i α k ...α 1, (5.57)where α + α k . . . α 1 = (αα k . . . α 1 ) and∑d α = ∂ α + yα+α i k ...α 1∂ α k...α 1i = ∂ α + yα∂ i i + yαµ∂ i µ i + · · ·|α k ...α 1|


<strong>Applied</strong> Jet <strong>Geom</strong>etry 823on J ∞ (X, Y ), with suitable notation for vector–fields, derivatives and differentialforms just as like as in the finite order case (see [Sardanashvily (1993);Sardanashvily (1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily(2000a); Sardanashvily (2002a)]).A vector–field u k on the k−-jet space J k (X, Y ) is called projectablevector–field if for any l < k there exists a vector–field u k on J l (X, Y ) → Xsuch thatu l ◦ π k l = T π k l ◦ u k .The tangent map T π k lsends projectable vector–fields on J k (X, Y ) onto theprojectable vector–fields on J l (X, Y ).Now consider projectable vector–fields u k which are extension to thehigher–order jet spaces of infinitesimal transformations of the fibre bundleY → X. The linear space of projectable vector–fields on J ∞ (X, Y )is defined as the limit of the inverse system of projectable vector–fieldson k−jet spaces. As a consequence, every projectable vector–field on thebundle Y → X,u = u α ∂ α + u i ∂ i ,induces a projectable vector–field u ∞ on J ∞ (X, Y ). We have its canonicaldecompositionu ∞ = u ∞ H + u ∞ V ,u ∞ H = u α ̂∂∞ α = u α (∂ α + yα∂ i i + ...), (5.59)∞∑u ∞ V = ̂∂ α k k...̂∂ α 1 1u i V ∂ α1...α ki ,k=0where u V is the vertical part of the splitting (5.14) of π 1∗0 u. In particular,u ∞ H is the canonical lift of the vector–field τ = uα ∂ α on X onto J ∞ (X, Y ).By the same limiting process we can introduce the notions of innerproduct of exterior forms and projectable vector–fields, the Lie bracket ofprojectable vector–fields and the Lie derivative of exterior forms by projectablevector–fields on J ∞ (X, Y ). All the usual identities are satisfied.In particular, the notion of contact forms is extended to the formŝdy i α 1...α r= dy i α 1...α r− y i α 1...α rνdx ν .Let Ω r,k denote the space of exterior forms on J ∞ (X, Y ) which are of theorder r in the horizontal forms dx ν and of the order k in the contact forms.


824 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThen, the space Ω n of exterior n−forms on J ∞ (X, Y ) admits the uniquedecompositionΩ n = Ω n,0 ⊕ Ω n−1,1 ⊕ . . . ⊕ Ω 0,n . (5.60)An exterior form is called a k−contact form if it belongs to the space Ω r,k .In particular, we have the k−contact projection h k : Ω n −→ Ω n−k,k . Forexample, the horizontal projection h 0 performs the replacement dy i α 1...α k−→ y i α 1...α k νdx ν .The exterior differential d on exterior forms on J ∞ (X, Y ) is decomposedinto the sumof the total differential operatorand the vertical differential operatord = d H + d V (5.61)d H φ = ̂∂ ∞ µ φ ... dx µ ∧ . . .d V φ = ∂φ ...∂y i α 1...α r̂dyiα1...α r∧ . . .These differentials satisfy the cohomology propertiesd H d H = 0, d V d V = 0, d V d H + d H d V = 0.Note that if σ is an exterior form on the k−jet space J k (X, Y ), thedecomposition (5.61) is reduced toπ r+1∗r dσ = d H σ + d V σ, which impliesh 0 (dσ) = d H h 0 (σ).5.6 Application: Jets and Non–AutonomousDynamicsAs complex nonlinear mechanics is the most exact basis of all complexnonlinear dynamical systems considered in this book, we give here the firstglimpse of mechanics on jet spaces.Recall that in ordinary (autonomous) mechanics we have a configurationmanifold M and the corresponding velocity phase–space manifold is


<strong>Applied</strong> Jet <strong>Geom</strong>etry 825its tangent bundle T M. However, in modern geometrical settings of non–autonomous (see [Giachetta et. al. (1997); Mangiarotti et. al. (1999); Mangiarottiand Sardanashvily (2000a); Saunders (1989); Sardanashvily (1993);Sardanashvily (1995); Sardanashvily (2002a)]), the configuration manifoldof time–dependent mechanics is a fibre bundle Q → R, called the configurationbundle, coordinated by (t, q i ), where t ∈ R is a Cartesian coordinateon the time axis R with the transition functions t ′ = t+const. The correspondingvelocity phase–space is the 1–jet space J 1 (R, Q), which admitsthe adapted coordinates (t, q i , qt). i It was proved in [Giachetta (1992);León et. al. (1996); Mangiarotti and Sardanashvily (1998)] that everydynamical equation ξ defines a connection on the affine jet bundleJ 1 (R, Q) → Q, and vice versa.Due to the canonical imbedding J 1 (R, Q) → T Q, every dynamical connectioninduces a nonlinear connection on the tangent bundle T Q → Q, andvice versa. As a consequence, every dynamical equation on Q induces anequivalent geodesic equation on the tangent bundle T Q → Q in accordancewith the following proposition. Given a configuration bundle Q → R, coordinatedby (t, q i ), and its 2–jet space J 2 (R, Q), coordinated by (t, q i , qt, i qtt),iany dynamical equation ξ on the configuration bundle Q → R,q i tt = ξ i (t, q i , q i t) (5.62)is equivalent to the geodesic equation with respect to a connection ˜K onthe tangent bundle T Q → Q,which fulfills the conditionsṫ = 1, ẗ = 0, ¨q i = ˜K i 0 + ˜K i j ˙q j ,˜K 0 α = 0, ξ i = ˜K i 0 + q j t ˜K i j |ṫ=1, ˙qi =q i t . (5.63)Recall that the 1–jet space J 1 (R, Q) is defined as the set of equivalenceclasses j 1 t c of sections c i : R → Q of the fibre bundle Q → R, which areidentified by their values c i (t) and the values of their partial derivatives∂ t c i = ∂ t c i (t) at time points t ∈ R. Also recall that there is the canonicalimbeddingλ : J 1 (R, Q) ↩→ T Q, locally given by λ = d t = ∂ t + q i t∂ i , (5.64)where d t denotes the total time derivative. From now on, we will identifyJ 1 (R, Q) with its image in the tangent bundle T Q. This is an affine bundlemodelled over the vertical tangent bundle V Q of Q → R.


826 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAs a consequence of (5.64), every connection Γ on a fibre bundle Q → R,Γ : Q → J 1 (R, Q), locally given by Γ = dt ⊗ (∂ t + Γ i ∂ i ), (5.65)is identified with the nowhere vanishing vector–field on Q [Mangiarotti andSardanashvily (1998); Mangiarotti et. al. (1999)],Γ : Q → J 1 (R, Q) ⊂ T Q, locally given by Γ = ∂ t + Γ i ∂ i . (5.66)This is the horizontal lift of the standard vector–field ∂ t on R by means ofthe connection (5.65). Conversely, any vector–field Γ on Q such that dt⌋Γ =1 defines a connection on Q → R. Therefore, the covariant differentialassociated with a connection Γ on Q → R readsD G : J 1 (R, Q) → V Q, locally given by ˙q i ◦ D G = q i t − Γ i .Let J 1 (R, J 1 (R, Q)) denote the (repeated) 1–jet space of the jet bundleJ 1 (R, Q) → R, coordinated by (t, q i , q i t, q i (t) , qi tt). The corresponding 2–jetspace J 2 (R, Q) of the fibre bundle Q → R is the holonomic subbundleq i t = q i (t) of J 1 (R, J 1 (R, Q)), coordinated by (t, q i , q i t, q i tt). There are theimbeddings˚λJ 2 (R, Q) −→ T J 1 (R, Q) −→ T λT T Q, with˚λ : (t, q i , qt, i qtt) i ↦→ (t, q i , qt, i ṫ = 1, ˙q i = qt, i ˙q t i = qtt). i (5.67)T λ ◦ ˚λ : (t, q i , q i t, q i tt) ↦→ (t, q i , ṫ = 1, ˙q i = q i t, ẗ = 0, ¨q i = q i tt), (5.68)where (t, q i , ˙q i , ¨q i ) are holonomic coordinates on the second tangent bundleT T Q. This global geometrical structure of time–dependent mechanics isdepicted in Figure 5.3.Therefore, a dynamical equation ξ on a configuration bundle Q → R,given in local coordinates by (5.62), is defined as the geodesic equationKer D ξ ⊂ J 2 (R, Q) for a holonomic connection ξ on the jet bundleJ 1 (R, Q) → R. Due to the map (5.67), a holonomic connection ξ is representedby the horizontal vector–field on J 1 (R, Q),ξ = ∂ t + q i t∂ i + ξ i (q µ , q i t)∂ t i. (5.69)A dynamical equation ξ is said to be conservative if there exists a trivializationQ ∼ = R×M such that the vector–field ξ (5.69) on J 1 (R, Q) ∼ = R×T Mis projectable onto T M. Then this projectionΞ ξ = ˙q i ∂ i + ξ i (q j , ˙q j ) ˙∂ i


<strong>Applied</strong> Jet <strong>Geom</strong>etry 827Fig. 5.3 Hierarchical geometrical structure of time–dependent mechanics. Note that(for simplicity) intermediate jet spaces, J 1 (R, J 1 (R, Q)) and T J 1 (R, Q), are not shown.is a second–order dynamical equation on a typical fibre M of Q → R,¨q i = Ξ i ξ. (5.70)Conversely, every second–order dynamical equation Ξ (5.70) on a manifoldM can be seen as a conservative dynamical equationξ Ξ = ∂ t + ˙q i ∂ i + u i ˙∂ion the trivial fibre bundle R × M → R.Now we can explore the fundamental relationship between the holonomicconnections ξ (5.69) on the 1−jet bundle J 1 (R, Q) → R and thedynamical connections γ on the affine 1−jet bundle J 1 (R, Q) → Q, givenbyγ = dq α ⊗ (∂ α + γ i α∂ t i), (q α ≡ (t, q i ), ∂ α ≡ (∂ t , ∂ i )). (5.71)Any dynamical connection γ (5.71) defines the holonomic connection ξ γon J 1 (R, Q) → R [Mangiarotti and Sardanashvily (1998)]ξ γ = ∂ t + q i t∂ i + (γ i 0 + q j t γ i j)∂ t i.Conversely, any holonomic connection ξ (5.69) on J 1 (R, Q) → R definesthe dynamical connectionγ ξ = dt ⊗ [∂ t + (ξ i − 1 2 qj t ∂ t jξ i )∂ t i] + dq j ⊗ [∂ j + 1 2 ∂t jξ i ∂ t i]. (5.72)


828 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIt follows that every dynamical connection γ (5.71) induces the dynamicalequation (5.62) on the configuration bundle Q → R, rewritten hereasq i tt = γ i 0 + q j t γ i j. (5.73)<strong>Diff</strong>erent dynamical connections may lead to the same dynamical equation(5.73). The dynamical connection γ ξ (5.72), associated with a dynamicalequation, possesses the propertyγ k i = ∂ t iγ k 0 + q j t ∂ t iγ k j ,which implies the relation ∂j tγk i = ∂t i γk j . Such a dynamical connection iscalled symmetric. Let γ be a dynamical connection (5.71) and ξ γ the correspondingdynamical equation (5.6). Then the connection (5.72), associatedwith ξ γ , takes the formγ ξγki = 1 2 (γk i + ∂ t iγ k 0 + q j t ∂ t iγ k j ), γ ξγk0 = ξ k − q i tγ ξγki .Note that γ = γ ξγiff γ is symmetric.To explore the relation between the connections γ (5.71) on the affinejet bundle J 1 (R, Q) → Q and the connectionsK = dq α ⊗ (∂ α + K β α ˙∂ β ) (5.74)on the tangent bundle T Q → Q, consider the diagramJ 1 (R, J 1 j 1 λ(R, Q)) ✲ J 1 (Q, T Q)γ❄J 1 (R, Q)λK❄✲ T Q(5.75)where J 1 (Q, T Q) is the 1–jet space of the tangent bundle T Q → Q, coordinatedby (q α , ˙q α , ˙q α µ). The jet prolongation j 1 λ of the canonical imbeddingλ (5.64) over Q readsWe havej 1 λ : (t, q i , q i t, q i µt) ↦→ (t, q i , ṫ = 1, ˙q i = q i t, ṫ µ = 0, ˙q i µ = q i µt).j 1 λ ◦ γ : (t, q i , q i t) ↦→ (t, q i , ṫ = 1, ˙q i = q i t, ṫ µ = 0, ˙q i µ = γ i µ),K ◦ λ : (t, q i , q i t) ↦→ (t, q i , ṫ = 1, ˙q i = q i 0, ṫ µ = K 0 µ, ˙q i µ = K i µ).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 829It follows that the diagram (5.75) can be commutative only if the componentsK 0 µ of the connection K on T Q → Q vanish. Since the transitionfunctions t → t ′ are independent of q i , a connection˜K = dq α ⊗ (∂ α + K i α ˙∂ i ) (5.76)with the components K 0 µ = 0 can exist on the tangent bundle T Q → Q. Itobeys the transformation lawK ′ iα = (∂ j x ′i Kµ j + ∂ µ ẋ ′i ) ∂qµ . (5.77)∂x′αNow the diagram (5.75) becomes commutative if the connections γ and ˜Kfulfill the relationγ i µ = K i µ(t, q i , ṫ = 1, ˙q i = q i t),which holds globally since the substitution of ˙q i = q i t into (5.77) restatesthe coordinate transformation law of γ.Every dynamical equation (5.62) on the configuration bundle Q → Rcan be written in the formq i tt = K i 0 ◦ λ + q j t K i j ◦ λ, (5.78)where ˜K is a connection (5.76) on the tangent bundle T Q → Q. Conversely,each connection ˜K (5.76) on T Q → Q defines the dynamical equation (5.78)on Q → R.Consider the geodesic equation (5.6) on T Q with respect to the connection˜K. Its solution is a geodesic curve c(t) which also satisfies thedynamical equation (5.62), and vice versa.From the physical viewpoint, a reference frame in mechanics on a configurationbundle Q → R sets a tangent vector at each point of Q whichcharacterizes the velocity of an ‘observer’ at this point. Then any connectionΓ on Q → R is said to be such a reference frame [Echeverría et. al.(1995); Mangiarotti and Sardanashvily (1998); Massa and Pagani (1994);Sardanashvily (1998)].Each connection Γ on a fibre bundle Q → R defines an atlas of localconstant trivializations of Q → R whose transition functions are independentof t, and vice versa. One finds Γ = ∂ t with respect to this atlas. Inparticular, there is 1–1 correspondence between the complete connectionsΓ (5.66) on Q → R and the trivializations of this bundle.


830 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionGiven a reference frame Γ, any connection K (5.74) on the tangentbundle T Q → Q defines the dynamical equationξ i = (K i α − Γ i K 0 α) ˙q α |ṫ=1, ˙q i =q i t . (5.79)Given a connection Γ on the fibre bundle Q → R and a connection Kon the tangent bundle T Q → Q, there is the connection ˜K on T Q → Qwith the components˜K 0 α = 0, ˜Ki α = K i α − Γ i K 0 α.5.6.1 GeodesicsIn this subsection we continue our study of non–autonomous, time–dependent mechanics on a configuration bundle Q → R, that we started insubsection 5.6 above. Recall that R is the time axis, while the correspondingvelocity phase–space manifold is the 1–jet space J 1 (R, Q) of sectionss : R −→ Q of the bundle Q → R. Also, recall that second–order dynamicalequation (dynamical equation, for short) on a fibre bundle Q → R is definedas a first–order dynamical equation on the jet bundle J 1 (R, Q) → R,given by a holonomic connection ξ on J 1 (R, Q) → R which takes its valuesin the 2–jet space J 2 (R, Q) ⊂ J 1 (R, J 1 (R, Q)) (see [León et. al. (1996);Mangiarotti and Sardanashvily (1998); Massa and Pagani (1994); Mangiarottiand Sardanashvily (1999)]). The global geometrical structure oftime–dependent mechanics is depicted in Figure 5.3 above.Since a configuration bundle Q → R is trivial, the existent formulationsof mechanics often imply its preliminary splitting Q = R × M [Cariñenaand F.Núñez (1993); Echeverría et. al. (1991); León et. al. (1996);Morandi et. al. (1990)]. This is not the case of mechanical systems subjectto time–dependent transformations, including inertial frame transformations.Recall that different trivializations of Q → R differ from eachother in projections Q → M. Since a configuration bundle Q → R hasno canonical trivialization in general, mechanics on Q → R is not a repetitionof mechanics on R × M, but implies additionally a connection onQ → R which is a reference frame [Mangiarotti and Sardanashvily (1998);Sardanashvily (1998)]. Considered independently on a trivialization ofQ → R, mechanical equations make the geometrical sense of geodesic equations.We now examine quadratic dynamical equations in details. In thiscase, the corresponding dynamical connection γ on J 1 (R, Q) → Q is


<strong>Applied</strong> Jet <strong>Geom</strong>etry 831affine, while the connection ˜K (5.63) on T Q → Q is linear. Then theequation for Jacobi vector–fields along the geodesics of the connection˜K can be considered. This equation coincides with the existent equationfor Jacobi fields of a Lagrangian system [Dittrich and Reuter (1994);Mangiarotti and Sardanashvily (1998)] in the case of non–degeneratequadratic Lagrangians, when they can be compared. We will considermore general case of quadratic Newtonian systems characterized by apair (ξ, µ) of a quadratic dynamical equation ξ and a Riemannian inertiatensor µ which satisfy a certain compatibility condition. Given areference frame, a Riemannian inertia tensor µ is extended to a Riemannianmetric on the configuration space Q. Then conjugate points of solutionsof the dynamical equation ξ can be examined in accordance withthe well–known geometrical criteria [Mangiarotti and Sardanashvily (1998);Sardanashvily (1998)].5.6.2 Quadratic Dynamical EquationsFrom the physical viewpoint, the most interesting dynamical equations arethe quadratic ones, i.e.,ξ i = a i jk(q µ )q j t q k t + b i j(q µ )q j t + f i (q µ ). (5.80)This property is coordinate–independent due to the affine transformationlaw of coordinates q i t. Then, it is clear that the corresponding dynamicalconnection γ ξ (5.72) is affine:γ = dq α ⊗ [∂ α + (γ i λ0(q ν ) + γ i λj(q ν )q j t )∂ t i],and vice versa. This connection is symmetric iff γ i λµ = γi µλ .There is 1–1 correspondence between the affine connections γ on theaffine jet bundle J 1 (R, Q) → Q and the linear connections ˜K (5.76) on thetangent bundle T Q → Q.This correspondence is given by the relationγ i µ = γ i µ0 + γ i µjq j t , γ i µλ = K µiα .In particular, if an affine dynamical connection γ is symmetric, so is thecorresponding linear connection ˜K.Any quadratic dynamical equationq i tt = a i jk(q µ )q j t q k t + b i j(q µ )q j t + f i (q µ ) (5.81)


832 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis equivalent to the geodesic equationṫ = 1, ẗ = 0,for the symmetric linear connection¨q i = a i jk(q µ ) ˙q i ˙q j + b i j(q µ ) ˙q j ṫ + f i (q µ )ṫṫ. (5.82)˜K = dq α ⊗ (∂ α + K µ αν(t, q i ) ˙q ν ˙∂µ )on T Q → Q, given by the componentsK 0 αν = 0, K 0i0 = f i , K 0ij = K ji0 = 1 2 bi j, K jik = a i jk. (5.83)Conversely, any linear connection K on the tangent bundle T Q → Qdefines the quadratic dynamical equationq i tt = K jik q j t q k t + (K 0ij + K ji0 )q j t + K 0i0 ,written with respect to a given reference frame (t, q i ) ≡ q µ .However, the geodesic equation (5.82) is not unique for the dynamicalequation (5.81). Any quadratic dynamical equation (5.80), being equivalentto the geodesic equation with respect to the linear connection ˜K (5.83), isalso equivalent to the geodesic equation with respect to an affine connectionK ′ on T Q → Q which differs from ˜K (5.83) in a soldering form σ onT Q → Q with the local componentsσ 0 α = 0, σ i k = h i k + (s − 1)h i k ˙q 0 , σ i 0 = −sh i k ˙q k − h i 0 ˙q 0 + h i 0,where s and h i α are local functions on Q.5.6.3 Equation of Free–MotionWe say that the dynamical equation (5.62), that is: qtt i = ξ i (t, q i , qt),iis a free motion equation iff there exists a reference frame (t, q i ) on theconfiguration bundle Q → R such that this equation readsq i tt = 0. (5.84)With respect to arbitrary bundle coordinates (t, q i ), a free motion equationtakes the formq i tt = d t Γ i + ∂ j Γ i (q j t − Γ j ) − ∂qi∂q µ ∂q µ∂q j ∂q k (qj t − Γ j )(q k t − Γ k ), (5.85)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 833where Γ i = ∂ t q i (t, q j ) is the connection associated with the initial frame(t, q i ). One can think of the r.h.s. of the equation (5.85) as being the generalcoordinate expression of an inertial force in mechanics. The correspondingdynamical connection γ on the affine jet bundle J 1 (R, Q) → Q readsγ i k = ∂ k Γ i − ∂qi∂q µ ∂q µ∂q j ∂q k (qj t − Γ j ), γ i 0 = ∂ t Γ i + ∂ j Γ i q j t − γ i kΓ k . (5.86)This affine dynamical connection defines a linear connection K on the tangentbundle T Q → Q whose curvature is necessarily zero. Thus, we cometo the following criterion for a dynamical equation to be a free motion equation:if ξ is a free motion equation, it is quadratic and the correspondinglinear symmetric connection (5.83) on the tangent bundle T Q → Q is flat.A free motion equation on a configuration bundle Q → R exists iff thetypical fibre M of Q admits a curvature–free linear symmetric connection.5.6.4 Quadratic Lagrangian and Newtonian SystemsRecall that a Lagrangian of an nD mechanical system on Q → R is definedas a function on the velocity phase–space J 1 (R, Q). In particular, let usconsider a non–degenerate quadratic LagrangianL = 1 2 µ ij(q µ )q i tq j t + k i (q µ )q i t + f(q µ ), (5.87)where µ ij (i, j = 1, ..., n) is a Riemannian fibre metric tensor in the verticaltangent bundle V Q, called the inertial metric tensor. As for quadraticdynamical equations, this property is coordinate–independent, namely onecan show that any quadratic polynomial on J 1 (R, Q) ⊂ T Q is extended toa bilinear form on T Q, so that the Lagrangian L (5.87) can be written asL = 1 2 γ αµq α t q µ t , (with q 0 t = 1),where γ is the (degenerate) fibre metric in the tangent bundle T Q, givenbyγ 00 = 2f, γ 0i = k i , γ ij = µ ij . (5.88)The associated Lagrangian equation takes the formq i tt = (µ −1 ) ik Γ λkν q α t q ν t , (5.89)where Γ λµν = − 1 2 (∂ αγ µν + ∂ ν γ µλ − ∂ µ γ λν )


834 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionare the Christoffel symbols of the first–kind of the metric γ αµ given incomponents by (5.88). The corresponding geodesic equation (5.82) on T Qreads¨q i = (µ −1 ) ik Γ λkν ˙q α ˙q ν , ṫ = 1, ẗ = 0, (5.90)such that ˜K (5.63) is a linear connection with the following components˜K 0 αν = 0, ˜Ki αν = (µ −1 ) ik Γ λkν .We have the relation˙q α (∂ α µ ij + K i αν ˙q ν ) = 0. (5.91)One can show that an arbitrary Lagrangian system on a configurationbundle Q → R is a particular Newtonian system on Q → R. The latteris defined as a pair (ξ, µ) of a dynamical equation ξ and a (degenerate)fibre metric µ in the fibre bundle V Q J 1 (R, Q) → J 1 (R, Q) which satisfythe symmetry condition ∂ t k µ ij = ∂ t j µ ik and the compatibility condition[Mangiarotti and Sardanashvily (1998); Mangiarotti et. al. (1999)]ξ⌋dµ ij + µ ik γ k j + µ jk γ k i = 0, (5.92)where γ ξ is the dynamical connection (5.72), i.e.,γ ξ = dt ⊗ [∂ t + (ξ i − 1 2 qj t ∂ t jξ i )∂ t i] + dq j ⊗ [∂ j + 1 2 ∂t jξ i ∂ t i].We restrict our consideration to non–degenerate quadratic Newtoniansystems when ξ is a quadratic dynamical equation (5.80) and µ is a Riemannianfibre metric in V Q, i.e., µ is independent of q i t and the symmetrycondition becomes trivial. In this case, the dynamical equation (5.81) isequivalent to the geodesic equation (5.82) with respect a symmetric linearconnection ˜K (5.83), while the compatibility condition (5.92) takes theform (5.91).Given a symmetric linear connection ˜K (5.83) on the tangent bundleT Q → Q, one can consider the equation for Jacobi vector–fields alonggeodesics of this connection, i.e., along solutions of the dynamical equation(5.81). If Q admits a Riemannian metric, the conjugate points of thesegeodesic can be investigated.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 8355.6.5 Jacobi FieldsLet us consider the quadratic dynamical equation (5.81) and the equivalentgeodesic equation (5.82) with respect to the symmetric linear connection˜K (5.83). Its Riemann curvature tensorR λµαβ = ∂ λ K µαβ − ∂ µ K α λ β + K γ λ βK µαγ − K µγβ K α λ γhas the temporal componentR λµ0β = 0. (5.93)Then the equation for a Jacobi vector–field u along a geodesic c reads˙q β ˙q µ (∇ β (∇ µ u α ) − R λµαβ u λ ) = 0, ∇ β ˙q α = 0, (5.94)where ∇ µ denotes the covariant derivative relative to the connection ˜K (see[Kobayashi and Nomizu (1963/9)]). Due to the relation (5.93), the equation(5.94) for the temporal component u 0 of a Jacobi field takes the formWe chose its solution˙q β ˙q µ (∂ µ ∂ β u 0 + K µγβ ∂ γ u 0 ) = 0.u 0 = 0, (5.95)because all geodesics obey the constraint ṫ = 0.Note that, in the case of a quadratic Lagrangian L, the equation (5.94)coincides with the Jacobi equationu j d 0 (∂ j ˙∂i L) + d 0 ( ˙u j ˙∂i ˙∂j L) − u j ∂ i ∂ j L = 0for a Jacobi field on solutions of the Lagrangian equations for L. Thisequation is the Lagrangian equation for the vertical extension L V of the LagrangianL (see [Dittrich and Reuter (1994); Mangiarotti and Sardanashvily(1998); Mangiarotti et. al. (1999)]).Let us consider a quadratic Newtonian system with a Riemannian inertiatensor µ ij . Given a reference frame (t, q i ) ≡ q α , this inertia tensor isextended to the following Riemannian metric on Qg 00 = 1, g 0i = 0, g ij = µ ij .However, its covariant derivative with respect to the connection ˜K (5.83)does not vanish in general. Nevertheless, due to the relations (5.91) and


836 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(5.95), we get the well–known formula for a Jacobi vector–field u along ageodesic c:∫ ba(gλµ ( ˙q α ∇ α u λ )( ˙q β ∇ β u µ ) + R λµαν u λ u α ˙q µ ˙q ν) dt+ g λµ ˙q α ∇ α u λ u ′µ | t=a − g λµ ˙q α ∇ α u λ u ′µ | t=b = 0.Therefore, the following assertions also remain true [Kobayashi and Nomizu(1963/9)]: (i) if the sectional curvature R λµαν u λ u α ˙q µ ˙q ν is positiveon a geodesic c, this geodesic has no conjugate points; (ii) if the sectionalcurvature R λµαν u λ u α v µ v ν , where u and v are arbitrary unit vectors ona Riemannian manifold Q less than k < 0, then, for every geodesic, thedistance between two consecutive conjugate points is at most π/ √ k.For example, let us consider a 1D motion described by the LagrangianL = 1 2 ( ˙q1 ) 2 − φ(q 1 ),where φ is a potential. The corresponding Lagrangian equation is equivalentto the geodesic one on the 2D Euclidean space R 2 with respect to theconnection ˜K whose non–zero component is ˜K 1 0 0 = −∂ 1 φ. The curvatureof ˜K has the non–zero componentR 1010 = ∂ 1 ˜K010 = −∂ 2 1φ.Choosing the particular Riemannian metric g with componentswe come to the formulag 11 = 1, g 01 = 0, g 00 = 1,∫ ba[( ˙q µ ∂ µ u 1 ) 2 − ∂ 2 1φ(u 1 ) 2 ]dt = 0for a Jacobi vector–field u, which vanishes at points a and b. Then we getthat, if ∂1φ 2 < 0 at points of c, this motion has no conjugate points. Inparticular, let us consider the oscillator φ = k(q 1 ) 2 /2. In this case, thesectional curvature is R 0101 = −k, while the half–period of this oscillatoris exactly π/ √ k.5.6.6 ConstraintsRecall that symplectic and Poisson manifolds give an adequate Hamiltonianformulation of classical and quantum conservative mechanics. This


<strong>Applied</strong> Jet <strong>Geom</strong>etry 837is also the case of presymplectic Hamiltonian systems. Recall that everypresymplectic form can be represented as a pull–back of a symplecticform by a coisotropic imbedding (see e.g., [Gotay (1982); Mangiarottiand Sardanashvily (1998)]), a presymplectic Hamiltonian systemscan be seen as Dirac constraint systems [Cariñena et. al. (1995);Mangiarotti and Sardanashvily (1998)]. An autonomous Lagrangian systemalso exemplifies a presymplectic Hamiltonian system where a presymplecticform is the exterior differential of the Poincaré–Cartan form,while a Hamiltonian is the energy function [Cariñena and Rañada (1993);León et. al. (1996); Mangiarotti and Sardanashvily (1998); Muñoz andRomán (1992)]. A generic example of conservative Hamiltonian mechanicsis a regular Poisson manifold (Z, w) where a Hamiltonian is a real functionH on Z. Given the corresponding Hamiltonian vector–field ϑ H = w ♯ (df),the closed subbundle ϑ H (Z) of the tangent bundle T Z is an autonomousfirst–order dynamical equation on a manifold Z, called the Hamiltonianequations. The evolution equation on the Poisson algebra C ∞ (Z) is theLie derivative L ϑH f = {H, f}, expressed into the Poisson bracket of theHamiltonian H and functions f on Z. However, this description cannot beextended in a straightforward manner to time–dependent mechanics subjectto time–dependent transformations.The existent formulations of time–dependent mechanics imply usually apreliminary splitting of a configuration space Q = R × M and a momentumphase–space manifold Π = R × Z, where Z is a Poisson manifold [Cariñenaand F.Núñez (1993); Chinea et. al. (1994); Echeverría et. al. (1991);Hamoui and Lichnerowicz (1984); Morandi et. al. (1990); León and Marrero(1993)]. From the physical viewpoint, this means that a certain referenceframe is chosen. In this case, the momentum phase–space Π is with thePoisson product of the zero Poisson structure on R and the Poisson structureon Z. A Hamiltonian is defined as a real function H on Π. Thecorresponding Hamiltonian vector–field ϑ H on Π is vertical with respect tothe fibration Π → R. Due to the canonical imbeddingone introduces the vector–fieldΠ × T R → T Π, (5.96)γ H = ∂ t + ϑ H , (5.97)where ∂ t is the standard vector–field on R [Hamoui and Lichnerowicz(1984)]. The first–order dynamical equation γ H (Π) ⊂ T Π on the mani-


838 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfold Π plays the role of Hamiltonian equations. The evolution equation onthe Poisson algebra C ∞ (Π) is given by the Lie derivativeL γH f = ∂ t f + {H, f}.This is not the case of mechanical systems subject to time–dependenttransformations. These transformations, including canonical and inertialframe transformations, violate the splitting R × Z. As a consequence, thereis no canonical imbedding (5.96), and the vector–field (5.97) is not welldefined. At the same time, one can treat the imbedding (5.96) as a trivialconnection on the bundle Π −→ R, while γ H (5.97) is the sum of thehorizontal lift onto Π of the vector–field ∂ t by this connection and of thevertical vector–field ϑ H .Let Q → R be a fibre bundle coordinated by (t, q i ), and J 1 (R, Q)its 1–jet space, equipped with the adapted coordinates (t, q i , qt).i Recallthat there is a canonical imbedding λ given by (5.64) onto the affinesubbundle of T Q → Q of elements υ ∈ T Q such that υ⌋dt = 1.This subbundle is modelled over the vertical tangent bundle V Q → Q.As a consequence, there is a 1–1 correspondence between the connectionsΓ on the fibre bundle Q → R, treated as sections of the affinejet bundle π 1 0 : J 1 (R, Q) → Q [Mangiarotti et. al. (1999)], and thenowhere vanishing vector–fields Γ = ∂ t + Γ i ∂ i on Q, called horizontalvector–fields, such that Γ⌋dt = 1 [Mangiarotti and Sardanashvily (1998);Mangiarotti et. al. (1999)]. The corresponding covariant differential readsD Γ = λ − Γ : J 1 (R, Q) −→ V Q, q i ◦ D Γ = q i t − Γ i .Let us also recall the total derivative d t = ∂ t + q i t∂ i + · · · and the exterioralgebra homomorphismh 0 : φdt + φ i dy i ↦→ (φ + φ i q i t)dt (5.98)which sends exterior forms on Q → R onto the horizontal forms onJ 1 (R, Q) → R, and vanishes on contact forms θ i = dy i − q i tdt.Lagrangian time–dependent mechanics follows directly Lagrangian fieldtheory (see [Giachetta (1992); Krupkova (1997); León et. al. (1997);Mangiarotti and Sardanashvily (1998); Massa and Pagani (1994)], as wellas subsection 5.9 below). This means that we have a configuration spaceQ → R of a mechanical system, and a Lagrangian is defined as a horizontaldensity on the velocity phase–space J 1 (R, Q),L = Ldt, with L : J 1 (R, Q) → R. (5.99)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 839A generic momentum phase–space manifold of time–dependent mechanicsis a fibre bundle Π −→ R with a regular Poisson structure whose characteristicdistribution belongs to the vertical tangent bundle V Π of Π −→ R[Hamoui and Lichnerowicz (1984)]. However, such a Poisson structure cannotgive dynamical equations. A first–order dynamical equation on Π −→ R,by definition, is a section of the affine jet bundle J 1 (R, Π) −→ Π, i.e., a connectionon Π −→ R. Being a horizontal vector–field, such a connectioncannot be a Hamiltonian vector–field with respect to the above Poissonstructure on Π.One can overcome this difficulty as follows. Let Q → R be a configurationbundle of time–dependent mechanics. The corresponding momentumphase–space is the vertical cotangent bundle Π = V ∗ Q → R, called theLegendre bundle, while the cotangent bundle T ∗ Q is the homogeneous momentumphase–space. T ∗ Q admits the canonical Liouville form Ξ and thesymplectic form dΞ, together with the corresponding non–degenerate Poissonbracket {, } T on the ring C ∞ (T ∗ Q). Let us consider the subring ofC ∞ (T ∗ Q) which comprises the pull–backs ζ ∗ f onto T ∗ Q of functions f onthe vertical cotangent bundle V ∗ Q by the canonical fibrationζ : T ∗ Q → V ∗ Q. (5.100)This subring is closed under the Poisson bracket {, } T , and V ∗ Q admits theregular Poisson structure {, } V such that [Vaisman (1994)]ζ ∗ {f, g} V = {ζ ∗ f, ζ ∗ g} T .Its characteristic distribution coincides with the vertical tangent bundleV V ∗ Q of V ∗ Q → R. Given a section h of the bundle (5.100), let us considerthe pull–back formsΘ = h ∗ (Ξ ∧ dt), Ω = h ∗ (dΞ ∧ dt) (5.101)on V ∗ Q, but these forms are independent of a section h and are canonicalexterior forms on V ∗ Q. The pull–backs h ∗ Ξ are called the Hamiltonianforms. With Ω, the Hamiltonian vector–field ϑ f for a function f on V ∗ Qis given by the relationϑ f ⌋Ω = −df ∧ dt,while the Poisson bracket (5.6.6) is written as{f, g} V dt = ϑ g ⌋ϑ f ⌋Ω.


840 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe pair (V ∗ Q, Ω) is the particular polysymplectic phase–space ofthe covariant Hamiltonian field theory (see [Cariñena et. al. (1991);Giachetta et. al. (1997); Gotay (1991a); Sardanashvily (1995)] for a survey).Following its general scheme, we can formulate the Hamiltonian time–dependent mechanics as follows [Mangiarotti and Sardanashvily (1998);Sardanashvily (1998)].Recall that connection γ on the Legendre bundle V ∗ Q −→ R is calledcanonical if the corresponding horizontal vector–field is canonical for thePoisson structure on V ∗ Q, i.e., the form γ⌋Ω is closed. Such a form isnecessarily exact. A canonical connection γ is a said to be a Hamiltonianconnection ifγ⌋Ω = dH, (5.102)where H is a Hamiltonian form on V ∗ Q. Every Hamiltonian form admitsa unique Hamiltonian connection γ H , while any canonical connection islocally a Hamiltonian one. Given a Hamiltonian form H, the kernel of thecovariant differential D γH , associated with the Hamiltonian connection γ H ,is a closed imbedded subbundle of the jet bundle J 1 (R, V ∗ Q) −→ R, andso is the system of first–order PDEs on the Legendre bundle V ∗ Q −→ R.These are the Hamiltonian equations in time–dependent mechanics, whilethe Lie derivativeL γH f = γ H ⌋df (5.103)defines the evolution equation on C ∞ (V ∗ Z). This Hamiltonian dynamicsis equivalent to the Lagrangian one for hyperregular Lagrangians, while adegenerate Lagrangian involves a set of associated Hamiltonian forms inorder to exhaust solutions of the Lagrangian equations [Giachetta et. al.(1997); Sardanashvily (1994); Sardanashvily (1995)].Since γ H is not a vertical vector–field, the r.h.s. of the evolution equation(5.103) is not expressed into the Poisson bracket in a canonical way,but contains a frame–dependent term. Every connection Γ on the configurationbundle Q → R is an affine section of the bundle (5.100), and definesthe Hamiltonian form H Γ = Γ ∗ Ξ on V ∗ Q. The corresponding Hamiltonianconnection is the canonical lift V ∗ Γ of Γ onto the Legendre bundle V ∗ Q [Giachettaet. al. (1997); Mangiarotti et. al. (1999)]. Then any Hamiltonianform H on V ∗ Q admits splittingsH = H Γ − ˜H Γ dt, with γ H = V ∗ Γ + ϑ eHΓ , (5.104)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 841where ϑ eHΓ is the vertical Hamiltonian field for the function ˜H Γ , which theenergy function with respect to the reference frame Γ. With the splitting(5.104), the evolution equation (5.103) takes the formL γH f = V ∗ Γ⌋H + { ˜H Γ , f} V . (5.105)Let the configuration bundle Q −→ R with an mD typical fibre M be coordinatedby (t, q i ). Then Legendre bundle V ∗ Q and the cotangent bundleT ∗ Q admit holonomic coordinates (t, q i , p i = ˙q i ) and (t, q i , p i , p), respectively.Relative to these coordinates, a Hamiltonian form H on V ∗ Q readsH = h ∗ Ξ = p i dq i − Hdt. (5.106)It is the well–known integral invariant of Poincaré–Cartan, where H is aHamiltonian in time–dependent mechanics. The expression (5.106) showsthat H fails to be a scalar under time–dependent transformations. Therefore,the evolution equation (5.105) takes the local formL γH = ∂ t f + {H, f} V , (5.107)but one should bear in mind that the terms in its r.h.s., taken separately,are not well–behaved objects under time–dependent transformations. Inparticular, the equality {H, f} V = 0 is not preserved under time–dependenttransformations.Every Lagrangian L defines the Legendre map̂L : J 1 (R, Q) −→ V ∗ Q, locally given by p i ◦ ̂L = π i , (5.108)whose image N L = ̂L(J 1 (R, Q)) ⊂ V ∗ Q is called the Lagrangian constraintspace. We state the comprehensive relationship between solutions of theLagrangian equations for an almost regular Lagrangian L and solutions inN L of the Hamiltonian equations for associated Hamiltonian forms.5.6.7 Time–Dependent Lagrangian DynamicsGiven a Lagrangian L on the velocity phase–space J 1 (R, Q), we followthe first variational formula of [Giachetta et. al. (1997); Mangiarotti andSardanashvily (1998); Sardanashvily (1997)], which gives the canonical decompositionof the Lie derivativeL j1 uL = (j 1 u⌋L) dt (5.109)


842 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionof L along a projectable vector–field u on Q −→ R. We havej 1 u⌋L = u V ⌋E L + d t (u⌋H L ),where u V = (u⌋θ i )∂ i ,H L = L + π i θ i , π i = ∂iL, t (5.110)is the Poincaré–Cartan form andE L : J 2 (R, Q) −→ V ∗ Q,E L = (∂ i − d t π i )Ldq iis the Euler–Lagrangian operator associated with L. The kernel Ker E L ⊂J 2 (R, Q) of E L defines the Lagrangian equations on Q, given by the coordinaterelations(∂ i − d t π i )L = 0. (5.111)On–shell, the first variational formula (5.109) leads to the weak identityL j 1 uL ≈ d t (u⌋H L )dt,and then, if L j1 uL = 0, to the weak conservation lawof the symmetry current J , given byd t (u⌋H L ) = −d t J ≈ 0J = −(u⌋H L ) = −π i (u t ˙q i − u i ) − u t L.Being the Lepagean equivalent of the Lagrangian L on J 1 (R, Q) (i.e.,L = h 0 (H L ) where h 0 is the map (5.98), see [Giachetta et. al. (1997); Mangiarottiand Sardanashvily (1998); Sardanashvily (1997)]), the Poincaré–Cartan form H L (5.110) is also the Lepagean equivalent of the Lagrangianon the repeated jet space J 1 (R, J 1 (R, Q)),L = ĥ0(H L ) = (L + ( ̂˙q i − ˙q i )π i )dt,ĥ 0 (dy i ) = ̂q i tdt,coordinated by (t, q i , ˙q i , ̂˙q i , ¨q i ). The Euler–Lagrangian operatorE L: J 1 (R, J 1 (R, Q)) → V ∗ J 1 (R, Q) for L is locally given byE L= (∂ i L − ̂d t π i + ∂ i π j ( ̂˙q j − ˙q j ))dq i + ∂ t iπ j ( ̂˙q j − ˙q j )d ˙q i , (5.112)with ̂dt = ∂ t + ̂˙q i ∂ i + ¨q i ∂ t i.Its kernel Ker E L⊂ J 1 (R, J 1 (R, Q)) defines the Cartan equations∂ t iπ j ( ̂˙q j − ˙q j ) = 0, ∂ i L − ̂d t π i + ( ̂˙q j − ˙q j )∂ i π j = 0. (5.113)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 843Since E L| J 2 (R,Q)= E L , the Cartan equations (5.113) are equivalent to theLagrangian equations (5.111) on integrable sections c = ċ of J 1 (R, Q) → R.These equations are equivalent in the case of regular Lagrangians.On sections c : R −→ J 1 (R, Q), the Cartan equations (5.113) are equivalentto the relationc ∗ (u⌋dH L ) = 0, (5.114)which is assumed to hold for all vertical vector–fields u on J 1 (R, Q) −→ R.With the Poincaré–Cartan form H L (5.110), we have the Legendre mapĤ L : J 1 (R, Q) −→ T ∗ Q, (p i , p) ◦ ĤL = (π i , L − π i ˙q i ).Let Z L = ĤL(J 1 (R, Q)) be an imbedded subbundle i L : Z L ↩→ T ∗ Q ofT ∗ Q → Q. It admits the pull–back de Donder form i ∗ LΞ. We haveH L = Ĥ∗ LΞ L = Ĥ∗ L(i ∗ LΞ).By analogy with the Cartan equations (5.114), the Hamilton–de Donderequations for sections r of T ∗ Q → R are written asr ∗ (u⌋dΞ L ) = 0 (5.115)where u is an arbitrary vertical vector–field on T ∗ Q → R [Lopez and Marsden(2003)].Let the Legendre map ĤL : J 1 (R, Q) −→ Z L be a submersion. Thena section c of J 1 (R, Q) −→ R is a solution of the Cartan equations (5.114)iff ĤL ◦ c is a solution of the Hamilton–de Donder equations (5.115), i.e.,Cartan and Hamilton–de Donder equations are quasi–equivalent [Gotay(1991a); Lopez and Marsden (2003)].5.6.8 Time–Dependent Hamiltonian DynamicsLet the Legendre bundle V ∗ Q → R be provided with the holonomic coordinates(t, q i , ˙q i ). Relative to these coordinates, the canonical 3–form Ω(5.101) and the canonical Poisson structure on V ∗ Q readΩ = dp i ∧ dq i ∧ dt, (5.116){f, g} V = ∂ i f∂ i g − ∂ i g∂ i f, (f, g ∈ C ∞ V ∗ Q). (5.117)The corresponding symplectic foliation coincides with the fibration V ∗ Q →R. The symplectic forms on the fibres of V ∗ Q → R are the pull–backsΩ t = dp i ∧ dq i of the canonical symplectic form on the typical fibre


844 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionT ∗ M of the Legendre bundle V ∗ Q → R with respect to trivializationmaps [Cariñena and Rañada (1989); Hamoui and Lichnerowicz (1984);Sardanashvily (1998)]. Given such a trivialization, the Poisson structure(5.117) is isomorphic to the product of the zero Poisson structure on R andthe canonical symplectic structure on T ∗ M.An automorphism ρ of the Legendre bundle V ∗ Q → R is a canonicaltransformation of the Poisson structure (5.117) iff it preserves the canonical3–form Ω (5.116). Let us emphasize that canonical transformationsare compatible with the fibration V ∗ Q → R, but not necessarily with thefibration π Q : V ∗ Q → Q.With respect to the Poisson bracket (5.117), the Hamiltonian vector–field ϑ f for a function f on the momentum phase–space manifold V ∗ Q isgiven byϑ f = ∂ i f∂ i − ∂ i f∂ i .A Hamiltonian vector–field, by definition, is canonical. Conversely, everyvertical canonical vector–field on the Legendre bundle V ∗ Q −→ R is locallya Hamiltonian vector–field.To prove this, let σ be a one–form on V ∗ Q. If σ ∧ dt is closed form, itis exact. Since V ∗ Q is diffeomorphic to R × T ∗ M, we have the de Rhamcohomology groupH 2 (V ∗ Q) = H 0 (R) ⊗ H 2 (T ∗ M) ⊕ H 1 (R) ⊗ H 1 (T ∗ M).The form σ ∧ dt belongs to its second item which is zero.If the two–form σ ∧ dt is exact, then σ ∧ dt = dg ∧ dt locally [Giachettaet. al. (1997)].Let γ = ∂ t + γ i ∂ i + γ i ∂ i be a canonical connection on the Legendrebundle V ∗ Q −→ R. Its components obey the relations∂ i γ j − ∂ j γ i = 0, ∂ i γ j − ∂ j γ i = 0, ∂ j γ i + ∂ i γ j = 0.Canonical connections constitute an affine space modelled over the vectorspace of vertical canonical vector–fields on V ∗ Q −→ R.If γ is a canonical connection, then the form γ⌋Ω is exact. Every connectionΓ on Q → R induces the connection on V ∗ Q → R,V ∗ Γ = ∂ t + Γ i ∂ i − p i ∂ j Γ i ∂ j ,which is a Hamiltonian connection for the frame Hamiltonian formV ∗ Γ⌋Ω = dH Γ , H Γ = p i dq i − p i Γ i dt. (5.118)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 845Thus, every canonical connection γ on V ∗ Q defines an exterior one–form H modulo closed forms so that dH = γ⌋Ω. Such a form is called alocally Hamiltonian form.Every locally Hamiltonian form on the momentum phase–space V ∗ Q islocally a Hamiltonian form modulo closed forms. Given locally Hamiltonianforms H γ and H γ ′, their difference σ = H γ − H γ ′ is a one–form on V ∗ Qsuch that the two–form σ ∧ dt is closed. The form σ ∧ dt is exact andσ = fdt + dg locally. Put H γ ′ = H Γ where Γ is a connection on V ∗ Q −→ R.Then H γ modulo closed forms takes the local form H γ = H Γ + fdt, andcoincides with the pull–back of the Liouville form Ξ on T ∗ Q by the localsection p = −p i Γ i + f of the fibre bundle (5.100).Conversely, each Hamiltonian form H on the momentum phase–spaceV ∗ Q admits a unique canonical connection γ H on V ∗ Q −→ R such that therelation (5.102) holds. Given a Hamiltonian form H, its exterior differentialdH = h ∗ dΞ = (dp i + ∂ i Hdt) ∧ (dq i − ∂ i Hdt)is a presymplectic form of constant rank 2m since the form(dH) m = (dp i ∧ dq i ) m − m(dp i ∧ dq i ) m−1 ∧ dH ∧ dtis nowhere vanishing. It is also seen that (dH) m ∧dt ≠ 0. It follows that thekernel of dH is a 1D distribution. Then the desired Hamiltonian connectionγ H = ∂ t + ∂ i H∂ i − ∂ i H∂ i (5.119)is a unique vector–field γ H on V ∗ Q such that γ H ⌋dH = 0 and γ H ⌋dt = 1.Hamiltonian forms constitute an affine space modelled over the vectorspace of horizontal densities fdt on V ∗ Q → R, i.e., over C ∞ (V ∗ Q).Therefore Hamiltonian connections γ H form an affine space modelled overthe vector space of Hamiltonian vector–fields. Every Hamiltonian form Hdefines the associated Hamiltonian mapĤ = j 1 π Q ◦ γ H : ∂ t + ∂ i H : V ∗ Q → J 1 (R, Q). (5.120)With the Hamiltonian map (5.120), we have another Hamiltonian formH bH = −Ĥ⌋Θ = p idq i − p i ∂ i H.Note that H bH = H iff H is a frame Hamiltonian form.Given a Hamiltonian connection γ H (5.119), the corresponding Hamiltonianequations D γH = 0 take the coordinate form˙q i = ∂ i H, ṗ i = −∂ i H. (5.121)


846 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionTheir classical solutions are integral sections of the Hamiltonian connectionγ H , i.e., ṙ = γ H ◦ r. On sections r of the Legendre bundle V ∗ Q −→ R, theHamiltonian equations (5.121) are equivalent to the relationr ∗ (u⌋dH) = 0 (5.122)which is assumed to hold for any vertical vector–field u on V ∗ Q −→ R.A Hamiltonian form H (5.118) is the Poincaré–Cartan form for theLagrangian on the jet space J 1 (R, V ) ∗ Q,L H = h 0 (H) = (p i ˙q i − H)ω. (5.123)Given a projectable vector–field u on the configuration bundle Q −→ R andits lift onto the Legendre bundle V ∗ Q −→ R,ũ = u t ∂ t + u i ∂ i − ∂ i u j p j ∂ i ,we haveL eu H = L J 1 euL H . (5.124)Note that the Hamiltonian equations (5.121) for H are exactly the Lagrangianequations for L H , i.e., they characterize the kernel of the Euler–Lagrangian operatorE H : J 1 (R, V ) ∗ Q → V ∗ V ∗ Q,E H = ( ˙q i − ∂ i H)dp i − (ṗ i + ∂ i H)dq ifor the Lagrangian L H , called the Hamiltonian operator for H.Using the relation (5.124), let us get the Hamiltonian conservation lawsin time–dependent mechanics. As in field theory, by gauge transformationsin time–dependent mechanics are meant automorphism of the configurationbundle Q → R, but only over translations of the base R. Then, projectablevector–fields on V ∗ Q → R,u = u t ∂ t + u i ∂ i , u⌋dt = u t = const, (5.125)can be seen as generators of local 1–parameter groups of local gauge transformations.Given a Hamiltonian form H (5.165), its Lie derivative (5.124)readsL eu H = L j1 euL H = (−u t ∂ t H + p i ∂ t u i − u i ∂ i H + ∂ j u i p i ∂ j H)dt. (5.126)The first variational formula (5.109) applied to the Lagrangian L H (5.123)leads to the weak identity L eu H ≈ d t (u⌋H)dt. If the Lie derivative (5.126)vanishes, we have the conserved symmetry currentJ u = u⌋dH = p i u i − u t H, (5.127)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 847along u. Every vector–field (5.125) is a superposition of a vertical vector–field and a reference frame on Q → R. If u is a vertical vector–field, J u isthe Nöther currentJ u (q) = u⌋q = p i u i , (q = p i dq i ∈ V ∗ Q). (5.128)The symmetry current along a reference frame Γ, given byJ Γ = p i Γ i − H = − ˜H Γ ,is the energy function with respect to the reference frame Γ, taken withthe sign minus [Echeverría et. al. (1995); Mangiarotti and Sardanashvily(1998); Sardanashvily (1998)]. Given a Hamiltonian form H, the energyfunctions ˜H Γ constitute an affine space modelled over the vector space ofNöther currents. Also, given a Hamiltonian form H, the conserved currents(5.127) form a Lie algebra with respect to the Poisson bracket{J u , J u ′} V = J [u,u′ ].The second of the above constructions enables us to represent the r.h.s.of the evolution equation (5.107) as a pure Poisson bracket. Given a Hamiltonianform H = h ∗ Ξ, let us consider its pull–back ζ ∗ H onto the cotangentbundle T ∗ Q. Note that the difference Ξ − ζ ∗ H is a horizontal one–form onT ∗ Q → R, whileis a function on T ∗ Q. Then the relationH ∗ = ∂ t ⌋(Ξ − ζ ∗ H) = p + H (5.129)ζ ∗ (L γH f) = {H ∗ , ζ ∗ f} T (5.130)holds for every function f ∈ C ∞ (V ∗ Q). In particular, given a projectablevector–field u (5.125), the symmetry current J u (5.127) is conserved iff{H ∗ , ζ ∗ J u } T = 0.Moreover, let ϑ H ∗ be the Hamiltonian vector–field for the function H ∗(5.129) with respect to the canonical Poisson structure {, } T on T ∗ Q. ThenT ζ(ϑ H ∗) = γ H .


848 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.6.9 Time–Dependent ConstraintsThe relation (5.130) enables us to extend the constraint algorithm of conservativemechanics and time–dependent mechanics on a product R × M (see[Chinea et. al. (1994); León and Marrero (1993)]) to mechanical systemssubject to time–dependent transformations.Let H be a Hamiltonian form on the momentum phase–space manifoldV ∗ Q. In accordance with the relation (5.130), a constraint f ∈ I N ispreserved if the bracket in (5.130) vanishes. It follows that the solutions ofthe Hamiltonian equations (5.121) do not leave the constraint space N if{H ∗ , ζ ∗ I N } T ⊂ ζ ∗ I N . (5.131)If the relation (5.131) fails to hold, let us introduce secondary constraints{H ∗ , ζ ∗ f} T , f ∈ I N , which belong to ζ ∗ C ∞ (V ∗ Q). If the collection ofprimary and secondary constraints is not closed with respect to the relation(5.131), let us add the tertiary constraints {H ∗ , {H ∗ , ζ ∗ f a } T } T and so on.Let us assume that N is a final constraint space for a Hamiltonianform H. If a Hamiltonian form H satisfies the relation (5.131), so is aHamiltonian formH f = H − fdt, (5.132)where f ∈ I ′ N is a first class constraint. Though Hamiltonian forms H andH f coincide with each other on the constraint space N, the correspondingHamiltonian equations have different solutions on the constraint space Nbecause dH| N ≠ dH f | N . At the same time, d(i ∗ N H) = d(i∗ N H f ). Therefore,let us introduce the constrained Hamiltonian formH N = i ∗ NH f , (5.133)which is the same for all f ∈ I ′ N. Note that H N (5.133) is not a trueHamiltonian form on N −→ R in general. On sections r of the fibre bundleN −→ R, we can write the equationsr ∗ (u N ⌋dH N ) = 0, (5.134)where u N is an arbitrary vertical vector–field on N −→ R. They are calledthe constrained Hamiltonian equations.For any Hamiltonian form H f (5.132), every solution of the Hamiltonianequations which lives in the constraint space N is a solution of theconstrained Hamiltonian equations (5.134). The constrained Hamiltonian


<strong>Applied</strong> Jet <strong>Geom</strong>etry 849equations can be written asr ∗ (u N ⌋di ∗ NH f ) = r ∗ (u N ⌋dH f | N ) = 0. (5.135)They differ from the Hamiltonian equations (5.122) for H f restricted to Nwhich readr ∗ (u⌋dH f | N ) = 0, (5.136)where r is a section of N → R and u is an arbitrary vertical vector–fieldon V ∗ Q → R. A solution r of the equations (5.136) satisfies the weakercondition (5.135).One can also consider the problem of constructing a generalized Hamiltoniansystem, similar to that for Dirac constraint system in conservativemechanics [Mangiarotti and Sardanashvily (1998)]. Let H satisfies the condition{H ∗ , ζ ∗ I ′ N} T ⊂ I N , whereas {H ∗ , ζ ∗ I ′ N} T ⊄ I N . The goal is to finda constraint f ∈ I N such that the modified Hamiltonian H − fdt wouldsatisfy both the conditions{H ∗ + ζ ∗ f, ζ ∗ I ′ N} T ⊂ ζ ∗ I N , {H ∗ + ζ ∗ f, ζ ∗ I N } T ⊂ ζ ∗ I N .The first of them is fulfilled for any f ∈ I N , while the latter is an equationfor a second–class constraint f.Note that, in contrast with the conservative case, the Hamiltonianvector–fields ϑ f for the first class constraints f ∈ I ′ N in time–dependentmechanics are not generators of gauge symmetries of a Hamiltonian form ingeneral. At the same time, generators of gauge symmetries define an idealof constraints as follows.5.6.10 Lagrangian ConstraintsLet us consider the Hamiltonian description of Lagrangian mechanical systemson a configuration bundle Q → R. If a Lagrangian is degenerate, wehave the Lagrangian constraint subspace of the Legendre bundle V ∗ Q anda set of Hamiltonian forms associated with the same Lagrangian. Given aLagrangian L (5.99) on the velocity phase–space J 1 (R, Q), a Hamiltonianform H on the momentum phase–space V ∗ Q is said to be associated withL if H satisfies the relationŝL ◦ Ĥ ◦ ̂L = ̂L, and H = H bH + Ĥ∗ L (5.137)


850 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere Ĥ and ̂L are the Hamiltonian map (5.120) and the Legendre map(5.108), respectively. Here, ̂L ◦ Ĥ is the projectorp i (z) = π i (t, q i , ∂ j H(z)), (z ∈ N L ), (5.138)from Π onto the Lagrangian constraint space N L = ̂L(J 1 (R, Q)). Therefore,Ĥ ◦ ̂L is the projector from J 1 (R, Q) onto Ĥ(N L). A Hamiltonian form iscalled weakly associated with a Lagrangian L if the condition (5.137) holdson the Lagrangian constraint space N L .If a bundle map Φ : V ∗ Q → J 1 (R, Q) obeys the relation (5.137), thenthe Hamiltonian form H = −Φ⌋Θ + Φ ∗ L is weakly associated with theLagrangian L. If Φ = Ĥ, then H is associated with L [Giachetta et. al.(1997)].Any Hamiltonian form H weakly associated with a Lagrangian L obeysthe relationH| NL = Ĥ∗ H L | NL , (5.139)where H L is the Poincaré–Cartan form (5.110). The relation (5.137) takesthe coordinate formH(z) = p i ∂ i H − L(t, q i , ∂ j H(z)), (z ∈ N L ). (5.140)Substituting (5.138) and (5.140) in (5.165), we get the relation (5.139).The difference between associated and weakly associated Hamiltonianforms lies in the following. Let H be an associated Hamiltonian form, i.e.,the equality (5.140) holds everywhere on V ∗ Q. The exterior differential ofthis equality leads to the relations∂ t H(z) = −(∂ t L) ◦ Ĥ(z),∂ iH(z) = −(∂ i L) ◦ Ĥ(z),(p i − (∂ t iL)(t, q i , ∂ j t H))∂ i t∂ a t H = 0, (z ∈ N L ).The last of them shows that the Hamiltonian form is not regular outsidethe Lagrangian constraint space N L . In particular, any Hamiltonian form isweakly associated with the Lagrangian L = 0, while the associated Hamiltonianforms are only H Γ .Here we restrict our consideration to almost regular Lagrangians L, i.e.,if: (i) the Lagrangian constraint space N L is a closed imbedded subbundlei N : N L −→ V ∗ Q of the bundle V ∗ Q −→ Q, (ii) the Legendre map ̂L :J 1 (R, Q) −→ N L is a fibre bundle, and (iii) the pre-image ̂L −1 (z) of anypoint z ∈ N L is a connected submanifold of J 1 (R, Q).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 851A Hamiltonian form H weakly associated with an almost regular LagrangianL exists iff the fibre bundle J 1 (R, V ) ∗ Q −→ N L admits a globalsection.The condition (iii) leads to the following property [Giachetta et. al.(1997); Mangiarotti and Sardanashvily (1998)]. The Poincaré–Cartan formH L for an almost regular Lagrangian L is constant on the connected pre–image ̂L −1 (z) of any point z ∈ N L .An immediate consequence of this fact is the following assertion [Giachettaet. al. (1997)]. All Hamiltonian forms weakly associated with analmost regular Lagrangian L coincide with each other on the Lagrangianconstraint space N L , and the Poincaré–Cartan form H L for L is the pull–backH L = ̂L ∗ H, π i ˙q i − L = H(t, q j , π j ),of any such a Hamiltonian form H.It follows that, given Hamiltonian forms H an H ′ weakly associatedwith an almost regular Lagrangian L, their difference is fdt, (f ∈ I N ).Above proposition enables us to connect Lagrangian and Cartan equationsfor an almost regular Lagrangian L with the Hamiltonian equations forHamiltonian forms weakly associated with L [Giachetta et. al. (1997)].Let a section r of V ∗ Q −→ R be a solution of the Hamiltonian equations(5.121) for a Hamiltonian form H weakly associated with an almost regularLagrangian L. If r lives in the constraint space N L , the section c = π Q ◦ rof Q −→ R satisfies the Lagrangian equations (5.111), while c = Ĥ ◦ r obeysthe Cartan equations (5.113).Given an almost regular Lagrangian L, let a section c of the jet bundleJ 1 (R, Q) −→ R be a solution of the Cartan equations (5.113). Let H be aHamiltonian form weakly associated with L, and let H satisfy the relationĤ ◦ ̂L ◦ c = j 1 (π 1 0 ◦ c). (5.141)Then, the section r = ̂L ◦ c of the Legendre bundle V ∗ Q −→ R is a solutionof the Hamiltonian equations (5.121) for H. Since Ĥ ◦ ̂L is a projectionoperator, the condition (5.141) implies that the solution s of the Cartanequations is actually an integrable section c = ċ where c is a solution of theLagrangian equations.Given a Hamiltonian form H weakly associated with an almost regularLagrangian L, let us consider the corresponding constrained Hamiltonianform H N (5.133). H N is the same for all Hamiltonian forms weakly associ-


852 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionated with L, and H L = ̂L ∗ H N .For any Hamiltonian form H weakly associated with an almost regularLagrangian L, every solution of the Hamiltonian equations which lives in theLagrangian constraint space N L is a solution of the constrained Hamiltonianequations (5.134).Using the equality H L = ̂L ∗ H N , one can show that the constrainedHamiltonian equations (5.134) are equivalent to the Hamilton–de Donderequations (5.115) and are quasi–equivalent to the Cartan equations (5.114)[Giachetta et. al. (1997); Mangiarotti and Sardanashvily (1998); Lopez andMarsden (2003)].5.6.11 Quadratic Degenerate Lagrangian SystemsGiven a configuration bundle Q → R, let us consider a quadratic LagrangianL which has the coordinate expressionL = 1 2 a ij ˙q i ˙q j + b i ˙q i + c, (5.142)where a, b and c are local functions on Q. This property is coordinate–independent due to the affine transformation law of the coordinates ˙q i .The associated Legendre mapp i ◦ ̂L = a ij ˙q j + b i (5.143)is an affine map over Q. It defines the corresponding linear mapL : V Q −→ V ∗ Q, p i ◦ L = a ij ˙q j . (5.144)Let the Lagrangian L (5.142) be almost regular, i.e., the matrix functiona ij is of constant rank. Then the Lagrangian constraint space N L =̂L(J 1 (R, Q)) is an affine subbundle of the bundle V ∗ Q −→ Q, modelled overthe vector subbundle N L (5.144) of V ∗ Q −→ Q. Hence, N L −→ Q has aglobal section. For the sake of simplicity, let us assume that it is the canonicalzero section ̂0(Q) of V ∗ Q −→ Q. Then N L = N L . Therefore, the kernelof the Legendre map (5.143) is an affine subbundle of the affine jet bundleJ 1 (R, Q) −→ Q, modelled over the kernel of the linear map L (5.144). Thenthere exists a connection Γ on the fibre bundle Q −→ R, given byΓ : Q −→ Ker ̂L ⊂ J 1 (R, Q), with a ij Γ j µ + b i = 0.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 853Connections Γ constitute an affine space modelled over the linear space ofvertical vector–fields υ on Q −→ R, satisfying the conditionsa ij υ j = 0 (5.145)and, as a consequence, the conditions υ i b i = 0. If the Lagrangian (5.142)is regular, the connection Γ is unique.There exists a linear bundle mapσ : V ∗ Q −→ V Q, ˙q i ◦ σ = σ ij p j ,such that L◦σ◦i N = i N . The map σ is a solution of the algebraic equationsThere exist the bundle splittinga ij σ jk a kb = a ib .V Q = Ker a ⊕ E ′ (5.146)and a nonholonomic atlas of this bundle such that transition functions ofKer a and E ′ are independent. Since a is a non–degenerate fibre metric inE ′ , there exists an atlas of E ′ such that a is brought into a diagonal matrixwith non–vanishing components a AA . Due to the splitting (5.146), we havethe corresponding bundle splittingV ∗ Q = (Ker a) ∗ ⊕ Im a. (5.147)Then the desired map σ is represented by a direct sum σ 1 ⊕ σ 0 of anarbitrary section σ 1 of the bundle ∨ 2 Ker a ∗ → Q and the section σ 0 of thebundle ∧ 2 E ′ → Q, which has non–vanishing components σ AA = (a AA ) −1with respect to the above atlas of E ′ . Moreover, σ satisfies the particularrelationsσ 0 = σ 0 ◦ L ◦ σ 0 , a ◦ σ 1 = 0, σ 1 ◦ a = 0. (5.148)The splitting (5.146) leads to the splittingJ 1 (R, Q) = S(J 1 (R, Q)) ⊕ F(J 1 (R, Q)) = Ker ̂L ⊕ Im(σ ◦ ̂L),(5.149)˙q i = S i + F i = [ ˙q i − σ ik0 (a kj ˙q j + b k )] + [σ ik0 (a kj ˙q j + b k )], (5.150)while the splitting (5.147) can be written asV ∗ Q = R(V ∗ Q) ⊕ P(V ∗ Q) = Ker σ 0 ⊕ N L , (5.151)p i = R i + P i = [p i − a ij σ jk0 p k] + [a ij σ jk0 p k]. (5.152)


854 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNote that, with respect to the coordinates S i α and F i α (5.150), the Lagrangian(5.142) readsL = 1 2 a ijF i F j + c ′ ,while the Lagrangian constraint space is given by the reducible constraintsR i = p i − a ij σ jk0 p k = 0.Given the linear map σ and the connection Γ as defined above, let usconsider the affine Hamiltonian mapΦ = ̂Γ + σ : V ∗ Q −→ J 1 (R, Q), Φ i = Γ i + σ ij p j , (5.153)and the Hamiltonian formH = H Φ + Φ ∗ L = p i dq i − [p i Γ i + 1 2 σ 0 ij p i p j + σ 1 ij p i p j − c ′ ]dt(5.154)= (R i + P i )dq i − [(R i + P i )Γ i + 1 2 σij 0 P iP j + σ ij1 p ip j − c ′ ]dt.In particular, if σ 1 is non–degenerate, so is the Hamiltonian form H.The Hamiltonian forms of the type H, parameterized by connections Γ,are weakly associated with the Lagrangian (5.142) and constitute a completeset. Then H is weakly associated with L. Let us write the correspondingHamiltonian equations (5.121) for a section r of the Legendre bundleV ∗ Q −→ R. They areċ = (̂Γ + σ) ◦ r, c = π Q ◦ r. (5.155)Due to the surjections S and F (5.150), the Hamiltonian equations (5.155)break in two partsS ◦ ċ = Γ ◦ c, ṙ i − σ ik (a kj ṙ j + b k ) = Γ i ◦ c, (5.156)F ◦ ċ = σ ◦ r, σ ik (a kj ṙ j + b k ) = σ ik r k . (5.157)Let c be an arbitrary section of Q −→ R, e.g., a solution of the Lagrangianequations. There exists a connection Γ such that the relation (5.156) holds,namely, Γ = S ◦ Γ ′ , where Γ ′ is a connection on Q −→ R which has c as anintegral section.If σ 1 = 0, then Φ = Ĥ and the Hamiltonian forms H are associated withthe Lagrangian (5.142). Thus, for different σ 1 , we have different completesets of Hamiltonian forms H, which differ from each other in the termυ i R i , where υ are vertical vector–fields (5.145). This term vanishes on the


<strong>Applied</strong> Jet <strong>Geom</strong>etry 855Lagrangian constraint space. The corresponding constrained Hamiltonianform H N = i ∗ NH and the constrained Hamiltonian equations (5.134) canbe written.For every Hamiltonian form H, the Hamiltonian equations (5.121) and(5.157) restricted to the Lagrangian constraint space N L are equivalent tothe constrained Hamiltonian equations.Due to the splitting (5.151), we have the corresponding splitting of thevertical tangent bundle V Q V ∗ Q of the bundle V ∗ Q −→ Q. In particular,any vertical vector–field u on V ∗ Q −→ R admits the decompositionu = [u − u T N ] + u T N , with u T N = u i ∂ i + a ij σ jk0 u k∂ i ,such that u N = u T N | NL is a vertical vector–field on the Lagrangian constraintspace N L −→ R. Let us consider the equationsr ∗ (u T N ⌋dH) = 0where r is a section of V ∗ Q −→ R and u is an arbitrary vertical vector–fieldon V ∗ Q −→ R. They are equivalent to the pair of equationsr ∗ (a ij σ jk0 ∂i ⌋dH) = 0, (5.158)r ∗ (∂ i ⌋dH) = 0. (5.159)Restricted to the Lagrangian constraint space, the Hamiltonian equationsfor different Hamiltonian forms H associated with the same quadraticLagrangian (5.142) differ from each other in the equations (5.156). Theseequations are independent of momenta and play the role of gauge–type conditions.5.6.12 Time–Dependent Integrable Hamiltonian SystemsRecall that the configuration space of a time–dependent mechanical systemis a fibre bundle M → R over the time axis R equipped with the bundlecoordinates q α ≡ (t, q k ), for k = 1, . . . , m. The corresponding momentumphase–space is the vertical cotangent bundle V ∗ M of M → R withholonomic bundle coordinates (t, q k , p k ).Recall that the cotangent bundle T ∗ M of M is coordinated by [Mangiarottiand Sardanashvily (1998); Giachetta et. al. (1997)](t, q k , p 0 = p, p k ),p ′ = p + ∂qk∂t p k, (5.160)


856 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand plays the role of the homogeneous momentum phase–space of time–dependent mechanics. It admits the canonical Liouville form Ξ = p α dq α ,the canonical symplectic form Ω T = dΞ, and the corresponding Poissonbracket{f, f ′ } T = ∂ α f∂ α f ′ − ∂ α f∂ α f ′ , (f, f ′ ∈ C ∞ (T ∗ M)). (5.161)There is a canonical 1D fibre bundleζ : T ∗ M → V ∗ M, (5.162)whose kernel is the annihilator of the vertical tangent bundle V M ⊂ T M.The transformation law (5.160) shows that it is a trivial affine bundle.Indeed, given a global section h of ζ, one can equip T ∗ M with the fibrecoordinate r = p − h possessing the identity transition functions.The fibre bundle (5.162) gives the vertical cotangent bundle V ∗ M withthe canonical Poisson structure {, } V such thatζ ∗ {f, f ′ } V = {ζ ∗ f, ζ ∗ f ′ } T , (5.163){f, f ′ } V = ∂ k f∂ k f ′ − ∂ k f∂ k f ′ , (5.164)for all f, f ′ ∈ C ∞ (V ∗ M). The corresponding symplectic foliation coincideswith the fibration V ∗ M → R.However, the Poisson structure (5.164) fails to give any dynamical equationon the momentum phase–space V ∗ M because Hamiltonian vector–fieldsϑ f = ∂ k f∂ k − ∂ k f∂ k , ϑ f ⌋df ′ = {f, f ′ } V , (f, f ′ ∈ C ∞ (V ∗ M)),of functions on V ∗ M are vertical. Hamiltonian dynamics of time–dependent mechanics is described in a different way as a particular Hamiltoniandynamics on fibre bundles [Mangiarotti and Sardanashvily (1998);Giachetta et. al. (1997)].A Hamiltonian on the momentum phase–space V ∗ M → R of time–dependent mechanics is defined as a global sectionh : V ∗ M → T ∗ M, p ◦ h = −H(t, q j , p j ),of the affine bundle ζ (5.162). It induces the pull–back Hamiltonian formH = h ∗ Ξ = p k dq k − Hdt, (5.165)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 857on V ∗ M. Given H (5.165), there exists a unique vector–field γ H on V ∗ Msuch thatThis vector–field readsγ H ⌋dt = 1, γ H ⌋dH = 0. (5.166)γ H = ∂ t + ∂ k H∂ k − ∂ k H∂ k . (5.167)It defines the first–order Hamiltonian equationṫ = 1, ˙q k = ∂ k H, ṗ k = −∂ k H (5.168)on V ∗ M, where (t, q k , p k , ṫ, ˙q k , ṗ k ) are holonomic coordinates on the tangentbundle T V ∗ M. Solutions of this equation are trajectories of the vector–fieldγ H . They assemble into a (regular) foliation of V ∗ M.A first integral of the Hamiltonian equation (5.168) is defined as asmooth real function F on V ∗ M whose Lie derivativeL γH F = γ H ⌋dF = ∂ t F + {H, F } Valong the vector–field γ H (5.167) vanishes, i.e., the function F is constant ontrajectories of the vector–field γ H . A time–dependent Hamiltonian system(V ∗ M, H) on V ∗ M is said to be completely integrable if the Hamiltonianequation (5.168) admits m first integrals F k which are in involution withrespect to the Poisson bracket {, } V (5.164) and whose differentials dF kare linearly independent almost everywhere. This system can be extendedto an autonomous completely integrable Hamiltonian system on T ∗ M asfollows.Let us consider the pull–back ζ ∗ H of the Hamiltonian form H = h ∗ Ξonto the cotangent bundle T ∗ M. Note that the difference Ξ − ζ ∗ h ∗ Ξ is ahorizontal 1–form on T ∗ M → R and thatH ∗ = ∂ t ⌋(Ξ − ζ ∗ h ∗ Ξ)) = p + H (5.169)is a function on T ∗ M [Sniatycki (1980)]. Let us regard H ∗ (5.169) asa Hamiltonian of an autonomous Hamiltonian system on the symplecticmanifold (T ∗ M, Ω T ). The Hamiltonian vector–field of H ∗ on T ∗ M readsγ T = ∂ t − ∂ t H∂ 0 + ∂ k H∂ k − ∂ k H∂ k .It is projected onto the vector–field γ H (5.167) on V ∗ M, and the relationζ ∗ (L γH f) = {H ∗ , ζ ∗ f} T ,(f ∈ C ∞ (V ∗ M)).


858 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionholds. An immediate consequence of this relation is the following.Let (V ∗ M, H; F k ) be a time–dependent completely integrable Hamiltoniansystem with first integrals {F k } on V ∗ M. Then (T ∗ M; H ∗ , ζ ∗ F k ) is anautonomous completely integrable Hamiltonian system on T ∗ M whose firstintegrals {H ∗ , ζ ∗ F k } are in involution with respect to the Poisson bracket{, } T (5.161). Furthermore, let N be a connected invariant manifold of thetime–dependent completely integrable Hamiltonian system (V ∗ M, H; F k ).Then h(N) ⊂ T ∗ M is a connected invariant manifold of the completely integrableHamiltonian system (T ∗ M; H ∗ , ζ ∗ F k ) on T ∗ M. If N contains nocritical points of first integrals F k , then {H ∗ , ζ ∗ F k } have no critical pointsin h(N).5.6.13 Time–Dependent Action–Angle CoordinatesLet us introduce time–dependent action–angle coordinates around an invariantmanifold N of a time–dependent completely integrable Hamiltoniansystem (V ∗ M, H) as those induced by the action–angle coordinates aroundthe invariant manifold h(N) of the autonomous completely integrable system(T ∗ M, H ∗ ).Let M ′ be a connected invariant manifold of an autonomous completelyintegrable system (F α ), α = 1, . . . , n, on a symplectic manifold (Z, Ω Z ), andlet the Hamiltonian vector–fields of first integrals F α on M ′ be complete.Let there exist a neighborhood U of M ′ such that F α have no critical pointsin U and the submersion ×F α : U → R n is a trivial fibre bundle over adomain V ′ ⊂ R n . Then U is isomorphic to the symplectic annulusW = V ′ × (R n−m × T m ),provided with the generalized action–angle coordinates(I 1 , . . . , I n ; x 1 , . . . , x n−m ; φ 1 , . . . , φ m )such that the symplectic form on W readsΩ Z = dI i ∧ x i + dI n−m+k ∧ dφ k ,and the first integrals F α are functions of the action coordinates (I α ) only.In particular, let M ′ be a compact invariant manifold of a completely integrablesystem {F α }, α = 1, . . . , n, on a symplectic manifold (Z, Ω Z ) whichdoes not contain critical points of the first integrals F α . Let the vector–field γ H (5.167) be complete. Let a connected invariant manifold N of atime–dependent completely integrable Hamiltonian system (V ∗ M, H; F k )


<strong>Applied</strong> Jet <strong>Geom</strong>etry 859contain no critical points of first integrals F k , and let its projection N 0onto the fibre V0 ∗ M along trajectories of γ H be compact. Then the invariantmanifold h(N) of the completely integrable Hamiltonian system(T ∗ M; H ∗ , ζ ∗ F k ) has an open neighborhood U.Now, the open neighborhood U of the invariant manifold h(N) of thecompletely integrable Hamiltonian system (T ∗ M; H ∗ , ζ ∗ F k ) is isomorphicto the symplectic annulusW ′ = V ′ × (R × T m ), V ′ = (−ε, ε) × V, (5.170)provided with the generalized action–angle coordinates(I 0 , . . . , I m ; t, φ 1 , . . . , φ m ). (5.171)Moreover, we find that J 0 = r, a α 0 = δ α 0 and, as a consequence,a 0 0 = ∂I 0∂J 0= 1,a 0 i = ∂I i∂J 0= 0,i.e., the action coordinate I 0 is linear in the coordinate r, while I i areindependent of r. With respect to the coordinates (5.171), the symplecticform on W ′ readsΩ T = dI 0 ∧ dt + dI k ∧ dφ k ,the Hamiltonian H ∗ is an affine function H ∗ = I 0 + H ′ (I j ) of the actioncoordinate I 0 , while the first integrals ζ ∗ F k depends only on the actioncoordinates I i . The Hamiltonian vector–field of the Hamiltonian H ∗ isγ T = ∂ t + ∂ i H ′ ∂ i . (5.172)Since the action coordinates I i are independent on the coordinate r, thesymplectic annulus W ′ (5.170) inherits the composite fibrationW ′ → V × (R × T m ) → R. (5.173)Therefore, one can regard W = V × (R × T m ) as a momentum phase–space of the time–dependent Hamiltonian system in question around theinvariant manifold N. It is coordinated by (I i , t, φ i ), which we agree to callthe time–dependent action–angle coordinates. By the relation similar to(5.163), W can be equipped with the Poisson structure{f, f ′ } W = ∂ i f∂ i f ′ − ∂ i f∂ i f ′ ,


860 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhile the global section h ′ : W → W ′ such that I 0 ◦ h ′ = −H ′ , of the trivialbundle ζ (5.173), gives W with the Hamiltonian formH ′ = I i dφ i − H ′ (I j )dt.The associated vector–field γ H (5.166) is exactly the projection onto Wof the Hamiltonian vector–field γ T (5.172), and takes the same coordinateform. It defines the Hamiltonian equation on W ,I i = const, ˙φi = ∂ i H ′ (I j ).One can think of this equation as being the Hamiltonian equation of a time–dependent Hamiltonian system around the invariant manifold N relative totime–dependent action–angle coordinates.5.6.14 Lyapunov StabilityThe notion of the Lyapunov stability of a dynamical equation on a smoothmanifold implies that this manifold is equipped with a Riemannian metric.At the same time, no preferable Riemannian metric is associated to a first–order dynamical equation. Here, we aim to study the Lyapunov stability offirst–order dynamical equations in non–autonomous mechanics with respectto different (time–dependent) Riemannian metrics.Let us recall that a solution s(t), for all t ∈ R, of a first–order dynamicalequation is said to be Lyapunov stable (in the positive direction) if fort 0 ∈ R and any ε > 0, there is δ > 0 such that, if s ′ (t) is another solutionand the distance between the points s(t 0 ) and s ′ (t 0 ) is inferior to δ, thenthe distance between the points s(t) and s ′ (t) for all t > t 0 is inferior to ε.In order to formulate a criterion of the Lyapunov stability with respect to atime–dependent Riemannian metric, we introduce the notion of a covariantLyapunov tensor as generalization of the well–known Lyapunov matrix.The latter is defined as the coefficient matrix of the variation equation[Gallavotti (1983); Hirsch and Smale (1974)], and fails to be a tensor undercoordinate transformations, unless they are linear and time–independent.On the contrary, the covariant Lyapunov tensor is a true tensor–field, butit essentially depends on the choice of a Riemannian metric. The followingwas shown in [Sardanashvily (2002b)]:(i) If the covariant Lyapunov tensor is negative definite in a tubularneighborhood of a solution s at points t ≥ t 0 , this solution is Lyapunovstable.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 861(ii) For any first–order dynamical equation, there exists a (time–dependent) Riemannian metric such that every solution of this equationis Lyapunov stable.(iii) Moreover, the Lyapunov exponent of any solution of a first–orderdynamical equation can be made equal to any real number with respectto the appropriate (time–dependent) Riemannian metric. It follows thatchaos in dynamical systems described by smooth (C ∞ ) first–order dynamicalequations can be characterized in full by time–dependent Riemannianmetrics.5.6.15 First–Order Dynamical EquationsLet R be the time axis provided with the Cartesian coordinate t andtransition functions t ′ = t+const. In geometrical terms [Mangiarotti andSardanashvily (1998)], a (smooth) first–order dynamical equation in non–autonomous mechanics is defined as a vector–field γ on a smooth fibrebundlewhich obeys the condition γ⌋dt = 1, i.e.,π : Y −→ R (5.174)γ = ∂ t + γ k ∂ k . (5.175)The associated first–order dynamical equation takes the formṫ = 1, ẏ k = γ k (t, y j )∂ k , (5.176)where (t, y k , ṫ, ẏ k ) are holonomic coordinates on T Y . Its solutions are trajectoriesof the vector–field γ (5.175). They assemble into a (regular) foliationF of Y . Equivalently, γ (5.175) is defined as a connection on the fibrebundle (5.174).A fibre bundle Y (5.174) is trivial, but it admits different trivializationsY ∼ = R × M, (5.177)distinguished by fibrations Y −→ M. For example, if there is a trivialization(5.177) such that, with respect to the associated coordinates, the componentsγ k of the connection γ (5.175) are independent of t, one says that γis a conservative first–order dynamical equation on M.Hereafter, the vector–field γ (5.175) is assumed to be complete, i.e.,there is a unique global solution of the dynamical equation γ through each


862 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionpoint of Y . For example, if fibres of Y −→ R are compact, any vector–fieldγ (5.175) on Y is complete.If the vector–field γ (5.175) is complete, there exists a trivialization(5.177) of Y , with an atlas Ψ = {(U; t, y a )} of a fibre bundle Y −→ R withtime–independent transition functions y ′a (y b ), such that any solution s ofγ readss a (t) = const, (for all t ∈ R),with respect to associated bundle coordinates (t, y a ). If γ is complete, thefoliation F of its trajectories is a fibration ζ of Y along these trajectoriesonto any fibre of Y , e.g., Y t=0∼ = M. This fibration induces a desiredtrivialization [Mangiarotti and Sardanashvily (1998)].One can think of the coordinates (t, y a ) as being the initial–date coordinatesbecause all points of the same trajectory differ from each other onlyin the temporal coordinate.Let us consider the canonical lift V γ of the vector–field γ (5.175) ontothe vertical tangent bundle V Y of Y −→ R. With respect to the holonomicbundle coordinates (t, y k , y k ) on V Y , it readsV γ = γ + ∂ j γ k y j ∂ k , where ∂ k = ∂∂y k .This vector–field obeys the condition V γ⌋dt = 1, and defines the first–orderdynamical equationṫ = 1, ẏ k = γ k (t, y i ), (5.178)˙ y tk= ∂j γ k (t, y i )y j (5.179)on V Y . The equation (5.178) coincides with the initial one (5.176). Theequation (5.179) is the well–known variation equation. Substituting a solutions of the initial dynamical equation (5.178) into (5.179), one gets alinear dynamical equation whose solutions s are Jacobi fields of the solutions. In particular, if Y −→ R is a vector bundle, there are the canonicalsplitting V Y ∼ = Y × Y and the map V Y −→ Y so that s + s obeys the initialdynamical equation (5.178) modulo the terms of order > 1 in s.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 8635.6.16 Lyapunov Tensor and Stability5.6.16.1 Lyapunov TensorThe collection of coefficientsl j k = ∂ j γ k (5.180)of the variation equation (5.179) is called the Lyapunov matrix. Clearly, itis not a tensor under bundle coordinate transformations of the fibre bundleY (5.174). Therefore, we introduce a covariant Lyapunov tensor as follows.Let a fibre bundle Y → R be provided with a Riemannian fibre metricg, defined as a section of the symmetrized tensor product∨ 2 V ∗ Y → Y (5.181)of the vertical cotangent bundle V ∗ Y of Y → R. With respect to theholonomic coordinates (t, y k , y k ) on V ∗ Y , it takes the coordinate formg = 1 2 g ij(t, y k )dy i ∨ dy j ,where {dy i } are the holonomic fibre bases for V ∗ Y .Given a first–order dynamical equation γ, letV ∗ γ = γ − ∂ j γ k y k ∂ j , where ∂ j = ∂∂y j. (5.182)be the canonical lift of the vector–field γ (5.175) onto V ∗ Y . It is a connectionon V ∗ Y −→ R. Let us consider the Lie derivative L γ g of the Riemannianfibre metric g along the vector–field V ∗ γ (5.182). It readsL ij = (L γ g) ij = ∂ t g ij + γ k ∂ k g ij + ∂ i γ k g kj + ∂ j γ k g ik . (5.183)This is a section of the fibre bundle (5.181) and, consequently, a tensor withrespect to any bundle coordinate transformation of the fibre bundle (5.174).We agree to call it the covariant Lyapunov tensor. If g is an Euclideanmetric, it becomes the following symmetrization of the Lyapunov matrix(5.180),L ij = ∂ i γ j + ∂ j γ i = l i j + l j i .Let us point the following two properties of the covariant Lyapunovtensor.


864 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(i) Written with respect to the initial–date coordinates, the covariantLyapunov tensor is given byL ab = ∂ t g ab .(ii) Given a solution s of the dynamical equation γ and a solution s ofthe variation equation (5.179), we haveL ij (t, s k (t))s i s j = ∂ t (g ij (t, s k (t))s i s j ).The definition of the covariant Lyapunov tensor (5.183) depends on thechoice of a Riemannian fibre metric on the fibre bundle Y .If the vector–field γ is complete, there is a Riemannian fibre metric onY such that the covariant Lyapunov tensor vanishes everywhere. Let uschoose the atlas of the initial–date coordinates. Using the fibration ζ : Y−→ Y t=0 , one can give Y with a time–independent Riemannian fibre metricg ab (t, y c ) = h(t)g ab (0, y c ) (5.184)where g ab (0, y c ) is a Riemannian metric on the fibre Y t=0 and h(t) is apositive smooth function on R. The covariant Lyapunov tensor with respectto the metric (5.184) is given byPutting h(t) = 1, we get L = 0.L ab = ∂ t hg ab .5.6.16.2 Lyapunov StabilityWith the covariant Lyapunov tensor (5.183), we get the following criterionof the stability condition of Lyapunov.Recall that, given a Riemannian fibre metric g on a fibre bundle Y−→ R, the instant–wise distance ρ t (s, s ′ ) between two solutions s and s ′ ofa dynamical equation γ on Y at an instant t is the distance between thepoints s(t) and s ′ (t) in the Riemannian space (Y t , g(t)).Let s be a solution of a first–order dynamical equation γ. If there existsan open tubular neighborhood U of the trajectory s where the covariantLyapunov tensor (5.183) is negative-definite at all instants t ≥ t 0 , thenthere exists an open tubular neighborhood U ′ of s such thatlim [ρt ′ t ′(s, →∞ s′ ) − ρ t (s, s ′ )] < 0


<strong>Applied</strong> Jet <strong>Geom</strong>etry 865for any t > t 0 and any solution s ′ crossing U ′ . Since the condition and thestatement are coordinate–independent, let us choose the following chart ofinitial–date coordinates that covering the trajectory s. Put t = 0 withouta loss of generality. There is an open neighborhood U 0 ⊂ Y 0 ∩ U ofs(0) in the Riemannian manifold (Y 0 , g(0)) which can be provided with thenormal coordinates (x a ) defined by the Riemannian metric g(0) in Y 0 andcentralized at s(0). Let us consider the open tubular U ′ = ζ −1 (U 0 ) withthe coordinates (t, x a ). It is the desired chart of initial–date coordinates.With respect to these coordinates, the solution s reads s a (t) = 0. Lets ′a (t) = u a = const be another solution crossing U ′ . The instant–wise distanceρ t (s, s ′ ), t ≥ 0, between solutions s and s ′ is the distance between thepoints (t, 0) and (t, u) in the Riemannian manifold (Y t , g(t)). This distancedoes not exceed the lengthof the curve[∫ 11/2ρ t (s, s ′ ) = g ab (t, τu c )u a u dτ] b (5.185)0x a = τu a , (τ ∈ [0, 1]) (5.186)in the Riemannian space (Y t , g(t)), while ρ 0 (s, s ′ ) = ρ 0 (s, s ′ ). The temporalderivative of the function ρ t (s, s ′ ) (5.185) reads∫∂ t ρ t (s, s ′ 11) =∂2(ρ t (s, s ′ )) 1/2 t g ab (t, τu c )u a u b dτ. (5.187)0Since the bilinear form ∂ t g ab = L ab , t ≥ 0, is negative-definite at all pointsof the curve (5.186), the derivative (5.187) at all points t ≥ t 0 is alsonegative. Hence, we getρ t>0 (s, s ′ ) < ρ t>0 (s, s ′ ) < ρ 0 (s, s ′ ) = ρ 0 (s, s ′ ).The solution s is Lyapunov stable with respect to the Riemannian fibremetric g. One can think of the solution s as being isometrically Lyapunovstable. Being Lyapunov stable with respect a Riemannian fibre metric g, asolution s need not be so with respect to another Riemannian fibre metricg ′ , unless g ′ results from g by a time–independent transformation.For any first–order dynamical equation defined by a complete vector–field γ (5.175) on a fibre bundle Y −→ R, there exists a Riemannian fibremetric on Y such that each solution of γ is Lyapunov stable. This property


866 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionobviously holds with respect to the Riemannian fibre metric (5.184) whereh = 1.Let λ be a real number. Given a dynamical equation γ defined by acomplete vector–field γ (5.175), there is a Riemannian fibre metric on Ysuch that the Lyapunov spectrum of any solution of γ reduces to λ. Toprove this, recall that the (upper) Lyapunov exponent of a solution s ′ withrespect to a solution s is defined as the limitK(s, s ′ ) =lim− 1t→∞ t ln(ρ t(s, s ′ )). (5.188)Let us give Y with the Riemannian fibre metric (5.184) where h = exp(λt).A simple computation shows that the Laypunov exponent (5.188) withrespect to this metric is exactly λ.If the upper limitλ =−limρ t=0 (s,s ′ )−→0 K(s, s′ )is negative, the solution s is said to be exponentially Lyapunov stable. Ifthere exists at least one positive Lyapunov exponent, one speaks aboutchaos in a dynamical system [Gutzwiller (1990)]. This shows that chaos insmooth dynamical systems can be characterized in full by time–dependentRiemannian metrics.ExampleHere is a simple example which shows that solutions of a smooth first–order dynamical equation can be made Lyapunov stable at will by the choiceof an appropriate time–dependent Riemannian metric.Let R be the time axis provided with the Cartesian coordinate t. Ingeometrical terms, a (smooth) first–order dynamical equation in non–autonomous mechanics is defined as a vector–field γ on a smooth fibrebundle Y −→ R which obeys the condition γ⌋dt = 1. With respect tobundle coordinates (t, y k ) on Y , this vector–field becomes (5.175). Theassociated first–order dynamical equation takes the formẏ k = γ k (t, y j )∂ k ,where (t, y k , ṫ, ẏ k ) are holonomic coordinates on the tangent bundle T Y ofY . Its solutions are trajectories of the vector–field γ (5.175).Let a fibre bundle Y → R be provided with a Riemannian fibre metric g,defined as a section of the symmetrized tensor product ∨ 2 V ∗ Y → Y of the


<strong>Applied</strong> Jet <strong>Geom</strong>etry 867vertical cotangent bundle V ∗ Y of Y → R. With respect to the holonomiccoordinates (t, y k , y k ) on V ∗ Y , it takes the coordinate formg = 1 2 g ij(t, y k )dy i ∨ dy j ,where {dy i } are the holonomic fibre bases for V ∗ Y .Recall that above we have proposed the following: Let λ be a realnumber. Given a dynamical equation defined by a complete vector–field γ(5.175), there exists a Riemannian fibre metric on Y such that the Lyapunovspectrum of any solution of γ is λ. The following example aims to illustratethis fact.Let us consider 1D motion on the axis R defined by the first–orderdynamical equationẏ = y (5.189)on the fibre bundle Y = R × R −→ R coordinated by (t, y). Solutions of theequation (5.189) reads(t) = c exp(t), (with c = const). (5.190)Let e yy = 1 be the standard Euclidean metric on R. With respect to thismetric, the instant–wise distance between two arbitrary solutionsof the equation (5.189) iss(t) = c exp(t), s ′ (t) = c ′ exp(t) (5.191)ρ t (s, s ′ ) e = |c − c ′ | exp(t).Hence, the Lyapunov exponent K(s, s ′ ) (5.188) equals 1, and so is theLyapunov spectrum of any solution (5.190) of the first–order dynamicalequation (5.189).Let now λ be an arbitrary real number. There exists a coordinatey ′ = y exp(−t) on R such that, written relative to this coordinate, thesolutions (5.190) of the equation (5.189) read s(t) = const. Let us choose theRiemannian fibre metric on Y → R which takes the form g y′ y ′ = exp(2λt)with respect to the coordinate y ′ . Then relative to the coordinate y, itreadsg yy = ∂y′ ∂y ′∂y ∂y g y ′ y ′ = exp(2(λ − 1)t). (5.192)


868 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe instant–wise distance between the solutions s and s ′ (5.191) with respectto the metric g (5.192) isρ t (s, s ′ ) = [g yy (s(t) − s ′ (t)) 2 ] 1/2 = |c − c ′ | exp(λt).One at once gets that the Lyapunov spectrum of any solution of the differentialequation (5.189) with respect to the metric (5.192) is λ.5.7 Application: Jets and Multi–Time RheonomicDynamicsRecall that a number of geometrical models in mechanics and physics arebased on the notion of ordinary, autonomous Lagrangian (i.e., a smoothreal function on R × T M). In this sense, we recall that a Lagrangian spaceL n = (M, L(x, y)) is defined as a pair which consists of a real, smooth,nD manifold M with local coordinates x i , (i = 1, ..., n) and a regular LagrangianL : T M → R. The geometry of Lagrangian spaces is now used invarious fields to study natural phenomena where the dependence on position,velocity or momentum is involved [Kamron and Olver (1989)]. Also,this geometry gives a model for both the gravitational and electromagneticfield theory, in a very natural blending of the geometrical structureof the space with the characteristic properties of the physical fields. Again,there are many problems in physics and variational calculus in which time–dependent Lagrangians are involved.In the context exposed in [Miron et. al. (1988); Miron and Anastasiei(1994)], the energy action functional E, attached to a given time–dependentLagrangian,L : R × T M → R, (t, x i , v i ) ↦→ L(t, x i , v i ), (i = 1, ..., n)not necessarily homogenous with respect to the direction {v i }, is of theformE(c) =∫ baL(t, x i (t), ẋ i (t)) dt, (5.193)where [a, b] ⊂ R, and c : [a, b] → M is a smooth curve, locally expressed byt ↦→ x i (t), and having the velocity ẋ = (ẋ i (t)). It is obvious that the non–homogeneity of the Lagrangian L, regarded as a smooth function on theproduct manifold R × T M, implies that the energy action functional E isdependent of the parametrizations of every curve c. In order to remove this


<strong>Applied</strong> Jet <strong>Geom</strong>etry 869difficulty, [Miron et. al. (1988); Miron and Anastasiei (1994)] regard thespace R × T M like a fibre bundle over M. In this context, the geometricalinvariance group of R × T M is given by¯t = t, ¯x i = ¯x i (x j ), ¯v i = ∂¯xi∂x j vj . (5.194)The structure of the gauge group (5.194) emphasizes the absolute characterof the time t from the classical rheonomic Lagrangian mechanics. At thesame time, we point out that the gauge group (5.194) is a subgroup of thegauge group of the configuration bundle J 1 (R, M), given as¯t = ¯t(t), ¯x i = ¯x i (x j ), ¯v i = ∂¯xi∂x j dtd¯t vj . (5.195)In other words, the gauge group (5.195) of the jet bundle J 1 (R, M), fromthe relativistic rheonomic Lagrangian mechanics is more general than thatused in the classical rheonomic Lagrangian mechanics, which ignores thetemporal reparametrizations. A deep exposition of the physical aspectsof the classical rheonomic Lagrangian geometry is done by [Ikeda (1990)],while the classical rheonomic Lagrangian mechanics is done by [Matsumoto(1982)].Therefore, to remove the parametrization dependence of E, they ignorethe temporal repametrizations on R × T M. Naturally, in these conditions,their energy functional becomes a well defined one, but their approachstands out by the ‘absolute’ character of the time t.In a more general geometrical approach, [Neagu and Udrişte (2000a);Udriste (2000); Neagu (2002)] tried to remove this inconvenience. Followingthis approach, we regard the mechanical 1–jet space J 1 (R, M) ≡ R × T Mas a fibre bundle over the base product–manifold R × M. The gauge groupof this bundle of configurations is given by 5.195. Consequently, our gaugegroup does not ignore the temporal reparametrizations, hence, it standsout by the relativistic character of the time t. In these conditions, using agiven semi–Riemannian metric h 11 (t) on R, we construct the more generaland natural energy action functional, settingE(c) =∫ baL(t, x i (t), ẋ i (t)) √ |h 11 | dt. (5.196)Obviously, E is well defined and is independent of the curve parametrizations.


870 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.7.1 Relativistic Rheonomic Lagrangian SpacesIn order to develop the time–dependent Lagrangian geometry, following[Neagu and Udrişte (2000a); Udriste (2000); Neagu (2002); Neagu andUdrişte (2000b); Neagu (2000)], we consider L : J 1 (R, M) → R to be asmooth Lagrangian function on the 1–jet bundle J 1 (R, M) → R, locallyexpressed by (t, x i , v i ) ↦→ L(t, x i , v i ). The so–called vertical fundamentalmetrical d−tensor of L is defined byR.(i)(j) = 1 ∂ 2 L2 ∂v i ∂v j . (5.197)G (1)(1)Let h = (h 11 ) be a semi–Riemannian metric on the temporal manifoldA Lagrangian function L : J 1 (R, M) → R whose vertical fundamentalmetrical d−tensor is of the formG (1)(1)(i)(j) (t, xk , v k ) = h 11 (t)g ij (t, x k , v k ), (5.198)where g ij (t, x k , v k ) is a d−tensor on J 1 (R, M), symmetric, of rank n andhaving a constant signature on J 1 (R, M), is called a Kronecker h−regularLagrangian function, with respect to the temporal semi–Riemannian metrich = (h 11 ).A pair RL n = (J 1 (R, M), L), where n = dim M, which consists ofthe 1−jet space J 1 (R, M) and a Kronecker h−regular Lagrangian functionL : J 1 (T, M) → R is called a relativistic rheonomic Lagrangian space.In our geometrization of the time–dependent Lagrangian function L thatwe will construct, all entities with geometrical or physical meaning willbe directly arisen from the vertical fundamental metrical d−tensor G (1)(1)(i)(j) .This fact points out the metrical character (see [Gotay et. al. (1998)])and the naturalness of the subsequent relativistic rheonomic Lagrangiangeometry.For example, suppose that the spatial manifold M is also equippedwith a semi–Riemannian metric g = (g ij (x)). Then, the time–dependentLagrangian function L 1 : J 1 (R, M) → R defined byL 1 = h 11 (t)g ij (x)v i v j (5.199)is a Kronecker h−regular time–dependent Lagrangian function. Consequently,the pair RL n = (J 1 (R, M), L 1 ) is a relativistic√rheonomic Lagrangianspace. We underline that the Lagrangian L 1 = L 1 |h11 | is exactly


<strong>Applied</strong> Jet <strong>Geom</strong>etry 871the energy Lagrangian whose extremals are the harmonic maps between thesemi–Riemannian manifolds (R, h) and (M, g). At the same time, this Lagrangianis a basic object in the physical theory of bosonic strings (comparewith subsection 6.7 below).In above notations, taking U (1)(i) (t, x) as a d−tensor–field on J 1 (R, M)and F : R × M → R a smooth map, the more general Lagrangian functionL 2 : J 1 (R, M) → R defined byL 2 = h 11 (t)g ij (x)v i v j + U (1)(i) (t, x)vi + F (t, x) (5.200)is also a Kronecker h−regular Lagrangian. The relativistic rheonomic Lagrangianspace RL n = (J 1 (R, M), L 2 ) is called the autonomous relativisticrheonomic Lagrangian space of electrodynamics because, in the particularcase h 11 = 1, we recover the classical Lagrangian space of electrodynamics[Miron et. al. (1988); Miron and Anastasiei (1994)] which governs the movementlaw of a particle placed concomitantly into a gravitational field and anelectromagnetic one. From a physical point of view, the semi–Riemannianmetric h 11 (t) (resp. g ij (x)) represents the gravitational potentials of thespace R (resp. M), the d−tensor U (1)(i)(t, x) stands for the electromagneticpotentials and F is a function which is called potential function. The nondynamicalcharacter of spatial gravitational potentials g ij (x) motivates usto use the term of ‘autonomous’.More general, if we consider g ij (t, x) a d−tensor–field on J 1 (R, M),symmetric, of rank n and having a constant signature on J 1 (R, M), we candefine the Kronecker h−regular Lagrangian function L 3 : J 1 (R, M) → R,settingL 3 = h 11 (t)g ij (t, x)v i v j + U (1)(i) (t, x)vi + F (t, x). (5.201)The pair RL n = (J 1 (R, M), L 3 ) is a relativistic rheonomic Lagrangian spacewhich is called the non–autonomous relativistic rheonomic Lagrangianspace of electrodynamics. Physically, we remark that the gravitational potentialsg ij (t, x) of the spatial manifold M are dependent of the temporalcoordinate t, emphasizing their dynamical character.5.7.2 Canonical Nonlinear ConnectionsLet us consider h = (h 11 ) a fixed semi–Riemannian metric on R and arheonomic Lagrangian space RL n = (J 1 (R, M), L), where L is a Kroneckerh−regular Lagrangian function. Let [a, b] ⊂ R be a compact interval in


872 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe temporal manifold R. In this context, we can define the energy actionfunctional of RL n , settingE : C ∞ (R, M) → R, E(c) =∫ baL(t, x i , v i ) √ |h|dt,where the smooth curve c is locally expressed by (t) → (x i (t)) and v i = dxidt .The extremals of the energy functional E verifies the Euler–Lagrangianequations2G (1)(1)(i)(j) ẍj +∂2 L∂x j ∂v i ẋj − ∂L∂x i + ∂2 L∂t∂v i + ∂L∂v i H1 11 = 0,(i = 1, ..., n),(5.202)where H 1 11 are the Christoffel symbols of the semi–Riemannian metric h 11 .Taking into account the Kronecker h−regularity of the Lagrangian functionL, it is possible to rearrange the Euler–Lagrangian equations (5.202)of the Lagrangian L = L √ |h|, in the Poisson form [Neagu and Udrişte(2000a)]∆ h x i + 2G i (t, x i , v i ) = 0, (i = 1, ..., n), where (5.203)∆ h x i = h 11 { ẍ i − H11v 1 i} , v i = ẋ i ,{2G i = gii ∂ 2 L2 ∂x j ∂v i vj − ∂L∂x i + ∂2 L∂t∂v i + ∂L}∂v i H1 11 + 2g ij h 11 H11v 1 j .Denoting G (r)(1)1 = h 11G r , the geometrical object G = (G (r)(1)1) is a spatialspray on J 1 (R, M). By a direct calculation, we deduce that the localgeometrical entities of J 1 (R, M)2S k = gki22H k = gki2{ ∂ 2 L∂x j ∂v i vj − ∂L }∂x i ,{ ∂ 2 L∂t∂v i + ∂L∂v i H1 11verify the following transformation rules2S p r ∂xp ∂xp d¯t ∂¯x= 2 ¯S l γ+ h11∂¯xr∂¯x l dt ∂x j vj ,2J p = 2J ¯r∂xp ∂xp d¯t ∂¯v l− h11∂¯xr∂¯x l dt ∂t .}, 2J k = h 11 H 1 11v j ,2H p = 2r ∂xp ∂xp¯H + h11∂¯xrd¯t∂¯x l dt∂¯v l∂t ,Consequently, the local entities 2G p = 2S p + 2H p + 2J p can be modified


<strong>Applied</strong> Jet <strong>Geom</strong>etry 873by the transformation laws2Ḡr = 2G p ∂¯xr ∂xp ∂¯x r µ− h11∂xp ∂¯x j ∂x p ¯vj . (5.204)The extremals of the energy functional attached to a Kroneckerh−regular Lagrangian function L on J 1 (R, M) are harmonic curves of thetime–dependent spray (H, G), with respect to the semi–Riemannian metrich, defined by the temporal componentsand the local spatial componentsG (i)(1)1 = h 11g ik4H (i)(1)1 = −1 2 H1 11(t)v i[ ∂ 2 L∂x j ∂v k vj − ∂L∂x k +∂2 L∂t∂v k + ∂L]∂x k H1 11 + 2h 11 H11g 1 kl v l .The time–dependent spray (H, G) constructed from the previous Theoremis called the canonical time–dependent spray attached to the relativisticrheonomic Lagrangian space RL n .In the particular case of an autonomous electrodynamics relativistic rheonomicLagrangian space (i.e., g ij (t, x k , v k ) = g ij (x k )), the canonical spatialspray G is given by the componentsG (i)(1)1 = 1 2 γi jkv j v k + h 11g li4where U (1)(i)j(1)(1)∂U ∂U (i)(j)=∂x− j ∂x. i⎡⎣U (1)(l)j vj +∂U(1)(l)∂t⎤+ U (1)(l) H1 11 − ∂F ⎦∂x l , (5.205)We have the following Theorem: The pair of local functions Γ =(M (i)(1)1 , N (i)(1)j), which consists of the temporal componentsand the spatial componentsM (i)(1)1 = 2H(i) (1)1 = −H1 11v i , (5.206)N (i)(1)j = ∂Gi (1)1∂v j , (5.207)where H (i)(1)1and G(i)(1)1are the components of the canonical time–dependentspray of RL n , represents a nonlinear connection on J 1 (R, M).


874 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe nonlinear connection Γ = (M (i)(1)1 , N (i)(1)j) from the preceding Theoremis called the canonical nonlinear connection of the relativistic rheonomicLagrangian space RL n .In the case of an autonomous electrodynamics relativistic rheonomic Lagrangianspace (i.e., g ij (t, x k , v k ) = g ij (x k )), the canonical nonlinear connectionbecomes Γ = (M (i)(1)1 , N (i)(1)j), whereM (i)(1)1 = −H1 11v i , N (i)4(1)j = γi jkv k + h 11g ik5.7.3 Cartan’s Canonical ConnectionsU (1)(k)j . (5.208)The main Theorem of this paper is the Theorem of existence of the Cartancanonical h−normal linear connection CΓ which allow the subsequentdevelopment of the relativistic rheonomic Lagrangian geometry of physicalfields, which will be exposed in the next sections.On the relativistic rheonomic Lagrangian space RL n = (J 1 (R, M), L)equipped with its canonical nonlinear connection Γ there is a uniqueh−normal Γ−linear connectionhaving the metrical properties:(i) g ij|k = 0, g ij | (1)(k) = 0,(ii) G k j1 = gki2CΓ = (H 1 11, G k j1, L i jk, C i(1)j(k) )δg ijδt , Lk ij = Lk ji , Ci(1) j(k) = Ci(1) k(j) .To prove this Theorem, let CΓ = (Ḡ1 11, G k j1 , Li jk , Ci(1) j(k)) be a h−normalΓ−linear connection whose coefficients are defined by Ḡ1 11 = H11, 1 G k j1 =g ki2δg ijδt , andL i jk = gim2(δgjmδx k+ δg kmδx j− δg )(jkδx m , C i(1)j(k) = gim ∂gjm2 ∂v k + ∂g km∂v j − ∂g )jk∂v m .By computations, one can verify that CΓ satisfies the conditions (i) and(ii).Conversely, let us consider ¯CΓ = (Ḡ 1 11, Ḡk j1 , ¯L i i(1)jk, ¯Cj(k) ) a h−normalΓ−linear connection which satisfies (i) and (ii). It follows directly thatḠ 1 11 = H 1 11, and Ḡ k j1 = gki2δg ijδt .


<strong>Applied</strong> Jet <strong>Geom</strong>etry 875The condition g ij|k = 0 is equivalent withδg ijδx k = g mj ¯L m ik + g im ¯Lm jk .Applying the Christoffel process to the indices {i, j, k}, we find¯L i jk = gim2(δgjmδx k+ δg kmδx j− δg )jkδx m .By analogy, using the relations C i(1)j(k) = Ci(1) k(j) and g ij| (1)(k)= 0, followinga Christoffel process applied to the indices {i, j, k}, we get(¯C i(1)j(k) = gim ∂gjm2 ∂v k + ∂g km∂v j − ∂g )jk∂v m .As a rule, the Cartan canonical connection of a relativistic rheonomicLagrangian space RL n verifies also the propertiesh 11/1 = h 11|k = h 11 | (1)(k) = 0 and g ij/1 = 0. (5.209)The torsion d−tensor T of the Cartan canonical connection of a relativisticrheonomic Lagrangian space is determined by only six local components,because the properties of the Cartan canonical connection imply therelations Tijm = 0 and S(i)(1)(1)(1)(j)(k)= 0. At the same time, we point out thatthe number of the curvature local d−tensors of the Cartan canonical connectionnot reduces. In conclusion, the curvature d−tensor R of the Cartancanonical connection is determined by five effective local d−tensors. Thetorsion and curvature d−tensors of the Cartan canonical connection of anRL n are called the torsion and curvature of RL n .All torsion d−tensors of an autonomous relativistic rheonomic Lagrangianspace of electrodynamics vanish, except⎡R (m)(1)1j = −h 11g mk⎣H 1411U (1)(k)j(k)j+ ∂U (1)∂t⎤⎦ ,R (m)(1)ij = rm ijkv k + h 11g mk []U (1)4(k)i|j + U (1)(k)j|i,where r m ijk are the curvature tensors of the semi–Riemannian metric g ij.


876 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.7.4 General Nonlinear ConnectionsRecall that a nonlinear connection (i.e., a supplementary–horizontal distributionof the vertical distribution of J 1 (R, M)) offers the possibility ofconstruction of the vector or covector adapted bases on J 1 (R, M)) [Neaguand Udrişte (2000a)]. A nonlinear connection Γ on J 1 (R, M) is determinedwhich modify by the trans-by a pair of local function sets M (i)(1)1formation laws¯M (j)(1)1d¯tdt = M (k) dt ∂¯x j(1)1d¯t ∂x k − ∂¯vj∂t ,A set of local functions M (i)(1)1and N(i)(1)j(j) ∂¯x¯N k(1)k∂x i = N (k) dt ∂¯x j(1)id¯t ∂x k − ∂¯vj∂x i .(5.210)(i)(resp. N(1)j ) on J 1 (R, M), which transformby the rules (5.210) is called a temporal nonlinear connection (resp.spatial nonlinear connection) on J 1 (R, M).For example, studying the transformation rules of the local componentsM (i)(1)1 = −H1 11v i , N (i)(1)j = γi jkv k ,where H11 1 (resp. γ i jk) are the Christoffel symbols of a temporal (resp.spatial) semi–Riemannian metric h (resp. ϕ), we conclude that Γ 0 =(M (i)(1)1 , N (i)(1)j ) represents a nonlinear connection on J 1 (R, M), which iscalled the canonical nonlinear connection attached to the metric pair (h, ϕ).If M (i)(1)1are the components of a temporal nonlinear connection, thenthe components H (i)(1)1 = 1 2 M (i)(1)1represent a temporal spray. Conversely, ifH (i)(i)(1)1are the components of a temporal spray, then M(1)1 = 2H(i) (1)1 arethe components of a temporal nonlinear connection. If G (i)(1)1are the componentsof a spatial spray, then the components N (i)(1)j = ∂Gi (1)1∂v jspatial nonlinear connection.Conversely, the spatial nonlinear connection N (i)(1)jrepresent ainduces the spatialspray 2G (i)(1)1 = N (i)(1)j vj .The previous theorems allow us to conclude that a time–dependentspray (H, G) induces naturally a nonlinear connection Γ on J 1 (R, M),which is called the canonical nonlinear connection associated to the time–dependent spray (H, G). We point out that the canonical nonlinear connectionΓ attached to the time–dependent spray (H, G) is a natural generalizationof the canonical nonlinear connection N induced by a time–dependentspray G from the classical rheonomic Lagrangian geometry [Miron et. al.(1988); Miron and Anastasiei (1994)].


<strong>Applied</strong> Jet <strong>Geom</strong>etry 877Let Γ = (M (i)(1)1 N (i)(1)j ) be a nonlinear connection on J 1 (R, M). Let usconsider the geometrical objects,δδt = ∂ (j) ∂−M∂t(1)1∂v j ,δδx i = ∂ (j) ∂−N∂xi (1)i∂v j ,δvi = dy i +M (i) (i)(1)1dt+N(1)j dxj .One can deduce that the set of vector–fields { δδt , δδx i ,∂∂v i }⊂ X (J 1 (R, M))and of covector–fields {dt, dx i , δv i } ⊂ X ∗ (J 1 (R, M)) are dual bases. Theseare called the adapted bases on J 1 (R, M), determined by the nonlinearconnection Γ. The big advantage of the adapted bases is that the transformationlaws of its elements are simple and natural. The transformation lawsof the elements of the adapted bases attached to the nonlinear connectionΓ areδδt = d¯t δdt δ¯t ,dt = dtd¯t d¯t,δδx i = ∂¯xj δ∂x i δ¯x j ,dx i = ∂xi∂¯x j d¯xj ,5.8 Jets and Action Principles∂∂v i = ∂¯xj dt δ∂x i d¯t δ¯v j ,δv i = ∂xi d¯t∂¯x j dt δ¯vj .Recall that in the classical calculus of variations one studies functionals ofthe form∫F L (z) = L(x, z, ∇z) dx, (with Ω ⊂ R n ), (5.211)Ωwhere x = (x 1 , . . . , x n ), dx = dx 1 ∧ · · · ∧ dx n , z = z(x) ∈ C 1 (¯Ω), and theLagrangian L = L(x, z, p) is a smooth function of x, z, and p = (p 1 , . . . , p n ).The corresponding Euler–Lagrangian equation, describing functions z(x)that are stationary for such a functional, is represented by the second–order PDE [Bryant et al. (2003)]∆z(x) = F ′ (z(x)).For example, we may identify a function z(x) with its graph N ⊂ R n+1 ,and take the LagrangianL = √ 1 + ||p|| 2 ,whose associated functional F L (z) equals the area of the graph, regardedas a hypersurface in Euclidean space. The Euler–Lagrangian equation describingfunctions z(x) stationary for this functional is H = 0, where H isthe mean curvature of the graph N.


878 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionTo study these Lagrangians and Euler–Lagrangian equations geometrically,we have to choose a class of admissible coordinate changes, andthere are four natural candidates. In increasing order of generality, theyare [Bryant et al. (2003)]:• Classical transformations, of the form x ′ = x ′ (x), z ′ = z ′ (z); in thissituation, we think of (x, z, p) as coordinates on the space J 1 (R n , R) of1−jets of maps R n → R.• Gauge transformations, of the form x ′ = x ′ (x), z ′ = z ′ (x, z); here, wethink of (x, z, p) as coordinates on the space of 1−jets of sections ofa bundle R n+1 → R n , where x = (x 1 , . . . , x n ) are coordinates on thebase R n and z ∈ R is a fibre coordinate.• Point transformations, of the form x ′ = x ′ (x, z), z ′ = z ′ (x, z); here, wethink of (x, z, p) as coordinates on the space of tangent hyperplanes{dz − p i dx i } ⊥ ⊂ T (x i ,z)(R n+1 )of the manifold R n+1 with coordinates (x 1 , . . . , x n , z).• Contact transformations, of the form x ′ = x ′ (x, z, p), z ′ = z ′ (x, z, p),p ′ = p ′ (x, z, p), satisfying the equation of differential 1−formsfor some function f(x, z, p) ≠ 0.dz ′ − p ′ idx i′ = f · (dz − p i dx i )Classical calculus of variations primarily concerns the following featuresof the functional F L (5.211).The first variation δF L (z) is analogous to the derivative of a function,where z = z(x) is thought of as an independent variable in an infinite–dimensional space of functions. The analog of the condition that a pointbe critical is the condition that z(x) be stationary for all fixed–boundaryvariations. Formally, we writeδF L (z) = 0,which will give us a second–order scalar PDE for the unknown functionz(x) of the form∂ z L − ∂ x i(∂ pi L) = 0, (5.212)namely the Euler–Lagrangian equation of the Lagrangian L(x, z, p).In this section we will study the PDE (5.212) in an invariant, geometricalsetting, following [Bryant et al. (2003)]. As a motivation for this


<strong>Applied</strong> Jet <strong>Geom</strong>etry 879geometrical approach, we note the fact that Lagrangian is invariant underthe large class of contact transformations. Also, note that the LagrangianL determines the functional F L , but not vice versa. To see this, observethat if we add to L(x, z, p) a divergence term and considerL ′ (x, z, p) = L(x, z, p) + ∑ (∂ x iK i (x, z) + ∂ z K i (x, z)p i )for functions K i (x, z), then by the Green’s Theorem, the functionals F Land F L ′ differ by a constant depending only on values of z on ∂Ω. L andL ′ have the same Euler–Lagrangian equations.Also, there is a relationship between symmetries of a Lagrangian Land conservation laws for the corresponding Euler–Lagrangian equations,described by the Noether Theorem. A subtlety here is that the group ofsymmetries of an equivalence class of Lagrangians may be strictly largerthan the group of symmetries of any particular representative. We willinvestigate how this discrepancy is reflected in the space of conservationlaws, in a manner that involves global topological issues.Finally, one considers the second variation δ 2 F L , analogous to the Hessianof a smooth function, usually with the goal of identifying local minimaof the functional. There has been a great deal of analytic work done in thisarea for classical variational problems, reducing the problem of local minimizationto understanding the behavior of certain Jacobi operators, butthe geometrical theory is not as well–developed as that of the first variationand the Euler–Lagrangian equations.Now we turn to multi–index notation [Griffiths (1983); Bryant et al.(2003); Choquet-Bruhat and DeWitt-Morete (1982)]. An exterior differentialsystem (EDS) is a pair (M, E) consisting of a smooth manifold Mand a homogeneous, differentially closed ideal E ⊆ Ω ∗ (M) in the algebraof smooth differential forms on M. Some of the EDSs that we study aredifferentially generated by the sections of a smooth subbundle I ⊆ T ∗ Mof the cotangent bundle of M; this subbundle, and sometimes its space ofsections, is called a Pfaffian system on M. It will be useful to use the notation{α, β, . . .} for the (two–sided) algebraic ideal generated by forms α,β,. . . , and to use the notation {I} for the algebraic ideal generated by thesections of a Pfaffian system I ⊆ T ∗ M. An integral manifold of an EDSdef(M, E) is a submanifold immersion ι : N ↩→ M for which ϕ N = ι ∗ ϕ = 0for all ϕ ∈ E. Integral manifolds of Pfaffian systems are defined similarly.A differential form ϕ on the total space of a fibre bundle π : E → Bis said to be semibasic if its contraction with any vector–field tangent to


880 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe fibers of π vanishes, or equivalently, if its value at each point e ∈ E isthe pull–back via π ∗ e of some form at π(e) ∈ B. Some authors call such aform horizontal. A stronger condition is that ϕ be basic, meaning that it islocally (in open subsets of E) the pull–back via π ∗ of a form on the base B[Bryant et al. (2003)].If (ω 1 , . . . , ω n ) is an ordered basis for a vector space V , then correspondingto a multi–index I = (i 1 , . . . , i k ) is the k−vectorω I = ω i1 ∧ · · · ∧ ω i k∈ ∧ k (V ).and for the complete multi–index we defineω = ω 1 ∧ · · · ∧ ω n .Letting (e 1 , . . . , e n ) be a dual basis for V ∗ , we also define the (n−k)−vectorω (I) = e I ⌋ω = e ik ⌋(e ik−1 ⌋ · · · (e i1 ⌋ω) · · · ).This ω (I) is, up to sign, just ω Ic , where I c is a multi–index complementaryto I.Recall that a contact manifold (M, I) is a smooth manifold M of dimension2n + 1, with a distinguished line subbundle I ⊂ T ∗ M of the cotangentbundle which is non–degenerate in the sense that for any local 1−form θgenerating I,θ ∧ (dθ) n ≠ 0.For example, A 1−jet is an equivalence class of functions having thesame value and the same first derivatives at some designated point of thedomain. On the space J 1 (R n , R) of 1−jets of functions, we can take coordinates(x i , z, p i ) corresponding to the jet at (x i ) ∈ R n of the linear functionf(¯x) = z + p i (¯x i − x i ). Then we define the contact formfor whichθ = dz − p i dx i ,dθ = −dp i ∧ dx i ,so the non–degeneracy condition θ ∧ (dθ) n ≠ 0 is apparent. In fact, thePfaff Theorem [Bryant et al. (2003)] implies that every contact manifold islocally isomorphic to this example; that is, every contact manifold (M, I)has local coordinates (x i , z, p i ) for which the form θ = dz − p i dx i generatesI.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 881Let (M, I) be a contact manifold of dimension 2n + 1, and assume thatI is generated by a global, non–vanishing section θ ∈ Γ(I); this assumptiononly simplifies our notation, and would in any case hold on a double–coverof M. Sections of I generate the contact differential idealI = {θ, dθ} ⊂ Ω ∗ (M)in the exterior algebra of differential forms on M. A Legendre submanifoldof M is an immersion ι : N ↩→ M of an nD submanifold N such thatι ∗ θ = 0 for any contact form θ ∈ Γ(I); in this case ι ∗ dθ = 0 as well,so a Legendre submanifold is the same thing as an integral manifold ofthe differential ideal I. In Pfaff coordinates with θ = dz − p i dx i , onesuch integral manifold is N 0 = {z = p i = 0}. To see other Legendresubmanifolds ‘near’ this one, note than any submanifold C 1 −close to N 0satisfies the independence condition [Bryant et al. (2003)]dx 1 ∧ · · · ∧ dx n ≠ 0,and can therefore be described locally as a graphIn this case, we haveN = {(x i , z(x), p i (x))}.θ| N = 0 iff p i (x) = ∂ x iz(x).Therefore, N is determined by the function z(x), and conversely, everyfunction z(x) determines such an N; we informally say that ‘the genericLegendre submanifold depends locally on one arbitrary function of n variables’.Legendre submanifolds of this form, with dx| N ≠ 0, are calledtransverse.Now, we are interested in functionals given by triples (M, I, Λ), where(M, I) is a (2n + 1)D contact manifold, and Λ ∈ Ω n (M) is a differentialform of degree n on M; such a Λ will be referred to as a Lagrangian on(M, I) [Bryant et al. (2003)]. We then define a functional on the set ofsmooth, compact Legendre submanifolds N ⊂ M, possibly with boundary∂N, by∫F Λ (N) = Λ.The classical variational problems described above may be recovered fromthis notion by taking M = J 1 (R n , R) ∼ = R 2n+1 with coordinates (x i , z, p i ),I generated by θ = dz − p i dx i , and Λ = L(x i , z, p i )dx. This formulationN


882 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionalso admits certain functionals depending on second derivatives of z(x),because there may be dp i −terms in Λ. Later, we will restrict attention to aclass of functionals which, possibly after a contact transformation, can beexpressed without second derivatives.Suppose given a Lagrangian Λ ∈ Ω n (M) on a contact manifold (M, I),and a fixed–boundary variation of Legendre submanifold F : N × [0, 1] →M; we wish to calculate d dt (∫ N tΛ).To do this, first recall the calculation of the Poincaré–Cartan form forthe equivalence class [Λ] ∈ ¯H n . Because I n+1 = Ω n+1 (M), we can write[Bryant et al. (2003)]and thendΛ = θ ∧ α + dθ ∧ β = θ ∧ (α + dβ) + d(θ ∧ β),Π = θ ∧ (α + dβ) = d(Λ − θ ∧ β). (5.213)We are looking for conditions on a Legendre submanifold f : N ↩→ M tobe stationary for [Λ] under all fixed–boundary variations, in the sense thatd ∣dt t=0( ∫ N tΛ) = 0 whenever F | t=0 = f. We calculate∫ ∫∫∫∂ t Λ = ∂ t (Λ − θ ∧ β) = L ∂t (Λ − θ ∧ β) = ∂ t ⌋Π.N t N t N t N tOne might express this result asδ(F Λ ) N (v) = int N v⌋f ∗ Π,where the variational vector–field v, lying in the space Γ 0 (f ∗ T M) of sectionsof f ∗ T M vanishing along ∂N, plays the role of ∂ t . The condition Π ≡0(mod{I}) allows us to write Π = θ ∧ Ψ for some n−form Ψ, not uniquelydetermined, and we have [Bryant et al. (2003)]∫ddt∣ Λ = g ft=0∫N ∗ Ψ,twhere g = (∂ t ⌋F ∗ θ)| t=0 . It was shown previously that this g could locallybe chosen arbitrarily in the interior N o , so the necessary and sufficientcondition for a Legendre submanifold f : N ↩→ M to be stationary for F Λis that f ∗ Ψ = 0.In the particular classical situation where M = {(x i , z, p i )}, θ = dz −p i dx i , and Λ = L(x, z, p)dx, we havedΛ = L z θ ∧ dx + L pi dp i ∧ dx = θ ∧ L z dx − dθ ∧ L pi dx (i) ,N


<strong>Applied</strong> Jet <strong>Geom</strong>etry 883so referring to (5.213),Π = θ ∧ (L z dx − d(L pi dx (i) )) = θ ∧ Ψ.Now, for a transverse Legendre submanifold N = {(x i , z(x), z x i(x))}, wehave Ψ| N = 0 iff (5.212) is valid along N.Later, (see section 5.10 below) we will extend the jet–action formalismpresented here – to the rigorous (and elegant) jet formulation of path–integrals in physical field systems.5.9 Application: Jets and Lagrangian Field TheoryIn this subsection we will apply the jet formalism defined in subsection4.14.12.5 above, and already applied for development of the time–dependentmechanics in subsection 5.6.1 above, to formulate the first–order Lagrangianfield theory on fibre bundles (see [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)] for details).Recall that the configuration space of the first–order Lagrangian fieldtheory on a fibre bundle Y → X, coordinated by (x α , y i , y i α), is the 1–jetspace J 1 (X, Y ) of the bundle Y → X, coordinated by (x α , y i , y i α). Therefore,a first–order Lagrangian L : J 1 (X, Y ) → ∧ n T ∗ X is defined as a horizontaldensity on J 1 (X, Y ),L = L(x α , y i , y i α)ω, with ω = dx 1 ∧...∧dx n , (n = dim X). (5.214)Let us follow the standard formulation of the variational problem on fibrebundles where deformations of sections of a fibre bundle Y → Xare induced by local 1–parameter groups of automorphisms of Y → Xover X (the so-called vertical gauge transformations). Here, we will notstudy the calculus of variations in depth, but apply in a straightforwardmanner the first variational formula (5.109) (for technical details,see [Sardanashvily (1993); Sardanashvily (1995); Giachetta et. al. (1997);Mangiarotti and Sardanashvily (2000a); Sardanashvily (2002a)]).Recall that a projectable vector–field u on a fibre bundle Y → X is aninfinitesimal generator of a local 1–parameter group of gauge transformationsof Y → X. Therefore, one can think of its jet prolongation j 1 u (5.9)as being the infinitesimal generator of gauge transformations of the configurationspace J 1 (X, Y ). Let the Lie derivative of a Lagrangian L along j 1 u


884 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbe given byL j 1 uL = [∂ α u α L + (u α ∂ α + u i ∂ i + (d α u i − y i µ∂ α u µ )∂ α i )L] ω. (5.215)The first variational formula (5.109) gives its canonical decomposition (inaccordance with the general variational problem), which readsL j 1 uL = u V ⌋E L + d H h 0 (u⌋H L ) (5.216)= (u i − y i µu µ )(∂ i − d α ∂ α i )Lω − d α [π α i (u µ y i µ − u i ) − u α L]ω.In the canonical decomposition (5.216), u V = (u⌋θ i )∂ i ; the mapE L : J 2 (X, Y ) → T ∗ Y ∧ (∧ n T ∗ X), given by E L = (∂ i L − d α π α i )θ i ∧ ω,(5.217)(with π α i = ∂i α L) is the Euler–Lagrangian operator associated to the LagrangianL; and the mapH L : J 1 (X, Y ) → M Y = T ∗ Y ∧ (∧ n−1 T ∗ X), given by (5.218)H L = L + π α i θ i ∧ ω α = π α i dy i ∧ ω α + (L − π α i y i α)ω, (5.219)is called the Poincaré–Cartan form.The kernel of the Euler–Lagrangian operator E L (5.217) defines thesystem of second–order Euler–Lagrangian equations, in local coordinatesgiven by(∂ i − d α ∂ α i )L = 0, (5.220)A solution of these equations is a section s : X −→ Y of the fibre bundle Y−→ X, whose second–order jet prolongation j 2 s lives in (5.220), i.e.,∂ i L ◦ s − (∂ α + ∂ α s j ∂ j + ∂ α ∂ µ s j ∂ µ j )∂α i L ◦ s = 0. (5.221)<strong>Diff</strong>erent Lagrangians L and L ′ can lead to the same Euler–Lagrangianoperator E L if their difference L 0 = L − L ′ is a variationally trivial Lagrangian,whose Euler–Lagrangian operator vanishes identically. A LagrangianL 0 is called variationally trivial iffL 0 = h 0 (ϕ), (5.222)where ϕ is a closed n−-form on Y . We have at least locally ϕ = dξ, andthenL 0 = h 0 (dξ) = d H (h 0 (ξ) = d α h 0 (ξ) α ω, h 0 (ξ) = h 0 (ξ) α ω α .


<strong>Applied</strong> Jet <strong>Geom</strong>etry 885The Poincaré–Cartan form H L (5.218) is called a Lepagean equivalentof a Lagrangian L if h 0 (H L ) = L. In contrast with other Lepagean forms(see [Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a)]), H Lis a horizontal form on the affine jet bundle J 1 (X, Y ) → Y .The fibre bundle M Y = T ∗ Y ∧ (∧ n−1 T ∗ X), figuring in the Poincaré–Cartan form (5.218) is called the homogeneous Legendre bundle. It hasholonomic local coordinates (x α , y i , p α i , p) with transition functionsp ′ αi= det( ∂xε∂x ′ ν ) ∂yj ∂x ′ α∂y ′ i∂x µ pµ j ,Relative to these coordinates, the map (5.218) reads(p µ i , p) ◦ H L = (π µ i , L − πµ i yi µ).p′ = det( ∂xε ∂yj ∂y ′ i∂x ′ ν )(p −∂y ′ i∂x µ pµ j ).(5.223)The transition functions (5.223) shows that M Yis a 1D affine bundleπ MΠ : M Y → Π (5.224)over the Legendre bundleΠ = ∧ n T ∗ X ⊗ V ∗ Y ⊗ T X = V ∗ Y ∧ (∧ n−1 T ∗ X), (5.225)with holonomic coordinates (x α , y i , p α i ). Then the composition̂L = π MΠ ◦H L : J 1 (X, Y ) −→ Π, (x α , y i , p α i )◦ ̂L = (x α , y i , π α i ), (5.226)is the well–known Legendre map. One can think of p α i as being the covariantmomenta of field functions, and the Legendre bundle Π (5.225) plays therole of a finite–dimensional momentum phase–space of fields in the covariantHamiltonian field theory (see subsection 5.10 below).The first variational formula (5.216) gives the standard procedure forthe study of differential conservation laws in Lagrangian field theory asfollows.Let u be a projectable vector–field on a fibre bundle Y → X treated asthe infinitesimal generator of a local 1–parameter group G u of gauge transformations.On–shell, i.e., on the kernel (5.220) of the Euler–Lagrangianoperator E L , the first variational formula (5.216) leads to the weak identityL j1 uL ≈ −d α J α ω, where (5.227)J = J α ω α , J α = π α i (u µ y i µ − u i ) − u α L, (5.228)


886 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis the symmetry current along the vector–field u. Let a Lagrangian L beinvariant under the gauge group G u . This implies that the Lie derivativeL j 1 uL (5.215) vanishes. Then we get the weak conservation lawd α J α ≈ 0 (5.229)of the symmetry current J (5.228). 2The weak conservation law (5.229) leads to the differential conservationlaw∂ α (J α ◦ s) = 0 (5.230)on solutions s : X −→ Y (5.221) of the Euler–Lagrangian equations (5.220).It implies the integral conservation law∫s ∗ J = 0, (5.231)∂Nwhere N is a compact nD submanifold of X with the boundary ∂N.In gauge theory, the symmetry current J (5.228) takes the formJ = W + d H U = (W α + d µ U µα )ω α , (5.232)where the term W depends only on the variational derivativesδ i L = (∂ i − d α ∂ α i )L, (5.233)i.e., W ≈ 0. The tensor–field U = U µα ω µα : J 1 (X, Y ) → ∧ n−2 T ∗ X isa horizontal (n − 2)−-form on J 1 (X, Y ) → X. Then one says that Jreduces to the superpotential U (see [Fatibene et. al. (1994); Giachetta et. al.(1997); Sardanashvily (1997)]). On–shell, such a symmetry current reducesto a d H −-exact form (5.232). In this way, the differential conservation law(5.230) and the integral conservation law (5.231) become tautological. Atthe same time, the superpotential form (5.232) of J implies the followingintegral relation∫s ∗ J =N n−1 ∫s ∗ U,∂N n−1 (5.234)2 The first variational formula defines the symmetry current (5.228) modulo the termsd µ(c µαi (yνu i ν − u i )), where c µαi are arbitrary skew–symmetric functions on Y [Giachettaet. al. (1997)]. Here, we set aside these boundary terms which are independent of aLagrangian L.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 887where N n−1 is a compact oriented (n − 1)D submanifold of X with theboundary ∂N n−1 . One can think of this relation as being a part of theEuler–Lagrangian equations written in an integral form.Let us consider conservation laws in the case of gauge transformationswhich preserve the Euler–Lagrangian operator E L , but not necessarily aLagrangian L. Let u be a projectable vector–field on Y → X, which isthe infinitesimal generator of a local 1–parameter group of such transformations,i.e.,L j2 uE L = 0,where j 2 u is the second–order jet prolongation of the vector–field u. Thereis the useful relation [Giachetta et. al. (1997)]Then, in accordance with (5.222), we have locallyIn this case, the weak identity (5.227) readsL j2 uE L = E Lj 1 uL. (5.235)L j 1 uL = d α h 0 (ξ) α ω. (5.236)d α (h 0 (ξ) α − J α ) ≈ 0, (5.237)where J is the symmetry current (5.228).Background fields, which do not live in the dynamical shell (5.220),violate conservation laws as follows. Let us consider the productY tot = Y × Y ′ (5.238)of a fibre bundle Y −→ X, coordinated by (x α , y i ), whose sections aredynamical fields and of a fibre bundle Y ′ −→ X, coordinated by (x α , y A ),whose sections are background fields that take the background valuesy B = φ B (x), and y B α = ∂ α φ B (x).A Lagrangian L of dynamical and background fields is defined on the totalconfiguration space J 1 (X, Y ) tot . Let u be a projectable vector–field onY tot which also projects onto Y ′ because gauge transformations of backgroundfields do not depend on dynamical fields. This vector–field takesthe coordinate formu = u α (x µ )∂ α + u A (x µ , y B )∂ A + u i (x µ , y B , y j )∂ i . (5.239)


888 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSubstitution of u (5.239) in the formula (5.216) leads to the first variationalformula in the presence of background fields:∂ α u α L + [u α ∂ α + u A ∂ A + u i ∂ i + (d α u A − y A µ ∂ α u µ )∂ α A(5.240)+(d α u i − y i µ∂ α u µ )∂ α i ]L = (u A − y A α u α )∂ A L + π α Ad α (u A − y A µ u µ )+(u i − yαu i α )δ i L − d α [π α i (u µ yµ i − u i ) − u α L]. (5.241)Then we have on the shell (5.220) the weak identity∂ α u α L + [u α ∂ α + u A ∂ A + u i ∂ i + (d α u A − y A µ ∂ α u µ )∂ α A + (d α u i − y i µ∂ α u µ )∂ α i ]L≈ (u A − y A α u α )∂ A L + π α Ad α (u A − y A µ u µ ) − d α [π α i (u µ y i µ − u i ) − u α L].If a total Lagrangian L is invariant under gauge transformations of Y tot , weget the weak identity(u A − y A µ u µ )∂ A L + π α Ad α (u A − y A µ u µ ) ≈ d α J α , (5.242)which is the transformation law of the symmetry current J in the presenceof background fields.5.9.1 Lagrangian Conservation LawsIn the first–order Lagrangian field theory, we have the following differentialtransformation and conservation laws on solutions s : X −→ Y (5.221) ofthe Euler–Lagrangian equations (5.220).Recall that given fibre coordinates (x α , y i ) of Y , the jet space J 1 (X, Y )is equipped with the adapted coordinates (x α , y i , y i α), while the first–orderLagrangian density on J 1 (X, Y ) is defined as the mapL : J 1 (X, Y ) → ∧ n T ∗ X,(n = dim X),L = L(x α , y i , y i α)ω, with ω = dx 1 ∧ ... ∧ dx n .The corresponding first–order Euler–Lagrangian equations for sections s :X −→ J 1 (X, Y ) of the jet bundle J 1 (X, Y ) → X read∂ α s i = s i α, ∂ i L − (∂ α + s j α∂ j + ∂ α s j α∂ α j )∂ α i L = 0. (5.243)We consider the Lie derivatives of Lagrangian densities in order to getdifferential conservation laws. Letu = u α (x)∂ α + u i (y)∂ i


<strong>Applied</strong> Jet <strong>Geom</strong>etry 889be a projectable vector–field on Y → X and u its jet lift (5.9) ontoJ 1 (X, Y ) → X. Given L, let us computer the Lie derivative L u L. Weget the identitys ∗ L u L ≈ − ddx α [πα i (u α s i α − u i ) − u α L]ω, π α i = ∂ α i L, (5.244)modulo the Euler–Lagrangian equations (5.243).Let L be a Lagrangian density on the jet space J 1 (X, Y ). For the sakeof simplicity, we shall denote the pull–back π 1∗0 L of L onto J 2 (X, Y ) by thesame symbol L.Let u be a projectable vector–field on Y −→ X and u its jet lift (5.9)onto the configuration bundle J 1 (X, Y ) −→ X. Recall that the vector–fieldu is associated with some 1–parameter group of transformations of Y .Let us calculate the Lie derivative L u L of the horizontal density L whenits Lepagian equivalent is chosen to be the Poincaré–Cartan form Ξ L , givenby the coordinate expressionΞ L = Lω + π α i (dy i − y i αdx α ) ∧ ω α . (5.245)In this case we recover the first variational formula (5.216) for projectablevector–fields on Y as (see [Giachetta and Mangiarotti (1990); Sardanashvily(1997)])L u L = u V ⌋E L + h 0 (du⌋Ξ L ). (5.246)Since the Poincaré–Cartan form Ξ L is a horizontal form on the jet bundleJ 1 (X, Y ) −→ Y , the formula (5.246) takes the formBeing restricted to the kernelL u L = u V ⌋E L + d H h 0 (u⌋Ξ L ). (5.247)[∂ i − (∂ α + y j α∂ j + y j αλ ∂α j )∂ α i ]L = 0of the Euler–Lagrangian operator E L (5.217), the equality (5.247) reducesto the weak identityL u L ≈ d H h 0 (u⌋Ξ L ), (5.248)∂ α u α L+[u α ∂ α +u i ∂ i +(∂ α u i +y j α∂ j u i −y i α∂ α u α )∂ α i ]L ≈ ̂∂ α [π α i (u i −u α y i α)+u α L],̂∂ α = ∂ α + y i α∂ i + y i αλ∂ α i .


890 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOn solutions s of the Euler–Lagrangian equations, the weak identity(5.248) becomes the weak differential transformation laws ∗ L u L ≈ d(s ∗ u⌋Ξ L ) (5.249)which takes the coordinate form (5.244).Note that, in order to get the differential transformation laws on solutionss of a given system of Euler–Lagrangian equations, one can examineother Lepagian equivalents ρ L of the Lagrangian density L, besides thePoincaré–Cartan form Ξ L . In this case, the first variational formula (5.246)and the corresponding weak identityL u L ≈ h 0 (du⌋ρ L )differ from relations (5.246) and (5.248) respectively in the strong identity0 = h 0 (du⌋ε) = d H h 0 (u⌋ε), (5.250)where ρ L = Ξ L +ε. From the physical point of view, it means that differentLepagian equivalents result in different superpotentials h 0 (u⌋ε).The form ε in the identity (5.250) has the coordinate expressionε = −(̂∂ ν c ανîdy i + c ανîdy ν) i ∧ ω α + χ.It is the general local expression for Lepagian equivalents of the zero Lagrangiandensity. We haveh 0 (u⌋ε) = ̂∂ ν [(u i − y i αu α )c ανi ]ω α .One can consider also other Lagrangian densities L ′ which possess thesame Euler–Lagrangian operator E L . Then the first variational formula andthe corresponding weak identity differ from relations (5.246) and (5.248)respectively in the strong identityL u h 0 (ɛ) = h 0 (du⌋ɛ) (5.251)where ɛ is some closed exterior form on Y . However, if the form h 0 (ɛ)possesses the same symmetries as the Lagrangian density L only, the contributionof the strong identity (5.251) into the weak identity (5.248) is nottautological.Note that the weak identity (5.248) is linear in the vector–field u, and wecan consider superposition of different weak identities (5.248) corresponding


<strong>Applied</strong> Jet <strong>Geom</strong>etry 891to different vector–fields u. For example, if u and u ′ are projectable vector–fields on the bundle Y −→ X which are projected onto the same vector–field on the base X, their difference u − u ′ is a vertical vector–field on Y−→ X. Therefore, the difference of the weak identity (5.248) with respectthe vector–fields u and u ′ results in the weak identity (5.248) with respectto the vertical vector–field u − u ′ .Now let us consider the case when a Lagrangian density L depends onbackground fields. We define such a Lagrangian density as the pull–back ofthe Lagrangian density L tot on the total configuration space by some fixedsections φ(x) describing background fields.Let us again consider the product (5.238), namely Y tot = Y × Y ′ , ofthe bundle Y whose sections are dynamical fields and the bundle Y ′ whosesections φ play the role of background fields. Let the bundles Y and Y ′ becoordinated by (x α , y i ) and (x α , y A ) respectively. The Lagrangian densityL tot is defined on the total configuration space J 1 (X, Y ) tot .Let u be a projectable vector–field on Y tot which is also projectable withrespect to projection Y × Y ′ → Y ′ . It has the coordinate formu = u α (x)∂ α + u A (x α , y B )∂ A + u i (x α , y B , y j )∂ i ,showing that transformations of background fields are independent on dynamicalfields.Calculating the Lie derivative of the Lagrangian density L tot by thisvector–field, we get the equality∂ α u α L tot + [u α ∂ α + u A ∂ A + u i ∂ i + (∂ α u A + y B α ∂ B u A − y A α ∂ α u α )∂ α A+(∂ α u i + y B α ∂ B u i + y j α∂ j u i − y i α∂ α u α )∂ α i ]L tot = ̂∂ α [π α i (u i − u α y i α) + u α L tot ]+(u i − y i αu α )(∂ i − ̂∂ α ∂ α i )L tot + (u A − y A α u α )∂ A L tot + π α A ̂∂ α (u A − y A α u α ),which can be rewritten as∂ α u α L tot + [u α (∂ α + y B α ∂ B + y B αλ∂ α B)L + u i ∂ i + (̂∂ α u i − y i α∂ α u α )∂ α i ]L tot= ̂∂ α [π α i (u i − u α y i α) + u α L tot ] + (u i − y i αu α )(∂ i − ̂∂ α ∂ α i )L tot .The pull–back of this equality to the bundle Y −→ X by sections φ A (x) ofthe bundle Y ′ which describe the background fields results in the familiarexpression (5.247) and the familiar weak identity (5.248) for the Lagrangiandensity L = φ ∗ L tot . Now the partial derivative ∂ α can be written as∂ α = ˜∂ α + ∂ α φ B ∂ B + ∂ α ∂ α φ B ∂ α B ,


892 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere ˜∂ α denote the partial derivatives with respect to the coordinates x αon which the Lagrangian density L tot depends explicitly.Note that Lagrangian densities of field models almost never depend explicitlyon the world coordinates x α . At the same time, almost all fieldmodels describe fields in the presence of a background world metric g onthe base manifold X, except topological field theories whose classical Lagrangiandensities are independent on g [Birmingham et. al. (1991)] andthe gravitation theory where a world metric g is a dynamical field.By a world metric on X is denoted a nondegenerate fibre metric g ανin cotangent and tangent bundles of X. In this case, the partial derivative∂ α L in the weak identity (5.248) contains the term∂L∂g∂ αν α g αν , so that themetric stress–energy–momentum tensor of fields (SEM–tensor, for short,see subsection 5.12.1 below)t αν√| g | = 2∂L∂g αν , | g |=| det(g αν) | .is called into play.The weak identity (5.248) and the weak transformation law (5.249) arebasic for our analysis of differential transformation and conservation lawsin field theory.In particular, one says that an isomorphism Φ of the fibre bundle Y−→ X is an invariant transformation if its jet prolongation j 1 Φ preservesthe Lagrangian density L, i.e.,j 1∗ ΦL = L.Let u be a projectable vector–field on Y −→ X. The corresponding local 1–parameter groups of isomorphisms of Y are invariant transformations iff thestrong equality: L u L = 0 holds. In this case, we have the correspondingweak conservation lawd(s ∗ u⌋Ξ L ) ≈ 0. (5.252)An isomorphism Φ of the bundle Y −→ X is called the generalizedinvariant transformation if it preserves the Euler–Lagrangian operator E L .Let u be a projectable vector–field on Y −→ X. The corresponding localisomorphisms of Y are generalized invariant transformations iff L u L =h 0 (ɛ), where ɛ is a closed n−-form on the bundle Y −→ X. In this case,the weak transformation law (5.249) readss ∗ ɛ ≈ d(s ∗ u⌋Ξ L )


<strong>Applied</strong> Jet <strong>Geom</strong>etry 893for every critical section s of Y −→ X. In particular, if ɛ = dε is an exactform, we get the weak conservation lawd(s ∗ (u⌋Ξ L − ε)) ≈ 0.In particular, gauge transformations in gauge theory on a 3D base X arethe invariant transformations if L is the Yang–Mills Lagrangian density andthey are the generalized invariant transformations if L is the Chern–SimonsLagrangian density.5.9.2 General Covariance ConditionNow we consider the class of bundles T −→ X which admit the canonicallift of vector–fields τ on X. They are called the bundles of geometricalobjects. In fact, such canonical lift is the particular case of the horizontallift of a field τ with respect to the suitable connection on the bundle T−→ X [Giachetta et. al. (2005)].Let τ = τ α ∂ α be a vector–field on the manifold X. There exists thecanonical lift˜τ = T τ = τ α ∂ α + ∂ ν τ α ẋ ν ∂∂ẋ α (5.253)of τ onto the tangent bundle T X of X. This lift consists with the horizontallift of τ by means the symmetric connection K on the tangent bundle whichhas τ as the integral section or as the geodesic field:∂ ν τ α + K α αντ α = 0.Generalizing the canonical lift (5.253), one can construct the canonicallifts of a vector–field τ on X onto the following bundles over X. For thesake of simplicity, we denote all these lifts by the same symbol ˜τ. We have:• the canonical lift of τ onto the cotangent bundle T ∗ X, given by˜τ = τ α ∂ α − ∂ β τ ν ∂ ẋ ν ;∂ẋ β• the canonical lift of τ onto the tensor bundle T k mX = (⊗ m T X) ⊗(⊗ k T ∗ X), given by˜τ = τ α ∂ α + [∂ ν τ α1 ẋ να2···αmβ 1···β k+ . . . − ∂ β1 τ ν ẋ α1···αmνβ 2···β k− . . .]∂;∂ẋ α1···αmβ 1···β k


894 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction• the canonical lift of τ onto the bundle C of the linear connections onT X, given by˜τ = τ α ∂ α + [∂ ν τ α k ν βα − ∂ β τ ν k α να − ∂ α τ ν k α βν − ∂ βα τ α ]∂∂k α βαOne can think of the vector–fields ˜τ on a bundle of geometrical objectsT as being the vector–fields associated with local 1–parameter groups ofthe holonomic isomorphisms of T induced by diffeomorphisms of its baseX. In particular, if T = T X they are the tangent isomorphisms. We callthese isomorphisms the general covariant transformations.Let T be the bundle of geometrical objects and L a Lagrangian densityon the configuration space J 1 (X, T ). Given a vector–field τ on the base Xand its canonical lift ˜τ onto T , one may use the first variational formula(5.247) in order to get the corresponding SEM transformation law. Theleft side of this formula can be simplified if the Lagrangian density satisfiesthe general covariance condition.Note that, if the Lagrangian density L depends on background fields, weshould consider the corresponding total bundle (5.238) and the Lagrangiandensity L tot on the total configuration space J 1 (X, T ) tot . We say that theLagrangian density L satisfies the general covariance condition if L tot isinvariant under 1–parameter groups of general covariant transformations ofT tot induced by diffeomorphisms of the base X. It takes place iff, for anyvector–field τ on X, the Lagrangian density L tot obeys the equalityL j 10 eτ L tot = 0 (5.254)where ˜τ is the canonical lift of τ onto T tot and j 1 0˜τ is the jet lift of ˜τ ontoJ 1 (X, T ) tot .If the Lagrangian density L does not depend on background fields, theequality (5.254) becomesL j 10 eτ L = 0. (5.255)Substituting it in the first variational formula (5.247), we get the weekconservation lawd H h 0 (˜τ⌋Ξ L ) ≈ 0. (5.256)One can show that the conserved quantity is reduced to a superpotentialterm.Here, we verify this fact in case of a tensor bundle T −→ X. Let it becoordinated by (x α , y A ) where the collective index A is employed. Given a.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 895vector–field τ on X, its canonical lift ˜τ on T reads˜τ = τ α ∂ α + u Aβ α∂ β τ α ∂ A .Let a Lagrangian density L on the configuration space J 1 (X, T ) beinvariant under general covarian transformations. Then, it satisfies theequality (5.255) which takes the coordinate form∂ α (τ α L) + u Aβ α∂ β τ α ∂ A L + ̂∂ α (u Aβ α∂ β τ α )∂AL α − yα A ∂ β τ α ∂ β AL = 0. (5.257)Due to the arbitrariness of the functions τ α , the equality (5.257) is equivalentto the system of the equalities∂ α L = 0,δ β αL + u Aβ α∂ A L + ̂∂ α (u Aβ α)∂AL α − yα A ∂ β AL = 0, (5.258)u Aβ α∂AL α + u Aα α∂ β AL = 0. (5.259)Note that the equality (5.258) can be brought into the formδ β αL + u Aβ αδ A L + ̂∂ α (u Aβ α∂AL) α = yα A ∂ β AL, (5.260)where δ A L are the variational derivatives of the Lagrangian density L.Substituting the relations (5.260) and (5.259) into the weak identitywe get the conservation laŵ∂ α [(u Aβ α∂ β τ α − y A α τ α )∂ α AL + τ α L] ≈ 0,̂∂ α [−u Aα αδ A Lτ α − ̂∂ α (u Aα α∂ α ALτ α )] ≈ 0, (5.261)where the conserved current is reduced to the superpotential termQ eτ α = −u Aα αδ A Lτ α − ̂∂ α (u Aα α∂ α ALτ α ). (5.262)For general field models, we have the product T × Y of a bundle T →X of geometrical objects and some other bundle Y → X. The lift of avector–field τ on the base X onto the corresponding configuration spaceJ 1 (X, T ) × J 1 (X, Y ) readsτ = j 1 0˜τ + τ α Γ i α∂ i + (∂ α (τ α Γ i α) + τ α y j α∂ j Γ i α − y i α∂ α τ α )∂ α iwhere Γ is a connection on the fibre bundle Y → X.In this case, we cannot say anything about the general covariance conditionindependently on the invariance of a Lagrangian density with respectto the internal symmetries.


896 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOn the other hand, in gauge theory (see subsection 5.11 below), severaltypes of gauge transformations are considered. To get the Noether conservationlaws, we restrict our consideration to vertical isomorphisms of theprincipal bundle P . These are the G−equivariant isomorphism Φ of P overId X , that is,r g ◦ Φ = Φ ◦ r g , (g ∈ G). (5.263)We call them the gauge isomorphisms. As is well–known, they yield thevertical isomorphisms of the bundle of principal connections C and theP −associated bundle E.For example, let P −→ X be a principal bundle with the structure Liegroup G. Let us consider general gauge isomorphisms Φ of this principalbundle over diffeomorphisms of the base X. They satisfy the relation(5.263). We denote by u G the projectable vector–fields on P correspondingto local 1–parameter groups of such isomorphisms. There is the 1–1 correspondence between these vector–fields and sections of the bundleT G P = T P/G. They are called the general principal vector–fields (see [Giachettaand Mangiarotti (1990)]). In particular, one can show that, given avector–field τ on the base X, its horizontal lift onto the principal bundle Pby means of a principal connection on P is a general principal vector–field.General gauge isomorphisms of the principal bundle P , as like as itsvertical isomorphisms, yield the corresponding isomorphisms of the associatedbundles E and the bundle of principal connections C. We denote bythe same symbol u G the corresponding general principal vector–fields onthese bundles.Consider the product S = C × E × T, where T → X is a bundle ofgeometrical objects. Let a Lagrangian density L on the corresponding configurationspace J 1 (T, S) be invariant under the isomorphisms of the bundleS which are general gauge isomorphisms of C × E over diffeomorphisms ofthe base X and the general covariant transformations of T induced by thesediffeomorphisms of X. In particular, vertical isomorphisms of S consist ofvertical isomorphisms of C × E only. It should be emphasized that thegeneral gauge isomorphisms of the bundle C × E and those of the bundleT taken separately are not the bundle isomorphisms of the product S becausethey must covering the same diffeomorphisms of the base X of Y .At the same time, one can say that the Lagrangian density L satisfies thegeneral covariance condition in the sense that it is invariant under generalisomorphisms of the bundle S [Giachetta and Mangiarotti (1990)].


<strong>Applied</strong> Jet <strong>Geom</strong>etry 897This is phrased in terms of the Lie derivatives as follows. Letu G = τ α ∂ α + u A ∂ Abe a general principal vector–field on the product C × E which is projectedonto the vector–field τ = τ α ∂ α on the base X. The corresponding generalprincipal vector–field on the bundle Y readsũ G = ˜τ + u A ∂ A , (5.264)where ˜τ is the canonical lift of τ onto the bundle of geometrical objectsT . A Lagrangian density L is invariant under general isomorphisms of thebundle S iffL j 10 eu GL = 0, (5.265)where the jet lift j 1 0ũ G of the vector–field ũ G takes the coordinate formj 1 0ũ G = j 1 0˜τ − y A α ∂ α u A ∂ α A + u A ∂ A + ̂∂ α u A ∂ α A.There are the topological field theories, besides the gravitation theory,where we can use the condition (5.265) (see subsection 5.11.8 below).5.10 Application: Jets and Hamiltonian Field TheoryRecall that the Hamiltonian counterpart of the classical Lagrangian fieldtheory (see subsection 5.9 above) is the covariant Hamiltonian field theory,in which momenta correspond to derivatives of fields with respect to allworld coordinates. It is well–known that classical Lagrangian and covariantHamiltonian field theories are equivalent in the case of a hyperregularLagrangian, and they are quasi–equivalent if a Lagrangian is almost–regular (see [Sardanashvily (1993); Sardanashvily (1995); Giachetta et. al.(1997); Giachetta et. al. (1999); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)]). Further, in order to quantize covariant Hamiltonianfield theory, one usually attempts to construct and quantize a multisymplecticgeneralization of the Poisson bracket. The path–integral quantizationof covariant Hamiltonian field theory was recently suggested in[Bashkirov and Sardanashvily (2004)].Recall that the symplectic Hamiltonian technique applied to field theoryleads to instantaneous Hamiltonian formalism on an infinite–dimensionalphase–space coordinated by field functions at some instant of time (see [Gotay(1991a)] for the strict mathematical exposition of this formalism). The


898 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiontrue Hamiltonian counterpart of classical first–order Lagrangian field theoryis covariant Hamiltonian formalism, where canonical momenta p µ i correspondto derivatives yµ i of fields y i with respect to all world coordinatesx µ . This formalism has been developed since the 1970s in its polysymplectic,multisymplectic and Hamilton–de Donder variants (see [Giachettaet. al. (1997); Lopez and Marsden (2003)]). In order to quantize covariantHamiltonian field theory, one usually attempts to construct multisymplecticgeneralization of the Poisson bracket with respect to the derivatives ∂/∂y iand ∂/∂p µ i[Kanatchikov (1998)].We can also quantize covariant Hamiltonian field theory in path–integralterms following [Bashkirov and Sardanashvily (2004)]. A polysymplecticHamiltonian system with a Hamiltonian H(x µ , y i , p µ i ) is equivalent to aLagrangian system with the LagrangianL H (x µ , y i , p µ i , yi α) = p α i y i α − H(x µ , y i , p µ i , yi α) (5.266)of the variables y i and p µ i . In subsection 6.3.10 below we will quantize thisLagrangian system in the framework of perturbative quantum field theory.Briefly, if there is no constraint and the matrix ∂ 2 H/∂p µ i ∂pν j is nondegenerateand positive–definite, this quantization is given by the generatingfunctional∫ ∫Z = N −1 exp{ (L H + Λ + iJ i y i + iJµp i µ i )dx} ∏ [dp(x)][dy(x)] (5.267)xof Euclidean Green functions, where Λ comes from the normalization condition∫ ∫exp{ (− 1 2 ∂i µ∂νHp j µ i pν j + Λ)dx} ∏ [dp(x)] = 1.xIf a Hamiltonian H is degenerate, the Lagrangian L H (5.266) may admitgauge symmetries. In this case, integration of a generating functional alonggauge group orbits must be finite. If there are constraints, the Lagrangiansystem with a Lagrangian L H (5.266) restricted to the constraint manifoldis quantized.In order to verify this path–integral quantization scheme, we apply itto Hamiltonian field systems associated to Lagrangian field systems withquadratic LagrangiansL = 1 2 aλµ ij yi αy j µ + b α i y i α + c, (5.268)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 899where a, b and c are functions of world coordinates x µ and field variablesy i . Note that, in the framework of perturbative quantum field theory, anyLagrangian is split into the sum of a quadratic Lagrangian (5.268) and aninteraction term quantized as a perturbation.For example, let the Lagrangian (5.268) be hyperregular, i.e., the matrixfunction a is nondegenerate. Then there exists a unique associated Hamiltoniansystem whose Hamiltonian H is quadratic in momenta p µ i , and sois the Lagrangian L H (5.266). If the matrix function a is positive–definiteon an Euclidean space–time, the generating functional (5.267) is a Gaussianintegral of momenta p µ i (x). Integrating Z with respect to pµ i (x), onerestarts the generating functional of quantum field theory with the originalLagrangian L (5.268). We extend this result to field theories withalmost–regular Lagrangians L (5.268), e.g., Yang–Mills gauge theory. Thekey point is that, though such a Lagrangian L induces constraints and admitsdifferent associated Hamiltonians H, all the Lagrangians L H coincideon the constraint manifold, and we have a unique constrained Hamiltoniansystem which is quasi–equivalent to the original Lagrangian one [Giachettaet. al. (1997)].5.10.1 Covariant Hamiltonian Field SystemsTo develop the covariant Hamiltonian field theory suitable for path–integral quantization, we start by following the geometrical formulationof classical field theory (see [Sardanashvily (1993); Sardanashvily (1995);Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a); Sardanashvily(2002a)]), in which classical fields are represented by sections offibre bundles. Let Y → X be a smooth fibre bundle provided with bundlecoordinates (x µ , y i ). Recall from subsection 5.9 above, that the configurationspace of Lagrangian field theory on Y is the 1–jet space J 1 (X, Y ) ofY . It is equipped with the bundle coordinates (x µ , y i , yµ) i compatible withthe composite fibrationJ 1 (X, Y ) π1 0−→ Yπ−→ X.Any section s : X −→ Y of a fibre bundle Y → X is prolonged to thesection j 1 s : X −→ J 1 (X, Y ) of the jet bundle J 1 (X, Y ) → X, such thaty i µ ◦ j 1 s = ∂ µ s i .Also, recall that the first–order Lagrangian is defined as a horizontal


900 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductiondensityL = Lω : J 1 (X, Y ) → ∧ n T ∗ X, (ω = dx 1 ∧ · · · dx n , n = dim X),(5.269)on the jet space J 1 (X, Y ). The corresponding Euler–Lagrangian equations(∂ i − d α ∂ α i )L = 0, d α = ∂ α + y i α∂ i + y i λµ∂ µ i , (5.270)represent the subset of the 2–jet space J 2 (X, Y ) of Y , coordinated by(x µ , y i , yα, i yλµ i ). A section s of Y → X is a solution of these equationsif its second jet prolongation j 2 s lives in the subset (5.270).The phase–space of covariant (polysymplectic) Hamiltonian field theoryon a fibre bundle Y −→ X is the Legendre bundle (see (5.225) above)Π = ∧ n T ∗ X ⊗ V ∗ Y ⊗ T X = V ∗ Y ∧(∧ n−1 T ∗ X), (5.271)where V ∗ Y is the vertical cotangent bundle of Y → X. The Legendrebundle Π is equipped with the holonomic bundle coordinates (x α , y i , p µ i )compatible with the composite fibrationΠ π Y−→ Yadmitting the canonical polysymplectic formΩ = dp α i ∧ dy i ∧ ω ⊗ ∂ α .π−→ X, (5.272)A covariant Hamiltonian H on Π (5.272) is defined as a section p = −H ofthe trivial line bundle (i.e., 1D fibre bundle)Z Y = T ∗ Y ∧ (∧ n−1 T ∗ X) → Π, (5.273)equipped with holonomic bundle coordinates (x α , y i , p µ i , p). This fibre bundleadmits the canonical multisymplectic Liouville formΞ = pω + p α i dy i ∧ ω α , with ω α = ∂ α ⌋ω.The pull–back of Ξ onto Π by a Hamiltonian H gives the Hamiltonian formH = H ∗ Ξ Y = p α i dy i ∧ ω α − Hω (5.274)on Π. The corresponding covariant Hamiltonian equations on Π,y i α = ∂ i αH, p α λi = −∂ i H, (5.275)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 901represent the closed submanifold of the jet space J 1 (X, Π) of Π. A sectionr of Π → X is a solution of these equations if its jet prolongation j 1 r livesin the submanifold (5.275).A section r of Π −→ X is a solution of the covariant Hamiltonian equations(5.275) iff it satisfies the condition r ∗ (u⌋dH) = 0 for any verticalvector–field u on Π −→ X.Alternatively, a section r of Π −→ X is a solution of the covariant Hamiltonianequations (5.275) iff it is a solution of the Euler–Lagrangian equationsfor the first–order Lagrangian L H on J 1 (X, Π),L H = h 0 (H) = L H ω = (p α i y i α − H)ω, (5.276)where h 0 sends exterior forms on Π onto horizontal exterior forms onJ 1 (X, Π) −→ X, using the rule h 0 (dy i ) = yαdx i α .Note that, for any section r of Π −→ X, the pull–backs r ∗ H and j 1 r ∗ L Hcoincide. This fact motivated [Bashkirov and Sardanashvily (2004)] toquantize covariant Hamiltonian field theory with a Hamiltonian H on Πas a Lagrangian system with the Lagrangian L H (5.276).Furthermore, let i N : N −→ Π be a closed imbedded subbundle of theLegendre bundle Π −→ Y which is regarded as a constraint space of a covariantHamiltonian field system with a Hamiltonian H. This Hamiltoniansystem is restricted to N as follows. Let H N = i ∗ N H be the pull–back of theHamiltonian form H (5.274) onto N. The constrained Hamiltonian formH N defines the constrained LagrangianL N = h 0 (H N ) = (j 1 i N ) ∗ L H (5.277)on the jet space J 1 (X, N L ) of the fibre bundle N L −→ X. The Euler–Lagrangian equations for this Lagrangian are called the constrained Hamiltonianequations.Note that, the Lagrangian L H (5.276) is the pull–back onto J 1 (X, Π)of the horizontal form L H on the bundle product Π × J 1 (X, Y ) over Y bythe canonical map J 1 (X, Π) → Π × J 1 (X, Y ). Therefore, the constrainedLagrangian L N (5.277) is the restriction of L H to N × J 1 (X, Y ).A section r of the fibre bundle N −→ X is a solution of constrainedHamiltonian equations iff it satisfies the condition r ∗ (u N ⌋dH) = 0 for anyvertical vector–field u N on N −→ X.Any solution of the covariant Hamiltonian equations (5.275) which livesin the constraint manifold N is also a solution of the constrained Hamiltonianequations on N. This fact motivates us to quantize covariant Hamil-


902 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiontonian field theory on a constraint manifold N as a Lagrangian system withthe pull–back Lagrangian L N (5.277).Since a constraint manifold is assumed to be a closed imbedded submanifoldof Π, there exists its open neighborhood U which is a fibre bundle U−→ N. If Π is a fibre bundle π N : Π −→ N over N, it is often convenient toquantize a Lagrangian system on Π with the pull–back Lagrangian π ∗ N L N ,but integration of the corresponding generating functional along the fibresof Π −→ N must be finite.In order to verify this quantization scheme, let us associate to a Lagrangianfield system on Y a covariant Hamiltonian system on Π, then letus quantize this Hamiltonian system and compare this quantization withthat of an original Lagrangian system.5.10.2 Associated Lagrangian and Hamiltonian SystemsIn order to relate classical Lagrangian and covariant Hamiltonian field theories,let us recall that, besides the Euler–Lagrangian equations, a LagrangianL (5.269) also induces the Cartan equations which are given bythe subset(y j µ − yµ)∂ j i α ∂ µ j L = 0, (5.278)∂ i L − d α ∂ α i L + (y j µ − y j µ)∂ i ∂ µ j L = 0, d α = ∂ α + y i α∂ i + y i λµ∂ µ i ,of the repeated jet space J 1 J 1 (X, Y ) coordinated by (x µ , y i , y i λ , yi α, y i λµ ). Asolution of the Cartan equations is a section s of the jet bundle J 1 (X, Y )−→ X whose jet prolongation j 1 s lives in the subset (5.278). Every solutions of the Euler–Lagrangian equations (5.270) defines the solution j 1 s ofthe Cartan equations (5.278). If s is a solution of the Cartan equationsand s = j 1 s, then s is a solution of the Euler–Lagrangian equations. If aLagrangian L is regular, the equations (5.270) and (5.278) are equivalent.Recall that any Lagrangian L (5.269) induces the Legendre map (5.226),i.e.,̂L : J 1 (X, Y ) −→ Π, p α i ◦ ̂L = ∂ α i L, (5.279)over Id Y whose image N L = ̂L(J 1 (X, Y )) is called the Lagrangian constraintspace. A Lagrangian L is said to be hyperregular if the Legendremap (5.279) is a diffeomorphism. A Lagrangian L is called almost–regular if the Lagrangian constraint space is a closed imbedded subbundlei N : N L → Π of the Legendre bundle Π → Y and the surjection


<strong>Applied</strong> Jet <strong>Geom</strong>etry 903̂L : J 1 (X, Y ) → N L is a submersion (i.e., a fibre bundle) whose fibres areconnected. Conversely, any Hamiltonian H induces the Hamiltonian mapĤ : Π → J 1 (X, Y ), y i α ◦ Ĥ = ∂i αH. (5.280)A Hamiltonian H on Π is said to be associated to a Lagrangian L onJ 1 (X, Y ) if H satisfies the following relations (with (x µ , y i , p µ i ) ∈ N L)̂L ◦ Ĥ ◦ ̂L = ̂L, p µ i = ∂µ i L(xµ , y i , ∂ j αH), (5.281)Ĥ ∗ L H = Ĥ∗ L, p µ i ∂i µH − H = L(x µ , y j , ∂ j αH). (5.282)If an associated Hamiltonian H exists, the Lagrangian constraint space N Lis given by the coordinate relations (5.281) and Ĥ ◦ ̂L is a projector fromΠ onto N L .For example, any hyperregular Lagrangian L admits a unique associatedHamiltonian H such thatĤ = ̂L −1 , H = p µ i ̂L i µ−1− L(x α , y i , ̂L i α−1).In this case, any solution s of the Euler–Lagrangian equations (5.270) definesthe solution r = ̂L ◦ j 1 s, of the covariant Hamiltonian equations(5.275). Conversely, any solution r of these Hamiltonian equations inducesthe solution s = π Y ◦ r of the Euler–Lagrangian equations (5.270).A degenerate Lagrangian need not admit an associated Hamiltonian. Ifsuch a Hamiltonian exists, it is not necessarily unique. Let us restrict ourconsideration to almost–regular Lagrangians. From the physical viewpoint,the most of Lagrangian field theories is of this type. From the mathematicalone, this notion of degeneracy is particularly appropriate for the studyof relations between Lagrangian and covariant Hamiltonian formalisms asfollows [Bashkirov and Sardanashvily (2004)].Let L be an almost–regular Lagrangian and H an associated Hamiltonian.Let a section r of Π −→ X be a solution of the covariant Hamiltonianequations (5.275) for H. If r lives in the constraint manifold N L , thens = π Y ◦ r satisfies the Euler–Lagrangian equations (5.270) for L, whiles = Ĥ ◦ r obeys the Cartan equations (5.278). Conversely, let s be asolution of the Cartan equations (5.278) for L. If H satisfies the relationĤ ◦ ̂L ◦ s = j 1 (π 1 0 ◦ s),the section r = ̂L ◦ s of the Legendre bundle Π −→ X is a solution of theHamiltonian equations (5.275) for H. If s = j 1 s, we get the relation between


904 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsolutions the Euler–Lagrangian equations and the covariant Hamiltonianones.Due to this Theorem, one need a set of different associated Hamiltoniansin order to recover all solutions of the Euler–Lagrangian and Cartan equationsfor an almost–regular Lagrangian L. We can overcome this ambiguityas follows.Let H, H ′ be two different Hamiltonians associated to an almost–regularLagrangian L. Let H, H ′ be the corresponding Hamiltonian forms (5.274).Their pull–backs i ∗ N H and i∗ N H′ onto the Lagrangian constraint manifoldN L coincide with each other.It follows that, if an almost–regular Lagrangian admits associatedHamiltonians H, it defines a unique constrained Hamiltonian form H N =i ∗ N H on the Lagrangian constraint manifold N L and a unique constrainedLagrangian L N = h 0 (H N ) (5.277) on the jet space J 1 (X, N L ) of the fibrebundle N L −→ X. For any Hamiltonian H associated to L, every solutionr of the Hamiltonian equations which lives in the Lagrangian constraintspace N L is a solution of the constrained Hamiltonian equations for L N .Let an almost–regular Lagrangian L admit associated Hamiltonians. Asection s of the jet bundle J 1 (X, Y ) → X is a solution of the Cartan equationsfor L iff ̂L◦s is a solution of the constrained Hamiltonian equations. Inparticular, any solution r of the constrained Hamiltonian equations givesthe solution s = Ĥ ◦ r of the Cartan equations. This Theorem showsthat the constrained Hamiltonian equations and the Cartan equations arequasi–equivalent. Thus, one can associate to an almost–regular LagrangianL (5.269) a unique constrained Lagrangian system on the constraint Lagrangianmanifold N L . Let us compare quantizations of these Lagrangiansystems on Y and N L ⊂ Π in the case of an almost–regular quadraticLagrangian L.5.10.3 Evolution OperatorRecall that the covariant Hamiltonian field theory is mainly developed inits multisymplectic and polysymplectic variants, related to the two differentLegendre maps in the first–order calculus of variations on fibre bundles[Sardanashvily (2002c)] (also, see [Gotay (1991a)] for a survey of symplecticformalism).Recall that, given a fibre bundle Y → X coordinated by (x α , y i ), everyfirst–order Lagrangian L : J 1 (X, Y ) → ∧ n T ∗ X is given by L = Lω, ω =dx 1 ∧ · · · dx n , (n = dim X), and induces the Legendre map of the 1–jet


<strong>Applied</strong> Jet <strong>Geom</strong>etry 905space J 1 (X, Y ) to the Legendre bundleΠ = ∧ n T ∗ X ⊗ V ∗ Y ⊗ T X, (5.283)coordinated by (x α , y i , p α i ).formThe Π admits the canonical polysymplecticΩ Π = dp α i ∧ dy i ∧ ω ⊗ ∂ α , (5.284)and is regarded as the polysymplectic phase–space of fields.The multisymplectic phase–space of fields is the homogeneous LegendrebundleZ = T ∗ Y ∧ (∧ n−1 T ∗ X), (5.285)coordinated by (x α , y i , p α i , p) and equipped with the canonical multisymplecticformΩ = dΞ = dp ∧ ω + dp α i ∧ dy i ∧ ω α , with ω α = ∂ α ⌋ω. (5.286)It is natural that one attempts to generalize a Poisson bracket on symplecticmanifolds to polysymplectic and multisymplectic manifolds in orderto get the covariant canonical quantization of field theory. <strong>Diff</strong>erent variantsof such a bracket have been suggested. However, it seems that no canonicalbracket corresponds to the T X−-valued polysymplectic form (5.284), unlessdim X = 1. On the contrary, using the exterior multisymplectic form(5.286), one can associate multivector–fields to exterior forms on the fibrebundle Z (6.212) (but not to all of them), and can introduce a desiredbracket of these forms via the Schouten–Nijenhuis bracket of multivector–fields.Note that no bracket determines the evolution operator in polysymplecticand multisymplectic Hamiltonian formalism, including the case ofdim X = 1 of the time–dependent mechanics (see subsection 5.6.1 above).Written as a bracket, the evolution operator can be quantized, and it determinesthe Heisenberg equation. 33 Recall that in the (Lorentz–invariant) Heisenberg quantum picture, the state vector|ψ > does not change with time, and an observable A satisfies the Heisenberg equationof motionȦ = (i) −1 [A, H] + (∂ tA) classic ,which becomes the classical Poisson equation if we replace its commutator [A, H] by thePoisson bracket A, H.


906 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall the following relationship between first–order dynamical equations,connections, multivector–fields and evolution operators on a fibrebundle.(i) Let π : Q → X be a fibre bundle coordinated by (x µ , q a ). As a sectionγ : Q −→ J 1 (X, Q) of the 1–jet bundle J 1 (X, Q) → Q, any connectionγ = dx µ ⊗ (∂ µ + γ a µ∂ a ), (5.287)on Q → X defines the first–order differential operatorD : J 1 (X, Q) → T ∗ X ⊗ V Q, (x µ , q a , q a µ) → (x µ , q a , q a µ − γ a µ(x ν , q b ))(5.288)on Q → X called the covariant differential with respect to γ. The kernel ofthis differential operator is a closed imbedded subbundle of J 1 (X, Q) → X,given by the first–order dynamical equationq a µ − γ a µ(x ν , q b ) = 0 (5.289)on a fibre bundle Q → X. Conversely, any first–order dynamical equationon Q → X is of this type.(ii) Let HQ ⊂ T Q be the horizontal distribution defined by a connectionγ. If X is orientable, there exists a nowhere vanishing global section of theexterior product ∧ n HQ → Q. It is a locally decomposable π−transversen−vector–field on Q. Conversely, every multivector–field of this type onQ → X induces a connection and, consequently, a first–order dynamicalequation on this fibre bundle [Echeverría et. al. (1998)].(iii) Given a first–order dynamical equation γ on a fibre bundle Q → X,the corresponding evolution operator d γ is defined as the pull–back d γ ontothe shell of the horizontal differentiald H = dx µ (∂ µ + q a µ∂ a )acting on smooth real functions on Q. It readsd γ f = (∂ µ + γ a µ∂ a )fdx µ , (f ∈ C ∞ (Q)). (5.290)This expression shows that d γ is projected onto Q, and it is a first–order differentialoperator on functions on Q. In particular, if a function f obeys theevolution equation d γ f = 0, it is constant on any solution of the dynamicalequation (5.289).In Hamiltonian dynamics on Q, a problem is to represent the evolutionoperator (5.290) as a bracket of f with some exterior form on Q.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 907First of all, let us study the case of fibre bundles Y −→ X over a 1Dorientable connected manifold X (i.e., X is either R or S 1 ). In this case, theLegendre bundle Π (5.283) is isomorphic to the vertical cotangent bundleV ∗ Y of Y −→ X coordinated by (x, y i , p i ), and the polysymplectic form Ω Π(5.284) on V ∗ Y readsΩ Π = dp i ∧ dy i ∧ dx ⊗ ∂ x . (5.291)Therefore, the homogeneous Legendre bundle (6.212) is the cotangent bundleT ∗ Y , coordinated by (x, y i , p, p i ), and the multisymplectic form (5.286)becomes the canonical symplectic form on T ∗ Y , given byΩ = dp ∧ dx + dp i ∧ dy i . (5.292)The vertical cotangent bundle V ∗ Ybracketadmits the canonical Poisson{f, f ′ } V = ∂ i f∂ i f ′ − ∂ i f∂ i f ′ , (f, f ′ ∈ C ∞ (V ∗ Y )), (5.293)given by the relationζ ∗ {f, f ′ } V = {ζ ∗ f, ζ ∗ f ′ },where {, } is the canonical Poisson bracket on T ∗ Y [Mangiarotti and Sardanashvily(1998); Sardanashvily (1998)]. However, the Poisson structure(5.293) fails to determine Hamiltonian dynamics on the fibre bundleV ∗ Q → X because all Hamiltonian vector–fields with respect to this structureare vertical. At the same time, in accordance with general polysymplecticformalism [Giachetta et. al. (1997)], a section h, p ◦ h = −H, of thefibre bundle V ∗ Y −→ T ∗ Y induces a polysymplectic Hamiltonian form onV ∗ Y ,H = p i dy i − Hdx. (5.294)The associated Hamiltonian connection on V ∗ Y −→ X with respect to thepolysymplectic form (5.291) isγ H = dx ⊗ (∂ x + ∂ i H∂ i − ∂ i H∂ i ). (5.295)It defines the Hamiltonian equation on V ∗ Y ,y i x = ∂ i H, p ix = −∂ i H.The corresponding evolution operator (5.290) takes the local formd γ f = (∂ x f + {H, f} V )dx, (f ∈ C ∞ (V ∗ Q)). (5.296)


908 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe bracket {H, f} V in this expression is not globally defined because H isnot a function on V ∗ Q. Therefore, the evolution operator (5.296) does notreduce to the Poisson bracket (5.293).Let us now consider the pull–back ζ ∗ H of the Hamiltonian form H(5.294) onto T ∗ Y . Then the differenceH ∗ = Ξ − ζ ∗ H = (p + H)dx (5.297)is a horizontal density on the fibre bundle T ∗ Y → X. It is a multisymplecticHamiltonian form. The corresponding Hamiltonian connection γ on T ∗ Y →X is given by the conditionwhere the mapγ(Ω) = dH ∗ , (5.298)γ(Ω) = dx ∧ [(∂ x + γ p ∂ p + γ i ∂ i + γ i ∂ i )⌋Ω]is induced by an endomorphism of T ∗ Y determined by the tangent–valuedform γ. We getγ = dx ⊗ (∂ x + γ p ∂ p + ∂ i H∂ i − ∂ i H∂ i ), (5.299)where the coefficient γ p is arbitrary. Note that this connection projects tothe connection γ H (5.295) on V ∗ Y → X. As a consequence, it defines theevolution operator whose restriction to the pull–back of functions on V ∗ Qis exactly the evolution operator (5.296). But now this operator locallyreduces to the Poisson bracket on T ∗ Y ,d γ f = {p + H, f}dx, (f ∈ C ∞ (V ∗ Y )). (5.300)However, this bracket is not globally defined, too, since p+H is a horizontaldensity, but not a function on T ∗ Y .Let us introduce the function E = ρ −1 (p + H) on T ∗ Y , where ρdx issome nowhere vanishing density on X. The Hamiltonian vector–field of Ewith respect to the symplectic form Ω on T ∗ Y readsϑ E = ρ −1 ∂ x − ∂ x E∂ p + ∂ i E∂ i − ∂ i E∂ i .This vector–field is horizontal with respect to the connection (5.299), whereγ p = −ρ∂ x E, and it determines this connection in the formγ = dx ⊗ (∂ x − ρ∂ x E∂ p + ρ∂ i E∂ i − ρ∂ i E∂ i ).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 909Therefore, the evolution operator (5.300) is rewritten asd γ f = ρ{E, f}dx, (5.301)The bracket {E, f} is well–defined, but d γ does not equal this bracket becauseof the factor ρ.The multisymplectic bracket with the horizontal density H ∗ (5.297) alsocannot help us since there is no Hamiltonian multivector–field associatedto H ∗ relative to the symplectic form Ω.The manifolds X = R and X = S 1 can be equipped with coordinates xpossessing transition functions x ′ = x+const, and one can always choose thedensity ρ = 1. Then the evolution operator (5.301) reduces to a Poissonbracket in full. If X = R, this is the case of time–dependent mechanicswhere time reparametrization is forbidden [Mangiarotti and Sardanashvily(2000b); Giachetta et. al. (2002a)].Now we turn to the general case of dim X > 1. In the frameworkof polysymplectic formalism [Giachetta et. al. (1997)], a polysymplecticHamiltonian form on the Legendre bundle Π (5.283) readsThe associated Hamiltonian connectionH = p α i dy i ∧ ω α − Hω. (5.302)γ H = dx α ⊗ (∂ α + γ i α∂ i + γ µ iλ ∂i µ)fails to be uniquely determined, but obeys the equationsγ i α = ∂ i αH, γ α iλ = −∂ i H.The values of these connections assemble into a closed imbedded subbubdley i α = ∂ i αH,p α iλ = −∂ i Hof the jet bundle J 1 (X, Π) −→ X which is the first–order polysymplecticHamiltonian equation on Π. This equation is not algebraically solved forthe highest order derivatives and, therefore, it is not a dynamical equation.As a consequence, the evolution operator depends on the jet coordinates p α iµand, therefore, it is not a differential operator on functions on Π. Clearly,no bracket on Π can determine such an operator.


910 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.10.4 Quadratic Degenerate SystemsGiven a fibre bundle Y → X, let us consider a quadratic LagrangianL (5.268), where a, b and c are local functions on Y . This property iscoordinate–independent since J 1 (X, Y ) → Y is an affine bundle modelledover the vector bundle T ∗ X ⊗ V Y , where V Y denotes the vertical tangentbundle of Y → X. The associated Legendre map (5.279) readsp α i ◦ ̂L = a αµij yj µ + b α i . (5.303)Let a Lagrangian L (5.268) be almost–regular, i.e., the matrix functiona is a linear bundle mapa : T ∗ X ⊗ V Y → Π, p α i = a αµij yj µ, (5.304)of constant rank, where (x α , y i , y i α) are bundle coordinates on T ∗ X ⊗ V Y .Then the Lagrangian constraint space N L (5.303) is an affine subbundle ofthe Legendre bundle Π → Y (5.283). Hence, N L → Y has a global section.For the sake of simplicity, let us assume that it is the canonical zero section̂0(Y ) of Π → Y . The kernel of the Legendre map (5.303) is also an affinesubbundle of the affine jet bundle J 1 (X, Y ) → Y . Therefore, it admits aglobal sectionΓ : Y → Ker ̂L ⊂ J 1 (X, Y ), a αµij Γj µ + b α i = 0, (5.305)which is a connection on Y → X. If the Lagrangian (5.268) is regular, theconnection (5.305) is unique.There exists a linear bundle mapσ : Π −→ T ∗ X ⊗ V Y,y i α ◦ σ = σ ijαµp µ j , (5.306)such thata ◦ σ ◦ a = a, a αµij σjk µαa ανkb = a ανib . (5.307)Note that σ is not unique, but it falls into the sum σ = σ 0 + σ 1 suchthatσ 0 ◦ a ◦ σ 0 = σ 0 , a ◦ σ 1 = σ 1 ◦ a = 0, (5.308)where σ 0 is uniquely defined. For example, there exists a nondegeneratemap σ (5.306).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 911Recall that there are the splittingsJ 1 (X, Y ) = S(J 1 (X, Y )) ⊕ F(J 1 (X, Y )) = Ker ̂L ⊕ Im(σ 0 ◦ ̂L), (5.309)y i α = S i α + F i α = [y i α − σ 0ikαµ (a αµkj yj µ + b α k )] + [σ 0ikαµ (a αµkj yj µ + b α k )],Π = R(Π) ⊕ P(Π) = Ker σ 0 ⊕ N L , (5.310)p α i = R α i + P α i = [p α i − a αµij σ 0 jkµαp α k ] + [a αµij σ 0 jkµαp α k ].The relations (5.308) lead to the equalitiesa αµij Sj µ = 0,σ 0jkµα R α k = 0, σ 1jkµα P α k = 0, R α i F i α = 0. (5.311)Due to these equalities, the Lagrangian (5.268) takes the formL = Lω,L = 1 2 aαµ ij F i αF j µ + c ′ . (5.312)One can show that, this Lagrangian admits a set of associated HamiltoniansH Γ = (R α i + P α i )Γ i α + 1 2 σ 0 ijαµP α i P µ j + 1 2 σ 1 ijαµR α i R µ j − c′ (5.313)indexed by connections Γ (5.305). Therefore, the Lagrangian constraintmanifold (5.303) is given by the reducible constraintsR α i = p α i − a αµij σ 0 jkµαp α k = 0. (5.314)Given a Hamiltonian H Γ , the corresponding Lagrangian (5.276) readsL HΓ= R α i (S i α − Γ i α) + P α i F i α − 1 2 σ 0 ijαµP α i P µ j − 1 2 σ 1 ijαµR α i R µ j + c′ . (5.315)Its restriction (5.277) to the Lagrangian constraint manifold N L (5.314) isL N = L N ω,L N = P α i F i α − 1 2 σ 0 ijαµP α i P µ j + c′ . (5.316)It is independent of the choice of a Hamiltonian (5.313). Note that theLagrangian L N may admit gauge symmetries due to the term Pi αF α.iThe Hamiltonian H Γ induces the Hamiltonian map ĤΓ (5.280) and theprojectorT = ̂L ◦ ĤΓ,p α i ◦ T = T αjiµ pµ j = aαν kjik σ 0 νµp µ j = Pα i , (5.317)from Π onto its summand N L in the decomposition (5.310). It obeys therelationsσ ◦ T = σ 0 , T ◦ a = a. (5.318)


912 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe projector T (5.317) is a linear map over Id Y . Therefore, T : Π → N Lis a vector bundle. Let us consider the pull–back L Π = T ∗ L N of theconstrained Lagrangian L N (5.316) onto Π. Due to the relations (5.311),it is given by the coordinate expressionL Π = L Π ω,L Π = p α i F i α − 1 2 σ 0 ijαµp α i p µ j + c′ . (5.319)This Lagrangian is gauge–invariant under the subgroup of the gauge groupof vertical automorphisms Φ of the affine bundle Π → Y such that T ◦ Φ =T . This subgroup coincides with the gauge group Aut Ker σ 0 of verticalautomorphisms of the vector bundle Ker σ 0 → Y .Note that the splittings (5.309) and (5.310) result from the splitting ofthe vector bundleT ∗ X ⊗ V Y = Ker a ⊕ E,which can be provided with the adapted coordinates (y a , y A ) such that a(5.304) is brought into a diagonal matrix with non–vanishing componentsa AA . Then the Legendre bundle Π → Y (5.283) admits the dual (nonholonomic)coordinates (p a , p A ), where p A are coordinates on the Lagrangianconstraint manifold N L , given by the irreducible constraints p a = 0. Writtenrelative to these coordinates, σ 0 becomes the diagonal matrixσ AA0 = (a AA ) −1 , σ aa0 = 0, (5.320)while σ Aa1 = σ AB1 = 0. Moreover, one can choose the coordinates y a (accordingly,p a ) and the map σ (5.306) such that σ 1 becomes a diagonalmatrix with non–vanishing positive components σ aa1 = V −1 , where Vω is avolume form on X. We further follow this choice of the adapted coordinates(p a , p A ). Let us writep a = M aiα p α i , p A = M Aiα p α i , (5.321)where M are the matrix functions on Y obeying the relationsM aiα a αµij = 0, M −1αai σ 0iα = 0, M aiα P α i = 0, M Aiα R α i = 0.Then the Lagrangian L N (5.316) with respect to the adapted coordinates(p a , p A ) takes the formL N = M −1αAi p A F i α − 1 2∑(a AA ) −1 (p A ) 2 + c ′ , (5.322)A


<strong>Applied</strong> Jet <strong>Geom</strong>etry 9135.11 Application: Gauge Fields on Principal Connections5.11.1 Connection StrengthGiven a principal G−bundle P → Q, the Frölicher–Nijenhuis bracket(4.140) on the space ∧ ∗ (P ) ⊗ V 1 (P ) of tangent–valued forms on P is compatiblewith the canonical action R G (4.31) of G on P , and induces theFrölicher–Nijenhuis bracket on the space ∧ ∗ (Q) ⊗ T G P (Q) of T G P −valuedforms on Q. Its coordinate form issues from the Lie bracket (4.36).Then any principal connection A ∈ ∧ 1 (Q) ⊗ T G P (Q) (5.42) sets theNijenhuis differentiald A : ∧ r (Q) ⊗ T G P (Q) → ∧ r+1 (Q) ⊗ V G P (Q),d A φ = [A, φ] F N , φ ∈ ∧ r (Q) ⊗ T G P (Q), (5.323)on the space ∧ ∗ (Q) ⊗ T G P (Q).The curvature R (5.34) can be equivalently defined as the NijenhuisdifferentialR : Y → ∧ 2 T ∗ Q ⊗ V Y, given by R = 1 2 d ΓΓ = 1 2 [Γ, Γ] F N . (5.324)Let us define the strength of a principal connection A, asF A = 1 2 d AA = 1 2 [A, A] F N ∈ ∧ 2 (Q)⊗V G P (Q), (5.325)F A = 1 2 F r λµdq α ∧ dq µ ⊗e r , F r λµ = ∂ α A r µ − ∂ µ A r α + c r pqA p αA q µ.(5.326)It is locally given by the expressionF A = dA + 1 [A, A] = dA + A ∧ A, (5.327)2where A is the local connection form (5.43). By definition, the strength(5.325) of a principal connection obeys the second Bianchi identityd A F A = [A, F A ] F N = 0. (5.328)It should be emphasized that the strength F A (5.325) is not the standardcurvature (5.34) of a connection on P , but there are the local relationsψ P ζ F A = zζ ∗Θ, where Θ = dà + 1 [Ã, Ã] (5.329)2


914 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis the g l −valued curvature form on P (see the expression (5.333) below).In particular, a principal connection is flat iff its strength vanishes.5.11.2 Associated BundlesGiven a principal G−bundle π P : P → Q, let V be a manifold providedwith an effective left action G × V → V, (g, v) ↦→ gv of the Lie groupG. Let us consider the quotientY = (P × V )/G (5.330)of the product P × V by identification of elements (p, v) and (pg, g −1 v) forall g ∈ G. We will use the notation (pG, G −1 v) for its points. Let [p] denotethe restriction of the canonical surjectionP × V → (P × V )/G (5.331)to the subset {p} × V so that [p](v) = [pg](g −1 v). Then the map Y → Q,[p](V ) ↦→ π P (p), makes the quotient Y (5.330) to a fibre bundle over Q.Let us note that, for any G−bundle, there exists an associated principalG−bundle [Steenrod (1951)]. The peculiarity of the G−bundle Y (5.330)is that it appears canonically associated to a principal bundle P . Indeed,every bundle atlas Ψ P = {(U α , z α )} of P determines a unique associatedbundle atlasΨ = {(U α , ψ α (q) = [z α (q)] −1 )}of the quotient Y (5.330), and each automorphism of P also induces thecorresponding automorphism (5.346) of Y .Every principal connection A on a principal bundle P → Q inducesa unique connection on the associated fibre bundle Y (5.330). Given thehorizontal splitting of the tangent bundle T P relative to A, the tangent mapto the canonical map (5.331) defines the horizontal splitting of the tangentbundle T Y of Y and, consequently, a connection on Y → Q [Kobayashi andNomizu (1963/9)]. This is called the associated principal connection or aprincipal connection on a P −associated bundle Y → Q. If Y is a vectorbundle, this connection takes the formA = dq α ⊗ (∂ α − A p αI pij y j ∂ i ), (5.332)where I p are generators of the linear representation of the Lie algebra g r in


<strong>Applied</strong> Jet <strong>Geom</strong>etry 915the vector space V . The curvature (5.34) of this connection readsR = − 1 2 F p λµ I p i jy j dq α ∧dq µ ⊗ ∂ i , (5.333)where F p λµare coefficients (5.326) of the strength of a principal connectionA.In particular, any principal connection A induces the associated linearconnection on the gauge algebra bundle V G P → Q. The correspondingcovariant differential ∇ A ξ (5.30) of its sections ξ = ξ p e p reads∇ A ξ : Q → T ∗ Q ⊗ V G P, ∇ A ξ = (∂ α ξ r + c r pqA p αξ q )dq α ⊗ e r .It coincides with the Nijenhuis differentiald A ξ = [A, ξ] F N = ∇ A ξ (5.334)of ξ seen as a V G P −valued 0–form, and is given by the local expressiongiven by the local expressionwhere A is the local connection form (5.43).∇ A ξ = dξ + [A, ξ], (5.335)5.11.3 Classical Gauge FieldsSince gauge potentials are represented by global sections of the connectionbundle C → Q (5.45), its 1–jet space J 1 (Q, C) plays the role of a configurationspace of classical gauge theory. The key point is that the jet spaceJ 1 (Q, C) admits the canonical splitting over C which leads to a uniquecanonical Yang–Mills Lagrangian density of gauge theory on J 1 (Q, C).Let us describe this splitting. One can show that the principalG−bundleJ 1 (Q, P ) → J 1 (Q, P )/G = C (5.336)is canonically isomorphic to the trivial pull–back bundleP C = C × P → C, (5.337)and that the latter admits the canonical principal connection [García(1977); Giachetta et. al. (1997)]A = dq α ⊗ (∂ α + a p αe p ) + da r α ⊗ ∂ α r ∈ O 1 (C) ⊗ T G (P C )(C). (5.338)


916 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSince C (5.45) is an affine bundle modelled over the vector bundleC = T ∗ Q ⊗ V G P → Q,the vertical tangent bundle of C possesses the canonical trivializationV C = C × T ∗ Q ⊗ V G P, where (5.339)V G P C = V G (C × P ) = C × V G P.Then the strength F A of the connection (5.338) is the V G P −valued horizontal2–form on C,F A = 1 2 d AA = 1 2 [A, A] F N ∈ ∧ 2 (C) ⊗ V G P (Q),F A = (da r µ ∧ dq µ + 1 2 cr pqa p αa q µdq α ∧ dq µ ) ⊗ e r . (5.340)Note that, given a global section connection A of the connection bundleC → Q, the pull–back A ∗ F A = F A is the strength (5.325) of the principalconnection A.Let us take the pull–back of the form F A onto J 1 (Q, C) with respect tothe fibration (5.336), and consider the V G P −valued horizontal 2–formF = h 0 (F A ) = 1 2 F r λµdq α ∧ dq µ ⊗ e r ,F r λµ = a r λµ − a r µλ + c r pqa p αa q µ, (5.341)where h 0 is the horizontal projection (5.58). Note thatF/2 : J 1 (Q, C) → C × ∧ 2 T ∗ Q ⊗ V G P (5.342)is an affine map over C of constant rank. Hence, its kernel C + = Ker Fis the affine subbundle of J 1 (Q, C) → C, and we have a desired canonicalsplittingJ 1 (Q, C) = C + ⊕ C − = C + ⊕ (C × ∧ 2 T ∗ Q⊗V G P ), (5.343)a r λµ = 1 2 (ar λµ + a r µλ − c r pqa p αa q µ) + 1 2 (ar λµ − a r µλ + c r pqa p αa q µ), (5.344)over C of the jet space J 1 (Q, C). The corresponding canonical projectionsareπ 1 = S : J 1 (Q, C) → C + , S r λµ = 1 2 (ar λµ + a r µλ − c r pqa p αa q µ), (5.345)and π 2 = F/2 given by (5.342).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 9175.11.4 Gauge TransformationsIn classical gauge theory, several classes of gauge transformations are examined[Giachetta et. al. (1997); Marathe and Martucci (1992); Socolovsky(1991)]. A most general gauge transformation is defined as an automorphismΦ P of a principal G−bundle P which is equivariant under the canonicalaction (4.31) of the group G on G, i.e.,R g ◦ Φ P = Φ P ◦ R g ,(g ∈ G).Such an automorphism of P induces the corresponding automorphismΦ Y : (pG, G −1 v) → (Φ P (p)G, G −1 v) (5.346)of the P −associated bundle Y (5.330) and the corresponding automorphismΦ C : J 1 (Q, P )/G → J 1 Φ P (J 1 (Q, P ))/G (5.347)of the connection bundle C.Every vertical automorphism of a principal bundle P is represented asΦ P (p) = pf(p), (p ∈ P ), (5.348)where f is a G−valued equivariant function on P , i.e.,f(pg) = g −1 f(p)g, (g ∈ G). (5.349)There is a 1–1 correspondence between these functions and the global sectionss of the group bundleP G = (P × G)/G, (5.350)whose typical fibre is the group G, subject to the adjoint representationof the structure group G. Therefore, P G (5.350) is also called the adjointbundle. There is the canonical fibre–to–fibre action of the group bundleP G on any P −associated bundle Y by the lawP G × Y → Y,((pG, G −1 gG), (pG, G −1 v)) ↦→ (pG, G −1 gv).Then the above–mentioned correspondence is given by the relationP G × P → P,(s(π P (p)), p) ↦→ pf(p),where P is a G−bundle associated to itself. Hence, the gauge group G(P )of vertical automorphisms of a principal G−bundle P → Q is isomorphicto the group of global sections of the P −associated group bundle (5.350).


918 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn order to study the gauge invariance of one or another object ingauge theory, it suffices to examine its invariance under an arbitrary 1–parameter subgroup [Φ P ] of the gauge group. Its infinitesimal generator isa G−invariant vertical vector–field ξ on a principal bundle P or, equivalently,a sectionξ = ξ p (x)e p (5.351)of the gauge algebra bundle V G P → Q (4.35). We will call it a gaugevector–field. One can think of its components ξ p (q) as being gauge parameters.Gauge vector–fields (5.351) are transformed under the infinitesimalgenerators of gauge transformations (i.e., other gauge vector–fields) ξ ′ bythe adjoint representationL ξ ′ξ = [ξ ′ , ξ] = c p rqξ ′ r ξ q e p ,(ξ, ξ ′ ∈ V G P (Q)).Therefore, gauge parameters are subject to the coadjoint representationξ ′ : ξ p ↦→ − c p rqξ ′ r ξ q . (5.352)Given a gauge vector–field ξ (5.351) seen as the infinitesimal generatorof a 1–parameter gauge group [Φ P ], let us get the gauge vector–fields on aP −associated bundle Y and the connection bundle C.The corresponding gauge vector–field on the P −associated vector bundleY → Q issues from the relation (5.346), and readsξ Y = ξ p I i p∂ i ,where I p are generators of the group G, acting on the typical fibre V of Y .The gauge vector–field ξ (5.351) acts on elements a (5.46) of the connectionbundle by the lawL ξ a = [ξ, a] F N = (−∂ α ξ r + c r pqξ p a q α)dq α ⊗ e r .In view of the vertical splitting (5.339), this quantity can be regarded asthe vertical vector–fieldξ C = (−∂ α ξ r + c r pqξ p a q α)∂ α r (5.353)on the connection bundle C, and is the infinitesimal generator of the 1–parameter group [Φ C ] of vertical automorphisms (5.347) of C, i.e., a desiredgauge vector–field on C.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 9195.11.5 Lagrangian Gauge TheoryClassical gauge theory of unbroken symmetries on a principal G−bundleP → Q deals with two types of fields. These are gauge potentials identifiedto global sections of the connection bundle C → Q (5.45) and matter fieldsrepresented by global sections of a P −associated vector bundle Y (5.330),called a matter bundle. Therefore, the total configuration space of classicalgauge theory is the product of jet bundlesJ 1 (X, Y ) tot = J 1 (X, Y ) × J 1 (Q, C). (5.354)Let us study a gauge–invariant Lagrangian on this configuration space.A total gauge vector–field on the product C × Y readsξ Y C = (−∂ α ξ r + c r pqξ p a q α)∂ α r + ξ p I i p∂ i = (u Aλp ∂ α ξ p + u A p ξ p )∂ A , (5.355)where we use the collective index A, and put the notationu Aλp ∂ A = −δ r p∂ α r , u A p ∂ A = c r qpa q α∂ α r + I i p∂ i .A Lagrangian L on the configuration space (5.354) is said to be gauge–invariant if its Lie derivative L J 1 ξ Y CL along any gauge vector–field ξ (5.351)vanishes. Then the first variational formula (5.109) leads to the strongequality0 = (u A p ξ p + u Aµp ∂ µ ξ p )δ A L + d α [(u A p ξ p + u Aµp ∂ µ ξ p )π α A], (5.356)where δ A L are the variational derivatives (5.233) of L and the total derivativereadsd α = ∂ α + a p λµ ∂µ p + y i α∂ i .Due to the arbitrariness of gauge parameters ξ p , this equality falls into thesystem of strong equalitiesu A p δ A L + d µ (u A p π µ A) = 0, (5.357)u Aµp δ A L + d α (u Aµp π α A) + u A p π µ A= 0, (5.358)u Aλp π µ A + uAµ p π α A = 0. (5.359)Substituting (5.358) and (5.359) in (5.357), we get the well–known constraintsu A p δ A L − d µ (u Aµp δ A L) = 0for the variational derivatives of a gauge–invariant Lagrangian L.


920 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionTreating the equalities (5.357) – (5.359) as the equations for a gauge–invariant Lagrangian, let us solve these equations for a LagrangianL = L(t, q i , a r µ, a r λµ)ω : J 1 (Q, C) → ∧ n T ∗ Q (5.360)without matter fields. In this case, the equations (5.357) – (5.359) readc r pq(a p µ∂ µ r L + a p λµ ∂λµ r L) = 0, (5.361)∂ q µ L + c r pqa p α∂ r µλ L = 0, (5.362)∂ p µλ L + ∂p λµ L = 0. (5.363)Let rewrite them relative to the coordinates (a q µ, Sµλ r , F µλ r ) (5.341) and(5.345), associated to the canonical splitting (5.343) of the jet spaceJ 1 (Q, C). The equation (5.363) reads∂L∂S r µλ= 0. (5.364)Then a simple computation brings the equation (5.362) into the form∂ µ q L = 0. (5.365)The equations (5.364) and (5.365) shows that the gauge–invariant Lagrangian(5.360) factorizes through the strength F (5.341) of gauge potentials.As a consequence, the equation (5.361) takes the form∂Lc r pqF p λµ∂Fλµr= 0.It admits a unique solution in the class of quadratic Lagrangians which isthe conventional Yang–Mills Lagrangian L Y M of gauge potentials on theconfiguration space J 1 (Q, C). In the presence of a background world metricg on the base Q, it readsL Y M = 14ε 2 aG pqg λµ g βν F p λβ F √µνq |g|ω, (where g = det(gµν )), (5.366)where a G is a G−invariant bilinear form on the Lie algebra of g r and ε isa coupling constant.5.11.6 Hamiltonian Gauge TheoryLet us consider gauge theory of principal connections on a principal bundleP −→ X with a structure Lie group G. Principal connections on P −→ X


<strong>Applied</strong> Jet <strong>Geom</strong>etry 921are represented by sections of the affine bundleC = J 1 (Q, P )/G −→ X, (5.367)modelled over the vector bundle T ∗ X ⊗ V G P [Giachetta et. al. (1997)].Here, V G P = V P/G is the fibre bundle in Lie algebras g of the group G.Given the basis {ε r } for g, we get the local fibre bases {e r } for V G P . Theconnection bundle C (5.367) is coordinated by (x µ , a r µ) such that, writtenrelative to these coordinates, sections A = A r µdx µ ⊗ e r of C −→ X are thefamiliar local connection 1–forms, regarded as gauge potentials.There is 1–1 correspondence between the sections ξ = ξ r e r of V G P−→ X and the vector–fields on P which are infinitesimal generators of 1–parameter groups of vertical automorphisms (gauge transformations) of P .Any section ξ of V G P −→ X induces the vector–field on C, given byu(ξ) = u k µ∂∂a r µ= (c r pqa p µξ q + ∂ µ ξ r ) ∂∂a r , (5.368)µwhere c r pq are the structure constants of the Lie algebra g.The configuration space of gauge theory is the jet space J 1 (Q, C)equipped with the coordinates (x α , a m α , a m µλ). It admits the canonical splitting(5.309) given by the coordinate expressiona r µλ = S r µλ + F r µλ = 1 2 (ar µλ + a r λµ − c r pqa p µa q α) + 1 2 (ar µλ − a r λµ + c r pqa p µa q α),where F is the strength of gauge fields up to the factor 1/2. The Yang–MillsLagrangian L Y M on the configuration space J 1 (Q, C) is given byL Y M = a G pqg λµ g βν F p λβ F q µν√|g|ω, (g = det(gµν )), (5.369)where a G is a non–degenerate G−invariant metric in the dual of the Liealgebra of g and g is a pseudo–Riemannian metric on X.The phase–space Π (5.271) of the gauge theory is with the canonicalcoordinates (x α , a p α, p µλq ). It admits the canonical splitting (5.310) given bythe coordinate expressionp µλm = R µλm + P mµλ= p (µλ)m + p [µλ]m = 1 2 (pµλ m + p λµm ) + 1 2 (pµλ m − p λµm ). (5.370)With respect to this splitting, the Legendre map induced by the Lagrangian(5.369) takes the formp (µλ)m ◦ ̂L Y M = 0, (5.371)p [µλ]m ◦ ̂L Y M = 4a G mng µα g λβ F n αβ√|g|. (5.372)


922 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe equalities (5.371) define the Lagrangian constraint space N L of Hamiltoniangauge theory. Obviously, it is an imbedded submanifold of Π, andthe Lagrangian L Y M is almost–regular.In order to construct an associated Hamiltonian, let us consider a connectionΓ (5.305) on the fibre bundle C → X which take their values intoKer ̂L, i.e.,Γ r λµ − Γ r µλ + c r pqa p λ aq µ = 0.Given a symmetric linear connection K on X and a principal connectionB on P → X, this connection readsΓ r λµ = 1 2 [∂ µB r α + ∂ α B r µ − c r pqa p αa q µ + c r pq(a p αB q µ + a p µB q α)] − K αβµ (a r β − B r β).The corresponding Hamiltonian (5.313) associated to L Y M is√H Γ = p λµr Γ r λµ + a mnG g µν g λβ p [µλ]m p n[νβ] |g|.Then we get the LagrangianL N = p [λµ]rF r λµ − a mnGg µν g λβ p [µλ]m√p [νβ]n |g|(5.316) on the Lagrangian constraint manifold (5.371) and its pull–back√L Π = L Π ω, L Π = p λµr Fλµ r − a mnG g µν g λβ p [µλ]m p n[νβ] |g|, (5.373)(5.319) onto Π.Both the Lagrangian L Y M (5.369) on C and the Lagrangian L Π (5.373)on Π are invariant under gauge transformations whose infinitesimal generatorsare the liftsj 1 u(ξ) = (c r pqa p µξ q + ∂ µ ξ r ) ∂∂a r + (c r pq(a p λµ ξq + a p µ∂ α ξ q ) + ∂ α ∂ µ ξ r )µu(ξ) = j 1 u(ξ) − c r pqp λµr ξ q ∂,∂p λµp∂∂a r λµof the vector–fields (5.368) onto J 1 (Q, C) and Π × J 1 (Q, C), respectively.We have the transformation lawsj 1 u(ξ)(F r λµ) = c r pqF p λµ ξq , j 1 u(ξ)(S r λµ) = c r pqS p λµ ξq +c r pqa p µ∂ α ξ q +∂ α ∂ µ ξ r .Therefore, one can choose the gauge conditionsg λµ S r λµ(x) − α r (x) = 1 2 gλµ (∂ α a r µ(x) + ∂ µ a r α(x)) − α r (x) = 0,,


<strong>Applied</strong> Jet <strong>Geom</strong>etry 923which are the familiar generalized Lorentz gauge.second-order differential operator (6.95) readsThe correspondingM r s ξ s = g λµ ( 1 2 cr pq(∂ α a r µ + ∂ µ a r α)ξ q + c r pqa p µ∂ α ξ q + ∂ α ∂ µ ξ r ).Passing to the Euclidean space and repeating the above quantization procedure,we come to the generating functional∫ ∫√Z = N −1 exp{ (p αµr Fαµ r − a mnG g µν g αβ p µαm p νβn |g|− 1 8 aG rsg αν g αµ (∂ α a r ν + ∂ ν a r α)(∂ α a s µ + ∂ µ a s α)− g αµ c r ( 1 2 cr pq(∂ α a r µ + ∂ µ a r α)c q + c r pqa p µc q α + c r αµ)+ iJ r µ a r µ + iJµαp r µαr )ω} ∏ [dc][dc][dp(x)][da(x)].xIts integration with respect to momenta restarts the familiar generatingfunctional of gauge theory.5.11.7 Gauge Conservation LawsOn–shell, the strong equality (5.356) becomes the weak Noether conservationlawof the Noether currentd α [(u A p ξ p + u Aµp ∂ µ ξ p )π α A] ≈ 0 (5.374)J α = −(u A p ξ p + u Aµp ∂ µ ξ p )π α A. (5.375)Therefore, the equalities (5.357) – (5.359) on–shell lead to the familiarNoether identitiesd µ (u A p π µ A) ≈ 0, (5.376)d α (u Aµp π α A) + u A p π µ A≈ 0, (5.377)u Aαp π µ A + uAµ p π α A = 0 (5.378)for a gauge–invariant Lagrangian L. They are equivalent to the weak equality(5.374) due to the arbitrariness of the gauge parameters ξ p (q).The expressions (5.374) and (5.375) shows that both the Noether conservationlaw and the Noether current depend on gauge parameters. Theweak identities (5.376) – (5.378) play the role of the necessary and sufficient


924 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionconditions in order that the Noether conservation law (5.374) is maintainedunder changes of gauge parameters. This means that, if the equality (5.374)holds for gauge parameters ξ, it does so for arbitrary deviations ξ + δξ ofξ. In particular, the Noether conservation law (5.374) is maintained undergauge transformations, when gauge parameters are transformed by thecoadjoint representation (5.352).It can be seen that the equalities (5.376) – (5.378) are not mutuallyindependent, but (5.376) is a corollary of (5.377) and (5.378). This propertyreflects the fact that, in accordance with the strong equalities (5.358) and(5.359), the Noether current (5.375) is brought into the superpotential formJ α = ξ p u Aαp δ A L − d µ (ξ p u Aµp π α A), U µα = −ξ p u Aµp π α A,(5.232). Since a matter field Lagrangian is independent of the jet coordinatesa p αµ, the Noether superpotential, U µα = ξ p π µαp , depends only ongauge potentials. The corresponding integral relation (5.234) reads∫s ∗ J α ω α =N n−1 ∫s ∗ (ξ p π µαp )ω µα ,∂N n−1 (5.379)where N n−1 is a compact oriented (n − 1)D submanifold of Q with theboundary ∂N n−1 . One can think of (5.379) as being the integral relationbetween the Noether current (5.375) and the gauge field, generated by thiscurrent. In electromagnetic theory seen as a U(1) gauge theory, the similarrelation between an electric current and the electromagnetic field generatedby this current is well known. However, it is free from gauge parametersdue to the peculiarity of Abelian gauge models.Note that the Noether current (5.375) in gauge theory takes the superpotentialform (5.232) because the infinitesimal generators of gauge transformations(5.355) depend on derivatives of gauge parameters.5.11.8 Topological Gauge TheoriesThe field models that we have investigated so far show that when a backgroundworld metric is present, the stress–energy–momentum (SEM) transformationlaw becomes the covariant conservation law of the metric SEM–tensor (see subsection on SEM–tensors 5.12.1 below). Topological gaugetheories exemplify the field models in the absence of a world metric [Giachettaet. al. (2005)].Let us consider the Chern–Simons gauge theory on a 3D base manifoldX 3 [Birmingham et. al. (1991); Witten (1988a)]. Physical interpretation


<strong>Applied</strong> Jet <strong>Geom</strong>etry 925of this fundamental model will be given in subsection 6.7.4 below.Let P −→ X 3 be a principal bundle with a structure semisimple Liegroup G and C the corresponding bundle of principal connections which iscoordinated by (x α , k m α ). The Chern–Simons Lagrangian density is givenby the coordinate expressionL CS = 12k aG mnε αλµ k m α (F n λµ + 1 3 cn pqk p αk q µ)d 3 x (5.380)where ε αλµ is the skew–symmetric Levi–Civita tensor.Note that the Lagrangian density (5.380) is not gauge–invariant andglobally defined. At the same time, it gives the globally defined Euler–Lagrangian operatorE LCS = 1 k aG mnε αλµ F n λµdk m α ω.Thus, the gauge transformations in the Chern–Simons model appear to bethe generalized invariant transformations which keep invariant the Euler–Lagrangian equations, but not the Lagrangian density. Solutions of theseequations are the curvature–free principal connections A on the principalbundle P −→ X 3 .Though the Chern–Simons Lagrangian density is not invariant undergauge transformations, we still have the Noether–type conservation law(5.374) in which the total conserved current is the standard Noether current(5.375) plus the additional term as follows.Let u g be the principal vector–field (5.353) on the bundle of principalconnections C. We calculate the Lie derivativeL ug L CS = 1 k aG mnε αλµ ∂ α (α m ∂ α A n µ)d 3 x.Hence, the Noether transformation law (5.374) becomes the conservationlawd α T CS α = d α (T α + 1 k aG mnε αλµ α m ∂ α A n µ) ≈ 0, where (5.381)T α = π µλn u gnµ = 1 k aG mnε αλµ A m α u gnµis the standard Noether current. After simplification, the conservation law(5.381) takes the formd α ( 1 k aG mnε αλµ α m F n αµ) ≈ 0.


926 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn the Chern–Simons model, the total conserved current T CS is equal tozero. At the same time, if we add the Chern–Simons Lagrangian densityto the Yang–Mills one, T CS plays the role of the massive term and makesthe contribution into the standard Noether current of the Yang–Mills gaugetheory.Let τ be a vector–field on the base X and τ B its lift onto the bundleC by means of a section B of C. Remind that the vector–fields τ B are thegeneral principal vector–fields associated with local 1–parameter groups ofgeneral gauge isomorphisms of C. We calculateL τ BL CS = 1 k aG mnε αλµ ∂ α (τ ν B m ν ∂ α A n µ) d 3 x.The corresponding SEM transformation law takes the formd α J CS α = d α (J α − 12k aG mnε αλµ τ ν B m ν ∂ α A n µ) ≈ 0, where (5.382)J α = π µλn [τ ν ∂ ν A m µ − τ ν (∂ µ B n ν − c n pqA p µB q ν) − ∂ µ τ ν (B n ν − A n ν )] − δ α ν τ ν L CSis the standard SEM–tensor relative to the lift τ B of the vector–field τ.Let A be a critical section. We consider the lift of the vector–field τon X onto C by means of the principal connection B = A. Then, theenergy–momentum conservation law (5.382) becomes the conservation lawd α [ 1 k aG mnτ ν ε αλµ A m α Fνµ−τ n α L CS ] = d α (− 16k τ α ε ανµ c npq A n αA p νA q µ) ≈ 0.(5.383)Note that, since the gauge symmetry of the Chern–Simons Lagrangiandensity is broken, the energy–momentum conservation law (5.383) fails tobe invariant under gauge transformations.Let us consider Lagrangian densities of topological gauge models whichare invariant under the general gauge isomorphisms of the bundle C.Though they imply the zero Euler–Lagrangian operators, the correspondingstrong identities may be used as the superpotential terms when such atopological Lagrangian density is added to the Yang–Mills one.Let P −→ X be a principal bundle with the structure Lie group G. Letus consider the bundle J 1 (Q, P ) −→ C. This also is a G principal bundle.Due to the canonical vertical splitting V P = P × g l , where g l is theleft Lie algebra of the group G, the complementary map (5.6) of J 1 (Q, P )defines the canonical G−valued one–form θ on J 1 (Q, P ). This form is theconnection form of the canonical principal connection on the principal bundleJ 1 (Q, P ) −→ C [García (1972)]. Moreover, if Γ A : P −→ J 1 (Q, P ) is


<strong>Applied</strong> Jet <strong>Geom</strong>etry 927a principal connection on P and A the corresponding connection form, wehave Γ ∗ A θ = A. If Ω and R A are the curvature 2–forms of the connectionsθ and A respectively, then Γ ∗ A Ω = R A.Local connection 1–forms on C associated with the canonical connectionθ are given by the coordinate expressions k m µ dx µ ⊗ I m . The correspondingcurvature two–form on C readsΩ C = (dk m µ ∧ dx µ − 1 2 cm nlk n µk l νdx ν ∧ dx µ ) ⊗ I m .Let I(g) be the algebra of real G−invariant polynomial on the Lie algebrag of the group G. Then, there is the well–known Weyl homomorphismof I(g) into the de Rham cohomology algebra H ∗ (C, R). Using this isomorphism,every k−linear element r ∈ I(g) is represented by the cohomologyclass of the closed characteristic 2k−form r(Ω C ) on C. If A is a section ofC, we have A ∗ r(Ω C ) = r(F ), where F is the strength of A and r(F ) isthe corresponding characteristic form on X.Let dim X be even and a characteristic n−form r(Ω C ) on C exist. Thisis a Lepagian form which defines a gauge–invariant Lagrangian densityL r = h 0 (r(Ω C ))on the jet space J 1 (Q, C). The Euler–Lagrangian operator associated withL r is equal to zero. Then, for any projectable vector–field u on C, we havethe strong relation (5.251):L u h 0 (r(Ω C )) = h 0 (du⌋r(Ω C )).If u is a general principal vector–field on C, this relation takes the formd H (u⌋r(Ω C )) = 0.For example, let dim X = 4 and the group G be semisimple. Then, thecharacteristic Chern–Pontryagin 4–formr(Ω C ) = a G mnΩ n C ∧ Ω m C .It is the Lepagian equivalent of the Chern–Pontryagin Lagrangian densityL = 1 k aG mnε αβµν F n αβF m µνd 4 xof the topological Yang–Mills theory.


928 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction5.12 Application: Modern <strong>Geom</strong>etrodynamicsIn this subsection we present some modern developments of the classicalEinstein–Wheeler geometrodynamics that we briefly reviewed as a motivationto our geometrical machinery.5.12.1 Stress–Energy–Momentum TensorsWhile in analytical mechanics there exists the conventional differential energyconservation law, in field theory it does not exist (see [Sardanashvily(1998)]). Let F be a smooth manifold. In time–dependent mechanics onthe phase–space R × T ∗ F coordinated by (t, y i , ẏ i ) and on the configurationspace R×T F coordinated by (t, y i , ẏ i ), the Lagrangian energy and theconstruction of the Hamiltonian formalism require the prior choice of a connectionon the bundle R × F −→ R. However, such a connection is usuallyhidden by using the natural trivial connection on this bundle. Therefore,given a Hamiltonian function H on the phase–space manifold R × T ∗ F , wehave the usual energy conservation lawdHdt ≈ ∂H∂t(5.384)where by ‘≈’ is meant the weak identity modulo the Hamiltonian equations.Given a Lagrangian function L on the configuration manifold R×T F , thereexists the fundamental identity∂L∂t + d dt (ẏi (t) ∂L − L) ≈ 0 (5.385)∂ẏi modulo the equations of motion. It is the energy conservation law in thefollowing sense. Let ̂L be the Legendre morphism given by ẏ i ◦ ̂L =∂ẏiL, and Q = Im ̂L the Lagrangian constraint manifold. Let H be aHamiltonian function associated with L and Ĥ the momentum morphism,ẏ i ◦ Ĥ = ∂ ẏ iH. Every solution r of the Hamiltonian equations of H whichlives on Q yields the solution Ĥ ◦ r of the Euler–Lagrangian equations ofL. Then, the identity (5.385) on Ĥ ◦ r recovers the energy transformationlaw (5.384) on r. 44 There are different Hamiltonian functions associated with the same singular Lagrangianfunction as a rule. Given such a Hamiltonian function, the Lagrangian constraintspace Q plays the role of the primary constraint space, and the Dirac procedurecan be used in order to get the final constraint space where a solution of the Hamiltonianequations exists [Gotay et. al. (1978); León and Marrero (1993)].


<strong>Applied</strong> Jet <strong>Geom</strong>etry 929Recall that in field theory, classical fields are described by sections ofa fibre bundle Y −→ X, while their dynamics is phrased in terms of jetspaces. We restrict ourselves to the first–order Lagrangian formalism whenthe configuration space is J 1 (X, Y ). Given fibred coordinates (x µ , y i ) of Y ,the jet space J 1 (X, Y ) is equipped with the adapted coordinates (x µ , y i , y i µ).Recall that the first–order Lagrangian density on J 1 (X, Y ) is defined to bethe morphismL : J 1 (X, Y ) → ∧ n T ∗ X,(n = dim X),L = L(x µ , y i , y i µ)ω, with ω = dx 1 ∧ ... ∧ dx n ,while the corresponding first–order Euler–Lagrangian equations for sectionss of the jet bundle J 1 (X, Y ) → X read∂ α s i = s i α, ∂ i L − (∂ α + s j α∂ j + ∂ α s j µ∂ µ j )∂α i L = 0. (5.386)As before, we consider the Lie derivatives of Lagrangian densities inorder to get differential conservation laws. Let u = u µ (x)∂ µ +u i (y)∂ i be aprojectable vector–field on Y → X and u its jet lift (5.9) onto J 1 (X, Y ) →X. Given L, let us calculate the Lie derivative L u L. We get the identitys ∗ L u L ≈ − ddx α [πα i (u µ s i µ − u i ) − u α L]ω, with π µ i = ∂µ i L, (5.387)modulo the Euler–Lagrangian equations (5.386). In particular, if u is avertical vector–field this identity becomes the current conservation law exemplifiedby the Noether identities in gauge theory [Sardanashvily (1994)].Let now τ = τ α ∂ α be a vector–field on X andτ Γ = τ µ (∂ µ + Γ i µ∂ i ) (5.388)its horizontal lift onto Y by a connection Γ on Y → X. In this case, theidentity (5.387) takes the forms ∗ L τ ΓL ≈ − ddx α [τ µ J Γαµ (s)]ω, (5.389)where J Γαµ (s) = [π α i (y i µ − Γ i µ) − δ α µL] ◦ s i µis the stress–energy–momentum (SEM) tensor on a field s relative to theconnection Γ. We here restrict ourselves to this particular case of SEM–tensors [Kijowski and Tulczyjew (1979)].


930 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFor example, let us choose the trivial local connection Γ i µ = 0. In thiscase, the identity (5.389) recovers the well–known conservation law∂L∂x α + ddx α J α µ(s) ≈ 0of the canonical energy–momentum tensorJ α µ(s) = π α i s i µ − δ α µL. (5.390)Physicists often lose sight of the fact that (5.390) fails to be a mathematicalwell–behaved object. The crucial point lies in the fact that the LiederivativeL τ ΓL = {∂ µ τ µ L+[τ µ ∂ µ +τ µ Γ i µ∂ i +(∂ α (τ µ Γ i µ)+τ µ y j α∂ j Γ i µ−y i µ∂ α τ µ )∂ α i ]L}ωis almost never equal to zero. Therefore, it is not obvious how to choosethe true energy–momentum tensor.The canonical energy–momentum tensor (5.390) in gauge theory is symmetrizedby hand in order to get the gauge–invariant one. In gauge theoryin the presence of a background world metric g, the identity (5.389) isbrought into the covariant conservation law for the metric SEM–tensor,∇ α t α µ ≈ 0. (5.391)In Einstein’s general relativity, the covariant conservation law (5.391)issues directly from the gravitation equations. It is concerned with thezero-spin matter in the presence of the gravitational field generated by thismatter, though the matter is not required to satisfy the motion equations.The total energy–momentum conservation law for matter and gravity isintroduced by hand. It is usually written asddx µ [(−g)N (t λµ + t g λµ) )] ≈ 0, (5.392)where the energy–momentum pseudotensor t g λµ of a metric gravitationalfield is defined to satisfy the relation(−g) N (t λµ + t g λµ ) ≈ 12κ ∂ σ∂ α [(−g) N (g λµ g σα − g σµ g λα ]modulo the Einstein equations. However, the conservation law (5.392) appearssatisfactory only in cases of asymptotic–flat gravitational fields andof a background metric.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 931Moreover, the covariant conservation law (5.391) fails to take place inthe affine–metric gravitation theory and in the gauge gravitation theory,e.g., in the presence of fermion fields.Thus, we have not any conventional energy–momentum conservationlaw in Lagrangian field theory. In particular, one may take different SEM–tensors for different field models and, moreover, different SEM–tensors fordifferent solutions of the same field equations. Just the latter in fact isthe above-mentioned symmetrization of the canonical energy–momentumtensor in gauge theory.Gauge theory exemplifies constraint field theories. Contemporary fieldmodels are almost always the constraint ones. To describe them, let us turnto the Hamiltonian formalism.When applied to field theory, the conventional Hamiltonian formalismtakes the form of the instantaneous Hamiltonian formalism where canonicalvariables are field functions at a given instant of time. The correspondingphase–space is infinite–dimensional, so that the Hamiltonian equations inthe bracket form fail to be differential equations.The true partners of the Lagrangian formalism in classical field theoryare polysymplectic and multisymplectic Hamiltonian machineries wherecanonical momenta correspond to derivatives of fields with respect to allworld coordinates, not only the temporal one [Cariñena et. al. (1991);Sardanashvily (1993)]. We here follow the multimomentum Hamiltonianformulation of field theory when the phase–space of fields is the Legendrebundle over YΠ = ∧ n T ∗ X ⊗ T X ⊗ V ∗ Y, (5.393)which is coordinated by (x α , y i , p α i ) [Sardanashvily (1993); Sardanashvily(1994)]. Every Lagrangian density L on J 1 (X, Y ) implies the Legendremorphism̂L : J 1 (X, Y ) −→ Π, p µ i ◦ ̂L = π µ i .The Legendre bundle (5.393) carries the polysymplectic formΩ = dp α i ∧ dy i ∧ ω ⊗ ∂ α . (5.394)Recall that one says that a connection γ on the fibred Legendre manifoldΠ → X is a Hamiltonian connection if the form γ⌋Ω is closed. Then, aHamiltonian form H on Π is defined to be an exterior form such thatdH = γ⌋Ω (5.395)


932 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionfor some Hamiltonian connection γ. The key point lies in the fact thatevery Hamiltonian form admits the following splittingH = p α i dy i ∧ ω α − p α i Γ i αω − ˜H Γ ω = p α i dy i ∧ ω α − Hω, ω α = ∂ α ⌋ω,(5.396)where Γ is a connection on Y → X.Given the splitting (5.396), the equality (5.395) becomes the Hamiltonianequations∂ α r i = ∂ i αH, ∂ α r α i = −∂ i H (5.397)for sections r of Π −→ X.The Hamiltonian equations (5.397) are the multimomentum generalizationof the standard Hamiltonian equations in mechanics. The correspondingmultimomentum generalization of the conventional energy conservationlaw (5.384) is the weak identityτ µ [(∂ µ + Γ i µ∂ i − ∂ i Γ j µrj α ∂λ) i ˜H Γ −ddx α T Γ α µ(r)] ≈ τ µ ri α Rλµ, i (5.398)αT Γ µ (r) = [ri α ∂µ i ˜H Γ − δ α µ(ri α ∂α i ˜H Γ − ˜H Γ )], (5.399)whereR i λµ = ∂ α Γ i µ − ∂ µ Γ i α + Γ j α∂ j Γ i µ − Γ j µ∂ j Γ i αis the curvature of the connection Γ. One can think of the tensor (5.399)as being the Hamiltonian SEM–tensor.If a Lagrangian density is regular, the multimomentum Hamiltonianformalism is equivalent to the Lagrangian formalism, otherwise in caseof degenerate Lagrangian densities. In field theory, if a Lagrangian densityis not regular, the Euler–Lagrangian equations become underdeterminedand require supplementary gauge–type conditions. In gauge theory,they are the familiar gauge conditions. However, in general case,the gauge–type conditions remain elusive. In the framework of themultimomentum Hamiltonian formalism, they appear automatically asa part of the Hamiltonian equations. The key point consists in thefact that, given a degenerate Lagrangian density, one must consider afamily of different associated Hamiltonian forms in order to exhaust allsolutions of the Euler–Lagrangian equations [Cariñena et. al. (1991);Sardanashvily (1993)].Lagrangian densities of all realistic field models are quadratic or affinein the velocity coordinates yµ. i Complete family of Hamiltonian forms associatedwith such a Lagrangian density always exists [Sardanashvily (1993);Sardanashvily (1994)]. Moreover, these Hamiltonian forms differ from each


<strong>Applied</strong> Jet <strong>Geom</strong>etry 933other only in connections Γ in the splitting (5.396). <strong>Diff</strong>erent connectionsare responsible for different gauge–type conditions mentioned above. Theyare also the connections which one should use in construction of the HamiltonianSEM–tensors (5.399).The identity (5.398) remains true in the first–order Lagrangian theoriesof gravity. In this work, we examine the metric-affine gravity whereindependent dynamical variables are world metrics and general linear connections.The energy–momentum conservation law in the affine–metricgravitation theory is not widely discussed. We construct the HamiltonianSEM–tensor for gravity. In case of the affine Hilbert–Einstein Lagrangiandensity, it is equal toT α µ = 12κ δα µR √ −gand the total conservation law (5.398) for matter and gravity is reducedto the conservation law for matter in the presence of a background worldmetric, otherwise in case of quadratic Lagrangian densities.Lagrangian SEM–TensorsGiven a Lagrangian density L, the jet space J 1 (X, Y ) carries the associatedPoincaré–Cartan form [Sardanashvily (1998)]and the Lagrangian polysymplectic formΞ L = π α i dy i ∧ ω α − π α i y i αω + Lω (5.400)Ω L = (∂ j π α i dy j + ∂ µ j πα i dy j µ) ∧ dy i ∧ ω ⊗ ∂ α .Using the pull–back of these forms onto the repeated jet space J 1 J 1 (X, Y ),one can construct the exterior generating form on J 1 J 1 (X, Y ),Λ L = dΞ L − λ⌋Ω L = [y i (λ) − yi α)dπ α i + (∂ i − ̂∂ α ∂ α i )Ldy i ] ∧ ω,(5.401)λ = dx α ⊗ ̂∂ α , ̂∂α = ∂ α + y i (λ) ∂ i + y i µλ∂ µ i .Its restriction to the sesquiholonomic jet space Ĵ 2 (X, Y ) defines the first–order Euler–Lagrangian operatorE ′ L : Ĵ 2 (X, Y ) −→ ∧ n+1 T ∗ Y,given byE ′ L = δ i Ldy i ∧ ω = [∂ i − (∂ α + y i α∂ i + y i µλ∂ µ i )∂α i ]Ldy i ∧ ω, (5.402)


934 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductioncorresponding to L. The restriction of the form (5.401) to the second–orderjet space J 2 (X, Y ) of Y recovers the familiar variational Euler–LagrangianoperatorE L : J 2 (X, Y ) −→ ∧ n+1 T ∗ Y,given by the expression (5.402), but with symmetric coordinates y i µλ = yi λµ .Let s be a section of the jet bundle J 1 (X, Y ) −→ X such that its jetprolongation j 1 s takes its values into Ker E ′ L given by the coordinate relations∂ i L − (∂ α + y j α∂ j + y j µλ ∂µ j )∂α i L = 0.Then, s satisfies the first–order Euler–Lagrangian equations (5.386). Theseequations are equivalent to the second–order Euler–Lagrangian equations∂ i L − (∂ α + ∂ α s j ∂ j + ∂ α ∂ µ s j ∂ µ j )∂α i L = 0. (5.403)for sections s of Y −→ X where s = j 1 s.We have the following differential conservation laws on solutions of thefirst–order Euler–Lagrangian equations.Given a Lagrangian density L on J 1 (X, Y ), let us consider its pull–backonto Ĵ 2 (X, Y ). Let u be a projectable vector–field on Y −→ X and u itsjet lift (5.9) onto J 1 (X, Y ) −→ X. Its pull–back onto J 1 J 1 (X, Y ) has thecanonical horizontal splitting (5.14) given by the expressionu = u H +u V = u α (∂ α +y i (λ) ∂ i +y i µλ∂ µ i )+[(ui −y i (λ) uα )∂ i +(u i µ −y i µλu α )∂ µ i ].Let us calculate the Lie derivative L u L. We haveL u L = ̂∂ α [π λ i (u i − u µ y i µ) + u α L]ω + u V ⌋E ′ L, (5.404)̂∂ α = ∂ α + y i α∂ i + y i µλ∂ µ i .Being restricted to Ker E ′ L, the equality (5.404) is written∂ α u α L + [u α ∂ α + u i ∂ i + (∂ α u i + y j α∂ j u i − y i µ∂ α u µ )∂ α i ]L (5.405)≈ ̂∂ α [π α i (u i − u µ y i µ) + u α L].On solutions s of the first–order Euler–Lagrangian equations, the weakidentity (5.405) becomes the differential conservation laws ∗ L u L ≈ d(u⌋Ξ L ◦ s),which takes the coordinate form (5.387).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 935In particular, let τ Γ be the horizontal lift (5.388) of a vector–field τ onX onto Y → X by a connection Γ on Y . In this case, the identity (5.405)is writtenτ µ {[∂ µ +Γ i µ∂ i +(∂ α Γ i µ+y j α∂ j Γ i µ)∂ α i ]L+ ̂∂ α [π α i (y i µ−Γ i µ)−δ α µL] ≈ 0. (5.406)On solutions s of the first–order Euler–Lagrangian equations, the identity(5.406) becomes the differential conservation law (5.389) where J Γα µ (s) arecoefficients of the T ∗ X−valued form on X,J Γ (s) = −(Γ⌋Ξ L ) ◦ s = [π α i (s i µ − Γ i µ) − δ α µL] dx µ ⊗ ω α . (5.407)This conservation law takes the coordinate formτ µ {[∂ µ + Γ i µ∂ i + (∂ α Γ i µ + s j α∂ j Γ i µ)∂ α i ]L +SEM Conservation Lawsddx α [πα i (s i µ − Γ i µ) − δ α µL] ≈ 0.Every projectable vector–field u on the bundle Y −→ X which coversa vector–field τ on the base X is represented as the sum of a verticalvector–field on Y −→ X and some lift of τ onto Y . Hence, any differentialtransformation law (5.249) can be represented as a superposition of sometransformation law associated with a vertical vector–field on the bundleY −→ X and the one induced by the lift of a vector–field on the base Xonto Y . Therefore, we can reduce our consideration to transformation lawsassociated with these two types of vector–fields on Y .Vertical vector–fields result in transformation and conservation laws ofNoether currents. In general case, a vector–field τ on a base X induces avector–field on Y only by means of some connection on the bundle Y −→ X.Such lifts result in the transformation laws of the SEM–tensors.Given a bundle Y → X, let τ be a vector–field on X andτ Γ = τ⌋Γ = τ µ (∂ µ + Γ i µ∂ i )its horizontal lift onto Y → X by means of a connection on Y , given byΓ = dx µ ⊗ (∂ µ + Γ i µ∂ i ).In this case, the weak identity (5.405) is written∂ µ τ µ L + [τ µ ∂ µ + τ µ Γ i µ∂ i + (∂ α (τ µ Γ i µ) + τ µ y j α∂ j Γ i µ − y i µ∂ α τ µ )∂ α i ]L− ̂∂ α [π α i (τ µ Γ i µ − τ µ y i µ) + δ α µτ µ L] ≈ 0. (5.408)


936 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOne can simplify it as follows:τ µ {[∂ µ + Γ i µ∂ i + (∂ α Γ i µ + y j α∂ j Γ i µ)∂ α i ]L − ̂∂ α [π α i (Γ i µ − y i µ) + δ α µL] ≈ 0.Let us emphasize that this relation takes place for arbitrary vector–field τon X. Therefore, it is equivalent to the system of the weak identities[∂ µ + Γ i µ∂ i + (∂ α Γ i µ + y j α∂ j Γ i µ)∂ α i ]L − ̂∂ α [π α i (Γ i µ − y i µ) + δ α µL] ≈ 0. (5.409)On solutions s of the Euler–Lagrangian equations, the weak identity(5.408) becomes the weak transformation laws ∗ L τ ΓL +ddx α [τ µ J Γαµ (s)]ω ≈ 0and to the equivalent system of the weak transformation laws[∂ µ +Γ i µ∂ i +(∂ α Γ i µ+∂ α s j ∂ j Γ i µ)∂ α i ]L+ ddx α [πα i (∂ µ s i −Γ i µ)−δ α µL] ≈ 0 (5.410)where J Γα µ (s) is the SEM–tensor given by the components of theT ∗ X−valued (n − 1)−form on X,J Γ (s) = −(Γ⌋Ξ L ) ◦ s = [π α i (∂ µ s i − Γ i µ) − δ α µL]dx µ ⊗ ω α .It is clear that the first and the second terms in (5.410) taken separatelyfail to be well–behaved objects. Therefore, only their combination mayresult in the satisfactory transformation or conservation law.For example, let a Lagrangian density L depend on a background metricg on the base X. In this case, we have∂ µ L = −t α β√|g| Γβµα , where t α β = g αγ t γβis the metric SEM–tensor (by definition), while Γ β µα are the Christoffelsymbols of the metric g. Then, the weak transformation law (5.410) takesthe form−t α β√|g|Γβµα + [Γ i µ∂ i + (∂ α Γ i µ + ∂ α s j ∂ j Γ i µ)∂ α i ]L+ ddx α [πα i (∂ µ s i − Γ i µ) − δ α µL] ≈ 0,and, under suitable conditions of symmetries of the Lagrangian density L,it may become the covariant conservation law ∇ α t α β = 0 where ∇ α denotesthe covariant derivative relative to the connection Γ β µα.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 937Note that, if we consider another Lepagian equivalent of the Lagrangiandensity L, the SEM transformation law takes the forms ∗ L τ ΓL +ddx α [τ µ J ′ Γ α µ(s)] ω ≈ 0,where J ′ Γ α µ = J Γαµ − ddx ν [(∂ µs i − Γ i µ)c λνi ],that is, the SEM–tensors J ′ Γ α αµ and J Γ µ differ from each other in thesuperpotential–type term: − ddx[(∂ ν µ s i − Γ i µ)c λνi ].In particular, if the bundle Y has a fibre metric a Y ij , one can choosec µνi = a Y ijg µα g νβ R j αβ ,where R is the curvature of the connection Γ on the bundle Y and g is ametric on X. In this case, the superpotential contribution into the SEM–tensor is equal to − ddx[a Y ν ij gλα g νβ (∂ µ s i − Γ i µ)R j αβ ].Let us now consider the weak identity (5.408) when a vector–field τ onthe base X induces a vector–field on Y by means of different connectionsΓ and Γ ′ on Y −→ X. Their difference result in the weak identity[τ µ σ i µ∂ i + (∂ α (τ µ σ i µ) + y j α∂ j (τ µ σ i µ))∂ α i ]L − ̂∂ α [π α i τ µ σ i µ] ≈ 0 (5.411)where σ = Γ ′ − Γ is a soldering form on the bundle Y −→ X andτ⌋σ = τ µ σ i µ∂ i (5.412)is a vertical vector–field. It is clear that the identity (5.411) is exactly theweak identity (5.405) in case of the vertival vector–field (5.412).It follows that every SEM transformation law contains a Noether transformationlaw. Conversely, every Noether transformation law associatedwith a vertical vector–field u V on Y −→ X can be get as the difference of twoSEM transformation laws if the vector–field u V takes the form u V = τ⌋σ,where σ is some soldering form on Y and τ is a vector–field on X. Infield theory, this representation fails to be unique. On the contrary, inNewtonian mechanics there is the 1–1 correspondence between the verticalvector–fields and the soldering forms on the bundle R × F −→ F.Note that one can consider the pull–back of the first–order Lagrangiandensity L and their Lepagian equivalents onto the infinite order jet spaceJ ∞ Y . In this case, there exists the canonical lift τ ∞ H (5.59) of a vector–fieldτ on X onto J ∞ Y . One can treat this lift as the horizontal lift of τ by


938 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmeans of the canonical connection on the bundle J ∞ Y −→ X, given byΓ ∞ = dx µ ⊗(∂ µ + y i ∂ i + y i α∂ α i + · · · ).Multimomentum Hamiltonian FormalismLet Π be the Legendre bundle (5.393) coordinated by (x α , y i , p α i ). ByJ 1 (X, Π) is meant the first–order jet space of Π −→ X. It is coordinated by(x α , y i , p α i , yi (µ) , pα iµ ). The Legendre manifold Π carries the generalized Liouvilleformθ = −p α i dy i ∧ ω ⊗ ∂ αand the polysymplectic form Ω (5.394).The Hamiltonian formalism in fibred manifolds is formulated intrinsicallyin terms of Hamiltonian connections which play the role similar tothat of Hamiltonian vector–fields in the symplectic geometry [Sardanashvily(1993)].We say that a jet field (resp. a connection)γ = dx α ⊗ (∂ α + γ i (λ) ∂ i + γ µ iλ ∂i µ)on the Legendre manifold Π −→ X is a Hamiltonian jet field (resp.Hamiltonian connection) if the following exterior form is closed:aγ⌋Ω = dp α i ∧ dy i ∧ ω α + γ α iλdy i ∧ ω − γ i (λ) dpα i ∧ ω.An exterior n−form H on the Legendre manifold Π is called a Hamiltonianform if, on an open neighborhood of each point of Π, there existsa Hamiltonian jet–field satisfying the equation γ⌋Ω = dH, i.e., if there existsa Hamiltonian connection satisfying the equation (5.395). Hamiltonianconnections constitute an affine subspace of connections on Π → X. Thefollowing construction shows that this subspace is not empty.Every connection Γ on Y → X is lifted to the connectionγ = ˜Γ = dx α ⊗ [∂ α + Γ i α(y)∂ i + (−∂ j Γ i α(y)p µ i − Kµ νλ(x)p ν j + K α αλ(x)p µ j )∂j µ]on Π → X, whereK = dx α ⊗ (∂ α + K µ ∂νλẋ µ )∂ẋ νis a linear symmetric connection on T ∗ X. We have the equality˜Γ⌋Ω = d(Γ⌋θ).


<strong>Applied</strong> Jet <strong>Geom</strong>etry 939This equality shows that ˜Γ is a Hamiltonian connection andH Γ = Γ⌋θ = p α i dy i ∧ ω α − p α i Γ i αωis a Hamiltonian form.Let H be a Hamiltonian form. For any exterior horizontal density ˜H =˜Hω on Π −→ X, the form H + ˜H is a Hamiltonian form. Conversely, ifH and H ′ are Hamiltonian forms, their difference H − H ′ is an exteriorhorizontal density on Π −→ X.Thus, Hamiltonian forms constitute an affine space modelled on a linearspace of the exterior horizontal densities on Π −→ X. It follows that everyHamiltonian form on Π can be given by the expression (5.396) where Γis some connection on Y −→ X. Moreover, a Hamiltonian form has thecanonical splitting (5.396) as follows.Every Hamiltonian form H implies the momentum mapĤ : Π −→ J 1 (X, Y ),y i α ◦ Ĥ = ∂i αH,and the associated connection Γ H = Ĥ ◦ ̂0 on Y where ̂0 is the global zerosection of Π → Y . As a consequence, we have the canonical splittingH = H ΓH − ˜H.The Hamiltonian operator E H of a Hamiltonian form H is defined to bethe first–order differential operator on Π → X,whereE H : j 1 Π → ∧ n+1 T ∗ Π,E H = dH − ̂Ω = [(y(λ) i − ∂i αH)dp α i − (p α iλ + ∂ i H)dy i ] ∧(5.413)ω̂Ω = dpαi ∧ dy i ∧ω α + p α iλdy i ∧ ω − y i (λ) dpα i ∧ ωis the pull–back of the multisymplectic form (5.394) onto j 1 Π.For any connection γ on Π → X, we haveE H ◦ γ = dH − γ⌋Ω.It follows that γ is a Hamiltonian connection for a Hamiltonian form H iffit takes its values into Ker E H given by the coordinate relationsy i (λ) = ∂i αH, p α iλ = −∂ i H. (5.414)Let a Hamiltonian connection γ has an integral section r of Π −→ X,that is, γ ◦ r = j 1 r. Then, the algebraic equations (5.414) are brought intothe first–order differential Hamiltonian equations (5.397).


940 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow we consider relations between Lagrangian and Hamiltonian formalisms.A Hamiltonian form H is defined to be associated with a Lagrangiandensity L if it satisfies the relationŝL ◦ Ĥ| Q = Id Q , Q = ̂L(J 1 (X, Y )),H = H bH + L ◦ Ĥ,which take the coordinate form∂ µ i L(xα , y j , ∂ j αH) = p µ i , L(xα , y j , ∂ j αH) = p µ i ∂i µH − H.Note that there are different Hamiltonian forms associated with the samesingular Lagrangian density.Bearing in mind physical application, we restrict our consideration toso–called semiregular Lagrangian densities L when the preimage ̂L −1 (q) ofeach point of q ∈ Q is the connected submanifold of J 1 (X, Y ). In this case,all Hamiltonian forms associated with a semiregular Lagrangian density Lcoincide on the Lagrangian constraint space Q, and the Poincaré–Cartanform Ξ L is the pull–backΞ L = H ◦ ̂L, π α i y i α − L = H(x µ , y i , π α i ),of any associated multimomentum Hamiltonian form H by the Legendremorphism ̂L [Zakharov (1992)]. Also the generating form (5.401) is thepull–back ofΛ L = E H ◦ J 1 ̂Lof the Hamiltonian operator (5.413) of any Hamiltonian form H associatedwith a semiregular Lagrangian density L. As a consequence, we getthe following correspondence between solutions of the Euler–Lagrangianequations and the Hamiltonian equations [Sardanashvily (1994); Zakharov(1992)].Let a section r of Π −→ X be a solution of the Hamiltonian equations(5.397) for a Hamiltonian form H associated with a semiregular Lagrangiandensity L. If r lives on the Lagrangian constraint space Q, the sections = Ĥ ◦ r of J 1 (X, Y ) −→ X satisfies the first–order Euler–Lagrangianequations (5.386). Conversely, given a semiregular Lagrangian density L,let s be a solution of the first–order Euler–Lagrangian equations (5.386).Let H be a Hamiltonian form associated with L so thatĤ ◦ ̂L ◦ s = s. (5.415)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 941Then, the section r = ̂L ◦ s of Π −→ X is a solution of the Hamiltonianequations (5.397) for H. For sections s and r, we have the relationss = j 1 s, and s = π ΠY ◦ r,where s is a solution of the second–order Euler–Lagrangian equations(5.403).We shall say that a family of Hamiltonian forms H associated with asemiregular Lagrangian density L is complete if, for each solution s of thefirst–order Euler–Lagrangian equations (5.386), there exists a solution r ofthe Hamiltonian equations (5.397) for some Hamiltonian form H from thisfamily so thatr = ̂L ◦ s, s = Ĥ ◦ r, s = J 1 (π ΠY ◦ r). (5.416)Such a complete family exists iff, for each solution s of the Euler–Lagrangianequations for L, there exists a Hamiltonian form H from this family so thatthe condition (5.415) holds.We do not discuss here existence of solutions of Euler–Lagrangian andHamiltonian equations. Note that, in contrast with mechanics, there aredifferent Hamiltonian connections associated with the same multimomentumHamiltonian form in general. Moreover, in field theory when the primaryconstraint space is the Lagrangian constraint space Q, there is afamily of Hamiltonian forms associated with the same Lagrangian densityas a rule. In practice, one can choose either the Hamiltonian equations orsolutions of the Hamiltonian equations such that these solutions live on theconstraint space.Hamiltonian SEM–TensorsLet H be a Hamiltonian form on the Legendre bundle Π over a fibrebundle Y −→ X. We have the following differential conservation law onsolutions of the Hamiltonian equations [Sardanashvily (1998)].Let r be a section of the fibred Legendre manifold Π −→ X. Given aconnection Γ on Y −→ X, we consider the T ∗ X−valued (n − 1)−formT Γ (r) = −(Γ⌋H) ◦ r, (5.417)T Γ (r) = [r α i (∂ µ r i − Γ i µ) − δ α µ(r α i (∂ α r i − Γ i α) − ˜H Γ )]dx µ ⊗ ω α ,on X where ˜H Γ is the Hamiltonian density in the splitting (5.396) of Hwith respect to the connection Γ.


942 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet τ = τ α ∂ α be a vector–field on X. Given a connection Γ on Y → X,it induces the projectable vector–field˜τ Γ = τ α ∂ α + τ α Γ i α∂ i + (−τ µ p α j ∂ i Γ j µ − p α i ∂ µ τ µ + p µ i ∂ µτ α )∂ i αon the Legendre bundle Π. Let us calculate the Lie derivative L eτ Γ˜HΓ on asection r. We have(L eτ Γ˜HΓ ) ◦ r = {∂ α τ α ˜HΓ + [τ α ∂ α+ τ α Γ i α∂ i + (−τ µ r α j ∂ i Γ j µ − r α i ∂ µ τ µ + r µ i ∂ µτ α )∂ i α] ˜H Γ }ω= τ µ r α i R i λµω + d(τ µ T Γαµ (r)ω α ) − (˜τ ΓV ⌋E H ) ◦ r, (5.418)where ˜τ ΓV is the vertical part of the canonical horizontal splitting (5.14)of the vector–field ˜τ V on Π over j 1 Π. If r is a solution of the Hamiltonianequations, the equality (5.418) becomes the conservation law (5.398). Theform (5.417) modulo the Hamiltonian equations readsT Γ (r) ≈ [r α i (∂ i µH − Γ i µ) − δ α µ(r α i ∂ i αH − H)]dx µ ⊗ ω α . (5.419)For example, if X = R and Γ is the trivial connection, we haveT Γ (r) = Hdt, where H is a Hamiltonian function. Then, the identity(5.398) becomes the conventional energy conservation law (5.384) in mechanics.Unless n = 1, the identity (5.398) cannot be regarded directly as theenergy–momentum conservation law. To clarify its physical meaning, weturn to the Lagrangian formalism.Let a Hamiltonian form H be associated with a semiregular Lagrangiandensity L. Let r be a solution of the Hamiltonian equations of H whichlives on the Lagrangian constraint space Q and s the associated solutionof the first–order Euler–Lagrangian equations of L so that they satisfy theconditions (5.416). Then, we haveT Γ (r) = J Γ ( ˜H ◦ r),T Γ (˜L ◦ s) = J Γ (s),where J Γ is the SEM–tensor (5.407).It follows that, on the Lagrangian constraint space Q, the form (5.419)can be treated the Hamiltonian SEM–tensor relative to the connection Γon Y −→ X.At the same time, the examples below show that, in several field models,the equality (5.398) is brought into the covariant conservation law (5.391)for the metric SEM–tensor.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 943In the Lagrangian formalism, the metric SEM–tensor is defined to be√ −gtαβ = 2 ∂L∂g αβ .In case of a background world metric g, this object is well–behaved. In theframework of the multimomentum Hamiltonian formalism, one can introducethe similar tensor√ −gtH αβ = 2 ∂H∂g αβ. (5.420)If a Hamiltonian form H is associated with a semiregular Lagrangiandensity L, there are the equalitiest H αβ (q) = −g αµ g βν t µν (x α , y i , ∂ i αH(q)),(q ∈ Q),t H αβ (x α , y i , π α i (z)) = −g αµ g βν t µν (z), Ĥ ◦ ̂L(z) = z.In view of these equalities, we can think of the tensor (5.420) restrictedto the Lagrangian constraint space Q as being the Hamiltonian metricSEM–tensor. On Q, the tensor (5.420) does not depend upon choice of aHamiltonian form H associated with L. Therefore, we shall denote it bythe common symbol t. Sett λ α = g αν t λν .In the presence of a background world metric g, the identity (5.398) takesthe formt λ α{ α λµ} √ −g + (Γ i µ∂ i − ∂ i Γ j µr α j ∂ i α) ˜H Γ ≈ddx α T Γ α µ + r α i R i αµ , (5.421)where by { α λµ} are meant the Christoffel symbols of the world metric g.SEM Tensors in Gauge TheoryIn this subsection, following [Sardanashvily (1998)] we consider thegauge theory of principal connections treated as gauge potentials. Here,the manifold X is assumed to be oriented and provided with a nondegeneratefibre metric g µν in the tangent bundle of X. We denote g = det(g µν ).Let P → X be a principal bundle with a structure Lie group G whichacts freely and transitively on P on the right: r g : p ↦→ pg, (p ∈ P, g ∈G).A principal connection A on P → X is defined to be a G-equivariant connectionon P such that j 1 r g ◦A = A◦r g for each canonical morphism r g .


944 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall that there is the 1–1 correspondence between the principal connectionson a principal bundle P → X and the global sections of the quotientbundleC = J 1 (X, P )/G → X. (5.422)The bundle (5.422) is the affine bundle modelled on the vector bundleC = T ∗ X ⊗ (V P/G). Given a bundle atlas Ψ P of P , the bundle C has thefibred coordinates (x µ , kµ m ) so that (kµm ◦ A)(x) = A m µ (x) are coefficientsof the local connection 1–form of a principal connection A with respect tothe atlas Ψ P . The 1–jet space J 1 (X, C) of the fibre bundle C −→ X iscoordinated by (x µ , kµ m , kµλ m ).There exists the canonical splitting over C, given byJ 1 (X, C) = C + ⊕ C − = (J 2 P/G) ⊕ (∧ 2 T ∗ X ⊗ V G P ), (5.423)k m µλ = 1 2 (km µλ + k m λµ + c m nlk n αk l µ) + 1 2 (km µλ − k m λµ − c m nlk n αk l µ).The corresponding surjections read:S : J 1 (X, C) → C + , S m λµ = k m µλ + k m λµ + c m nlk n αk l µ,F : J 1 (X, C) → C − , F m λµ = k m µλ − k m λµ − c m nlk n αk l µ.On the configuration space (5.423), the conventional Yang–Mills Lagrangiandensity L Y M of gauge potentials in the presence of a backgroundworld metric is given by the expressionL Y M = 1√4ε 2 aG mng λµ g βν FλβF m µνn |g|ω, (5.424)where a G is a nondegenerate G−invariant metric in the Lie algebra g of G.The Legendre morphism associated with the Lagrangian density (5.424)takes the formp (µλ)m ◦ ̂L Y M = 0, (5.425)p [µλ]m ◦ ̂L Y M = ε −2 a G mng λα g µβ F n αβ√|g|. (5.426)The equation (5.425) defines the constraint space of gauge theory.Given a symmetric connection K on the tangent bundle T X, everyprincipal connection B on P induces the connectionΓ m µλ = ∂ µ B m α − c m nlk n µB l α − K β µλ(B m β − k m β ) (5.427)on the bundle of principal connections C.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 945Let τ be a vector–field on the base X andτ BK = τ α {∂ α + [∂ µ B m α − c m nlk n µB l α − K β µλ(B m β − k m β )]∂ µ m} (5.428)its horizontal lift onto C by means of the connection (5.427). For everyvector–field τ, one can choose the connection K on the tangent bundle T Xwhich has τ as the geodesic field. In this case, the horizontal lift (5.428) ofthe vector–field τ becomes its canonical liftτ B = τ α ∂ α + [τ α (∂ µ B m α − c m nlk n µB l α) + ∂ µ τ α (B m α − k m α )]∂ µ m , (5.429)by means of the principal connection B on the principal bundle P [Giachettaand Mangiarotti (1990)]. The vector–field (5.429) is just the generalprincipal vector–field on C that has been mentioned in the previous Section.Hence, the Lie derivative of the Lagrangian density (5.424) by the jetlift τ B of the field τ B becomesL τ BL Y M = (∂ α τ α L Y M + τ α ∂ α L Y M − F m µν∂ α τ µ π νλm )ω.The corresponding SEM transformation law takes the form√∂ α τ α L Y M − τ µ t α β |g|Γβµα − Fµν∂ m α τ µ π νλm ≈ (5.430)̂∂ α [π νλm (τ µ (∂ ν Bµ m − c m nlkν n Bµ) l + ∂ ν τ µ (Bµ m − kµ m ) − τ µ kνµ) m + δ α µτ µ L Y M ],wheret α β = 1 √|g|(π ναm F m βν − δ α βL Y M )is the metric SEM–tensor of gauge potentials.Note that, in general case of the principal connection B, the correspondingSEM transformation law (5.430) differs from the covariant conservationlaw in the Noether conservation laŵ∂ α (π νλm u gmν ) ≈ 0,whereu g = (∂ ν α m + c m nlk l να n )∂ ν m, α m = τ µ (B m µ − A m µ )is the principal vector–field (5.353) on C.Following the general procedure [Sardanashvily (1993); Sardanashvily(1994)], let us consider connections on the fibre bundle C −→ X which taketheir values into Ker ̂L Y M :Γ : C → C + , Γ m µλ − Γ m λµ − c m nlk n αk l µ = 0. (5.431)


946 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionMoreover, we can restrict ourselves to connections of the following type.Every principal connection B on P induces the connection Γ B (5.431) onC such thatΓ B ◦ B = S ◦ j 1 B,Γ Bmµλ = 1 2 [cm nlk n αk l µ + ∂ µ B m α + ∂ α B m µ − c m nl(k n µB l α + k n αB l µ)] − Γ β µλ (Bm β − k m β ).For all these connections, the following Hamiltonian formsH B = p µλm dk m µ ∧ ω α − p µλm Γ Bmµλ ω − ˜H Y M ω, (5.432)˜H Y M = ε24 amn Gg µν g λβ p [µλ]mp [νβ]n |g| −1/2 ,are associated with the Lagrangian density L Y M and constitute a completefamily. The corresponding Hamiltonian equations for sections r of Π −→ Xread∂ α p µλm = −c n lmkνp l n[µν] + c n mlBνp l (µν)n − Γ µ λν p(λν) m , (5.433)∂ α kµ m + ∂ µ kα m m= 2Γ B(µλ) , (5.434)plus the equation (5.426). The equations (5.426) and (5.433) restricted tothe constraint space (5.425) are the familiar Yang–Mills equations. <strong>Diff</strong>erentHamiltonian forms (5.432) lead to the different equations (5.434). Theequation (5.434) is independent of canonical momenta and plays the role ofthe gauge–type condition. Its solution is k(x) = B.Let A be a solution of the Yang–Mills equations. There exists the Hamiltonianform H B=A (5.432) such that r A = ̂L Y M ◦ A is a solution of thecorresponding Hamiltonian equations (5.433), (5.434) and (5.426) on theconstraint space (5.425).On the solution r A , the curvature of the connection Γ A is reduced toR m λαµ = 1 2 (∂ αF m αµ − c m qnk q αF n αµ − Γ β αλ F m βµ − Γ β µλ F m αβ) =12 [(∂ αF m λµ − c m qnk q αF n λµ − Γ β λα F m µβ) − (∂ µ F m λα − c m qnk q µF n λα − Γ β λµ F m αβ)]where F = F ◦ A is the strength of A. If we setthen we haveS α µ = p [αλ]m ∂αµ m ˜H Y M = ε22 √ |g| amn Gg µν g αβ p [αλ]S α µ = 1 2 p[αλ] F m µα, ˜HY M = 1 2 Sα α.mp [βν]n ,


<strong>Applied</strong> Jet <strong>Geom</strong>etry 947Using (5.425), (5.426) and (5.433), we get the relations∂ β nΓ Amαµ p αλm ∂ n βλ ˜H Y M = Γ β αµS α β,r A[λα]m R m λαµ = ∂ α S α µ(r A )−Γ β µλ Sα β(r A )and we find thatt α µ√|g| = 2Sαµ − 1 2 δα µS α α, T ΓAαµ (r A ) = S α µ(r A ) − 1 2 δα µS α α(r A ),t α µ(r A ) √ |g| = T α Γ A µ(r A ) + S λ µ(r A ).Hence, the identity (5.421) in gauge theory is brought into the covariantenergy–momentum conservation law∇ α t α µ(r A ) ≈ 0.The Lagrangian partner of the Hamiltonian SEM–tensor T ΓA (r A ) is theSEM–tensor J ΓA (A) (5.407) on the solution A relative to the connectionΓ A on the bundle C. This is exactly the familiar symmetrized canonicalenergy–momentum tensor of gauge potentials.SEM Tensors of Matter FieldsIn gauge theory, matter fields possessing only internal symmetries aredescribed by sections of a vector bundle Y = (P × V )/G, associated witha principal bundle P [Sardanashvily (1998)]. It has a G−invariant fibremetric a Y . Because of the canonical vertical splitting V Y = Y × Y , themetric a Y is a fibre metric in the vertical tangent bundle V Y → X. Everyprincipal connection A on a principal bundle P yields the associatedconnectionΓ = dx α ⊗ [∂ α + A m µ (x)I mij y j ∂ i ], (5.435)where A m µ (x) are coefficients of the local connection 1–form and I m aregenerators of the structure group G on the standard fibre V of the bundleY .On the configuration space J 1 (X, Y ), the regular Lagrangian density ofmatter fields in the presence of a background connection Γ on Y readsL (m) = 1 2 aY ij[g µν (y i µ − Γ i µ)(y j ν − Γ j ν) − m 2 y i y j ] √ |g|ω. (5.436)The Legendre bundle of the vector bundle Y −→ X is Π = ∧ n T ∗ X ⊗T X ⊗ Y ∗ . The unique Hamiltonian form on Π associated with the La-


948 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongrangian density L (m) (5.436) is writtenH (m) = p α i dy i ∧ ω α − p α i Γ i αω − 1 2 (aij Y g µνp µ i pν j |g| −1 + m 2 a Y ijy i y j ) √ |g|ω,(5.437)where a Y is the fibre metric in V ∗ Y dual to a Y . There is the 1–1 correspondencebetween the solutions of the first–order Euler–Lagrangian equationsof the regular Lagrangian density (5.436) and the solutions of the Hamiltonianequations of the Hamiltonian form (5.437).To examine the conservation law (5.421), let us take the same HamiltonianSEM–tensor relative to the connection Γ (5.435) for all solutions rof the Hamiltonian equations. The following equality motivates the optionabove. We haveT α Γ µ(r) = t α µ(r) √ |g| = [a ijY g µνr α i p ν j |g| −1− 1 2 δα µ(a ijY g ανr α i r ν j |g| −1 + m 2 a Y ijr i r j )] √ |g|.The gauge invariance condition I mj i r α j ∂i α ˜H = 0 also takes place. Then, itcan be observed that the identity (5.421) reduces to the familiar covariantenergy–momentum conservation law√|g|∇α t α µ(r) ≈ −r α i F m λµI mij y j .SEM Tensors in Affine–Metric Gravitation TheoryNow we can apply the Hamiltonian SEM–tensor machinery to gravitationtheory [Sardanashvily (1998); Giachetta and Sardanashvily (1996)].Here, X 4 is a 4D world manifold which obeys the well–known topologicalconditions in order that a gravitational field exists on X 4 .Recall that the contemporary concept of gravitational interaction isbased on the gauge gravitation theory with two types of gravitationalfields: tetrad gravitational fields and Lorentz gauge potentials. In absenceof fermion matter, one can choose by gravitational variables a pseudo–Riemannian metric g on a world space–time manifold X 4 and a generallinear connections K on the tangent bundle of X 4 . We call them a worldmetric and a world connection respectively. Here we are not concernedwith the matter interacting with a general linear connection, for it is non–Lagrangian and hypothetical as a rule.Let LX −→ X 4 be the principal bundle of linear frames in the tangentspaces to X 4 . Its structure group is GL + (4, R). The world connections are


<strong>Applied</strong> Jet <strong>Geom</strong>etry 949associated with the principal connections on the principal bundle LX −→X 4 . Hence, there is the 1–1 correspondence between the world connectionsand the global sections of the quotient bundleC = J 1 (X 4 , LX)/GL + (4, R). (5.438)We therefore can apply the standard procedure of the Hamiltonian gaugetheory in order to describe the configuration and phase–spaces of worldconnections [Sardanashvily (1993); Sardanashvily (1994)].Also, there is the 1–1 correspondence between the world metrics g onX 4 and the global sections of the bundle Σ of pseudo–Riemannian bilinearforms in tangent spaces to X 4 . This bundle is associated with theGL 4 −principal bundle LX. The 2–fold covering of the bundle Σ is thequotient bundle LX/SO(3, 1).The total configuration space of the affine–metric gravitational variablesis the productJ 1 (X 4 , C) × J 1 (X 4 , Σ). (5.439)coordinated by (x µ , g αβ , k α βµ, g αβ α, k α βµλ). Also, the total phase–spaceΠ of the affine–metric gravity is the product of the Legendre bundles overthe above–mentioned bundles C and Σ. It has the corresponding canonicalcoordinates (x µ , g αβ , k α βµ, p α αβ , p βµλ α ).On the configuration space (5.439), the Hilbert–Einstein Lagrangiandensity of general relativity readsL HE = − 12κ gβλ F α √βαλ −gω, with (5.440)F α βνλ = k α βλν − k α βνλ + k α ενk ε βλ − k α ελk ε βν.The corresponding Legendre morphism is given by the expressionsp αβ α ◦ ̂L HE = 0,p α βνλ ◦ ̂L HE = π α βνλ = 12κ (δν αg βλ − δ α αg βν ) √ −g, (5.441)which define the constraint space of general relativity in the affine–metricvariables.Now, let us consider the following connections on the bundle C × Σ inorder to construct a complete family of Hamiltonian forms associated withthe Lagrangian density (5.440).


950 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLet K be a world space–time connection andΓ Kαβνλ = 1 2 [kα ενk ε βλ − k α ελk ε βν + ∂ α K α βν + ∂ ν K α βλ−2K ε (νλ)(K α βε − k α βε) + K ε βλk α εν + K ε βνk α ελ − K α ελk ε βν − K α ενk ε βλ]be the corresponding connection on the bundle C (5.438). Let K ′ be anothersymmetric world connection. Building on these connections, we setup the following connection on the bundle C × Σ,Γ αβ α = −K ′ αελg εβ − K ′ βελg αε ,Γ α βνλ = Γ Kαβνλ − 1 2 Rα βνλ , (5.442)where R α βνλ is the Riemann curvature tensor of K.For all connections (5.442), the following Hamiltonian forms are associatedwith the Lagrangian density L HE and constitute a complete family:H HE = (p αβ α dg αβ + p α βνλ dk α βν) ∧ ω α − H HE ω,H HE = −p αβ α (K ′ αελg εβ + K ′ βελg αε )+ p α βνλ Γ Kαβνλ − 1 2 Rα βνλ(p α βνλ − π α βνλ )= −p αβ α (K ′ αελg εβ + K ′ βελg αε ) + p α βνλ Γ α βνλ + ˜H HE ,˜H HE = 12κ R√ −g. (5.443)Given the Hamiltonian form H HE (5.443) plus a Hamiltonian form H Mfor matter, we have the corresponding Hamiltonian equations∂ α g αβ + K ′ αελg εβ + K ′ βελg αε = 0, (5.444)∂ α k α βν = Γ Kαβνλ − 1 2 Rα βνλ, (5.445)∂ α p α αβ = p σ εβ K ′ εασ + p σ εα K ′ εβσ (5.446)− 12κ (R αβ − 1 2 g αβR) √ −g − ∂H M∂g αβ ,∂ α p α βνλ = −p α ε[νγ] k β εγ + p ε β[νγ] k ε αγ − p α βεγ K ν (εγ)−p α ε(νγ) K β εγ + p ε β(νγ) K ε αγ , (5.447)plus the motion equations of matter. The Hamiltonian equations (5.444)and (5.445) are independent of canonical momenta and so, reduce to thegauge–type conditions. The equation (5.445) breaks into the following two


<strong>Applied</strong> Jet <strong>Geom</strong>etry 951parts,F α βλν = R α βνλ, and (5.448)∂ ν (K α βλ − k α βλ) + ∂ α (K α βν − k α βν) − 2K ε (νλ)(K α βε − k α βε)+ K ε βλk α εν + K ε βνk α ελ − K α ελk ε βν − K α ενk ε βλ = 0, (5.449)where F is the curvature of the connection k(x). It is clear that the gauge–type conditions (5.444) and (5.445) are satisfied byk(x) = K, K ′ αβλ = Γ α βλ. (5.450)When restricted to the constraint space (5.441), the Hamiltonian equations(5.446) and (5.447) become1κ (R αβ − 1 2 g αβR) √ −g = − ∂H M, (5.451)∂gαβ D α ( √ −gg νβ ) − δ ν αD α ( √ −gg λβ ) + √ −g[g νβ (k α αλ − k α λα)+ g λβ (k ν λα − k ν αλ) + δ ν αg λβ (k µ µλ − k µ λµ)] = 0, (5.452)where D α g αβ = ∂ α g αβ + k α µλg µβ + k β µλg αµ .Substituting the equation (5.448) into the equation (5.451), we get theEinstein equations1κ (F αβ − 1 2 g αβF ) = −t αβ , (5.453)where t αβ is the metric SEM–tensor of matter. The equations(5.452) and(5.453) are the familiar equations of affine–metric gravity. In particular,the former is the equation for torsion and nonmetricity terms of the generallinear connection k(x). In the absence of matter sources of a general linearconnection, it admits the well–known solutionk α βν = Γ α βν − 1 2 δα ν V β , D α g βγ = V α g βγ ,where V α is an arbitrary covector–field corresponding to the well–knownprojective freedom.Let s = (k(x), g(x)) be a solution of the Euler–Lagrangian equationsof the first–order Hilbert–Einstein Lagrangian density (5.440) and r thecorresponding solution of the Hamiltonian equations of the Hamiltonianform (5.443) where K and K ′ are given by the expressions (5.450). For thissolution r, let us take the SEM–tensor T s (5.399) relative to the connection


952 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(5.442) where K and K ′ are given by the expressions (5.450). It readsT sαµ = δ α µ ˜H HE = 12κ δα µR √ −gand the identity (5.398) takes the form(∂ µ + Γ αβ µ∂ αβ + Γ i µ∂ i − ∂ i Γ j µp α j ∂ i λ)( ˜H HE + ˜H M )≈ddx α (T s α µ + T Mαµ ) + p α βνλ R α βνλµ + p α i R i λµ (5.454)where T M is the SEM–tensor for matter.One can verify that the SEM–tensor T s meets the condition(∂ µ + Γ αβ µ∂ αβ ) ˜H HE = ddx α T s α µ, (5.455)so that on solutions (5.450), the curvature of the connection (5.442) vanishes.Hence, the identity (5.454) is reduced to the conservation law (5.421)of matter in the presence of a background metric. The gravitation SEM–tensor is eliminated from the conservation law because the Hamiltonianform H HE is affine in all canonical momenta. Note that only gauge–typeconditions (5.444), (5.445) and the motion equations of matter have beenused.At the same time, since the canonical momenta p αβ α of the world metricare equal to zero, the Hamiltonian equation (5.446) on the Lagrangianconstraint space becomes∂ αβ ( ˜H HE + ˜H M ) = 0.Hence, the equality (5.454) takes the formπ βνλ α ∂ µ R α βνλ+(∂ µ +Γ i µ∂ i −∂ i Γ j µp α j ∂λ) i ˜H M ≈ddx α (T s α λµ+T M µ )+p α i Rλµ.i(5.456)This is the form of the energy–momentum conservation law which we observealso in case of quadratic Lagrangian densities of affine–metric gravity.Substituting the equality (5.455) into (5.456), we get the above result.As a test case of quadratic Lagrangian densities of affine–metric gravity,let us examine the sumL = (− 12κ gβλ F α βαλ + 1 4ε g αγg βσ g νµ g λε F α βνλF γ σµε) √ −gω (5.457)


<strong>Applied</strong> Jet <strong>Geom</strong>etry 953of the Hilbert–Einstein Lagrangian density and the Yang–Mills one. Thecorresponding Legendre map readsp αβ α ◦ ̂L = 0, (5.458)p α β(νλ) ◦ ̂L = 0, (5.459)p α β[νλ] ◦ ̂L = π α βνλ + 1 ε g αγg βσ g νµ g λε F γ σεµ√−g. (5.460)The relations (5.458) and (5.459) defines the Lagrangian constraint space.Let us consider two connections on the bundle C × Σ,Γ αβ α = −K ′ αελg εβ − K ′ βελg αε , and Γ α βνλ = Γ Kαβνλ , (5.461)where the notations of the expression (5.442) are used. The correspondingHamiltonian formsH = (p αβ α dg αβ + p α βνλ dk α βν) ∧ ω α − Hω,H = −p αβ α (K ′ αελg εβ + K ′ βελg αε ) + p α βνλ Γ Kαβνλ + ˜H,˜H = ε 4 gαγ g βσ g νµ g λε (p α β[νλ] − π α βνλ )(p γ σ[µε] − π γ βµε ), (5.462)are associated with the Lagrangian density (5.457) and constitute a completefamily.Given the Hamiltonian form (5.462) plus the Hamiltonian form H M formatter, we have the corresponding Hamiltonian equations∂ α g αβ + K ′ αελg εβ + K ′ βελg αε = 0, (5.463)∂ α k α βν = Γ α K βνλ + εg αγ g βσ g νµ g λε (p σ[µε] γ − π βµε γ ), (5.464)∂ α p α αβ = − ∂H∂g αβ − ∂H M,∂gαβ (5.465)∂ α p α βνλ = −p α ε[νγ] k β εγ + p ε β[νγ] k ε αγ−p α βεγ K ν (εγ) − p α ε(νγ) K β εγ + p ε β(νγ) K ε αγ (5.466)plus the motion equations for matter. The equation (5.464) breaks into theequation (5.460) and the gauge–type condition (5.449). The gauge–typeconditions (5.463) and (5.449) have the solution (5.450). Substituting theequation (5.464) into the equation (5.465) on the constraint space (5.458),we get the quadratic Einstein equations. Substitution of the equations(5.459) and (5.460) into the equation (5.466) results into the Yang–Millsgeneralization of the equation (5.452),∂ α p α βνλ + p α ε[νγ] k β εγ − p ε β[νγ] k ε αγ = 0.


954 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionConsider now the splitting of the Hamiltonian form (5.462) with respectto the connection (5.442) and the Hamiltonian density˜H Γ = ˜H + 1 2 p α βνλ R α βνλ.Let s = (k(x), g(x)) be a solution of the Euler–Lagrangian equationsof the Lagrangian density (5.457) and r the corresponding solution of theHamiltonian equations of the Hamiltonian form (5.462) where K and K ′are given by the expressions (5.450). For this solution r, let us take theSEM–tensor T s (5.399) relative to the connection (5.442) where K and K ′are given by the expressions (5.450). It readsT sαµ = 1 2 p α β[νλ] R α βνµ + ε 2 gαγ g βσ g νδ g µε p α β[νλ] (p γ σ[δε] − π γ σδε )and is equal to−δ α µ( ˜H + ε 2 gαγ g βσ g νδ g τε π α βντ (p γ σ[δε] − π γ σδε ))1ε R α βνλ R α βνµ + π βνλ α R α βνµ − δ α µ( 1 4ε R α βνλ R α βνλ + 12κ R).The weak identity (5.398) now becomes(∂ µ + Γ αβ µ∂ αβ + Γ i µ∂ i − ∂ i Γ j µp α j ∂ i λ − p αβνλ ∂≈and can be simplified toα∂k σ Γ K βνµγδddx α (T s α µ + T Mαµ ) + p α βνλ R α βνλµ + p α i R i λµ∂∂Pγδλ )( ˜H Γ + ˜H M )σ(∂ µ + Γ i µ∂ i − ∂ i Γ j µp α j ∂λ) i ˜H βνλ ∂ α ∂M − p α∂k σ Γ K βνµγδ ∂p ˜H γδλ Γσ≈ddx α (T s α αµ + T M µ ) + p α i Rλµ i , where (5.467)βνλ ∂ α ∂p α Γ K βνµ∂p ˜H γδλ Γ (5.468)σ∂k σ γδ= 1 κ kγ βµ(g βν R α γαν − g αν R β ανγ) √ −g − k γ (βµ)p α νβλ R α νγλ.Let us choose the local geodetic coordinate system at a point x ∈ X.Relative to this coordinate system, the equality (5.467) at x becomes theconservation law(∂ µ + Γ i µ∂ i − ∂ i Γ j µp α j ∂ i λ) ˜H M ≈ ddx α (T s α µ + T Mαµ ) + p α i R i λµ.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 955For example, in gauge theory, we haveddx α (T Γ α µ + t Mαµ ) = 0,where t M is the metric SEM–tensor of matter.5.12.2 Gauge Systems of Gravity and Fermion FieldsIn physical reality, one observes three types of field systems: gravitationalfields, fermion fields, and gauge fields associated with internal symmetries(see [Giachetta and Sardanashvily (1997)]). If the gauge invariance underinternal symmetries is kept in the presence of a gravitational field, Lagrangiandensities of gauge fields must depend on a metric gravitationalfield only.In the gauge gravitation theory, gravity is represented by pairs (h, A h )of gravitational fields h and associated Lorentz connections A h [Hehl et. al.(1995); Sardanashvily (1992)]. The connection A h is usually identified withboth a connection on a world manifold X and a spinor connection on thespinor bundle S h → X whose sections describe Dirac fermion fields ψ h inthe presence of the gravitational field h. The problem arises when Diracfermion fields are described in the framework of the affine–metric gravitationtheory. In this case, the fact that a world connection is some Lorentzconnection may result from the field equations, but it cannot be assumedin advance. There are models where the world connection is not a Lorentzconnection [Hehl et. al. (1995)]. Moreover, it may happen that a worldconnection is the Lorentz connection with respect to different gravitationalfields [Thompson (1993)]. At the same time, a Dirac fermion field can beregarded only in a pair (h, ψ h ) with a certain gravitational field h.One has to define the representation of cotangent vectors to X by theDirac’s γ−matrices in order to construct the Dirac operator. Given a tetradgravitational field h(x), we have the representationγ h : dx µ ↦→ ̂dx µ = h µ aγ a .However, different gravitational fields h and h ′ yield the nonequivalent representationsγ h and γ h ′.It follows that fermion–gravitation pairs (h, ψ h ) are described by sectionsof the composite spinor bundleS → Σ → X, (5.469)


956 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere Σ → X is the bundle of gravitational fields h where values of hplay the role of parameter coordinates, besides the familiar world coordinates[Sardanashvily (1992)]. In particular, every spinor bundle S h → Xis isomorphic to the restriction of S → Σ to h(X) ⊂ Σ. Performing thisrestriction, we come to the familiar case of a field model in the presenceof a gravitational field h(x). The feature of the dynamics of field systemson the composite bundle (5.469) lies in the fact that we have the modifiedcovariant differential of fermion fields which depend on derivatives ofgravitational fields h.As a consequence, we get the following covariant derivative of Diracfermion fields in the presence of a gravitational field h(x):˜D α = ∂ α − 1 2 Aabc µ(∂ α h µ c + K µ νλh ν c )I ab , (5.470)A abc µ = 1 2 (ηca h b µ − η cb h a µ),where K is a general linear connection on a world manifold X, 5 η is theMinkowski metric, and I ab = 1 4 [γ a, γ b ] are generators of the spinor Liegroup L s = SL(2, C).The covariant derivative (5.470) has been considered by [Aringazinand Mikhailov (1991); Ponomarev and Obukhov (1982); Tucker and Wang(1995)]. The relation (5.472) correspond to the canonical decomposition ofthe Lie algebra of the general linear group. By the well–known Theorem[Kobayashi and Nomizu (1963/9)], every general linear connection beingprojected onto the Lie algebra of the Lorentz group induces a Lorentz connection.In our opinion, the advantage of the covariant derivative (5.470), consistsin the fact that, being derived in the framework of the gauge gravitationtheory, it may be also applied to the affine–metric gravitation theory and5 The connectioneK ab α = A abc µ (∂αhµ c + Kµ νλh ν c ) (5.471)is not the connectionK k mλ = h k µ(∂ αh µ m + K µ νλh ν m) = K ab α(η am δ k b − η bm δk a )written with respect to the reference frame h a = h a α dxα , but there is the relationeK ab α = 1 2 (Kab α − K ba α). (5.472)If K is a Lorentz connection A h , then the connection e K given by (5.471) is consistentwith K itself.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 957the conventional Einstein’s gravitation theory. We are not concerned herewith the general problem of equivalence of metric, affine and affine–metrictheories of gravity [Ferraris and Kijowski (1982)]. At the same time, whenK is the Levi–Civita connection of h, the Lagrangian density of fermionfields which uses the covariant derivative (5.470) becomes that in the Einstein’sgravitation theory. It follows that the configuration space of metric(or tetrad) gravitational fields and general linear connections may play therole of the universal configuration space of realistic gravitational models.In particular, one then can think of the generalized Komar superpotentialas being the universal superpotential of energy–momentum of gravity[Giachetta and Sardanashvily (1995)].We follow [Giachetta and Sardanashvily (1997)] in the geometrical approachto field theory when classical fields are described by global sectionsof a fibre bundle Y −→ X over a smooth world space–time manifold X.Their dynamics is phrased in terms of jet spaces [Sardanashvily (1993);Saunders (1989)]. Recall that a kth–order differential operator on sectionsof a fibre bundle Y −→ X is defined to be a bundle morphism of the jetbundle J k (X, Y ) −→ X to a vector bundle over X.In particular, given bundle coordinates (x µ , y i ) of a fibre bundle Y −→X, the 1–jet space J 1 (X, Y ) of Y has the adapted coordinates (x µ , y i , yµ),iwhere yµ(j i xs) 1 = ∂ µ s i (x).There is the 1–1 correspondence between the connections on the fibrebundle Y → X and the global sections Γ = dx α ⊗ (∂ α + Γ i α∂ i ) of the affinejet bundle J 1 (X, Y ) → Y . Every connection Γ on Y → X induces thefirst–order differential operator on Y ,D Γ : J 1 (X, Y ) −→ T ∗ X ⊗ V Y, D Γ = (y i α − Γ i α)dx α ⊗ ∂ i ,which is called the covariant differential relative to the connection Γ.Recall that in the first–order Lagrangian formalism, the 1–jet spaceJ 1 (X, Y ) of Y plays the role of the finite–dimensional configuration spaceof fields represented by sections s of a bundle Y → X. A first–order Lagrangiandensity L : J 1 (X, Y ) −→ ∧ n T ∗ X is defined to be a horizontaldensity L = L(x µ , y i , yµ)ω i on the jet bundle J 1 (X, Y ) → X, whereω = dx 1 ∧ ... ∧ dx n , (n = dim X). Since the jet bundle J 1 (X, Y ) → Y isaffine, every polynomial Lagrangian density of field theory factors throughL : J 1 D(X, Y ) −→ T ∗ X ⊗ V Y → ∧ n T ∗ X, where D is the covariant differentialon Y , and V Y is the vertical tangent bundle of Y .Let us consider the gauge theory of gravity and fermion fields. By X


958 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis further meant an oriented 4D world manifold which satisfies the well–known topological conditions in order that gravitational fields and spinorstructure can exist on X. To summarize these conditions, we assume thatX is not compact and that the tangent bundle of X is trivial [Giachettaand Sardanashvily (1997)].Let LX be the principal bundle of oriented linear frames in tangentspaces to X. In gravitation theory, its structure group GL + (4, R) is reducedto the connected Lorentz group L = SO(1, 3). It means that there exists areduced subbundle L h X of LX whose structure group is L. In accordancewith the well–known Theorem, there is the 1–1 correspondence between thereduced L subbundles L h X of LX and the global sections h of the quotientbundleΣ = LX/L −→ X. (5.473)These sections h describe gravitational fields on X, for the bundle (5.473)is the 2–folder covering of the bundle of pseudo–Riemannian metrics on X.Given a section h of Σ, let Ψ h be an atlas of LX such that the correspondinglocal sections z h ξ of LX take their values into Lh X. With respectto Ψ h and a holonomic atlas Ψ T = {ψ T ξ } of LX, a gravitational field h canbe represented by a family of GL 4 −valued tetrad functionsh ξ = ψ T ξ ◦ z h ξ , dx α = h α a (x)h a . (5.474)By the Lorentz connections A h associated with a gravitational field hare meant the principal connections on the reduced subbundle L h X of LX.They give rise to principal connections on LX and to spinor connectionson the L s −lift P h of L h X.Given a Minkowski space M, let Cl 1,3 be the complex Clifford algebra 6generated by elements of M. A spinor space V is defined to be a minimal6 Recall that Clifford algebras are a type of associative algebra, named after Englishgeometer W. Clifford. They can be thought of as one of the possible generalizationsof the complex numbers and quaternions. The theory of Clifford algebras is intimatelyconnected with the theory of quadratic forms and orthogonal transformations. Themost important Clifford algebras are those over R and C equipped with nondegeneratequadratic forms. Recall that every nondegenerate quadratic form on a finite–dimensionalreal vector space is equivalent to the standard diagonal form Q(x) = x 2 1 +· · ·+x2 p −x2 p+1 −· · · − x 2 p+q , where n = p + q is the dimension of the vector space. The pair of integers(p, q) is called the signature of the quadratic form. Similarly, one can define Cliffordalgebras on complex vector spaces. Every nondegenerate quadratic form on a complexvector space is equivalent to the standard diagonal form Q(z) = z1 2 + z2 2 + · · · + z2 n, sothere is essentially only one Clifford algebra in each dimension. One can show that thecomplex Clifford algebra may be obtained as the complexification of the real one.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 959left ideal of Cl 1,3 on which this algebra acts on the left. We have therepresentation γ : M ⊗ V → V of elements of the Minkowski spaceM ⊂ Cl 1,3 by Dirac’s matrices γ on V .Let us consider a bundle of complex Clifford algebras Cl 1,3 over Xwhose structure group is the Clifford group of invertible elements of Cl 1,3 .Its subbundles are both a spinor bundle S M −→ X and the bundle Y M−→ X of Minkowski spaces of generating elements of Cl 1,3 . To describeDirac fermion fields on a world manifold X, one must require Y M to beisomorphic to the cotangent bundle T ∗ X of X. It takes place if there existsa reduced L subbundle L h X such thatThen, the spinor bundleY M = (L h X × M)/L.S M = S h = (P h × V )/L s (5.475)is associated with the L s −lift P h of L h X. In this case, there exists therepresentationγ h : T ∗ X ⊗S h = (P h ×(M ⊗V ))/L s −→ (P h ×γ(M ×V ))/L s = S h (5.476)of cotangent vectors to a world manifold X by Dirac’s γ−matrices on elementsof the spinor bundle S h . As a shorthand, one can writêdx α = γ h (dx α ) = h α a (x)γ a .Given the representation (5.476), we shall say that sections of the spinorbundle S h describe Dirac fermion fields in the presence of the gravitationalfield h. Let a principal connection on S h be given byA h = dx α ⊗ (∂ α + 1 2 Aab αI abAB ψ B ∂ A ).Given the corresponding covariant differential D and the representation γ h(5.476), one can construct the Dirac operator on the spinor bundle S h , asD h = γ h ◦ D : J 1 S h → T ∗ X ⊗ V S h → V S h , (5.477)ẏ A ◦ D h = h α a γ aA B(y B α − 1 2 Aab αI abAB y B ).<strong>Diff</strong>erent gravitational fields h and h ′ define nonequivalent representationsγ h and γ h ′. It follows that a Dirac fermion field must be regarded onlyin a pair with a certain gravitational field. There is the 1–1 correspondencebetween these pairs and sections of the composite spinor bundle (5.469).


960 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall that we have a composite bundleY → Σ → X (5.478)of a bundle Y → X denoted by Y Σ and a bundle Σ → X. It is coordinatedby (x α , σ m , y i ) where (x µ , σ m ) are coordinates of Σ and y i are the fibrecoordinates of Y Σ . We further assume that Σ has a global section.The application of composite bundles to field theory is founded on thefollowing [Sardanashvily (1992)]. Given a global section h of Σ, the restrictionY h of Y Σ to h(X) is a subbundle of Y → X. There is the 1–1correspondence between the global sections s h of Y h and the global sectionsof the composite bundle (5.478) which cover h. Therefore, one canthink of sections s h of Y h as describing fermion fields in the presence of abackground parameter field h, whereas sections of the composite bundle Ydescribe all the pairs (s h , h). The configuration space of these pairs is the1–jet space J 1 (X, Y ) of the composite bundle Y .Every connectionA Σ = dx α ⊗ (∂ α + Ãi α∂ i ) + dσ m ⊗ (∂ m + A i m∂ i )on the bundle Y Σ induces the horizontal splittingV Y = V Y Σ ⊕ (Y × V Σ),locally given byẏ i ∂ i + ˙σ m ∂ m = (ẏ i − A i m ˙σ m )∂ i + ˙σ m (∂ m + A i m∂ i ).Using this splitting, one can construct the first–order differential operator(5.41) on the composite bundle Y , namely˜D : J 1 (X, Y ) → T ∗ X ⊗V Y Σ , ˜D = dx α ⊗(y i α−Ãi α−A i mσ m α )∂ i . (5.479)This operator possess the following property. Given a global section h ofΣ, let Γ be a connection on Σ whose integral section is h, that is, Γ ◦ h =j 1 h. Note that the differential (5.479) restricted to J 1 (X, Y ) h ⊂ J 1 (X, Y )becomes the familiar covariant differential relative to the connection on Y h ,A h = dx α ⊗ [∂ α + (A i m∂ α h m + Ãi α)∂ i ].Thus, it is ˜D that we may use in order to construct a Lagrangian densityL : J 1 (X, Y )eD−→ T ∗ X ⊗ V Y Σ → ∧ n T ∗ Xfor sections of the composite bundle Y .In particular, in gravitation theory, we have the composite bundleLX → Σ → X, where Σ is the quotient bundle (5.473) and LX Σ =


<strong>Applied</strong> Jet <strong>Geom</strong>etry 961LX → Σ is the L−principal bundle. Let P Σ be the L s −principal lift ofLX Σ such that P Σ /L s = Σ and LX Σ = r(P Σ ). In particular, there isthe imbedding of the L s −lift P h of L h X onto the restriction of P Σ to h(X)[Giachetta and Sardanashvily (1997)].Let us consider the composite spinor bundle (5.469) where S Σ = (P Σ ×V )/L s is associated with the L s −principal bundle P Σ . Note that, given aglobal section h of Σ, the restriction S Σ to h(X) is the spinor bundle S h(5.475) whose sections describe Dirac fermion fields in the presence of thegravitational field h.Let us give the principal bundle LX with a holonomic atlas {ψ T ξ , U ξ }and the principal bundles P Σ and LX Σ with associated atlases {zɛ s , U ɛ } and{z ɛ = r ◦ zɛ s }. With respect to these atlases, the composite spinor bundleis equipped with the bundle coordinates (x α , σ µ a, ψ A ) where (x α , σ µ a) arecoordinates of the bundle Σ such that σ µ a are the matrix components of thegroup element (ψ T ξ ◦ z ɛ )(σ), σ ∈ U ɛ , π ΣX (σ) ∈ U ξ . Given a section h of Σ,we have (σ α a ◦h)(x) = h α a (x), where h α a (x) are the tetrad functions (5.474).Let us consider the bundle of Minkowski spaces (LX × M)/L → Σ associatedwith the L−principal bundle LX Σ . Since LX Σ is trivial, it isisomorphic to the pull–back Σ × T ∗ X which we denote by the same symbolT ∗ X. Then, one can define the bundle morphism γ Σ over Σ, given byγ Σ : T ∗ X ⊗ S Σ → S Σ , ̂dx α = γ Σ (dx α ) = σ α a γ a . (5.480)When restricted to h(X) ⊂ Σ, the map (5.480) becomes the morphismγ h (5.476). We use this morphism in order to construct the total Diracoperator on the composite spinor bundle S (5.469).Letà = dx α ⊗ (∂ α + ÃB α ∂ B ) + dσ µ a ⊗ (∂ a µ + A Ba µ∂ B )be a principal connection on the bundle S Σ and ˜D the corresponding differential(5.479). We have the first–order differential operator on S, givenbyD = γ Σ ◦ ˜D : J 1 S → T ∗ X ⊗ V S Σ → V S Σ ,˙ψ A ◦ D = σ α a γ aA B(ψ B α − ÃB α − A Ba µσ µ aλ ).One can think of it as being the total Dirac operator since, for every sectionh, the restriction of D to J 1 S h ⊂ J 1 S becomes the Dirac operator D h


962 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(5.477) relative to the connection on the bundle S h , given byA h = dx α ⊗ [∂ α + (ÃB α + A Ba µ∂ α h µ a)∂ B ].In order to construct the differential ˜D (5.479) on J 1 (X, S) in explicitform, let us consider the principal connection on the bundle LX Σ which isgiven by the local connection formà = (Ãab µdx µ + A abc µdσ µ c ) ⊗ I ab , (5.481)à ab µ = 1 2 Kν λµσ α c (η ca σ b ν − η cb σ a ν),A abc µ = 1 2 (ηca σ b µ − η cb σ a µ), (5.482)where K is a general linear connection on T X and (5.482) correspondsto the canonical left–invariant connection on the bundle GL + (4, R) −→GL + (4, R)/L.Therefore, the differential ˜D relative to the connection (5.481) reads˜D = dx α ⊗ [∂ α − 1 2 Aabc µ(σ µ cλ + Kµ νλσ ν c )I abAB ψ B ∂ A ]. (5.483)Given a section h, the connection à (5.481) is reduced to the Lorentzconnection ˜K (5.471) on L h X, and the differential (5.483) leads to thecovariant derivatives of fermion fields (5.470). We will use the differential(5.483) in order to construct a Lagrangian density of Dirac fermion fields.Their Lagrangian density is defined on the configuration space J 1 (X, S ⊕S + ) coordinated by (x µ , σ µ a, ψ A , ψ + A , σµ aλ , ψA α , ψ + Aλ). It readsL ψ = { i 2 [ψ+ A (γ0 γ α ) A B(ψ B α − 1 2 Aabc µ(σ µ cλ + Kµ νλσ ν c )I abBC ψ C )− (ψ + Aλ − 1 2 Aabc µ(σ µ cλ + Kµ νλσ ν c )ψ + C I+ abCA )(γ 0 γ α ) A Bψ B ] (5.484)− mψ + A (γ0 ) A Bψ B }σ −1 ω,where γ µ = σ µ aγ a , and σ = det(σ µ a), while ψ + A (γ0 ) A Bψ B is the Lorentz–invariant fibre metric in the bundle S ⊕ S ∗ [Crawford (1991)].One can show that∂L ψ∂K µ νλ+ ∂L ψ∂K µ λν= 0.Hence, the Lagrangian density (5.484) depends on the torsion of the generallinear connection K only. In particular, it follows that, if K is the


<strong>Applied</strong> Jet <strong>Geom</strong>etry 963Levi–Civita connection of a gravitational field h(x), after the substitutionσ ν c = h ν c (x), the Lagrangian density (5.484) becomes the familiar Lagrangiandensity of fermion fields in the Einstein’s gravitation theory.5.12.3 Hawking–Penrose Quantum Gravity and BlackHolesIn their search for quantum gravity, S. Hawking and R. Penrose use thestraightforward application of quantum theory to general relativity [Hawkingand Israel (1979); Penrose (1989); Hawking and Penrose (1996)], ratherthan following the more fashioned string theory approach (described below).According to Hawking, “Einstein’s general relativity is a beautiful theorythat agrees with every observation that has been made so far. It mightrequire modifications on the Planck scale, and it might be only a low energyapproximation to some more fundamental theory, like e.g., superstringtheory, but it will not affect many of the predictions that can be get fromgravity...” [Hawking and Israel (1979)].Space–Time Manifold, Gravity, Black Holes and Big BangThe crucial technique for investigating Hawking–Penrose singularitiesand black holes, has been the study of the global causal structure of space–time [Hawking and Israel (1979)]. Define I + (p) to be the set of all points ofthe space–time manifold M that can be reached from the point p by futuredirected time like curves. One can think of I + (p) as the set of all events thatcan be influenced by what happens at p. One now considers the boundaryI ˙ + (S) of the future of a set S. It is easy to see that this boundary cannotbe time–like. For in that case, a point q just outside the boundary wouldbe to the future of a point p just inside. Nor can the boundary of the futurebe space–like, except at the set S itself. For in that case every past directedcurve from a point q, just to the future of the boundary, would cross theboundary and leave the future of S. That would be a contradiction withthe fact that q is in the future of S. Therefore, the boundary of the futureis null apart from at S itself.To show that each generator of the boundary of the future has a pastend point on the set, one has to impose some global condition on the causalstructure. The strongest and physically most important condition is that of


964 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionglobal space–time hyperbolicity. 7 The significance of global hyperbolicity forsingularity theorems stems from the following [Hawking and Israel (1979);Hawking and Penrose (1996)]. Let U be globally hyperbolic and let p andq be points of U that can be joined by a time like or null curve. Then thereis a time–like or null–geodesic between p and q which maximizes the lengthof time like or null curves from p to q. The method of proof is to showthe space of all time like or null curves from p to q is compact in a certaintopology. One then shows that the length of the curve is an upper semi–continuous function on this space. It must therefore attain its maximumand the curve of maximum length will be a geodesic because otherwise asmall variation will give a longer curve.One can now consider the second variation of the length of a geodesicγ. One can show that γ can be varied to a longer curve if there is aninfinitesimally neighboring geodesic from p which intersects γ again at apoint r between p and q. The point r is said to be conjugate to p. One canillustrate this by considering two points p and q on the surface of the Earth.Without loss of generality one can take p to be at the north pole. Becausethe Earth has a positive definite metric rather than a Lorentzian one, thereis a geodesic of minimal length, rather than a geodesic of maximum length.This minimal geodesic will be a line of longitude running from the northpole to the point q. But there will be another geodesic from p to q whichruns down the back from the north pole to the south pole and then up toq. This geodesic contains a point conjugate to p at the south pole where allthe geodesics from p intersect. Both geodesics from p to q are stationarypoints of the length under a small variation. But now in a positive definitemetric the second variation of a geodesic containing a conjugate point cangive a shorter curve from p to q. Thus, on the Earth, the geodesic that goesdown to the south pole and then comes up is not the shortest curve from pto q.The reason one gets conjugate points in space–time is that gravity is anattractive force. It therefore curves space–time in such a way that neighboringgeodesics are bent towards each other rather than away. One can7 Recall that an open set U is said to be globally hyperbolic if:(1) For every pair of points p and q in U the intersection of the future of p and the pastof q has compact closure. In other words, it is a bounded diamond shaped region.(2) Strong causality holds on U. That is, there are no closed or almost closed time–likecurves contained in U.


<strong>Applied</strong> Jet <strong>Geom</strong>etry 965see this from the Newman–Penrose equationdρdv = ρ2 + σ ij σ ij + 1 n R αβl α l β , (α, β = 0, 1, 2, 3)where n = 2 for null geodesics and n = 3 for time–like geodesics. Here v isan affine parameter along a congruence of geodesics, with tangent vector l αwhich are hypersurface orthogonal. The quantity ρ is the average rate ofconvergence of the geodesics, while σ measures the shear. The term R αβ l α l βgives the direct gravitational effect of the matter on the convergence of thegeodesics. By the Einstein equation (3.1), it will be non–negative for anynull vector l α if the matter obeys the so–called weak energy condition, whichsays that the energy density T 00 is non–negative in any frame, i.e.,T αβ v α v β ≥ 0, (5.485)for any time–like vector v α , is obeyed by the classical SEM–tensor ofany reasonable matter [Hawking and Israel (1979); Hawking and Penrose(1996)].Suppose the weak energy condition holds, and that the null geodesicsfrom a point p begin to converge again and that ρ has the positive valueρ 0 . Then the Newman–Penrose equation would imply that the convergenceρ would become infinite at a point q within an affine parameter distance1ρ 0if the null geodesic can be extended that far. If ρ = ρ 0 at v = v 0 thenρ ≥1ρ −1 +v 0−v . Thus there is a conjugate point before v = v 0 + ρ −1 .Infinitesimally neighboring null geodesics from p will intersect at q. Thismeans the point q will be conjugate to p along the null geodesic γ joiningthem. For points on γ beyond the conjugate point q there will be a variationof γ that gives a time like curve from p. Thus γ cannot lie in the boundaryof the future of p beyond the conjugate point q. So γ will have a future endpoint as a generator of the boundary of the future of p.The situation with time–like geodesics is similar, except that the strongenergy condition [Hawking and Israel (1979); Hawking and Penrose (1996)],T αβ v α v β ≥ 1 2 vα v α T, (5.486)that is required to make R αβ l α l β non–negative for every time like vectorl α , is rather stronger than the weak energy condition (5.485). However,it is still physically reasonable, at least in an averaged sense, in classicaltheory. If the strong energy condition holds, and the time like geodesicsfrom p begin converging again, then there will be a point q conjugate to p.


966 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFinally there is the generic energy condition, which says:(1) The strong energy condition holds.(2) Every time–like or null geodesic has a point where l [a R b]cd[e l f] l c l d ≠ 0.One normally thinks of a space–time singularity as a region in whichthe curvature becomes unboundedly large. However, the trouble with thisdefinition is that one could simply leave out the singular points and saythat the remaining manifold was the whole of space–time. It is thereforebetter to define space–time as the maximal manifold on which the metric issuitably smooth. One can then recognize the occurrence of singularities bythe existence of incomplete geodesics that cannot be extended to infinitevalues of the affine parameter.Hawking–Penrose Singularity TheoremsHawking–Penrose Singularity is defined as follows [Hawking and Israel(1979); Penrose (1989); Hawking and Penrose (1996)]:A space–time manifold is singular if it is time–like or null geodesicallyincomplete but cannot be embedded in a larger space–time manifold.This definition reflects the most objectionable feature of singularities,that there can be particles whose history has a beginning or end at a finitetime. There are examples in which geodesic incompleteness can occur withthe curvature remaining bounded, but it is thought that generically thecurvature will diverge along incomplete geodesics. This is important if oneis to appeal to quantum effects to solve the problems raised by singularitiesin classical general relativity.Singularity Theorems include:(1) Energy condition (i.e., weak (5.485), strong (5.486), or generic (5.12.3)).(2) Condition on global structure (e.g., there should not be any closedtime–like curves).(3) Gravity strong enough to trap a region (so that nothing could escape).The various singularity theorems show that space–time must be time likeor null geodesically incomplete if different combinations of the three kindsof conditions hold. One can weaken one condition if one assumes strongerversions of the other two. The Hawking–Penrose Singularity theorems havethe generic energy condition, the strongest of the three energy conditions.The global condition is fairly weak, that there should be no closed time like


<strong>Applied</strong> Jet <strong>Geom</strong>etry 967curves. And the no escape condition is the most general, that there shouldbe either a trapped surface or a closed space like three surface.The theorems predict singularities in two situations. One is in the futurein the gravitational collapse of stars and other massive bodies. Suchsingularities would be an end of time, at least for particles moving on the incompletegeodesics. The other situation in which singularities are predictedis in the past at the beginning of the present expansion of the universe.The prediction of singularities means that classical general relativity isnot a complete theory. Because the singular points have to be cut out ofthe space–time manifold one cannot define the field equations there andcannot predict what will come out of a singularity. With the singularityin the past the only way to deal with this problem seems to be to appealto quantum gravity. But the singularities that are predicted in the futureseem to have a property that Penrose has called, Cosmic Censorship. Thatis they conveniently occur in places like black holes that are hidden fromexternal observers. So any break down of predictability that may occurat these singularities will not affect what happens in the outside world, atleast not according to classical theory.Hawking Cosmic Censorship Hypothesis says: “Nature abhors a nakedsingularity” [Hawking and Israel (1979); Hawking and Penrose (1996)].However, there is unpredictability in the quantum theory. This is relatedto the fact that gravitational fields can have intrinsic entropy which is notjust the result of coarse graining. Gravitational entropy, and the fact thattime has a beginning and may have an end, are the two main themes ofHawking’s research, because they are the ways in which gravity is distinctlydifferent from other physical fields.The fact that gravity has a quantity that behaves like entropy was firstnoticed in the purely classical theory. It depends on Penrose’s CosmicCensorship Conjecture. This is unproved but is believed to be true forsuitably general initial data and state equations.One makes the approximation of treating the region around a collapsingstar as asymptotically flat. Then, as Penrose showed, one can conformallyembed the space–time manifold M in a manifold with boundary ¯M. Theboundary ∂M will be a null surface and will consist of two components,future and past null infinity, called I + and I − . One says that weak CosmicCensorship holds if two conditions are satisfied. First, it is assumed that thenull geodesic generators of I + are complete in a certain conformal metric.This implies that observers far from the collapse live to an old age and arenot wiped out by a thunderbolt singularity sent out from the collapsing


968 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionstar. Second, it is assumed that the past of I + is globally hyperbolic.This means there are no naked singularities that can be seen from largedistances. Penrose has also a stronger form of Cosmic Censorship whichassumes that the whole space–time is globally hyperbolic.Weak Cosmic Censorship Hypothesis reads:(1) I + and I − are complete.(2) I − (I + ) is globally hyperbolic.If weak Cosmic Censorship holds, the singularities that are predictedto occur in gravitational collapse cannot be visible from I + . This meansthat there must be a region of space–time that is not in the past of I + .This region is said to be a black hole because no light or anything else canescape from it to infinity. The boundary of the black hole region is calledthe event horizon. Because it is also the boundary of the past of I + theevent horizon will be generated by null–geodesic segments that may havepast end points but don’t have any future end points. It then follows thatif the weak energy condition holds the generators of the horizon cannot beconverging. For if they were they would intersect each other within a finitedistance [Hawking and Israel (1979); Penrose (1989); Hawking and Penrose(1996)].This implies that the area of a cross section of the event horizon cannever decrease with time and in general will increase. Moreover if twoblack holes collide and merge together the area of the final black hole willbe greater than the sum of the areas of the original black holes. This isvery similar to the behavior of entropy according to the Second Law ofThermodynamics: 8 .Second Law of Black Hole Mechanics: δA ≥ 0.Second Law of Thermodynamics: δS ≥ 0.The similarity with thermodynamics is increased by what is called theFirst Law of Black Hole Mechanics, which relates the change in mass of ablack hole to the change in the area of the event horizon and the changein its angular momentum and electric charge. One can compare this to theFirst Law of Thermodynamics which gives the change in internal energy interms of the change in entropy and the external work done on the system[Hawking and Israel (1979); Hawking and Penrose (1996)]:First Law of Black Hole Mechanics: δE = κ8πδA + ΩδJ + ΦδQ.First Law of Thermodynamics: δE = T δS + P δV.8 Recall that Second Law of Thermodynamics states: Entropy can never decrease andthe entropy of a total system is greater than the sum of its constituent parts


<strong>Applied</strong> Jet <strong>Geom</strong>etry 969One sees that if the area A of the event horizon is analogous to entropyS then the quantity analogous to temperature is what is called the surfacegravity of the black hole κ. This is a measure of the strength of the gravitationalfield on the event horizon. The similarity with thermodynamicsis further increased by the so–called Zeroth Law of Black Hole Mechanics:the surface gravity is the same everywhere on the event horizon of a timeindependent black hole [Hawking and Israel (1979)].Zeroth Law of Black Hole Mechanics:κ is the same everywhere on the horizon of a time independent black hole.Zeroth Law of Thermodynamics:T is the same everywhere for a system in thermal equilibrium.Encouraged by these similarities Bekenstein proposed that some multipleof the area of the event horizon actually was the entropy of a blackhole. He suggested a generalized Second Law: the sum of this black holeentropy and the entropy of matter outside black holes would never decrease(see [Strominger and Vafa (1996)]).Generalized Second Law: δ(S + cA) ≥ 0.However, this proposal was not consistent. If black holes have an entropyproportional to horizon area A they should also have a non zerotemperature proportional to surface gravity.Path–Integral Model for Black HolesRecall that the fact that gravity is attractive means that it will tendto draw the matter in the universe together to form objects like stars andgalaxies. These can support themselves for a time against further contractionby thermal pressure, in the case of stars, or by rotation and internalmotions, in the case of galaxies. However, eventually the heat or the angularmomentum will be carried away and the object will begin to shrink.If the mass is less than about one and a half times that of the Sun thecontraction can be stopped by the degeneracy pressure of electrons or neutrons.The object will settle down to be a white dwarf or a neutron starrespectively. However, if the mass is greater than this limit there is nothingthat can hold it up and stop it continuing to contract. Once it has shrunkto a certain critical size the gravitational field at its surface will be sostrong that the light cones will be bent inward [Hawking and Israel (1979);Hawking and Penrose (1996)].If the Cosmic Censorship Conjecture is correct the trapped surface andthe singularity it predicts cannot be visible from far away. Thus there must


970 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbe a region of space–time from which it is not possible to escape to infinity.This region is said to be a black hole. Its boundary is called the eventhorizon and it is a null surface formed by the light rays that just fail to getaway to infinity. As we saw in the last subsection, the area A of a crosssection of the event horizon can never decrease, at least in the classicaltheory. This, and perturbation calculations of spherical collapse, suggestthat black holes will settle down to a stationary state.Recall that the Schwarzschild metric form, given byds 2 = −(1 − 2M r )dt2 + (1 − 2M r )−1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ),represents the gravitational field that a black hole would settle down to ifit were non rotating. In the usual r and t coordinates there is an apparentsingularity at the Schwarzschild radius r = 2M. However, this is justcaused by a bad choice of coordinates. One can choose other coordinatesin which the metric is regular there.Now, if one performs the Wick rotation, t = iτ, one gets a positivedefinite metric, usually called Euclidean even though they may be curved.In the Euclidean–Schwarzschild metric( ) 2 ( ) dτ rds 2 = x 2 2 2+4M 4M 2 dx 2 + r 2 (dθ 2 + sin 2 θdφ 2 )there is again an apparent singularity at r = 2M. However, one can definea new radial coordinate x to be 4M(1 − 2Mr −1 ) 1 2 .The metric in the x − τ plane then becomes like the origin of polarcoordinates if one identifies the coordinate τ with period 8πM. Similarly,other Euclidean black hole metrics will have apparent singularities on theirhorizons which can be removed by identifying the imaginary time coordinatewith period 2π κ .To see the significance of having imaginary time identified with someperiod β, let us consider the amplitude to go from some field configurationφ 1 on the surface t 1 to a configuration φ 2 on the surface t 2 . This will begiven by the matrix element of e iH(t2−t1) . However, one can also representthis amplitude as a path integral over all fields φ between t 1 and t 2 whichagree with the given fields φ 1 and φ 2 on the two surfaces,∫< φ 2 , t 2 |φ 1 , t 1 >=< φ 2 | exp(−iH(t 2 − t 1 ))|φ 1 >= D[φ] exp(iA[φ]).One now chooses the time separation (t 2 − t 1 ) to be pure imaginary andequal to β. One also puts the initial field φ 1 equal to the final field φ 2 and


<strong>Applied</strong> Jet <strong>Geom</strong>etry 971sums over a complete basis of states φ n . On the left one has the expectationvalue of e −βH summed over all states. This is just the thermodynamicpartition function Z at the temperature T = β −1 ,Z = ∑ ∫< φ n | exp(−βH)|φ n > = D[φ] exp(−A[φ]). (5.487)On the r.h.s. of this equation one has a path integral (see chapter 6).One puts φ 1 = φ 2 and sums over all field configurations φ n . This meansthat effectively one is doing the path integral over all fields φ on a space–time that is identified periodically in the imaginary time direction withperiod β. Thus the partition function for the field φ at temperature Tis given by a path integral over all fields on a Euclidean space–time. Thisspace–time is periodic in the imaginary time direction with period β = T −1[Hawking and Israel (1979); Hawking and Penrose (1996)].If one calculates the path integral in flat space–time identified withperiod β in the imaginary time direction one gets the usual result for thepartition function of black body radiation. However, as we have just seen,the Euclidean–Schwarzschild solution is also periodic in imaginary timewith period 2π κ. This means that fields on the Schwarzschild backgroundwill behave as if they were in a thermal state with temperature κ2π .The periodicity in imaginary time explained why the messy calculationof frequency mixing led to radiation that was exactly thermal. However,this derivation avoided the problem of the very high frequencies that takepart in the frequency mixing approach. It can also be applied when thereare interactions between the quantum fields on the background. The factthat the path integral is on a periodic background implies that all physicalquantities like expectation values will be thermal. This would have beenvery difficult to establish in the frequency mixing approach [Hawking andIsrael (1979); Hawking and Penrose (1996)].One can extend these interactions to include interactions with the gravitationalfield itself. One starts with a background metric g 0 such as theEuclidean–Schwarzschild metric that is a solution of the classical field equations.One can then expand the action A in a power series in the perturbationsδg about g 0 , asA[g] = A[g 0 ] + A 2 (δg) 2 + A 3 (δg) 3 + ...Here, the linear term vanishes because the background is a solution of thefield equations. The quadratic term can be regarded as describing gravitonson the background while the cubic and higher terms describe interactions


972 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbetween the gravitons. The path integral over the quadratic terms are finite.There are non renormalizable divergences at two loops in pure gravity butthese cancel with the fermions in super–gravity theories. It is not knownwhether super–gravity theories have divergences at three loops or higherbecause no one has been brave or foolhardy enough to try the calculation.Some recent work indicates that they may be finite to all orders. But evenif there are higher loop divergences they will make very little differenceexcept when the background is curved on the scale of the Planck length(10 −33 cm).More interesting than the higher order terms is the zeroth order term,the action of the background metric g 0 [Hawking and Israel (1979); Hawkingand Penrose (1996)],A = − 116π∫R(−g) 1 2 d 4 x + 1 ∫8πK(±h) 1 2 d 3 x.Recall that the usual Einstein–Hilbert action for general relativity is thevolume integral of the scalar curvature R. This is zero for vacuum solutionsso one might think that the action of the Euclidean-Schwarzschild solutionwas zero. However, there is also a surface term in the action proportional tothe integral of K, the trace of the second fundamental form of the boundarysurface. When one includes this and subtracts off the surface term for flatspace one finds the action of the Euclidean–Schwarzschild metric isβ216πwhere β is the period in imaginary time at infinity. Thus the dominantcontribution to the path integral for the partition function Z given by(5.487), is e −β216π ,Z = ∑ )exp(−βE n ) = exp(− β2.16πIf one differentiates log Z with respect to the period β one gets theexpectation value of the energy, or in other words, the mass,< E > = − ddβ (log Z) = β 8π .So this gives the mass M = β8π. This confirms the relation between the massand the period, or inverse temperature, that we already knew. However,one can go further. By standard thermodynamic arguments, the log ofthe partition function is equal to minus the free energy F divided by thetemperature T , i.e., log Z = − F T. And the free energy is the mass or energyplus the temperature times the entropy S, i.e., F = < E > + T S. Putting


<strong>Applied</strong> Jet <strong>Geom</strong>etry 973all this together one sees that the action of the black hole gives an entropyof 4πM 2 ,S = β216π = 4πM 2 = 1 4 A.This is exactly what is required to make the laws of black holes the sameas the laws of thermodynamics [Hawking and Israel (1979); Hawking andPenrose (1996)]. The reason why does one get this intrinsic gravitationalentropy which has no parallel in other quantum field theories, is that gravityallows different topologies for the space–time manifold.In the case we are considering the Euclidean–Schwarzschild solutionhas a boundary at infinity that has topology S 2 × S 1 . The S 2 is a largespace like two sphere at infinity and the S 1 corresponds to the imaginarytime direction which is identified periodical. One can fill in this boundarywith metrics of at least two different topologies. One is the Euclidean–Schwarzschild metric. This has topology R 2 × S 2 , that is the Euclideantwo plane times a two sphere. The other is R 3 × S 1 , the topology ofEuclidean flat space periodically identified in the imaginary time direction.These two topologies have different Euler numbers. The Euler numberof periodically identified flat space is zero, while that of the Euclidean–Schwarzschild solution is two,Total action = M(τ 2 − τ 1 ).The significance of this is as follows: on the topology of periodically identifiedflat space one can find a periodic time function τ whose gradient isno where zero and which agrees with the imaginary time coordinate onthe boundary at infinity. One can then work out the action of the regionbetween two surfaces τ 1 and τ 2 . There will be two contributions to theaction, a volume integral over the matter Lagrangian, plus the Einstein–Hilbert Lagrangian and a surface term. If the solution is time independentthe surface term over τ = τ 1 will cancel with the surface term over τ = τ 2 .Thus the only net contribution to the surface term comes from the boundaryat infinity. This gives half the mass times the imaginary time interval(τ 2 − τ 1 ). If the mass is non–zero there must be non–zero matter fields tocreate the mass. One can show that the volume integral over the matterLagrangian plus the Einstein–Hilbert Lagrangian also gives 1 2 M(τ 2 − τ 1 ).Thus the total action is M(τ 2 − τ 1 ). If one puts this contribution to thelog of the partition function into the thermodynamic formulae one finds


974 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe expectation value of the energy to be the mass, as one would expect.However, the entropy contributed by the background field will be zero.However, the situation is different with the Euclidean–Schwarzschildsolution, which says:Total action including corner contribution = M(τ 2 − τ 1 )Total action without corner contribution = 1 2 M(τ 2 − τ 1 )Because the Euler number is two rather than zero one cannot find atime function τ whose gradient is everywhere non–zero. The best one cando is choose the imaginary time coordinate of the Schwarzschild solution.This has a fixed two sphere at the horizon where τ behaves like an angularcoordinate. If one now works out the action between two surfacesof constant τ the volume integral vanishes because there are no matterfields and the scalar curvature is zero. The trace K surface term at infinityagain gives 1 2 M(τ 2 − τ 1 ). However there is now another surface termat the horizon where the τ 1 and τ 2 surfaces meet in a corner. One canevaluate this surface term and find that it also is equal to 1 2 M(τ 2 − τ 1 ).Thus the total action for the region between τ 1 and τ 2 is M(τ 2 − τ 1 ).If one used this action with τ 2 − τ 1 = β one would find that the entropywas zero. However, when one looks at the action of the EuclideanSchwarzschild solution from a 4−dimensional point of view rather than a3 + 1, there is no reason to include a surface term on the horizon becausethe metric is regular there. Leaving out the surface term on the horizonreduces the action by one quarter the area of the horizon, which is just theintrinsic gravitational entropy of the black hole [Hawking and Israel (1979);Hawking and Penrose (1996)].Quantum CosmologyAccording to Hawking, cosmology used to be considered a pseudo–science and the preserve of physicists who may have done useful work intheir earlier years but who had gone mystic in their dotage. There is a seriousobjection that cosmology cannot predict anything about the universeunless it makes some assumption about the initial conditions. Without suchan assumption, all one can say is that things are as they are now becausethey were as they were at an earlier stage. Yet many people believe thatscience should be concerned only with the local laws which govern how the


<strong>Applied</strong> Jet <strong>Geom</strong>etry 975universe evolves in time. They would feel that the boundary conditionsfor the universe that determine how the universe began were a questionfor metaphysics or religion rather than science [Hawking and Israel (1979);Hawking and Penrose (1996)].Hawking–Penrose theorems showed that according to general relativitythere should be a singularity in our past. At this singularity the field equationscould not be defined. Thus classical general relativity brings about itsown downfall: it predicts that it cannot predict the universe. For Hawkingthis sounds rally disturbing: If the laws of physics could break down at thebeginning of the universe, why couldn’t they break down any where. Inquantum theory it is a principle that anything can happen if it is not absolutelyforbidden. Once one allows that singular histories could take partin the path integral they could occur any where and predictability woulddisappear completely. If the laws of physics break down at singularities,they could break down any where.The only way to have a scientific theory is if the laws of physics holdeverywhere including at the beginning of the universe. One can regard thisas a triumph for the Principle of Democracy: Why should the beginningof the universe be exempt from the laws that apply to other points. If allpoints are equal one cannot allow some to be more equal than others.To implement the idea that the laws of physics hold everywhere, oneshould take the path integral only over non–singular metrics. One knowsin the ordinary path integral case that the measure is concentrated on non–differentiable paths. But these are the completion in some suitable topologyof the set of smooth paths with well defined action. Similarly, one wouldexpect that the path integral for quantum gravity should be taken over thecompletion of the space of smooth metrics. What the path integral cannotinclude is metrics with singularities whose action is not defined.In the case of black holes we saw that the path integral should be takenover Euclidean, that is, positive definite metrics. This meant that thesingularities of black holes, like the Schwarzschild solution, did not appearon the Euclidean metrics which did not go inside the horizon. Insteadthe horizon was like the origin of polar coordinates. The action of theEuclidean metric was therefore well defined. One could regard this as aquantum version of Cosmic Censorship: the break down of the structure ata singularity should not affect any physical measurement.It seems, therefore, that the path integral for quantum gravity should betaken over non–singular Euclidean metrics. But what should the boundaryconditions be on these metrics. There are two, and only two, nat-


976 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionural choices. The first is metrics that approach the flat Euclidean metricoutside a compact set. The second possibility is metrics on manifoldsthat are compact and without boundary. Therefore, the natural choicesfor path integral for quantum gravity are [Hawking and Israel (1979);Hawking and Penrose (1996)]: (i) asymptotically Euclidean metrics, and(ii) compact metrics without boundary. The first class of asymptoticallyEuclidean metrics is appropriate for scattering calculations. In these onesends particles in from infinity and observes what comes out again to infinity.All measurements are made at infinity where one has a flat backgroundmetric and one can interpret small fluctuations in the fields as particles inthe usual way. One doesn’t ask what happens in the interaction region inthe middle. That is why one does a path integral over all possible historiesfor the interaction region, that is, over all asymptotically Euclidean metrics.However, in cosmology one is interested in measurements that are made ina finite region rather than at infinity. We are on the inside of the universenot looking in from the outside. To see what difference this makes let usfirst suppose that the path integral for cosmology is to be taken over allasymptotically Euclidean metrics.The so–called No Boundary Proposal of Hartle and Hawking reads[Hawking and Israel (1979); Hawking and Penrose (1996)]: The path integralfor quantum gravity should be taken over all compact Euclideanmetrics. One can paraphrase this as: the boundary condition of the universeis that it has no boundary. According to Hawking, this no boundaryproposal seems to account for the universe we live in. That is an isotropicand homogeneous expanding universe with small perturbations. We can observethe spectrum and statistics of these perturbations in the fluctuationsin the microwave background. The results so far agree with the predictionsof the no boundary proposal. It will be a real test of the proposal and thewhole Euclidean quantum gravity program when the observations of themicrowave background are extended to smaller angular scales.In order to use the no boundary proposal to make predictions, it isuseful to introduce a concept that can describe the state of the universe atone time:∫Probability of induced metric h ij on Σ =d[g] exp(−A[g]).metrics on M thatinduce h ij on ΣConsider the probability that the space–time manifold M contains anembedded three dimensional manifold Σ with induced metric h ij . Thisis given by a path integral over all metrics g ab on M that induce h ij


<strong>Applied</strong> Jet <strong>Geom</strong>etry 977on Σ. If M is simply–connected, which we will assume, the surface Σwill divide M into two parts M + and M − [Hawking and Israel (1979);Hawking and Penrose (1996)],Probability of h ij = Ψ + (h ij ) × Ψ − (h ij ), where∫Ψ + (h ij ) =d[g] exp(−A[g]).metrics on M + thatinduce h ij on ΣIn this case, the probability for Σ to have the metric h ij can be factorized.It is the product of two wave functions Ψ + and Ψ − . These are given bypath integrals over all metrics on M + and M − respectively, that inducethe given three metric h ij on Σ. In most cases, the two wave functions willbe equal and we will drop the superscripts + and +. Ψ is called the wavefunction of the universe. If there are matter fields φ, the wave functionwill also depend on their values φ 0 on Σ. But it will not depend explicitlyon time because there is no preferred time coordinate in a closed universe.The no boundary proposal implies that the wave function of the universe isgiven by a path integral over fields on a compact manifold M + whose onlyboundary is the surface Σ. The path integral is taken over all metrics andmatter fields on M + that agree with the metric h ij and matter fields φ 0 onΣ.One can describe the position of the surface Σ by a function τ of threecoordinates x i on Σ. But the wave function defined by the path integralcannot depend on τ or on the choice of the coordinates x i . This impliesthat the wave function Ψ has to obey four functional differential equations.Three of these equations are called the momentum constraint One candescribe the position of the surface Σ by a function τ of three coordinatesx i on Σ. But the wave function defined by the path integral cannot dependon τ or on the choice of the coordinates x i . This implies that the wavefunction Ψ has to obey four functional differential equations. ( Three of∂Ψthese equations are called the momentum constraint equation:∂h ij);j =0. They express the fact that the wave function should be the same fordifferent 3 metrics h ij that can be get from each other by transformationsof the coordinates x i . The fourth equation is called the Wheeler–DeWittequation∂(G 2)ijkl − h 1 2 3 R Ψ = 0.∂h ij ∂h klIt corresponds to the independence of the wave function on τ. One can


978 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthink of it as the Schrödinger equation for the universe. But there is notime derivative term because the wave function does not depend on timeexplicitly.In order to estimate the wave function of the universe, one can use thesaddle point approximation to the path integral as in the case of black holes.One finds a Euclidean metric g 0 on the manifold M + that satisfies the fieldequations and induces the metric h ij on the boundary Σ. One can thenexpand the action A in a power series around the background metric g 0 ,A[g] = A[g 0 ] + 1 2 δgA 2δg + ...As before, the term linear in the perturbations vanishes. The quadraticterm can be regarded as giving the contribution of gravitons on the backgroundand the higher order terms as interactions between the gravitons.These can be ignored when the radius of curvature of the background islarge compared to the Planck scale. Therefore, according to [Hawking andIsrael (1979); Hawking and Penrose (1996)] we have1Ψ ≈(det A 2 ) 1 2exp(−A[g o ]).Consider now a situation in which there are no matter fields but there is apositive cosmological constant Λ. Let us take the surface Σ to be a threesphere and the metric h ij to be the round three sphere metric of radius a.Then the manifold M + bounded by Σ can be taken to be the four ball. Themetric that satisfies the field equations is part of a four sphere of radius 1 Hwhere H 2 = Λ 3 ,A = 1 ∫(R − 2Λ)(−g) 1 2 d 4 x + 1 ∫16π8πK(±h) 1 2 d 3 x.For a 3–sphere Σ of radius less than 1 Hthere are two possible Euclideansolutions: either M + can be less than a hemisphere or it can be more.However there are arguments that show that one should pick the solutioncorresponding to less than a hemisphere.One can interpret the wave function Ψ as follows. The real time solutionof the Einstein equations with a Λ term and maximal symmetry is de Sitterspace (see, e.g., [Witten (1998b)]). This can be embedded as a hyperboloidin five dimensional Minkowski space. Here, we have two choices:


<strong>Applied</strong> Jet <strong>Geom</strong>etry 979(1) Lorentzian–de Sitter metric,ds 2 = −dt 2 + 1H 2 coshHt(dr2 + sin 2 r(dθ 2 + sin 2 θdφ 2 )).One can think of it as a closed universe that shrinks down from infinitesize to a minimum radius and then expands again exponentially. Themetric can be written in the form of a Friedmann universe with scalefactor coshHt. Putting τ = it converts the cosh into cos giving theEuclidean metric on a four sphere of radius 1 H .(2) Euclidean metric,ds 2 = dτ 2 + 1H 2 cos Hτ(dr2 + sin 2 r(dθ 2 + sin 2 θdφ 2 )).Thus one gets the idea that a wave function which varies exponentiallywith the three metric h ij corresponds to an imaginary time Euclideanmetric. On the other hand, a wave function which oscillates rapidlycorresponds to a real time Lorentzian metric.Hawking says: “The Euclidean path integral over all topologically trivialmetrics can be done by time slicing and so is unitary when analyticallycontinued to the Lorentzian. On the other hand, the path integral over alltopologically non–trivial metrics is asymptotically independent of the initialstate. Thus the total path integral is unitary and information is not lostin the formation and evaporation of black holes. The way the informationgets out seems to be that a true event horizon never forms, just an apparenthorizon.”Like in the case of the pair creation of black holes, one can describe thespontaneous creation of an exponentially expanding universe. One joins thelower half of the Euclidean four sphere to the upper half of the Lorentzianhyperboloid.Unlike the black hole pair creation, one couldn’t say that the de Sitteruniverse was created out of field energy in a pre–existing space. Instead, itwould quite literally be created out of nothing: not just out of the vacuumbut out of absolutely nothing at all because there is nothing outside theuniverse. In the Euclidean regime, the de Sitter universe is just a closedspace like the surface of the Earth but with two more dimensions ([Witten(1998b)]). If the cosmological constant is small compared to the Planckvalue, the curvature of the Euclidean four sphere should be small. This willmean that the saddle point approximation to the path integral should be


980 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongood, and that the calculation of the wave function of the universe will notbe affected by our ignorance of what happens in very high curvatures.One can also solve the field equations for boundary metrics that aren’texactly the round three sphere metric. If the radius of the three sphere isless than 1 H, the solution is a real Euclidean metric. The action will be realand the wave function will be exponentially damped compared to the roundthree sphere of the same volume. If the radius of the three sphere is greaterthan this critical radius there will be two complex conjugate solutions andthe wave function will oscillate rapidly with small changes in h ij .Any measurement made in cosmology can be formulated in terms ofthe wave function. Thus the no boundary proposal makes cosmology intoa science because one can predict the result of any observation. The casewe have just been considering of no matter fields and just a cosmologicalconstant does not correspond to the universe we live in. Nevertheless, itis a useful example, both because it is a simple model that can be solvedfairly explicitly and because, as we shall see, it seems to correspond to theearly stages of the universe.Although it is not obvious from the wave function, a de Sitter universehas thermal properties rather like a black hole. One can see this by writingthe de Sitter metric in a static form (rather like the Schwarzschild solution)ds 2 = −(1 − H 2 r 2 )dt 2 + (1 − H 2 r 2 ) −1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ).There is an apparent singularity at r = 1 H. However, as in the Schwarzschildsolution, one can remove it by a coordinate transformation and it correspondsto an event horizon.If one returns to the static form of the de Sitter metric and put τ = itone gets a Euclidean metric. There is an apparent singularity on thehorizon. However, by defining a new radial coordinate and identifyingτ with period 2π H, one gets a regular Euclidean metric which is just thefour sphere. Because the imaginary time coordinate is periodic, de Sitterspace and all quantum fields in it will behave as if they were at atemperature H 2π . As we shall see, we can observe the consequences ofthis temperature in the fluctuations in the microwave background. Onecan also apply arguments similar to the black hole case to the actionof the Euclidean–de Sitter solution [Witten (1998b)]. One finds that itπhas an intrinsic entropy ofH, which is a quarter of the area of the2event horizon. Again this entropy arises for a topological reason: theEuler number of the four sphere is two. This means that there cannotbe a global time coordinate on Euclidean–de Sitter space. One can in-


<strong>Applied</strong> Jet <strong>Geom</strong>etry 981terpret this cosmological entropy as reflecting an observers lack of knowledgeof the universe beyond his event horizon [Hawking and Israel (1979);Hawking and Penrose (1996)]:{Euclidean metric periodic with period 2π Temperature T = H ,H ⇒ 2πArea A of event horizon = 4π ,H 2Entropy S = π .H 2


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Chapter 6<strong>Geom</strong>etrical Path Integrals and TheirApplicationsThe machinery of applied differential geometry, as presented so far, is: (i)rigorous, (ii) elegant, and (iii) powerful – as a tool for understanding, predictionand control of complex nonlinear systems. However, due to itssmooth nature, it is limited to modelling of deterministic and continuous–time dynamical systems only. Naturally, the question arises: is it possibleto extend this smooth machinery, so to be able to effectively deal alsowith probabilistic and discrete–time dynamical systems, like e.g., Markovchains? And the answer is: Yes. Namely, in the very core of the XX Centurygeometrodynamics, there is a powerful conceptual and computationaltool that is ‘by default’ used as a starting point for virtually every new physicaltheory – the celebrated Feynman path integral. In the path–integralformalism, we first formulate the specific classical action of a new theory,and subsequently perform its quantization by means of the associatedamplitude. This action–amplitude picture is the core structure in any newphysical theory. Unlike mathematical manifolds, bundles and jets, the pathintegral is an invention of the physical mind of Richard (Dick) Feynman.Its virtual paths are in general neither deterministic not smooth, althoughthey include bundles and jets of deterministic and smooth paths, as well asMarkov chains. Yet, it is essentially an applied differential geometry, withits Riemannian and symplectic versions, among many others. At the beginning,it worked only for conservative physical systems. Today it includesalso dissipative structures, as well as various sources and sinks. Its smoothpart reveals all celebrated equations of the 20th Century, both classicaland quantum. It is the core of modern quantum gravity and string theory.It is arguably the most important construct of mathematical physics. Atthe edge of a new millennium, if you asked a typical theoretical physicist:what will be your main research tool in the new millennium, he/she would983


984 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmost probably say: path integral. And today, we see it moving out fromphysics, into the realm of social sciences. Finally, since Feynman’s fairlyintuitive invention of the path integral [Feynman (1951)], a lot of researchhas been done to make it mathematically rigorous (see e.g., [Loo (1999);Loo (2000); Albeverio et. al. (1986); Klauder (1997); Klauder (2000);Shabanov and Klauder (1998)]).6.1 Intuition Behind a Path Integral6.1.1 Classical Probability ConceptRecall that a random variable X is defined by its distribution function f(x).Its probabilistic description is based on the following rules: (i) P (X = x i )is the probability that X = x i ; and (ii) P (a ≤ X ≤ b) is the probabilitythat X lies in a closed interval [a, b]. Its statistical description is basedon: (i) µ X or E(X) is the mean or expectation of X; and (ii) σ X is thestandard deviation of X. There are two cases of random variables: discreteand continuous, each having its own probability (and statistics) theory.6.1.2 Discrete Random VariableA discrete random variable X has only a countable number of values {x i }.Its distribution function f(x i ) has the following properties:∑P (X = x i ) = f(x i ), f(x i ) ≥ 0, f(x i ) dx = 1.Statistical description of X is based on its discrete mean value µ X andstandard deviation σ X , given respectively byµ X = E(X) = ∑ √x i f(x i ), σ X = E(X 2 ) − µ 2 X .ii6.1.3 Continuous Random VariableHere f(x) is a piecewise continuous function such that:P (a ≤ X ≤ b) =∫ bf(x) dx, f(x) ≥ 0,∫ ∞∫f(x) dx = f(x) dx = 1.a−∞RStatistical description of X is based on its continuous mean µ X and


<strong>Geom</strong>etrical Path Integrals and Their Applications 985standard deviation σ X , given respectively byµ X = E(X) =∫ ∞−∞xf(x) dx, σ X =√E(X 2 ) − µ 2 X .Now, let us observe the similarity between the two descriptions. Thesame kind of similarity between discrete and continuous quantum spectrumstroke Dirac∫when he suggested the combined integral approach, that hedenoted by Σ – meaning ‘both integral and sum at once’: summing overdiscrete spectrum and integration over continuous spectrum.To emphasize this similarity even further, as well as to set–up the stagefor the path integral, recall the notion of a cumulative distribution functionof a random variable X, that is a function F : R −→ R, defined byF (a) = P (X) ≤ a.In particular, suppose that f(x) is the distribution function of X. ThenF (x) = ∑ x i≤xf(x i ), or F (x) =∫ ∞−∞f(t) dt,according to as x is a discrete or continuous random variable. In eithercase, F (a) ≤ F (b) whenever a ≤ b. Also,lim F (x) = 0 and lim F (x) = 1,x−→−∞ x−→∞that is, F (x) is monotonic and its limit to the left is 0 and the limit to theright is 1. Furthermore, its cumulative probability is given byP (a ≤ X ≤ b) = F (b) − F (a),and the Fundamental Theorem of Calculus tells us that, in the continuumcase,f(x) = ∂ x F (x).6.1.4 General Markov Stochastic DynamicsRecall that Markov stochastic process is a random process characterized bya lack of memory, i.e., the statistical properties of the immediate futureare uniquely determined by the present, regardless of the past [Gardiner(1985)].For example, a random walk is an example of the Markov chain, i.e.,a discrete–time Markov process, such that the motion of the system in


986 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionconsideration is viewed as a sequence of states, in which the transition fromone state to another depends only on the preceding one, or the probabilityof the system being in state k depends only on the previous state k − 1.The property of a Markov chain of prime importance in biodynamics isthe existence of an invariant distribution of states: we start with an initialstate x 0 whose absolute probability is 1. Ultimately the states should bedistributed according to a specified distribution.Between the pure deterministic dynamics, in which all DOF of the systemin consideration are explicitly taken into account, leading to classicaldynamical equations, for example in Hamiltonian form (3.34), i.e.,˙q i = ∂ pi H,ṗ i = −∂ q iH– and pure stochastic dynamics (Markov process), there is so–called hybriddynamics, particularly Brownian dynamics, in which some of DOF are representedonly through their stochastic influence on others. As an example,suppose a system of particles interacts with a viscous medium. Insteadof specifying a detailed interaction of each particle with the particles ofthe viscous medium, we represent the medium as a stochastic force actingon the particle. The stochastic force reduces the dimensionally of thedynamics.Recall that the Brownian dynamics represents the phase–space trajectoriesof a collection of particles that individually obey Langevin rate equationsin the field of force (i.e., the particles interact with each other viasome deterministic force). For a free particle, the Langevin equation reads[Gardiner (1985)]:m ˙v = R(t) − βv,where m denotes the mass of the particle and v its velocity. The right–handside represent the coupling to a heat bath; the effect of the random forceR(t) is to heat the particle. To balance overheating (on the average), theparticle is subjected to friction β. In humanoid dynamics this is performedwith the Rayleigh–Van der Pol’s dissipation. Formally, the solution to theLangevin equation can be written asv(t) = v(0) exp(− β )m t + 1 ∫ texp[−(t − τ)β/m] R(τ) dτ,mwhere the integral on the right–hand side is a stochastic integral and thesolution v(t) is a random variable. The stochastic properties of the solutiondepend significantly on the stochastic properties of the random force0


<strong>Geom</strong>etrical Path Integrals and Their Applications 987R(t). In the Brownian dynamics the random force R(t) is Gaussian distributed.Then the problem boils down to finding the solution to theLangevin stochastic differential equation with the supplementary condition(mean zero and variance)< R(t) > = 0, < R(t) R(0) > = 2βk B T δ(t),where < . > denotes the mean value, T is temperature, k B −equipartition(i.e., uniform distribution of energy) coefficient, Dirac δ(t)−function.Algorithm for computer simulation of the Brownian dynamics (for asingle particle) can be written as [Heermann (1990)]:(1) Assign an initial position and velocity.(2) Draw a random number from a Gaussian distribution with mean zeroand variance.(3) Integrate the velocity to get v n+1 .(4) Add the random component to the velocity.Another approach to taking account the coupling of the system to a heatbath is to subject the particles to collisions with virtual particles [Heermann(1990)]. Such collisions are imagined to affect only momenta of the particles,hence they affect the kinetic energy and introduce fluctuations in the totalenergy. Each stochastic collision is assumed to be an instantaneous eventaffecting only one particle.The collision–coupling idea is incorporated into the Hamiltonian modelof dynamics (3.34) by adding a stochastic force R i = R i (t) to the ṗ equation˙q i = ∂ pi H,ṗ i = −∂ q iH + R i (t).On the other hand, the so–called Ito stochastic integral represents akind of classical Riemann–Stieltjes integral from linear functional analysis,which is (in 1D case) for an arbitrary time–function G(t) defined as themean square limit∫ tG(t)dW (t) = ms lim { ∑n G(t i−1 [W (t i ) − W (t i−1 ]}.tn→∞ 0i=1Now, the general ND Markov process can be defined by Ito stochasticdifferential equation (SDE),dx i (t) = A i [x i (t), t]dt + B ij [x i (t), t] dW j (t),x i (0) = x i0 , (i, j = 1, . . . , N)


988 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionor corresponding Ito stochastic integral equationx i (t) = x i (0) +∫ t0ds A i [x i (s), s] +∫ t0dW j (s) B ij [x i (s), s],in which x i (t) is the variable of interest, the vector A i [x(t), t] denotes deterministicdrift, the matrix B ij [x(t), t] represents continuous stochastic diffusionfluctuations, and W j (t) is an N−variable Wiener process (i.e., generalizedBrownian motion) [Wiener (1961)], and dW j (t) = W j (t + dt) − W j (t).Now, there are three well–known special cases of the Chapman–Kolmogorov equation (see [Gardiner (1985)]):(1) When both B ij [x(t), t] and W (t) are zero, i.e., in the case of puredeterministic motion, it reduces to the Liouville equation∂ t P (x ′ , t ′ |x ′′ , t ′′ ) = − ∑ i∂∂x i {A i[x(t), t] P (x ′ , t ′ |x ′′ , t ′′ )} .(2) When only W (t) is zero, it reduces to the Fokker–Planck equation∂ t P (x ′ , t ′ |x ′′ , t ′′ ) = − ∑ i+ 1 ∑2ij∂∂x i {A i[x(t), t] P (x ′ , t ′ |x ′′ , t ′′ )}∂ 2∂x i ∂x j {B ij[x(t), t] P (x ′ , t ′ |x ′′ , t ′′ )} .(3) When both A i [x(t), t] and B ij [x(t), t] are zero, i.e., the state–space consistsof integers only, it reduces to the Master equation of discontinuousjumps∫∂ t P (x ′ , t ′ |x ′′ , t ′′ ) =dx {W (x ′ |x ′′ , t) P (x ′ , t ′ |x ′′ , t ′′ ) − W (x ′′ |x ′ , t) P (x ′ , t ′ |x ′′ , t ′′ )} .The Markov assumption can now be formulated in terms of the conditionalprobabilities P (x i , t i ): if the times t i increase from right to left,the conditional probability is determined entirely by the knowledge of themost recent condition. Markov process is generated by a set of conditionalprobabilities whose probability–density P = P (x ′ , t ′ |x ′′ , t ′′ ) evolution obeys


<strong>Geom</strong>etrical Path Integrals and Their Applications 989the general Chapman–Kolmogorov integro–differential equation∂ t P = − ∑ i+ 1 ∑2ij∂∂x i {A i[x(t), t] P }∂ 2∂x i ∂x j {B ij[x(t), t] P } +∫dx {W (x ′ |x ′′ , t) P − W (x ′′ |x ′ , t) P }including deterministic drift, diffusion fluctuations and discontinuousjumps (given respectively in the first, second and third terms on the r.h.s.).It is this general Chapman–Kolmogorov integro–differential equation,with its conditional probability density evolution, P = P (x ′ , t ′ |x ′′ , t ′′ ), thatwe are going to model by various forms of the Feynman path integral,providing us with the physical insight behind the abstract (conditional)probability densities.6.1.5 Quantum Probability ConceptAn alternative concept of probability, the so–called quantum probability, isbased on the following physical facts (elaborated in detail in this section):(1) The time–dependent Schrödinger equation represents a complex–valuedgeneralization of the real–valued Fokker–Planck equation for describingthe spatio–temporal probability density function for the system exhibitingcontinuous–time Markov ∫stochastic process.(2) The Feynman path integral Σ is a generalization of the time–dependentSchrödinger equation, including both continuous–time and discrete–time Markov stochastic processes.(3) Both Schrödinger equation and path integral give ‘physical description’of any system they are modelling in terms of its physical energy, insteadof an abstract probabilistic description of the Fokker–Planck equation.∫Therefore, the Feynman path integral Σ, as a generalization of the time–dependent Schrödinger equation, gives a unique physical description forthe general Markov stochastic process, in terms of the physically basedgeneralized probability density functions, valid both for continuous–timeand discrete–time Markov systems.Basic consequence: a different way for calculating probabilities. Thedifference is rooted in the fact that sum of squares is different from thesquare of sums, as is explained in the following text.


990 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNamely, in Dirac–Feynman quantum formalism, each possible routefrom the initial system state A to the final system state B is called ahistory. This history comprises any kind of a route (see Figure 6.1), rangingfrom continuous and smooth deterministic (mechanical–like) paths tocompletely discontinues and random Markov chains (see, e.g., [Gardiner(1985)]). Each history (labelled by index i) is quantitatively described bya complex number 1 z i called the ‘individual transition amplitude’. Its absolutesquare, |z i | 2 , is called the individual transition probability. Now,the total transition amplitude is the sum of all individual transition amplitudes,∑ i z i, called the sum–over–histories. The absolute square of thissum–over–histories, | ∑ i z i| 2 , is the total transition probability.In this way, the overall probability of the system’s transition from someinitial state A to some final state B is given not by adding up the probabilitiesfor each history–route, but by ‘head–to–tail’ adding up the sequenceof amplitudes making–up each route first (i.e., performing the sum–over–histories) – to get the total amplitude as a ‘resultant vector’, and thensquaring the total amplitude to get the overall transition probability.Fig. 6.1 Two ways of physical transition from an initial state A to the correspondingfinal state B. (a) Classical physics proposes a single deterministictrajectory, minimizing the total system’s energy. (b) Quantum physics proposesa family of Markov stochastic histories, namely all possible routes from A toB, both continuous–time and discrete–time Markov chains, each giving an equalcontribution to the total transition probability.1 Recall that a complex number z = x + iy, where i = √ −1 is the imaginary unit, xis the real part and y is the imaginary part, can be represented also in its polar form,z = r(cos θ + i sin θ), where the radius vector in the complex plane, r = |z| = p x 2 + y 2 ,is the modulus or amplitude, and angle θ is the phase; as well as in its exponential formz = re iθ . In this way, complex numbers actually represent 2D vectors with usual vector‘head–to–tail’ addition rule.


<strong>Geom</strong>etrical Path Integrals and Their Applications 9916.1.6 Quantum Coherent StatesRecall that a quantum coherent state is a specific kind of quantum stateof the quantum harmonic oscillator whose dynamics most closely resemblethe oscillating behavior of a classical harmonic oscillator. It was the firstexample of quantum dynamics when Erwin Schrödinger derived it in 1926while searching for solutions of the Schrödinger equation that satisfy thecorrespondence principle. The quantum harmonic oscillator and hence, thecoherent state, arise in the quantum theory of a wide range of physicalsystems. For instance, a coherent state describes the oscillating motion ofthe particle in a quadratic potential well. In the quantum electrodynamicsand other bosonic quantum field theories they were introduced by the2005 Nobel Prize winning work of Roy Glauber in 1963 [Glauber (1963a);Glauber (1963b)]. Here the coherent state of a field describes an oscillatingfield, the closest quantum state to a classical sinusoidal wave such as acontinuous laser wave.In classical optics, light is thought of as electromagnetic waves radiatingfrom a source. Specifically, coherent light is thought of as light that isemitted by many such sources that are in phase. For instance, a lightbulb radiates light that is the result of waves being emitted at all thepoints along the filament. Such light is incoherent because the process ishighly random in space and time. On the other hand, in a laser, light isemitted by a carefully controlled system in processes that are not randombut interconnected by stimulation and the resulting light is highly ordered,or coherent. Therefore a coherent state corresponds closely to the quantumstate of light emitted by an ideal laser. Semi–classically we describe sucha state by an electric field oscillating as a stable wave. Contrary to thecoherent state, which is the most wave–like quantum state, the Fock state(e.g., a single photon) is the most particle–like state. It is indivisible andcontains only one quanta of energy. These two states are examples of theopposite extremes in the concept of wave–particle duality. A coherent statedistributes its quantum–mechanical uncertainty equally, which means thatthe phase and amplitude uncertainty are approximately equal. Conversely,in a single–particle state the phase is completely uncertain.Formally, the coherent state |α〉 is defined to be the eigenstate of theannihilation operator a, i.e., a|α〉 = α|α〉. Note that since a is not Hermitian,α = |α|e iθ is complex. |α| and θ are called the amplitude and phaseof the state.Physically, a|α〉 = α|α〉 means that a coherent state is left unchanged


992 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionby the detection (or annihilation) of a particle. Consequently, in a coherentstate, one has exactly the same probability to detect a second particle.Note, this condition is necessary for the coherent state’s Poisson detectionstatistics. Compare this to a single–particle’s Fock state: Once one particleis detected, we have zero probability of detecting another.Now, recall that a Bose–Einstein condensate (BEC) is a collection ofboson atoms that are all in the same quantum state. An approximatetheoretical description of its properties can be derived by assuming theBEC is in a coherent state. However, unlike photons, atoms interact witheach other so it now appears that it is more likely to be one of the squeezedcoherent states (see [Breitenbach et. al. (1997)]). In quantum field theoryand string theory, a generalization of coherent states to the case of infinitelymany degrees of freedom is used to define a vacuum state with a differentvacuum expectation value from the original vacuum.6.1.7 Dirac’s < bra | ket > Transition AmplitudeNow, we are ready to move–on into the realm of quantum mechanics. Recallthat [Dirac (1982)] described behavior of quantum systems in terms ofcomplex–valued ket–vectors |A > living in the Hilbert space H, and theirduals, bra–covectors < B| (i.e., 1–forms) living in the dual Hilbert spaceH ∗ . 2 The Hermitian inner product of kets and bras, the bra–ket < B|A >,is a complex number, which is the evaluation of the ket |A > by the bra< B|. This complex number, say re iθ represents the system’s transition2 Recall that a norm on a complex vector space H is a mapping from H into thecomplex numbers, ‖·‖ : H → C; h ↦→ ‖h‖, such that the following set of norm–axiomshold:(N1) ‖h‖ ≥ 0 for all h ∈ H and ‖h‖ = 0 implies h = 0 (positive definiteness);(N2) ‖λ h‖ = |λ| ‖h‖ for all h ∈ H and λ ∈ C (homogeneity); and(N3) ‖h 1 + h 2 ‖ ≤ ‖h 1 ‖ + ‖h 2 ‖ for all h 1 , h 2 ∈ H (triangle inequality). The pair(H, ‖·‖) is called a normed space.A Hermitian inner product on a complex vector space H is a mapping 〈·, ·〉 : H×H → Csuch that the following set of inner–product–axioms hold:(IP1) 〈h h 1 + h 2 〉 = 〈h h 1 + h h 2 〉 ;(IP2) 〈α h, h 1 〉 = α 〈 h, h 1 〉 ;(IP3) 〈h 1 , h 2 〉 = 〈h 1 , h 2 〉 (so 〈h, h〉 is real);(IP4) 〈h, h〉 ≥ 0 and 〈h, h〉 = 0 provided h = 0.The standard inner product on the product space C n = C × · · · × C is defined by〈z, w〉 = P ni=1 z iw i , and axioms (IP1)–(IP4) are readily checked. Also C n is a normedspace with ‖z‖ 2 = P ni=1 |z i| 2 . The pair (H, 〈·, ·〉) is called an inner product space.Let (H, ‖·‖) be a normed space. If the corresponding metric d is complete, we say(H, ‖·‖) is a Banach space. If (H, ‖·‖) is an inner product space whose correspondingmetric is complete, we say (H, ‖·‖) is a Hilbert space.


<strong>Geom</strong>etrical Path Integrals and Their Applications 993amplitude 3 from its initial state A to its final state B 4 , i.e.,T ransition Amplitude =< B|A >= re iθ .That is, there is a process that can mediate a transition of a system frominitial state A to the final state B and the amplitude for this transitionequals < B|A >= re iθ . The absolute square of the amplitude, | < B|A > | 2represents the transition probability. Therefore, the probability of a transitionevent equals the absolute square of a complex number, i.e.,T ransition P robability = | < B|A > | 2 = |re iθ | 2 .These complex amplitudes obey the usual laws of probability: when atransition event can happen in alternative ways then we add the complexnumbers,< B 1 |A 1 > + < B 2 |A 2 >= r 1 e iθ1 + r 2 e iθ2 ,and when it can happen only as a succession of intermediate steps then wemultiply the complex numbers,< B|A >=< B|c >< c|A >= (r 1 e iθ1 )(r 2 e iθ2 ) = r 1 r 2 e i(θ1+θ2) .In general,(1) The amplitude for n mutually alternative processes equals the sum∑ nk=1 r ke iθ kof the amplitudes for the alternatives; and(2) If transition from A to B occurs in a sequence of m steps, then the totaltransition amplitude equals the product ∏ mj=1 r je iθj of the amplitudesof the steps.Formally, we have the so–called expansion principle, including both3 Transition amplitude is otherwise called probability amplitude, or just amplitude.4 Recall that in quantum mechanics, complex numbers are regarded as the vacuum–state, or the ground–state, and the entire amplitude < b|a > is a vacuum–to–vacuumamplitude for a process that includes the creation of the state a, its transition to b, andthe annihilation of b to the vacuum once more.


994 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionproducts and sums, 5 < B|A >=n∑< B|c i >< c i |A > . (6.1)i=16.1.8 Feynman’s Sum–over–HistoriesNow, iterating the Dirac’s expansion principle (6.1) over a complete set ofall possible states of the system, leads to the simplest form of the Feynmanpath integral, or, sum–over–histories. Imagine that the initial and finalstates, A and B, are points on the vertical lines x = 0 and x = n + 1,respectively, in the x − y plane, and that (c(k) i(k) , k) is a given point onthe line x = k for 0 < i(k) < m (see Figure 6.2). Suppose that thesum of projectors for each intermediate state is complete 6 Applying thecompleteness iteratively, we get the following expression for the transitionamplitude:< B|A >= ∑ ∑ ... ∑ < B|c(1) i(1) >< c(1) i(1) |c(2) i(2) > ... < c(n) i(n) |A >,where the sum is taken over all i(k) ranging between 1 and m, and k rangingbetween 1 and n. Each term in this sum can be construed as a combinatorialroute from A to B in the 2D space of the x − y plane. Thus the transitionamplitude for the system going from some initial state A to some final stateB is seen as a summation of contributions from all the routes connectingA to B.Feynman used this description to produce his celebrated path integralexpression for a transition amplitude (see, e.g., [Grosche and Steiner(1998)]). His path integral takes the form∫T ransition Amplitude =< B|A >= Σ, D[x] e iS[x] , (6.2)5 In Dirac’s language, the completeness of intermediate states becomes the statementthat a certain sum of projectors is equal to the identity. Namely, suppose that P i |c i >= 1 for each i. Then< b|a >=< b||a >=< b| X i|c i >< c i ||a >= X i< b|c i >< c i |a > .6 We assume that following sum is equal to one, for each k from 1 to n − 1:|c(k) 1 >< c(k) 1 | + ... + |c(k) m >< c(k) m| = 1.


<strong>Geom</strong>etrical Path Integrals and Their Applications 995Fig. 6.2 Analysis of all possible routes from the source A to the detector B is simplifiedto include only double straight lines (in a plane).∫where the sum–integral Σ is taken over all possible routes x = x(t) from theinitial point A = A(t ini ) to the final point B = B(t fin ), and S = S[x] is theclassical action for a particle to travel from A to B along a given extremalpath x. In this way, Feynman took seriously Dirac’s conjecture interpretingthe exponential of the classical action functional (De iS ), resembling acomplex number (re iθ ), as an elementary amplitude. By integrating thiselementary amplitude, De iS , over the infinitude of all possible histories, weget the total system’s transition amplitude. 77 For the quantum physics associated with a classical (Newtonian) particle the actionS is given by the integral along the given route from a to b of the difference T − V whereT is the classical kinetic energy and V is the classical potential energy of the particle.The beauty of Feynman’s approach to quantum physics is that it shows the relationshipbetween the classical and the quantum in a particularly transparent manner. Classicalmotion corresponds to those regions where all nearby routes contribute constructively tothe summation. This classical path occurs when the variation of the action is null. Toask for those paths where the variation of the action is zero is a problem in the calculusof variations, and it leads directly to Newton’s equations of motion (derived using theEuler–Lagrangian equations). Thus with the appropriate choice of action, classical andquantum points of view are unified.Also, a discretization of the Schrodinger equationi dψdt = − 2 d 2 ψ2m dx 2 + V ψ,leads to a sum–over–histories that has a discrete path integral as its solution. Therefore,the transition amplitude is equivalent to the wave ψ. The particle travelling on thex−axis is executing a one–step random walk, see Figure 6.3.


996 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 6.3Random walk (a particular case of Markov chain) on the x−axis.6.1.9 The Basic Form of a Path IntegralIn Feynman’s version of non–relativistic quantum mechanics, the time evolutionψ(x ′ , t ′ ) ↦→ ψ(x ′′ , t ′′ ) of the wave function ψ = ψ(x, t) of the elementary1D particle may be described by the integral equation [Grosche andSteiner (1998)]∫ψ(x ′′ , t ′′ ) = K(x ′′ , x ′ ; t ′′ , t ′ ) ψ(x ′ , t ′ ), (6.3)Rwhere the propagator or Feynman kernel K = K(x ′′ , x ′ ; t ′′ , t ′ ) is definedthrough a limiting procedure,N−1∏∫K(x ′′ , x ′ ; t ′′ , t ′ ) = lim A −Nɛ→0k=1dx k e i P N−1j=0 ɛL(xj+1,(xj+1−xj)/ɛ) . (6.4)The time interval t ′′ − t ′ has been discretized into N steps of length ɛ =(t ′′ − t ′ )/N, and the r.h.s. of (6.4) represents an integral over all piecewiselinear paths x(t) of a ‘virtual’ particle propagating from x ′ to x ′′ , illustratedin Figure 6.4.The prefactor A −N is a normalization and L denotes the Lagrangianfunction of the particle. Knowing the propagator G is tantamount to havingsolved the quantum dynamics. This is the simplest instance of a pathintegral, and is often written schematically as∫K(x ′ , t ′ ; x ′′ , t ′′ ) = Σ, D[x(t)] e iS[x(t)] ,where D[x(t)] is a functional measure on the ‘space of all paths’, and theexponential weight depends on the classical action S[x(t)] of a path. Recallalso that this procedure can be defined in a mathematically clean way if


<strong>Geom</strong>etrical Path Integrals and Their Applications 997A piecewise linear particle path contributing to the discrete Feynman propa-Fig. 6.4gator.we Wick–rotate the time variable t to imaginary values t ↦→ τ = it, therebymaking all integrals real [Reed and Simon (1975)].6.1.10 Application: Adaptive Path IntegralNow, we can extend the Feynman sum–over–histories (6.2), by adding thesynaptic–like weights w i = w i (t) into the measure D[x], to get the adaptivepath integral:Adaptive T ransition Amplitude =< B|A > w =∫Σ, D[w, x] e iS[x] , (6.5)where the adaptive measure D[w, x] is defined by the weighted product (ofdiscrete time steps)n∏D[w, x] = lim w i (t) dx i (t). (6.6)n−→∞In (6.6) the synaptic weights w i = w i (t) are updated by the unsupervisedHebbian–like learning rule [Hebb (1949)]:t=1w i (t + 1) = w i (t) + σ η (wi d(t) − w i a(t)), (6.7)where σ = σ(t), η = η(t) represent local signal and noise amplitudes,respectively, while superscripts d and a denote desired and achieved systemstates, respectively. Theoretically, equations (6.5–6.7) define an∞−dimensional complex–valued neural network. 8 Practically, in a com-8 For details on complex–valued neural networks, see e.g., complex–domain extensionof the standard backpropagation learning algorithm [Georgiou and Koutsougeras (1992);Benvenuto and Piazza (1992)].


998 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionputer simulation we can use 10 7 ≤ n ≤ 10 8 , approaching the number of neuronsin the brain. Such equations are usually solved using Markov–ChainMonte–Carlo methods on parallel (cluster) computers (see, e.g., [Wehnerand Wolfer (1983a); Wehner and Wolfer (1983b)]).6.2 Path Integral History6.2.1 Extract from Feynman’s Nobel LectureIn his Nobel Lecture, December 11, 1965, Richard (Dick) Feynman saidthat he and his PhD supervisor, John Wheeler, had found the action A =A[x; t i , t j ], directly involving the motions of the charges only, 9∫A[x; t i , t j ] = m i (ẋ i µẋ i µ) 1 2 dti + 1 ∫ ∫2 e ie j δ(Iij) 2 ẋ i µ(t i )ẋ j µ(t j ) dt i dt jwith (i ≠ j) (6.8)I 2 ij = [ x i µ(t i ) − x j µ(t j ) ] [ x i µ(t i ) − x j µ(t j ) ] ,where x i µ = x i µ(t i ) is the four–vector position of the ith particle as a functionof the proper time t i , while ẋ i µ(t i ) = dx i µ(t i )/dt i is the velocity four–vector.The first term in the action A[x; t i , t j ] (6.8) is the integral of the propertime t i , the ordinary action of relativistic mechanics of free particles ofmass m i (summation over µ). The second term in the action A[x; t i , t j ](6.8) represents the electrical interaction of the charges. It is summed overeach pair of charges (the factor 1 2is to count each pair once, the term i = jis omitted to avoid self–action). The interaction is a double integral over adelta function of the square of space–time interval I 2 between two points onthe paths. Thus, interaction occurs only when this interval vanishes, thatis, along light cones (see [Wheeler and Feynman (1949)]).Feynman comments here: “The fact that the interaction is exactly one–half advanced and half–retarded meant that we could write such a principleof least action, whereas interaction via retarded waves alone cannot bewritten in such a way. So, all of classical electrodynamics was contained inthis very simple form.”“...The problem is only to make a quantum theory, which has as its classicalanalog, this expression (6.8). Now, there is no unique way to make aquantum theory from classical mechanics, although all the textbooks make9 Wheeler–Feynman Idea [Wheeler and Feynman (1949)] “The energy tensor can beregarded only as a provisional means of representing matter. In reality, matter consistsof electrically charged particles.”


<strong>Geom</strong>etrical Path Integrals and Their Applications 999believe there is. What they would tell you to do, was find the momentumvariables and replace them by (/i)(∂/∂x), but I couldn’t find a momentumvariable, as there wasn’t any.”“The character of quantum mechanics of the day was to write things inthe famous Hamiltonian way (in the form of Schrödinger equation), whichdescribed how the wave function changes from instant to instant, and interms of the Hamiltonian operator H. If the classical physics could bereduced to a Hamiltonian form, everything was all right. Now, least actiondoes not imply a Hamiltonian form if the action is a function of anythingmore than positions and velocities at the same moment. If the action is ofthe form of the integral of the Lagrangian L = L(ẋ, x), a function of thevelocities and positions at the same time t,∫S[x] = L(ẋ, x) dt, (6.9)then you can start with the Lagrangian L and then create a HamiltonianH and work out the quantum mechanics, more or less uniquely. But theaction A[x; t i , t j ] (6.8) involves the key variables, positions (and velocities),at two different times t i and t j and therefore, it was not obvious what todo to make the quantum–mechanical analogue...”So, Feynman was looking for the action integral in quantum mechanics.He says: “...I simply turned to Professor Jehle and said, ‘Listen, do youknow any way of doing quantum mechanics, starting with action – wherethe action integral comes into the quantum mechanics?” ‘No”, he said, ‘butDirac has a paper in which the Lagrangian, at least, comes into quantummechanics.” What Dirac said was the following: There is in quantum mechanicsa very important quantity which carries the wave function fromone time to another, besides the differential equation but equivalent to it,a kind of a kernel, which we might call K(x ′ , x), which carries the wavefunction ψ(x) known at time t, to the wave function ψ(x ′ ) at time t + ε,∫ψ(x ′ , t + ε) = K(x ′ , x) ψ(x, t) dx.Dirac points out that this function K was analogous to the quantity inclassical mechanics that you would calculate if you took the exponential of[iε multiplied by the Lagrangian L(ẋ, x)], imagining that these two positionsx, x ′ corresponded to t and t + ε. In other words,K(x ′ , x) is analogous to e iεL( x′ −xε ,x)/ .


1000 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSo, Feynman continues: “What does he mean, they are analogous; whatdoes that mean, analogous? What is the use of that?” Professor Jehle said,‘You Americans! You always want to find a use for everything!” I said thatI thought that Dirac must mean that they were equal. ‘No”, he explained,‘he doesn’t mean they are equal.” ‘Well”, I said, ‘Let’s see what happens ifwe make them equal.”“So, I simply put them equal, taking the simplest example where theLagrangian isL = 1 2 Mẋ2 − V (x),but soon found I had to put a constant of proportionality N in, suitablyadjusted. When I substituted for K to get∫ [ ]iεψ(x ′ − x, t + ε) = N exp L(x′ , x) ψ(x, t) dx (6.10)εand just calculated things out by Taylor series expansion, out came theSchrödinger equation. So, I turned to Professor Jehle, not really understanding,and said, ‘Well, you see, Dirac meant that they were proportional.”Professor Jehle’s eyes were bugging out – he had taken out a littlenotebook and was rapidly copying it down from the blackboard, and said,‘No, no, this is an important discovery. You Americans are always trying tofind out how something can be used. That’s a good way to discover things!”So, I thought I was finding out what Dirac meant, but, as a matter of fact,had made the discovery that what Dirac thought was analogous, was, infact, equal. I had then, at least, the connection between the Lagrangian andquantum mechanics, but still with wave functions and infinitesimal times.”“It must have been a day or so later when I was lying in bed thinkingabout these things, that I imagined what would happen if I wanted tocalculate the wave function at a finite interval later. I would put one ofthese factors e iεL in here, and that would give us the wave functions thenext moment, t + ε, and then I could substitute that back into (6.10) toget another factor of e iεL and give us the wave function the next moment,t + 2ε, and so on and so on. In that way I found myself thinking of a largenumber of integrals, one after the other in sequence. In the integrand wasthe product of the exponentials, which was the exponential of the sum ofterms like εL. Now, L is the Lagrangian and ε is like the time interval dt, sothat if you took a sum of such terms, that’s exactly like an integral. That’slike Riemann’s formula for the integral ∫ Ldt, you just take the value at


<strong>Geom</strong>etrical Path Integrals and Their Applications 1001each point and add them together. We are to take the limit as ε → 0.Therefore, the connection between the wave function of one instant andthe wave function of another instant a finite time later could be get by aninfinite number of integrals (because ε goes to zero), of exponential whereS is the action expression (6.9). At last, I had succeeded in representingquantum mechanics directly in terms of the action S[x].”Fully satisfied, Feynman comments: “This led later on to the idea of thetransition amplitude for a path: that for each possible way that the particlecan go from one point to another in space–time, there’s an amplitude. Thatamplitude is e to the power of [i/ times the action S[x] for the path], i.e.,e iS[x]/ . Amplitudes from various paths superpose by addition. This thenis another, a third way, of describing quantum mechanics, which looks quitedifferent from that of Schrödinger or Heisenberg, but which is equivalent tothem.”“...Now immediately after making a few checks on this thing, what wewanted to do, was to substitute the action A[x; t i , t j ] (6.8) for the otherS[x] (6.9). The first trouble was that I could not get the thing to workwith the relativistic case of spin one–half. However, although I could dealwith the matter only non–relativistically, I could deal with the light orthe photon interactions perfectly well by just putting the interaction termsof (6.8) into any action, replacing the mass terms by the non–relativisticLdt = 1 2 Mẋ2 dt,A[x; t i , t j ] = 1 ∑∫m i (ẋ i2µ) 2 dt i + 1 2i∑i,j(i≠j)e i e j∫ ∫δ(I 2 ij) ẋ i µ(t i )ẋ j µ(t j ) dt i dt j .When the action has a delay, as it now had, and involved more than onetime, I had to lose the idea of a wave function. That is, I could no longerdescribe the program as: given the amplitude for all positions at a certaintime to calculate the amplitude at another time. However, that didn’tcause very much trouble. It just meant developing a new idea. Insteadof wave functions we could talk about this: that if a source of a certainkind emits a particle, and a detector is there to receive it, we can give theamplitude that the source will emit and the detector receive, e iA[x;ti,tj]/ .We do this without specifying the exact instant that the source emits orthe exact instant that any detector receives, without trying to specify thestate of anything at any particular time in between, but by just finding theamplitude for the complete experiment. And, then we could discuss howthat amplitude would change if you had a scattering sample in between, as


1002 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionyou rotated and changed angles, and so on, without really having any wavefunctions...It was also possible to discover what the old concepts of energyand momentum would mean with this generalized action. And, so I believedthat I had a quantum theory of classical electrodynamics – or rather of thisnew classical electrodynamics described by the action A[x; t i , t j ] (6.8)...”6.2.2 Lagrangian Path IntegralDirac and Feynman first developed the Lagrangian approach to functionalintegration. To review this approach, we start with the time–dependentSchrödinger equationi ∂ t ψ(x, t) = −∂ x 2ψ(x, t) + V (x) ψ(x, t)appropriate to a particle of mass m moving in a potential V (x), x ∈ R. Asolution to this equation can be written as an integral (see e.g., [Klauder(1997); Klauder (2000)]),∫ψ(x ′′ , t ′′ ) =K(x ′′ , t ′′ ; x ′ , t ′ ) ψ(x ′ , t ′ ) dx ′ ,which represents the wave function ψ(x ′′ , t ′′ ) at time t ′′ as a linear superpositionover the wave function ψ(x ′ , t ′ ) at the initial time t ′ , t ′ < t ′′ . Theintegral kernel K(x ′′ , t ′′ ; x ′ , t ′ ) is known as the propagator, and accordingto Feynman [Feynman (1948)] it may be given by∫K(x ′′ , t ′′ ; x ′ , t ′ ) = ND[x] e (i/) ∫[(m/2) ẋ 2 (t)−V (x(t))] dt ,which is a formal expression symbolizing an integral over a suitable setof paths. This integral is supposed to run over all continuous paths x(t),t ′ ≤ t ≤ t ′′ , where x(t ′′ ) = x ′′ and x(t ′ ) = x ′ are fixed end points for allpaths. Note that the integrand involves the classical Lagrangian for thesystem.To overcome the convergence problems, Feynman adopted a lattice regularizationas a procedure to yield well–defined integrals which was thenfollowed by a limit as the lattice spacing goes to zero called the continuumlimit. With ε > 0 denoting the lattice spacing, the details regarding the


<strong>Geom</strong>etrical Path Integrals and Their Applications 1003lattice regularization procedure are given by∫K(x ′′ , t ′′ ; x ′ , t ′ ) = lim (m/2πiε) (N+1)/2 · · ·ε→0∫N∑· · · exp{(i/) [(m/2ε)(x l+1 − x l ) 2 − ε V (x l ) ]}l=0N∏dx l ,where x N+1 = x ′′ , x 0 = x ′ , and ε ≡ (t ′′ − t ′ )/(N + 1), N ∈ {1, 2, 3, . . . }. Inthis version, at least, we have an expression that has a reasonable chance ofbeing well defined, provided, that one interprets the conditionally convergentintegrals involved in an appropriate manner. One common and fullyacceptable interpretation adds a convergence factor to the exponent of thepreceding integral in the form −(ε 2 /2) ∑ Nl=1 x2 l, which is a term thatformally makes no contribution to the final result in the continuum limitsave for ensuring that the integrals involved are now rendered absolutelyconvergent.l=16.2.3 Hamiltonian Path IntegralIt is necessary to retrace history at this point to recall the introductionof the phase–space path integral by Feynman [Feynman (1951); Groscheand Steiner (1998)]. In Appendix B to this article, Feynman introduceda formal expression for the configuration or q−space propagator given by(see e.g., [Klauder (1997); Klauder (2000)])∫K(q ′′ , t ′′ ; q ′ , t ′ ) = M D[p] D[q] exp{(i/) ∫ [ p ˙q − H(p, q) ] dt}.In this equation one is instructed to integrate over all paths q(t), t ′ ≤ t ≤ t ′′ ,with q(t ′′ ) ≡ q ′′ and q(t ′ ) ≡ q ′ held fixed, as well as to integrate over allpaths p(t), t ′ ≤ t ≤ t ′′ , without restriction.It is widely appreciated that the phase–space path integral is more generallyapplicable than the original, Lagrangian, version of the path integral.For example, the original configuration space path integral is satisfactoryfor Lagrangians of the general formL(x) = 1 2 mẋ2 + A(x) ẋ − V (x) ,but it is unsuitable, for example, for the case of a relativistic particle withthe LagrangianL(x) = −m qrt1 − ẋ 2


1004 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionexpressed in units where the speed of light is unity. For such a system – aswell as many more general expressions – the phase–space form of the pathintegral is to be preferred. In particular, for the relativistic free particle,the phase–space path integral∫MD[p] D[q] exp{(i/) ∫ [ p ˙q − qrtp 2 + m 2 ] dt},is readily evaluated and induces the correct propagator.6.2.4 Feynman–Kac FormulaThrough his own research, M. Kac was fully aware of Wiener’s theory ofBrownian motion and the associated diffusion equation that describes thecorresponding distribution function. Therefore, it is not surprising that hewas well prepared to give a path integral expression in the sense of Feynmanfor an equation similar to the time–dependent Schrödinger equation savefor a rotation of the time variable by -π/2 in the complex plane, namely,by the change t −→ −it (see e.g., [Klauder (1997); Klauder (2000)]). Inparticular, Kac [Kac (1951)] considered the equation∂ t ρ(x, t) = ∂ x 2ρ(x, t) − V (x) ρ(x, t). (6.11)This equation is analogous to Schrödinger equation but differs from it in certaindetails. Besides certain constants which are different, and the changet −→ −it, the nature of the dependent variable function ρ(x, t) is quite differentfrom the normal quantum mechanical wave function. For one thing,if the function ρ is initially real it will remain real as time proceeds. Lessobvious is the fact that if ρ(x, t) ≥ 0 for all x at some time t, then thefunction will continue to be nonnegative for all time t. Thus we can interpretρ(x, t) more like a probability density; in fact in the special casethat V (x) = 0, then ρ(x, t) is the probability density for a Brownian particlewhich underlies the Wiener measure. In this regard, ν is called thediffusion constant.The fundamental solution of (6.11) with V (x) = 0 is readily given asW (x, T ; y, 0) =(1qrt2πνT exp −)(x − y)2,2νTwhich describes the solution to the diffusion equation subject to the initial


<strong>Geom</strong>etrical Path Integrals and Their Applications 1005conditionlim W (x, T ; y, 0) = δ(x − y) .T →0 +Moreover, it follows that the solution of the diffusion equation for a generalinitial condition is given by∫ρ(x ′′ , t ′′ ) = W (x ′′ , t ′′ ; x ′ , t ′ ) ρ(x ′ , t ′ ) dx ′ .Iteration of this equation N times, with ɛ = (t ′′ − t ′ )/(N + 1), leads to theequationρ(x ′′ , t ′′ ) = N ′ ∫∫· · ·e −(1/2νɛ) P N∏Nl=0 (x l+1−x l ) 2l=1dx l ρ(x ′ , t ′ ) dx ′ ,where x N+1 ≡ x ′′ and x 0 ≡ x ′ . This equation features the imaginary timepropagator for a free particle of unit mass as given formally as∫∫W (x ′′ , t ′′ ; x ′ , t ′ ) = N D[x] e −(1/2ν) ẋ 2 dt ,where N denotes a formal normalization factor.The similarity of this expression with the Feynman path integral [forV (x) = 0] is clear, but there is a profound difference between these equations.In the former (Feynman) case the underlying measure is only finitelyadditive, while in the latter (Wiener) case the continuum limit actually definesa genuine measure, i.e., a countably additive measure on paths, whichis a version of the famous Wiener measure. In particular,∫W (x ′′ , t ′′ ; x ′ , t ′ ) = dµ ν W (x),where µ ν W denotes a measure on continuous paths x(t), t′ ≤ t ≤ t ′′ , forwhich x(t ′′ ) ≡ x ′′ and x(t ′ ) ≡ x ′ . Such a measure is said to be a pinnedWiener measure, since it specifies its path values at two time points, i.e.,at t = t ′ and at t = t ′′ > t ′ .We note that Brownian motion paths have the property that with probabilityone they are concentrated on continuous paths. However, it is alsotrue that the time derivative of a Brownian path is almost nowhere defined,which means that, with probability one, ẋ(t) = ±∞ for all t.


1006 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWhen the potential V (x) ≠ 0 the propagator associated with (6.11) isformally given by∫∫ ∫W (x ′′ , t ′′ ; x ′ , t ′ ) = N D[x]e −(1/2ν) ẋ 2 dt− V (x) dt ,an expression which is well defined if V (x) ≥ c, -∞ < c < ∞. A mathematicallyimproved expression makes use of the Wiener measure and reads∫ ∫W (x ′′ , t ′′ ; x ′ , t ′ ) = e − V (x(t)) dt dµ ν W (x).This is an elegant relation in that it represents a solution to the differentialequation (6.11) in the form of an integral over Brownian motion pathssuitably weighted by the potential V . Incidentally, since the propagatoris evidently a strictly positive function, it follows that the solution of thedifferential equation (6.11) is nonnegative for all time t provided it is nonnegativefor any particular time value.6.2.5 Itô FormulaItô [Ito (1960)] proposed another version of a continuous–time regularizationthat resolved some of the troublesome issues. In essence, the proposal ofItô takes the form given by∫lim N ν D[x] exp{(i/) ∫ [ 1ν→∞ 2 mẋ2 − V (x)] dt} exp{−(1/2ν) ∫ [ẍ 2 + ẋ 2 ] dt}.Note well the alternative form of the auxiliary factor introduced as a regulator.The additional term ẍ 2 , the square of the second derivative ofx, acts to smooth out the paths sufficiently well so that in the case of(21) both x(t) and ẋ(t) are continuous functions, leaving ẍ(t) as the termwhich does not exist. However, since only x and ẋ appear in the restof the integrand, the indicated path integral can be well defined; thisis already a positive contribution all by itself (see e.g., [Klauder (1997);Klauder (2000)]).6.3 Standard Path–Integral Quantization6.3.1 Canonical versus Path–Integral QuantizationRecall that in the usual, canonical formulation of quantum mechanics, thesystem’s phase–space coordinates, q, and momenta, p, are replaced by the


<strong>Geom</strong>etrical Path Integrals and Their Applications 1007corresponding Hermitian operators in the Hilbert space, with real measurableeigenvalues, which obey Heisenberg commutation relations.The path–integral quantization is instead based directly on the notion ofa propagator K(q f , t f ; q i , t i ) which is defined such that (see [Ryder (1996);Cheng and Li (1984); Gunion (2003)])∫ψ(q f , t f ) = K(q f , t f ; q i , t i ) ψ(q i , t i ) dq i , (6.12)i.e., the wave function ψ(q f , t f ) at final time t f is given by a Huygensprinciple in terms of the wave function ψ(q i , t i ) at an initial time t i , wherewe have to integrate over all the points q i since all can, in principle, sendout little wavelets that would influence the value of the wave function atq f at the later time t f . This equation is very general and is an expressionof causality. We use the normal units with = 1.According to the usual interpretation of quantum mechanics, ψ(q f , t f )is the probability amplitude that the particle is at the point q f and thetime t f , which means that K(q f , t f ; q i , t i ) is the probability amplitude fora transition from q i and t i to q f and t f . The probability that the particleis observed at q f at time t f if it began at q i at time t i isP (q f , t f ; q i , t i ) = |K(q f , t f ; q i , t i )| 2 .Let us now divide the time interval between t i and t f into two, with tas the intermediate time, and q the intermediate point in space. Repeatedapplication of (6.12) gives∫ ∫ψ(q f , t f ) = K(q f , t f ; q, t) dq K(q, t; q i , t i ) ψ(q i , t i ) dq i ,from which it follows that∫K(q f , t f ; q i , t i ) =dq K(q f , t f ; q, t) K(q, t; q i , t i ).This equation says that the transition from (q i , t i ) to (q f , t f ) may be regardedas the result of the transition from (q i , t i ) to all available intermediatepoints q followed by a transition from (q, t) to (q f , t f ). This notionof all possible paths is crucial in the path–integral formulation of quantummechanics.Now, recall that the state vector |ψ, t〉 Sin the Schrödinger picture isrelated to that in the Heisenberg picture |ψ〉 Hby|ψ, t〉 S= e −iHt |ψ〉 H,


1008 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionor, equivalently,We also define the vector|ψ〉 H= e iHt |ψ, t〉 S.|q, t〉 H= e iHt |q〉 S,which is the Heisenberg version of the Schrödinger state |q〉. Then, we canequally well writeψ(q, t) = 〈q, t |ψ〉 H. (6.13)By completeness of states we can now write∫〈q f , t f |ψ〉 H= 〈q f , t f |q i , t i 〉 H〈q i , t i |ψ〉 Hdq i ,which with the definition of (6.13) becomes∫ψ(q f , t f ) = 〈q f , t f |q i , t i 〉 Hψ(q i , t i ) dq i .Comparing with (6.12), we getK(q f , t f ; q i , t i ) = 〈q f , t f |q i , t i 〉 H.Now, let us calculate the quantum–mechanics propagator〈〈q ′ , t ′ |q, t〉 H= q ′ |e −iH(t−t′) |q〉using the path–integral formalism that will incorporate the direct quantizationof the coordinates, without Hilbert space and Hermitian operators.The first step is to divide up the time interval into n + 1 tiny pieces:t l = lε + t with t ′ = (n + 1)ε + t. Then, by completeness, we can write(dropping the Heisenberg picture index H from now on)∫ ∫〈q ′ , t ′ |q, t〉 = dq 1 (t 1 )... dq n (t n ) 〈q ′ , t ′ |q n , t n 〉 ×× 〈q n , t n |q n−1 , t n−1 〉 ... 〈q 1 , t 1 |q, t〉 . (6.14)The integral ∫ dq 1 (t 1 )...dq n (t n ) is an integral over all possible paths, whichare not trajectories in the normal sense, since there is no requirement ofcontinuity, but rather Markov chains.Now, for small ε we can write〈〈q ′ , ε |q, 0〉 = q ′ |e −iεH(P,Q) |q〉 = δ(q ′ − q) − iε 〈q ′ |H(P, Q) |q〉 ,


<strong>Geom</strong>etrical Path Integrals and Their Applications 1009where H(P, Q) is the Hamiltonian (e.g., H(P, Q) = 1 2 P 2 + V (Q), whereP, Q are the momentum and coordinate operators). Then we have (see[Ryder (1996); Cheng and Li (1984); Gunion (2003)])∫(dp〈q ′ |H(P, Q) |q〉 =−q) H p, 1 )2π eip(q′ 2 (q′ + q) .Putting this into our earlier form we get∫ [ dp〈q ′ , ε |q, 0〉 ≃2π exp i{p(q ′ − q) − εH(p, 1 )}]2 (q′ + q) ,where the 0th order in ε → δ(q ′ − q) and the 1st order in ε →−iε 〈q ′ |H(P, Q) |q〉. If we now substitute many such forms into (6.14) wefinally get∫〈q ′ , t ′ |q, t〉 = limn→∞⎧⎨ n+1× exp⎩ i ∑[p j (q j − q j−1 )] − Hj=1n∏i=1n+1∏dq ik=1dp k2π × (6.15)(p j , 1 2 (q j + q j+1 )⎫)⎬(t j − t j−1 )]⎭ ,with q 0 = q and q n+1 = q ′ . Roughly, the above formula says to integrateover all possible momenta and coordinate values associated with a smallinterval, weighted by something that is going to turn into the exponentialof the action e iS in the limit where ε → 0. It should be stressed thatthe different q i and p k integrals are independent, which implies that p k forone interval can be completely different from the p k ′ for some other interval(including the neighboring intervals). In principle, the integral (6.15) shouldbe defined by analytic continuation into the complex plane of, for example,the p k integrals.Now, if we go to the differential limit where we call t j − t j−1 ≡ dτ andwrite (qj−qj−1)(t j−t j−1)≡ ˙q, then the above formula takes the form∫{ ∫ }t′〈q ′ , t ′ |q, t〉 = D[p]D[q] exp i [p ˙q − H(p, q)] dτ ,where we have used the shorthand notation∫∫ ∏ dq(τ)dp(τ)D[p]D[q] ≡.2πτt


1010 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNote that the above integration is an integration over the p and q values atevery time τ. This is what we call a functional integral. We can think of agiven set of choices for all the p(τ) and q(τ) as defining a path in the 6Dphase–space. The most important point of the above result is that we haveget an expression for a quantum–mechanical transition amplitude in termsof an integral involving only pure complex numbers, without operators.We can actually perform the above integral for Hamiltonians of thetype H = H(P, Q). We use square completion in the exponential for this,defining the integral in the complex p plane and continuing to the physicalsituation. In particular, we have∫ ∞−∞{dp2π exp iε(p ˙q − 1 }2 p2 ] = √ 1 [ ] 1exp2πiε 2 iε ˙q2 ,(see [Ryder (1996); Cheng and Li (1984); Gunion (2003)]) which, substitutinginto (6.15) gives∫ ∏〈q ′ , t ′ |q, t〉 = limn→∞in+1dq√ i∑exp{iε2πiεj=1[ 1 2 (q j − q j−1ε) 2 −V ( q j + q j+1)]}.2This can be formally written as∫〈q ′ , t ′ |q, t〉 =D[q] e iS[q] ,where∫∫ ∏D[q] ≡idq i√2πiε,while∫ t′S[q] = L(q, ˙q) dτis the standard action with the LagrangiantL = 1 2 ˙q2 − V (q).


<strong>Geom</strong>etrical Path Integrals and Their Applications 1011Generalization to many degrees of freedom is straightforward:∫{ ∫ [t′ N] }∑〈q ′ 1 ...q ′ N , t ′ |q 1 ...q N , t〉 = D[p]D[q] exp i p n ˙q n − H(p n , q n ) dτ ,with∫∫D[p]D[q] =N∏n=1dq n dp n.2πHere, q n (t) = q n and q n (t ′ ) = q n ′ for all n = 1, ..., N, and we are allowingfor the full Hamiltonian of the system to depend upon all the N momentaand coordinates collectively.tn=16.3.2 Application: Particles, Sources, Fields and Gauges6.3.2.1 Particles(i) Consider first〈q ′ , t ′ |Q(t 0 )|q, t〉∫ ∏= dqi (t i ) 〈q ′ , t ′ |q n , t n 〉 ... 〈q i0 , t i0 |Q(t 0 )|q i−1 , t i−1 〉 ... 〈q 1 , t 1 |q, t〉 ,where we choose one of the time interval ends to coincide with t 0 , i.e.,t i0 = t 0 . If we operate Q(t 0 ) to the left, then it is replaced by its eigenvalueq i0 = q(t 0 ). Aside from this one addition, everything else is evaluated justas before and we will obviously get∫{ ∫ }t′〈q ′ , t ′ |Q(t 0 )|q, t〉 = D[p]D[q] q(t 0 ) exp i [p ˙q − H(p, q)]dτ .(ii) Next, suppose we want a path–integral expression for〈q ′ , t ′ |Q(t 1 )Q(t 2 )|q, t〉 in the case where t 1 > t 2 . For this, we have to insertas intermediate states |q i1 , t i1 〉 〈q i1 , t i1 | with t i1 = t 1 and |q i2 , t i2 〉 〈q i2 , t i2 |with t i2 = t 2 and since we have ordered the times at which we do theinsertions we must have the first insertion to the left of the 2nd insertionwhen t 1 > t 2 . Once these insertions are done, we evaluate 〈q i1 , t i1 | Q(t 1 ) =〈q i1 , t i1 | q(t 1 ) and 〈q i2 , t i2 | Q(t 2 ) = 〈q i2 , t i2 | q(t 2 ) and then proceed as beforeand get∫{ ∫ }t′〈q ′ , t ′ |Q(t 1 )Q(t 2 )|q, t〉 = D[p]D[q] q(t 1 ) q(t 2 ) exp i [p ˙q − H(p, q)]dτ .tt


1012 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow, let us ask what the above integral is equal to if t 2 > t 1 ? It is obviousthat what we get for the above integral is 〈q ′ , t ′ |Q(t 2 )Q(t 1 )|q, t〉 . Clearly,this generalizes to an arbitrary number of Q operators.(iii) When we enter into quantum field theory, the Q’s will be replacedby fields, since it is the fields that play the role of coordinates in the 2ndquantization conditions.6.3.2.2 SourcesThe source is represented by modifying the Lagrangian:L → L + J(t)q(t).Let us define |0, t〉 J as the ground state (vacuum) vector (in the movingframe, i.e., with the e iHt included) in the presence of the source. Therequired transition amplitude isZ[J] ∝ 〈0, +∞|0, −∞〉 J ,where the source J = J(t) plays a role analogous to that of an electromagneticcurrent, which acts as a source of the electromagnetic field. In otherwords, we can think of the scalar product J µ A µ , where J µ is the currentfrom a scalar (or Dirac) field acting as a source of the potential A µ . Inthe same way, we can always define a current J that acts as the sourcefor some arbitrary field φ. Z[J] (otherwise denoted by W [J]) is a functionalof the current J, defined as (see [Ryder (1996); Cheng and Li (1984);Gunion (2003)])∫{ ∫ }t′Z[J] ∝ D[p]D[q] exp i [p(τ) ˙q(τ) − H(p, q) + J(τ)q(τ)]dτ ,twith the normalization condition Z[J = 0] = 1. Here, the argument ofthe exponential depends upon the functions q(τ) and p(τ) and we thenintegrate over all possible forms of these two functions. So the exponentialis a functional that maps a choice for these two functions into a number.For example, for a quadratically completable H(p, q), the p integral can beperformed as a q integral∫ { ∫ +∞Z[J] ∝ D[q] exp i(L + Jq + 1 ) }2 iεq2 dτ ,−∞where the addittion to H was chosen in the form of a convergence factor- 1 2 iεq2 .


<strong>Geom</strong>etrical Path Integrals and Their Applications 10136.3.2.3 FieldsLet us now treat the abstract scalar field φ(x) as a coordinate in the sensethat we imagine dividing space up into many little cubes and the averagevalue of the field φ(x) in that cube is treated as a coordinate for that littlecube. Then, we go through the multi–coordinate analogue of the procedurewe just considered above and take the continuum limit. The final result is∫ { ∫Z[J] ∝ D[φ] exp id 4 x(L (φ(x)) + J(x)φ(x) + 1 2 iεφ2 )},where for L we would employ the Klein–Gordon Lagrangian form. In theabove, the dx 0 integral is the same as dτ, while the d 3 ⃗x integral is summingover the sub–Lagrangians of all the different little cubes of space and thentaking the continuum limit. L is the Lagrangian density describing theLagrangian for each little cube after taking the many–cube limit (see [Ryder(1996); Cheng and Li (1984); Gunion (2003)]) for the full derivation).We can now introduce interactions, L I . Assuming the simple form ofthe Hamiltonian, we have∫ { ∫Z[J] ∝ D[φ] exp i}d 4 x (L (φ(x)) + L I (φ(x)) + J(x)φ(x)) ,again using the normalization factor required for Z[J = 0] = 1.For example of Klein Gordon theory, we would useL = L 0 + L I , L 012 [∂ µφ∂ µ φ − µ 2 φ 2 ], L I = L I (φ),where ∂ µ ≡ ∂ x µ and we can freely manipulate indices, as we are working inEuclidean space R 3 . In order to define the above Z[J], we have to includea convergence factor iεφ 2 ,L 0 → 1 2 [∂ µφ∂ µ φ − µ 2 φ 2 + iεφ 2 ], so that∫∫Z[J] ∝ D[φ] exp{i d 4 x( 1 2 [∂ µφ∂ µ φ − µ 2 φ 2 + iεφ 2 ] + L I (φ(x)) + J(x)φ(x))}is the appropriate generating function in the free field theory case.6.3.2.4 GaugesIn the path integral approach to quantization of the gauge theory, we implementgauge fixing by restricting in some manner or other the path integral


1014 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionover gauge fields ∫ D[A µ ]. In other words we will write instead∫∫Z[J] ∝ D[A µ ] δ (some gauge fixing condition) exp{i d 4 xL (A µ )}.A common approach would be to start with the gauge conditionL = − 1 4 F µνF µν − 1 2 (∂µ A µ ) 2where the electrodynamic field tensor is given by F µν = ∂ µ A ν − ∂ ν A µ , andcalculate∫{ ∫}Z[J] ∝ D[A µ ] exp i d 4 x [L(A µ (x)) + J µ (x)A µ (x)]as the generating function for the vacuum expectation values of time orderedproducts of the A µ fields. Note that J µ should be conserved (∂ µ J µ = 0)in order for the full expression L(A µ ) + J µ A µ to be gauge–invariant underthe integral sign when A µ → A µ + ∂ µ Λ. For a proper approach, see [Ryder(1996); Cheng and Li (1984); Gunion (2003)].6.3.3 Riemannian–Symplectic <strong>Geom</strong>etriesIn this section, following [Shabanov and Klauder (1998)], we describe pathintegral quantization on Riemannian–symplectic manifolds. Let ˆq j be aset of Cartesian coordinate canonical operators satisfying the Heisenbergcommutation relations [ˆq j , ˆq k ] = iω jk . Here ω jk = −ω kj is the canonicalsymplectic structure. We introduce the canonical coherent states as |q〉 ≡e iqj ω jk ˆq k |0〉, where ω jn ω nk = δ k j , and |0〉 is the ground state of a harmonicoscillator with unit angular frequency. Any state |ψ〉 is given as a functionon phase–space in this representation by 〈q|ψ〉 = ψ(q). A general operator can be represented in the form  = ∫ dq a(q)|q〉〈q|, where a(q) is thelower symbol of the operator and dq is a properly normalized form of theLiouville measure. The function A(q, q ′ ) = 〈q|Â|q′ 〉 is the kernel of theoperator.The main object of the path integral formalism is the integral kernel ofthe evolution operatorK t (q, q ′ ) = 〈q|e −itĤ|q ′ 〉 =q(t)=q∫D[q] e i R t0 dτ( 1 2 qj ω jk ˙q k −h) . (6.16)q(0)=q ′


<strong>Geom</strong>etrical Path Integrals and Their Applications 1015Here Ĥ is the Hamiltonian, and h(q) its symbol. The measure formally impliesa sum over all phase-space paths pinned at the initial and final points,and a Wiener measure regularization implies the following replacementD[q] → D[µ ν (q)] = D[q] e − 1 R t2ν 0 dτ ˙q2 = N ν (t) dµ ν W (q) . (6.17)The factor N ν (t) equals 2πe νt/2 for every degree of freedom, dµ ν W (q) standsfor the Wiener measure, and ν denotes the diffusion constant. We denoteby Kt ν (q, q ′ ) the integral kernel of the evolution operator for a finite ν.The Wiener measure determines a stochastic process on the flat phase–space. The integral of the symplectic 1–form ∫ qωdq is a stochastic integralthat is interpreted in the Stratonovich sense. Under general coordinatetransformations q = q(¯q), the Wiener measure describes the same stochasticprocess on flat space in the curvilinear coordinates dq 2 = dσ(¯q) 2 , so that thevalue of the integral is not changed apart from a possible phase term. Afterthe calculation of the integral, the evolution operator kernel is get by takingthe limit ν → ∞. The existence of this limit, and also the covariance undergeneral phase-space coordinate transformations, can be proved through theoperator formalism for the regularized kernel Kt ν (q, q ′ ).Note that the integral (6.16) with the Wiener measure inserted can beregarded as an ordinary Lagrangian path integral with a complex action,where the configuration space is the original phase–space and the Hamiltonianh(q) serves as a potential. Making use of this observation it is nothard to derive the corresponding Schrödinger–like equation[ (∂ t Kt ν (q, q ′ ν) = ∂2 q j + i ) ]22 ω jkq k − ih(q) Kt ν (q, q ′ ) , (6.18)subject to the initial condition Kt=0(q, ν q ′ ) = δ(q − q ′ ), 0 < ν < ∞. One canshow that ˆK tν → ˆK t as ν → ∞ for all t > 0. The covariance under generalcoordinate transformations follows from the covariance of the ‘kinetic’energy of the Schrödinger operator in (6.18): The Laplace operator is replacedby the Laplace–Beltrami operator in the new curvilinear coordinatesq = q(¯q), so the solution is not changed, but written in the new coordinates.This is similar to the covariance of the ordinary Schrödinger equation andthe corresponding Lagrangian path integral relative to general coordinatetransformations on the configuration space: The kinetic energy operator(the Laplace operator) in the ordinary Schrödinger equation gives a termquadratic in time derivatives in the path integral measure which is sufficientfor the general coordinate covariance. We remark that the regularization


1016 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionprocedure based on the modified Schrödinger equation (6.18) applies to farmore general Hamiltonians than those quadratic in canonical momenta andleading to the conventional Lagrangian path integral.6.3.4 Euclidean Stochastic Path IntegralRecall that modern stochastic calculus permits development of three alternativedescriptions of complex stochastic systems:(1) Langevin rate ODEs [Gardiner (1985)],(2) Fokker–Planck PDEs [Gardiner (1985)], and(3) Euclidean stochastic path–integrals [Wehner and Wolfer (1983a);Wehner and Wolfer (1983b); Graham (1978); Langouche et. al. (1980)].L. Ingber [Ingber (1997)] has successfully applied a stochastic path–integral of the Euclidean form∫〈f|i〉 = D[w] e −A[w] ,Ωto in–depth analysis of three completely different non–equilibrium nonlinearmultivariate Gaussian–Markovian systems:(1) Statistical mechanical descriptions of neocortex (short–term memoryand EEG),(2) Analysis of financial markets (interest–rate and trading models), and(3) Combat analysis (baselining simulations to exercise data).The core of all his work are of context–specific mesoscopic order parametersM G . Ingber starts with Langevin rate equations (with zero–mean Gaussianwhite noise η j (t)), as order parameter equations (see [Haken (1983); Haken(1993)])Ṁ G = f G + ĝ G j η j , (G = 1, ..., Λ), (j = 1, ..., N), (6.19)< η j (t) > η = 0, < η j (t), η j′ (t ′ ) > η = δ jj′ δ(t − t ′ ),where f G and ĝjG are generally nonlinear functions of mesoscopic order parametersM G , j is a microscopic index indicating the source of fluctuations,and N ≥ Λ.Via a somewhat lengthy calculation, involving an intermediate derivationof a corresponding Fokker–Planck or Schrödinger–type equation forthe conditional probability distribution P [M(t)|M(t 0 )], equation (6.19) is


<strong>Geom</strong>etrical Path Integrals and Their Applications 1017developed into the more useful probability distribution for the order parametersM G at long–time macroscopic time event t = (u + 1)θ + t 0 , interms of a Stratonovich–Riemannian path–integral over mesoscopic Gaussianconditional probabilities. Here, macroscopic variables are defined asthe long–time limit of the evolving mesoscopic system.The corresponding Schrödinger–type equation is∂ t P = 1 2 (gGG′ P ), GG ′ −(g G P ), G +V, (6.20)g GG′ = k T δ jk ĝ G j ĝ G′k ,g G = f G + 1 2 δjk ĝ G′j ĝ G k,G ′,[.], G = ∂[.]/∂M G .This is properly referred to as a Fokker–Planck equation when V = 0.Note that although the partial differential equation (6.20) contains equivalentinformation regarding the order parameters M G as in the stochasticdifferential equation (6.19), all references to j have been properly averagedover, i.e., ĝjG in (6.19) is an entity with parameters in both microscopic andmesoscopic spaces, but M is a purely mesoscopic variable, and this is moreclearly reflected in (6.20).Now, the path integral representation is given in terms of the LagrangianL = L(Ṁ G , M G , t), as∫P [M t |M t0 ] dM(t) = D[M] exp(−A[M])δ[M(t 0 ) = M 0 ]δ[M(t) = M t ],where the action A[M] is given by(6.21)∫ tA[M] = k −1Tmin dt ′ L(Ṁ G , M G , t),t 0and the path measure D[M] is equal toD[M] = limu→∞The Lagrangian L is given byu+1∏ρ=1g 1/2 ∏ G(2πθ) −1/2 dM G ρ .L(Ṁ G , M G , t) = 1 2 (Ṁ G − h G )g GG ′(Ṁ G′ − h G′ ) + 1 2 hG ;G + R/6 − V,


1018 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere h G = g G − 1 2 g−1/2 (g 1/2 g GG′ ) ,G ′, comma ‘,’ denotes partial derivative,and g GG ′ is the inverse metric tensor g GG ′ = (g GG′ ) −1 , with determinantg = det(g GG ′). The covariant derivative h G ;G is defined classicallythrough the Christoffel symbols Γ F GF ,h G ;G = h G ,G + Γ F GF h G = g −1/2 (g 1/2 h G ) ,G ,Γ F GF = g LF (g JL,K + g KL,J − g JK,L ),while the Riemannian curvature is defined asR = g JL R JL = g JL g JK R F JKL ,R F JKL = 1 2 (g F K,JL − g JK,F L − g F L,JK + g JL,F K ) + g MN (Γ M F KΓ N F L − Γ N JK).Therefore, mesoscopic variables have been defined as M G in theLangevin and Fokker–Planck representations, in terms of their developmentfrom the microscopic system labelled by j. The Riemannian curvature termR arises from nonlinear g GG ′, which is a genuine metric of this space [Graham(1978)]. Even if a stationary solution, i.e., Ṁ G = 0, is ultimatelysought, a necessarily prior stochastic treatment of Ṁ G terms gives rise tothese Riemannian ‘corrections’. Even for a constant metric, the term h G ;Gcontributes to L for a nonlinear mean h G . V may include terms such as∑T ′ J T ′GM G , where the Lagrange multipliers J T ′G are constraints on M G ,which are advantageously modelled as extrinsic sources in this representation;they too may be time–dependent. Using the variational principle, J T Gmay also be used to constrain M G to regions where they are empiricallybound. With respect to a steady state ¯P , when it exists, the informationgain in state P is defined by∫I[P ] = D[M ′ ]P ln(P/ ¯P ), where D[M ′ ] = D[M]/dM u+1 .A robust and accurate histogram–based (i.e., non–Monte Carlo) path–integral algorithm to calculate the long–time probability distribution hasbeen developed to handle nonlinear Lagrangians (see [Wehner and Wolfer(1983a); Wehner and Wolfer (1983b); Graham (1978); Langouche et. al.(1980)]).In the economics literature, there appears to be sentiment to define(6.19) by the Ito, rather than the Stratonovich prescription. It shouldbe noted that virtually all investigations of other physical systems, whichare also continuous time models of discrete processes, conclude that theStratonovich interpretation coincides with reality, when multiplicative noise


<strong>Geom</strong>etrical Path Integrals and Their Applications 1019with zero correlation time, modelled in terms of white noise η j , is properlyconsidered as the limit of real noise with finite correlation time (see [Gardiner(1985)]). The path integral succinctly demonstrates the differencebetween the two: The Ito prescription corresponds to the the so–calledprepoint discretization of L, whereinθṀ(t) → M ρ+1 − M ρ and M(t) → M ρ .The Stratonovich prescription corresponds to the midpoint discretization ofL, whereinθṀ(t) → M ρ+1 − M ρ and M(t) → 1 2 (M ρ+1 + M ρ ).In terms of the functions appearing in the Fokker–Planck equation (6.20),the Ito prescription of the prepoint discretized Lagrangian, L I , is relativelysimple:L I (Ṁ G , M G , t) = 1 2 (Ṁ G − g G )g GG ′(Ṁ G′ − g G′ ) − V,however, this is deceptive because of its nonstandard calculus [Ingber(1997)].Now, as L possesses a variational principle, sets of contour graphs, atdifferent long–time epochs of the path–integral of P over its variables at allintermediate times, give a visually intuitive and accurate decision–aid toview the dynamic evolution of the scenario. For example, this Lagrangianapproach permits a quantitative assessment of the following concepts usuallyonly loosely defined:‘Momentum’ = Π G = LṀ G‘Mass’ = g GG ′ = LṀ G Ṁ G′‘Force’ = F = L M G‘F = ma’ : 0 = δL = L M G − ∂ t LṀ GThese physical entities provide another form of intuitive, but quantitativelyprecise, presentation of these analyzes. For example, daily newspapers usethis terminology to discuss the movement of security prices. Π G serve ascanonical momenta indicators (CMI) for these systems [Ingber (1997)].


1020 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction6.3.5 Application: Stochastic Optimal ControlA path–integral based optimal control model for nonlinear stochastic systemshas recently been developed in [Kappen (2006)]. The author addressedthe role of noise and the issue of efficient computation in stochastic optimalcontrol problems. He considered a class of nonlinear control problems thatcan be formulated as a path integral and where the noise plays the role oftemperature. The path integral displays symmetry breaking and there exista critical noise value that separates regimes where optimal control yieldsqualitatively different solutions. The path integral can be computed efficientlyby Monte Carlo integration or by Laplace approximation, and cantherefore be used to solve high dimensional stochastic control problems.Recall that optimal control of nonlinear systems in the presence of noiseis a very general problem that occurs in many areas of science and engineering.It underlies autonomous system behavior, such as the control ofmovement and planning of actions of animals and robots, but also optimizationof financial investment policies and control of chemical plants.The problem is stated as: given that the system is in this configuration atthis time, what is the optimal course of action to reach a goal state at somefuture time. The cost of each time course of actions consists typically of apath contribution, that specifies the amount of work or other cost of thetrajectory, and an end cost, that specifies to what extend the trajectoryreaches the goal state.Also recall that in the absence of noise, the optimal control problemcan be solved in two ways: using (i) the Pontryagin Maximum Principle(PMP, see previous subsection), which represents a pair of ordinary differentialequations that are similar to the Hamiltonian equations; or (ii) theHamilton–Jacobi–Bellman (HJB) equation, which is a partial differentialequation (PDE) [Bellman and Kalaba (1964)].In the presence of Wiener noise, the PMP formalism is replaced by a setof stochastic differential equations (SDEs), which become difficult to solve(compare with [Yong and Zhou (1999)]). The inclusion of noise in the HJBframework is mathematically quite straightforward, yielding the so–calledstochastic HJB equation [Stengel (1993)]. However, its solution requires adiscretization of space and time and the computation becomes intractablein both memory requirement and CPU time in high dimensions. As a result,deterministic control can be computed efficiently using the PMP approach,but stochastic control is intractable due to the curse of dimensionality.For small noise, one expects that optimal stochastic control resembles


<strong>Geom</strong>etrical Path Integrals and Their Applications 1021optimal deterministic control, but for larger noise, the optimal stochasticcontrol can be entirely different from the deterministic control [Russell andNorvig (2003)]. However, there is currently no good understanding hownoise affects optimal control.In this subsection, we address both the issue of efficient computationand the role of noise in stochastic optimal control. We consider a class ofnonlinear stochastic control problems, that can be formulated as a statisticalmechanics problem. This class of control problems includes arbitrarydynamical systems, but with a limited control mechanism. It containslinear–quadratic [Stengel (1993)] control as a special case. We show thatunder certain conditions on the noise, the HJB equation can be written asa linear PDE− ∂ t ψ = Hψ, (6.22)with H a (non–Hermitian) operator. Equation (6.22) must be solved subjectto a boundary condition at the end time. As a result of the linearity of(6.22), the solution can be obtained in terms of a diffusion process evolvingforward in time, and can be written as a path integral. The path–integralhas a direct interpretation as a free energy, where noise plays the role oftemperature.This link between stochastic optimal control and a free energy has animmediate consequence that phenomena that allow for a free energy description,typically display phase transitions. [Kappen (2006)] has arguedthat for stochastic optimal control one can identify a critical noise valuethat separates regimes where the optimal control has been qualitativelydifferent. He showed how the Laplace approximation can be combinedwith Monte Carlo sampling to efficiently calculate the optimal control.6.3.5.1 Path–Integral FormalismLet x i be an nD stochastic variable that is subject to the SDEdx i = (b i (x i , t) + u i )dt + dξ i (6.23)with dξ i being an nD Wiener process with 〈 dξ i dξ j〉= νij dt, and functionsν ij independent of x i , u i and time t. The term b i (x i , t) is an arbitrary nDfunction of x i and t, and u i represents an nD vector of control variables.Given the value of x i at an initial time t, the stochastic optimal control


1022 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionproblem is to find the control path u i (·) that minimizes〈∫ tfC(x i , t, u i (·)) = φ(x i (t f )) + dτ( 1 〉t 2 u i(τ)Ru i (τ) + V (x i (τ), τ)) ,x i(6.24)with R a matrix, V (x i , t) a time–dependent potential, and φ(x i ) the endcost. The brackets 〈〉 x i denote expectation value with respect to the stochastictrajectories (6.23) that start at x i .One defines the optimal cost–to–go function from any time t and statex i asJ(x i , t) = minu i (·) C(xi , t, u i (·)).J satisfies the following stochastic HJB equation [Kappen (2006)]( 1− ∂ t J(x i , t) = minu i 2 u iRu i +V +(b i +u i )∂ x iJ(x i , t)+ 1 )2 ν ij∂ xi x j J(xi , t)= − 1 2 R−1 ∂ x iJ(x i , t)∂ x iJ+V +b i ∂ x iJ(x i , t)+ 1 2 ν ij∂ x i x j J(xi , t),where b i = (b i ) T , and u i = (u i ) T , and(6.25)u i = −R −1 ∂ x iJ(x i , t) (6.26)is the optimal control at the point (x i , t). The HJB equation is nonlinearin J and must be solved with end boundary condition J(x i , t f ) = φ(x i ).Let us define ψ(x i , t) through the Log Transformand assume that there exists a scalar λ such thatJ(x i , t) = −λ log ψ(x i , t), (6.27)λδ ij = (Rν) ij , (6.28)with δ ij the Kronecker delta. In the one dimensional case, such a λ canalways be found. In the higher dimensional case, this restricts the matricesR ∝ (ν ij ) −1 . Equation (6.28) reduces the dependence of optimalcontrol on the nD noise matrix to a scalar value λ that will play the role oftemperature, while (6.25) reduces to the linear equation (6.22) withH = − V λ + b i∂ x i + 1 2 ν ij∂ xi x j J(xi , t).


<strong>Geom</strong>etrical Path Integrals and Their Applications 1023Let ρ(y i , τ|x i , t) with ρ(y i , t|x i , t) = δ(y i − x i ) describe a diffusion processfor τ > t defined by the Fokker–Planck equation∂ τ ρ = H † ρ = − V λ ρ − ∂ x i(b iρ) + 1 2 ν ij∂ x i x j J(xi , t)ρ (6.29)with H † the Hermitian–conjugateof H. Then A(τ) = ∫ dy i ρ(y i , τ|x i , t)ψ(y i , τ) is independent of τ and inparticular A(t) = A(t f ). It immediately follows that∫ψ(x i , t) = dy i ρ(y i , t f |x i , t) exp(−φ(y i )/λ) (6.30)We arrive at the important conclusion that ψ(x i , t) can be computed eitherby backward integration using (6.22) or by forward integration of a diffusionprocess given by (6.29).We can write the integral in (6.30) as a path integral. Following [Kappen(2006)] we can divide the time interval t → t f in n 1 intervals and writeρ(y i , t f |x i , t) = ∏ n 1i=1 ρ(xi i , t i|x i i−1 , t i−1) and let n 1 → ∞. The result is∫(ψ(x i , t) = [dx i ] x i exp − 1 )λ S(xi (t → t f )) (6.31)with ∫ [dx i ] x i an integral over all paths x i (t → t f ) that start at x i and with∫ tfS(x i (t → t f )) = φ(x i (t f )+ dτ( 1t 2 (ẋi −b i (x i , τ))R(ẋ i −b i (x i , τ))+V (x i , τ))(6.32)the Action associated with a path. From (6.27) and (6.31), the cost–to–goJ(x, t) becomes a log partition sum (i.e., a free energy) with temperatureλ.6.3.5.2 Monte Carlo SamplingThe path integral (6.31) can be estimated by stochastic integration from tto t f of the diffusion process (6.29) in which particles get annihilated at arate V (x i , t)/λ [Kappen (2006)]:x i = x i + b i (x i , t)dt + dξ i , with probability 1 − V dt/λx i = †, with probability V dt/λ (6.33)where † denotes that the particle is taken out of the simulation. Denotethe trajectories by x i α(t → t f ), (α = 1, . . . , N). Then, ψ(x i , t) and u i are


1024 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionestimated asˆψ(x i , t) =∑α∈alivew α , u i dt =1ˆψ(x i , t)N∑α∈alivew α dξ i α(t), (6.34)withw α = 1 N exp(−φ(xi α(t f ))/λ),where ‘alive’ denotes the subset of trajectories that do not get killed alongthe way by the † operation. The normalization 1/N ensures that the annihilationprocess is properly taken into account. Equation (6.34) states thatoptimal control at time t is obtained by averaging the initial directions ofthe noise component of the trajectories dξ i α(t), weighted by their success att f .The above sampling procedure can be quite inefficient, when many trajectoriesget annihilated. One of the simplest procedures to improve itis by importance sampling. We replace the diffusion process that yieldsρ(y i , t f |x i , t) by another diffusion process, that will yield ρ ′ (y i , t f |x i , t) =exp(−S ′ /λ). Then (6.31) becomes,∫ψ(x i , t) = [dx i ] x i exp (−S ′ /λ) exp (−(S − S ′ )/λ) .The idea is to chose ρ ′ such as to make the sampling of the path integralas efficient as possible. Following [Kappen (2006)], here we use the Laplaceapproximation, which is given by the k deterministic trajectories x β (t → t f )that minimize the ActionJ(x i , t) ≈ −λ logk∑exp(−S(x i β(t → t f )/λ).β=1The Laplace approximation ignores all fluctuations around the modes andbecomes exact in the limit λ → 0. The Laplace approximation can becomputed efficiently, requiring O(n 2 m 2 ) operations, where m is the numberof time discretization.For each Laplace trajectory, we can define a diffusion processes ρ ′ β accordingto (6.33) with b i (x i , t) = x i β (t). The estimators for ψ and ui aregiven again by (6.34), but with weightsw α = 1 N exp ( − ( S(x i α(t → t f )) − S ′ β(x i α(t → t f )) ) /λ ) .S is the original Action (6.32) and Sβ ′ is the new Action for the Laplaceguided diffusion. When there are multiple Laplace trajectories one should


<strong>Geom</strong>etrical Path Integrals and Their Applications 1025include all of these in the sample.6.3.6 Application: Nonlinear Dynamics of Option PricingClassical theory of option pricing is based on the results found in 1973by Black and Scholes [Black and Scholes (1973)] and, independently, Merton[Merton (1973)]. Their pioneering work starts from the basic assumptionthat the asset prices follow the dynamics of a particular stochasticprocess (geometrical Brownian motion), so that they have a lognormaldistribution [Hull (2000); Paul and Baschnagel (1999)]. In the case ofan efficient market with no arbitrage possibilities, no dividends and constantvolatilities, they found that the price of each financial derivative isruled by an ordinary partial differential equation, known as the (Nobel–Prize winning) Black–Scholes–Merton (BSM) formula. In the most simplecase of a so–called European option, the BSM equation can be explicitlysolved to get an analytical formula for the price of the option [Hull (2000);Paul and Baschnagel (1999)]. When we consider other financial derivatives,which are commonly traded in real markets and allow anticipatedexercise and/or depend on the history of the underlying asset, the BSMformula fails to give an analytical result. Appropriate numerical procedureshave been developed in the literature to price exotic financial derivativeswith path–dependent features, as discussed in detail in [Hull (2000);Wilmott et. al. (1993); Potters et. al. (2001)]. The aim of this work is togive a contribution to the problem of efficient option pricing in financialanalysis, showing how it is possible to use path integral methods to developa fast and precise algorithm for the evaluation of option prices.Following recent studies on the application of the path integral approachto the financial market as appeared in the econophysics literature(see [Matacz (2002)] for a comprehensive list of references), in [Montagnaet. al. (2002)] the authors proposed an original, efficient path integral algorithmto price financial derivatives, including those with path–dependentand early exercise features, and to compare the results with those get withthe standard procedures known in the literature.6.3.6.1 Theory and Simulations of Option PricingClassical Theory and Path–Dependent OptionsThe basic ingredient for the development of a theory of option pricing is


1026 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona suitable model for the time evolution of the asset prices. The assumptionof the BSM model is that the price S of an asset is driven by a Brownianmotion and verifies the stochastic differential equation (SDE) [Hull (2000);Paul and Baschnagel (1999)]dS = µSdt + σSdw, (6.35)which, by means of the Itô lemma, can be cast in the form of an arithmeticBrownian motion for the logarithm of Sd(ln S) = Adt + σdw, (6.36)where σ is the volatility, A = ( µ − σ 2 /2 ) , µ is the drift parameter and wis the realization of a Wiener process. Due to the properties of a Wienerprocess, (6.36) may be written asd(ln S) = Adt + σɛ √ dt, (6.37)where ɛ follows from a standardized normal distribution with mean 0and variance 1. Thus, in terms of the logarithms of the asset pricesz ′ = ln S ′ , z = ln S, the conditional transition probability p(z ′ |z) tohave at the time t ′ a price S ′ under the hypothesis that the pricewas S at the time t < t ′ is given by [Paul and Baschnagel (1999);Bennati et. al. (1999)]p(z ′ |z) ={1√2π(t′ − t)σ exp − [z′ − (z + A(t ′ − t))] 2 }2 2σ 2 (t ′ , (6.38)− t)which is a gaussian distribution with mean z +A(t ′ −t) and variance σ 2 (t ′ −t). If we require the options to be exercised only at specific times t i , i =1, · · · , n, the asset price, between two consequent times t i−1 and t i , willfollow (6.37) and the related transition probability will be{1p(z i |z i−1 ) = √ exp − [z i − (z i−1 + A∆t)] 2 }2π∆tσ2 2σ 2 , (6.39)∆twith ∆t = t i − t i−1 .A time–evolution model for the asset price is strictly necessary in atheory of option pricing because the fair price at time t = 0 of an optionO, without possibility of anticipated exercise before the expiration date ormaturity T (a so–called European option), is given by the scaled expectation


<strong>Geom</strong>etrical Path Integrals and Their Applications 1027value [Hull (2000)]O(0) = e −rT E[O(T )], (6.40)where r is the risk–free interest and E[·] indicates the mean value, whichcan be computed only if a model for the asset underlying the option isunderstood. For example, the value O of an European call option at thematurity T will be max{S T −X, 0}, where X is the strike price, while for anEuropean put option the value O at the maturity will be max{X − S T , 0}.It is worth emphasizing, for what follows, that the case of an Europeanoption is particularly simple, since in such a situation the price of theoption can be evaluated by means of analytical formulae, which are get bysolving the BSM partial differential equation with the appropriate boundaryconditions [Hull (2000); Paul and Baschnagel (1999)]. On the other hand,many further kinds of options are present in the financial markets, suchas American options (options which can be exercised at any time up tothe expiration date) and exotic options [Hull (2000)], i.e., derivatives withcomplicated payoffs or whose value depend on the whole time evolutionof the underlying asset and not just on its value at the end. For suchoptions with path-dependent and early exercise features no exact solutionsare available and pricing them correctly is a great challenge.In the case of options with possibility of anticipated exercise before theexpiration date, the above discussion needs to be generalized, by introducinga slicing of the time interval T . Let us consider, for definiteness, thecase of an option which can be exercised within the maturity but only atthe times t 1 = ∆t, t 2 = 2∆t, . . . , t n = n∆t = T. At each time slice t i−1the value O i−1 of the option will be the maximum between its expectationvalue at the time t i scaled with e −r∆t and its value in the case of anticipatedexercise Oi−1 Y . If S i−1 denotes the price of the underlying asset atthe time t i−1 , we can thus write for each i = 1, . . . , nO i−1 (S i−1 ) = max { O Y i−1(S i−1 ), e −r∆t E[O i |S i−1 ] } , (6.41)where E[O i |S i−1 ] is the conditional expectation value of O i , i.e., its expectationvalue under the hypothesis of having the price S i−1 at the time t i−1 .In this way, to get the actual price O 0 , it is necessary to proceed backwardin time and calculate O n−1 , . . . , O 1 , where the value O n of the optionat maturity is nothing but On Y (S n ). It is therefore clear that evaluatingthe price of an option with early exercise features means to simulate theevolution of the underlying asset price (to get the Oi Y ) and to calculate a


1028 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(usually large) number of expectation conditional probabilities.Standard Numerical ProceduresTo value derivatives when analytical formulae are not available, appropriatenumerical techniques have to be advocated. They involve the useof Monte Carlo (MC) simulation, binomial trees (and their improvements)and finite–difference methods [Hull (2000); Wilmott et. al. (1993)].A natural way to simulate price paths is to discretize (6.37) asln S(t + ∆t) − ln S(t) = A∆t + σɛ √ ∆t,or, equivalently,[S(t + ∆t) = S(t) exp A∆t + σɛ √ ]∆t , (6.42)which is correct for any ∆t > 0, even if finite. Given the spot price S 0 , i.e.,the price of the asset at time t = 0, one can extract from a standardizednormal distribution a value ɛ k , (k = 1, . . . , n) for the random variable ɛ tosimulate one possible path followed by the price by means of (6.42):√ ]S(k∆t) = S((k − 1)∆t) exp[A∆t + σɛ k ∆t .Iterating the procedure m times, one can simulate m price paths{(S 0 , S (j)1 , S(j) 2 ,. . . , S n(j) ≡ S (j)T) : j = 1, . . . , m} and evaluate the price of the option. Insuch a MC simulation of the stochastic dynamics of asset price (MonteCarlo random walk) the mean values E[O i |S i−1 ], i = 1, . . . , n are given byE[O i |S i−1 ] = O(1) i + O (2)i+ · · · + O (m)iwith no need to calculate transition probabilities because, through the extractionof the possible ɛ values, the paths are automatically weighted accordingto the probability distribution function of (6.39). Unfortunately,this method leads to an estimated value whose numerical error is proportionalto m −1/2 . Thus, even if it is powerful because of the possibilityto control the paths and to impose additional constrains (as it is usuallyrequired by exotic and path-dependent options), the MC random walk isextremely time consuming when precise predictions are required and appropriatevariance reduction procedures have to be used to save CPU time[Hull (2000)]. This difficulty can be overcome by means of the method of them,


<strong>Geom</strong>etrical Path Integrals and Their Applications 1029binomial trees and its extensions (see [Hull (2000)] and references therein),whose main idea stands in a deterministic choice of the possible paths tolimit the number of intermediate points. At each time step the price S iis assumed to have only two choices: increase to the value uS i , u > 1 ordecrease to dS i , 0 < d < 1, where the parameters u and d are given interms of σ and ∆t in such a way to give the correct values for the mean andvariance of stock price changes over the time interval ∆t. Also finite differencemethods are known in the literature [Hull (2000)] as an alternative totime-consuming MC simulations. They give the value of the derivative bysolving the differential equation satisfied by the derivative, by converting itinto a difference equation. Although tree approaches and finite differencemethods are known to be faster than the MC random walk, they are difficultto apply when a detailed control of the history of the derivative is requiredand are also computationally time consuming when a number of stochasticvariables is involved [Hull (2000)]. It follows that the development ofefficient and fast computational algorithms to price financial derivatives isstill a key issue in financial analysis.6.3.6.2 Option Pricing via Path IntegralsRecall that the path integral method is an integral formulation of the dynamicsof a stochastic process. It is a suitable framework for the calculationof the transition probabilities associated to a given stochastic process,which is seen as the convolution of an infinite sequence of infinitesimalshort-time steps [Bennati et. al. (1999)]. For the problem of option pricing,the path–integral method can be employed for the explicit calculationof the expectation values of the quantities of financial interest, given byintegrals of the form [Bennati et. al. (1999)]∫E[O i |S i−1 ] = dz i p(z i |z i−1 )O i (e zi ), (6.43)where z = ln S and p(z i |z i−1 ) is the transition probability. E[O i |S i−1 ]is the conditional expectation value of some functional O i of the stochasticprocess. For example, for an European call option at the maturityT the quantity of interest will be max {S T − X, 0}, X being the strikeprice. As already emphasized, and discussed in the literature [Hull (2000);Wilmott et. al. (1993); Potters et. al. (2001); Rosa-Clot and Taddei (2002);Matacz (2002)], the computational complexity associated to this calculationis generally great: in the case of exotic options, with path-dependent


1030 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand early exercise features, integrals of the type (6.43) cannot be analyticallysolved. As a consequence, we demand two things from a path integralframework: a very quick way to estimate the transition probability associatedto a stochastic process (6.37) and a clever choice of the integrationpoints with which evaluate the integrals (6.43). In particular, our aim isto develop an efficient calculation of the probability distribution withoutlosing information on the path followed by the asset price during its timeevolution.Transition ProbabilityThe probability distribution function related to a SDE verifies theChapman–Kolmogorov equation [Paul and Baschnagel (1999)]∫p(z ′′ |z ′ ) = dzp(z ′′ |z)p(z|z ′ ), (6.44)which states that the probability (density) of a transition from the valuez ′ (at time t ′ ) to the value z ′′ (at time t ′′ ) is the ‘summation’ over all thepossible intermediate values z of the probability of separate and consequenttransitions z ′ → z, z → z ′′ . As a consequence, if we consider a finite timeinterval [t ′ , t ′′ ] and we apply a time slicing, by considering n+1 subintervalsof length ∆t = (t ′′ − t ′ )/n + 1, we can write, by iteration of (6.44)p(z ′′ |z ′ ) =∫ +∞−∞· · ·∫ +∞−∞dz 1 · · · dz n p(z ′′ |z n )p(z n |z n−1 ) · · · p(z 1 |z ′ ),which, thanks to (6.38), can be written as [Montagna et. al. (2002)]∫ +∞· · · (6.45)−∞∫ +∞· · ·−∞{1dz 1 · · · dz n √(2πσ2 ∆t) exp n+1− 1n+1∑2σ 2 ∆tk=1[z k − (z k−1 + A∆t)] 2 }.In the limit n → ∞, ∆t → 0 such that (n + 1)∆t = (t ′′ − t ′ ) (infinitesequence of infinitesimal time steps), the expression (6.45), as explicitlyshown in [Bennati et. al. (1999)], exhibits a Lagrangian structure and it ispossible to express the transition probability in the path integral formalismas a convolution of the form [Bennati et. al. (1999)]∫{p(z ′′ , t ′′ |z ′ , t ′ ) = D[σ −1˜z] ∫ }t′′exp − L(˜z(τ), ˙˜z(τ); τ)dτ ,Ct ′


<strong>Geom</strong>etrical Path Integrals and Their Applications 1031where L is the Lagrangian, given byL(˜z(τ), ˙˜z(τ); τ) = 12σ 2 [ ˙˜z(τ) − A ] 2,and the integral is performed (with functional measure D[·]) over thepaths ˜z(·) belonging to C, i.e., all the continuous functions with constrains˜z(t ′ ) ≡ z ′ , ˜z(t ′′ ) ≡ z ′′ . As carefully discussed in [Bennati et. al. (1999)],a path integral is well defined only if both a continuous formal expressionand a discretization rule are given. As done in many applications, the Itôprescription is adopted here (see subsection 6.2.5 above).A first, naïve evaluation of the transition probability (6.45) can be performedvia Monte Carlo simulation, by writing (6.45) as∫ +∞−∞p(z ′′ , t ′′ |z ′ , t ′ ) =∫ +∞ n∏{1· · · dg i √2πσ2 ∆t exp − 1}2σ 2 ∆t [z′′ − (z n + A∆t)] 2 , (6.46)−∞iin terms of the variables g i defined by the relationdg k ={dz√ k2πσ2 ∆t exp − 1}2σ 2 ∆t [z k − (z k−1 + A∆t)] 2 , (6.47)and extracting each g i from a gaussian distribution of mean z k−1 + A∆tand variance σ 2 ∆t. However, as we will see, this method requires a largenumber of calls to get a good precision. This is due to the fact that eachg i is related to the previous g i−1 , so that this implementation of the pathintegral approach can be seen to be equivalent to a naïve MC simulation ofrandom walks, with no variance reduction.By means of appropriate manipulations [Schulman (1981)] of the integrandentering (6.45), it is possible, as shown in the following, to get a pathintegral expression which will contain a factorized integral with a constantkernel and a consequent variance reduction. If we define z ′′ = z n+1 andy k = z k − kA∆t, k = 1, . . . , n, we can express the transition probabilitydistribution as∫ +∞ ∫ {+∞1· · · dy 1 · · · dy n √(2πσ2 ∆t) ·exp − 1n+1∑n+1 2σ 2 [y k − y k−1 ]},2∆t−∞−∞(6.48)in order to get rid of the contribution of the drift parameter. Now let usk=1


1032 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionextract from the argument of the exponential function a quadratic formn+1∑[y k − y k−1 ] 2 = y0 2 − 2y 1 y 0 + y1 2 + y1 2 − 2y 1 y 2 + . . . + yn+12k=1= y t My + [y 2 0 − 2y 1 y 0 + y 2 n+1 − 2y n y n+1 ], (6.49)by introducing the nD array y and the nxn matrix M defined as [Montagnaet. al. (2002)]⎛ ⎞⎛⎞y 12 −1 0 · · · · · · 0y 2−1 2 −1 0 · · · 0y =.0 −1 2 −1 · · · 0, M =, (6.50)⎜ ⎟0 · · · −1 2 −1 0⎝ . ⎠⎜⎟⎝ 0 · · · · · · −1 2 −1 ⎠y n 0 · · · · · · · · · −1 2where M is a real, symmetric, non singular and tridiagonal matrix. Interms of the eigenvalues m i of the matrix M, the contribution in (6.49) canbe written asy t My = w t O t MOw = w t M d w =n∑m i wi 2 , (6.51)by introducing the orthogonal matrix O which diagonalizes M, with w i =O ij y j . Because of the orthogonality of O, the Jacobian∣ J = detdw i ∣∣∣∣ = det |O ki |,dy kof the transformation y k → w k equals 1, so that ∏ ni=1 dw i = ∏ ni=1 dy i.After some algebra, (6.49) can be written asn+1∑[y k − y k−1 ] 2 =k=1i=1n∑m i wi 2 + y0 2 − 2y 1 y 0 + yn+1 2 − 2y n y n+1 =i=1n∑i=1[m i w i − (y ] 20O 1i + y n+1 O ni )+ y0 2 + yn+1 2 −m in∑i=1(y 0 O 1i + y n+1 O ni ) 2m i.(6.52)


<strong>Geom</strong>etrical Path Integrals and Their Applications 1033Now, if we introduce new variables h i obeying the relation√{midh i =2πσ 2 ∆t exp −m [i2σ 2 w i − (y ] } 20O 1i + y n+1 O ni )dw i ,∆tm i(6.53)it is possible to express the finite–time probability distribution p(z ′′ |z ′ ) as[Montagna et. al. (2002)]=Z +∞−∞Z +∞−∞Z +∞· · ·−∞Z +∞· · ·−∞nYi=1nYi=1(× exp − 12σ 2 ∆tZ +∞Z +∞ nY= · · ·−∞× exp(−∞i=1− 12σ 2 ∆t()1dy i p(2πσ2 ∆t) exp − 1n+1X[y n+1 2σ 2 k − y k−1 ] 2∆tk=1dw i1p(2πσ2 ∆t) n+1 e−(y2 0 +y2 n+1 )/2σ2 ∆tnXi=1"m i„w i −dh i1p2πσ2 ∆t det(M)"y 2 0 + y 2 n+1 +« #)2(y0O1i + yn+1Oni) (y0O1i + yn+1Oni)2−m im i#)nX (y 0O 1i + y n+1O ni) 2.m ii=1The probability distribution function, as given by (6.54), is an integralwhose kernel is a constant function (with respect to the integration variables)and which can be factorized into the n integrals∫ +∞{dh i exp − 1 (y 0 O 1i + y n+1 O ni ) 2 }2σ 2 , (6.55)∆t m i−∞given in terms of the h i , which are gaussian variables that can be extractedfrom a normal distribution with mean (y 0 O 1i + y n+1 O ni ) 2 /m i and varianceσ 2 ∆t/m i . <strong>Diff</strong>erently to the first, naïve implementation of the path integral,now each h i is no longer dependent on the previous h i−1 , and importancesampling over the paths is automatically accounted for.It is worth noticing that, by means of the extraction of the randomvariables h i , we are creating price paths, since at each intermediate time t ithe asset price is given byS i = exp {(6.54)n∑O ik h k + iA∆t}. (6.56)k=1Therefore, this path integral algorithm can be easily adapted to the cases inwhich the derivative to be valued has, in the time interval [0, T ], additional


1034 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionconstraints, as in the case of interesting path–dependent options, such asAsian and barrier options [Hull (2000)].Integration PointsThe above illustrated method represents a powerful and fast tool tocalculate the transition probability in the path integral framework and itcan be employed if we need to value a generic option with maturity Tand with possibility of anticipated exercise at times t i = i∆t (n∆t = T )[Montagna et. al. (2002)]. As a consequence of this time slicing, one mustnumerically evaluate n − 1 mean values of the type (9), in order to checkat any time t i , and for any value of the stock price, whether early exerciseis more convenient with respect to holding the option for a future time. Tokeep under control the computational complexity and the time of execution,it is mandatory to limit as far as possible the number of points for theintegral evaluation. This means that we would like to have a linear growth ofthe number of integration points with the time. Let us suppose to evaluateeach mean value∫E[O i |S i−1 ] = dz i p(z i |z i−1 )O i (e zi ),with p integration points, i.e., considering only p fixed values for z i . To thisend, we can create a grid of possible prices, according to the dynamics ofthe stochastic process as given by (6.37)z(t + ∆t) − z(t) = ln S(t + ∆t) − ln S(t) = A∆t + ɛσ √ ∆t. (6.57)Starting from z 0 , we thus evaluate the expectation value E[O 1 |S 0 ] withp = 2m+1, m ∈ N values of z 1 centered on the mean value E[z 1 ] = z 0 +A∆tand which differ from each other of a quantity of the order of σ √ ∆tz j 1 = z 0 + A∆t + jσ √ ∆t,(j = −m, . . . , +m).Going on like this, we can evaluate each expectation value E[O 2 |z j 1 ] get fromeach one of the z 1 ’s created above with p values for z 2 centered around themean valueE[z 2 |z j 1 ] = zj 1 + A∆t = z 0 + 2A∆t + jσ √ ∆t.Iterating the procedure until the maturity, we create a deterministicgrid of points such that, at a given time t i , there are (p − 1)i + 1 values of


<strong>Geom</strong>etrical Path Integrals and Their Applications 1035z i , in agreement with the request of linear growth. This procedure of selectionof integration points, together with the calculation of the transitionprobability previously described, is the basis of the path integral simulationof the price of a generic option.By applying the results derived above, we have at disposal an efficientpath integral algorithm both for the calculation of transition probabilitiesand the evaluation of option prices. In [Montagna et. al. (2002)] the applicationof the above path–integral method to European and Americanoptions in the BSM model was illustrated and comparisons with the resultswere get with the standard procedures known in the literature were shown.First, the path integral simulation of the probability distribution of thelogarithm of the stock prices, p(lnS), as a function of the logarithm of thestock price, for a BSM–like stochastic model, was given by (6.36). Once thetransition probability has been computed, the price of an option could becomputed in a path integral approach as the conditional expectation valueof a given functional of the stochastic process. For example, the price of anEuropean call option was given by∫ +∞C = e −r(T −t) dz f p(z f , T |z i , t) max[e z f− X, 0], (6.58)−∞while for an European put it will be∫ +∞P = e −r(T −t) dz f p(z f , T |z i , t) max[X − e z f, 0], (6.59)−∞where r is the risk–free interest rate. Therefore just 1D integrals need to beevaluated and they can be precisely computed with standard quadraturerules.6.3.6.3 Continuum Limit and American OptionsIn the specific case of an American option, the possibility of exercise at anytime up to the expiration date allows to develop, within the path integralformalism, a specific algorithm, which, as shown in the following, is preciseand very quick [Montagna et. al. (2002)].Given the time slicing considered above, the case of American optionsrequires the limit ∆t −→ 0 which, putting σ −→ 0, leads to a delta–liketransition probabilityp(z, t + ∆t|z t , t) ≈ δ(z − z t − A∆t).


1036 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis means that, apart from volatility effects, the price z i at time t i willhave a value remarkably close to the expected value ¯z = z i−1 + A∆t, givenby the drift growth. In order to take care of the volatility effects, a possiblesolution is to estimate the integral of interest, i.e.,E[O i |S i−1 ] =∫ +∞−∞dz p(z|z i−1 )O i (e z ), (6.60)by inserting in (6.60) the analytical expression for the p(z|z i−1 ) transitionprobability{1p(z|z i−1 ) = √ exp − (z − z i−1 − A∆t) 2 }2π∆tσ2 2σ 2 ∆t{ }1= √ exp (z − ¯z)2−2π∆tσ2 2σ 2 ,∆ttogether with a Taylor expansion of the kernel function O i (e z ) = f(z)around the expected value ¯z. Hence, up to the second–order in z − ¯z, thekernel function becomeswhich inducesf(z) = f(¯z) + (z − ¯z)f ′ (¯z) + 1 2 f ′′ (¯z)(z − ¯z) 2 + O((z − ¯z) 3 ),E[O i |S i−1 ] = f(¯z) + σ22 f ′′ (¯z), + . . . ,since the first derivative does not give contribution to (6.60), being theintegral of an odd function over the whole z range. The second derivativecan be numerically estimated asf ′′ (¯z) = 1δ 2 [f(¯z + δ σ ) − 2f(¯z) + f(¯z − δ σ )],σwith δ σ = O(σ √ ∆t), as dictated by the dynamics of the stochastic process.6.3.7 Application: Nonlinear Dynamics of Complex NetsRecall that many systems in nature, such as neural nets, food webs,metabolic systems, co–authorship of papers, the worldwide web, etc. canbe represented as complex networks, or small–world networks (see, e.g.,[Watts and Strogatz (1998); Dorogovtsev and Mendes (2003)]). In particular,it has been recognized that many networks have scale–free topology;the distribution of the degree obeys the power law, P (k) ∼ k −γ . The study


<strong>Geom</strong>etrical Path Integrals and Their Applications 1037of the scale–free network now attracts the interests of many researchers inmathematics, physics, engineering and biology [Ichinomiya (2004)].Another important aspect of complex networks is their dynamics, describinge.g., the spreading of viruses in the Internet, change of populationsin a food web, and synchronization of neurons in a brain. In particular,[Ichinomiya (2004)] studied the synchronization of the random network ofoscillators. His work follows the previous studies (see [Strogatz (2000)]) thatshowed that mean–field type synchronization, that Kuramoto observed inglobally–coupled oscillators [Kuramoto (1984)], appeared also in the small–world networks.6.3.7.1 Continuum Limit of the Kuramoto NetIchinomiya started with the standard network with N nodes, described bya variant of the Kuramoto model. Namely, at each node, there exists anoscillator and the phase of each oscillator θ i is evolving according to˙θ i = ω i + K ∑ ja ij sin(θ j − θ i ), (6.61)where K is the coupling constant, a ij is 1 if the nodes i and j are connected,and 0 otherwise; ω i is a random number, whose distribution is given by thefunction N(ω).For the analytic study, it is convenient to use the continuum limit equation.We define P (k) as the distribution of nodes with degree k, andρ(k, ω; t, θ) the density of oscillators with phase θ at time t, for given ωand k. We assume that ρ(k, ω; t, θ) is normalized as∫ 2π0ρ(k, ω; t, θ)dθ = 1.For simplicity, we also assume N(ω) = N(−ω). Thus, we suppose that thecollective oscillation corresponds to the stable solution, ˙ρ = 0.Now we construct the continuum limit equation for the network of oscillators.The evolution of ρ is determined by the continuity equation ∂ t ρ =−∂ θ (ρv), where v is defined by the continuum limit of the r.h.s of (6.61).Because one randomly selected edge connects to the node of degree k, frequencyω, phase θ with the probability kP (k)N(ω)ρ(k, ω; t, θ)/ ∫ dkkP (k),


1038 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionρ(k, ω; t, θ) obeys the equation∂ t ρ(k, ω; t, θ) = −∂ θ [ρ(k, ω; t, θ) (ω ++ Kk ∫ dω ′ ∫ dk ′ ∫ dθ ′ N(ω ′ )P (k ′ )k ′ ρ(k ′ , ω ′ ; t, θ ′ ) sin(θ − θ ′ )∫dk′ P (k ′ )k ′ )].The mean–field solution of this equation was studied by [Ichinomiya (2004)].6.3.7.2 Path–Integral Approach to Complex NetsRecently, [Ichinomiya (2005)] introduced the path–integral (see subsection4.4.6 above) approach in studying the dynamics of complex networks. Heconsidered the stochastic generalization of the Kuramoto network (6.61),given byẋ i = f i (x i ) +N∑a ij g(x i , x j ) + ξ i (t), (6.62)j=1where f i = f i (x i ) and g ij = g(x i , x j ) are functions of network activations x i ,ξ i (t) is a random force that satisfies 〈ξ i (t) = 0〉, 〈ξ i (t)ξ j (t ′ )〉 = δ ij δ(t−t ′ )σ 2 .He assumed x i = x i,0 at t = 0. In order to discuss the dynamics of thissystem, he introduced the so–called Matrin–Siggia–Rose (MSR) generatingfunctional Z given by [de Dominicis (1978)]( ) 〈 NNt ∫ 1 ∏ NZ[{l ik }, {¯l ik }] =πwhere the action S is given by∏N ti=1 k=0dx ik d¯x ik e −S exp(l ik x ik + ¯l ik ¯x ik )J〉,S = ∑ ik[ σ2 ∆t¯x 22ik+i¯x ik {x ik −x i,k−1 −∆t(f i (x i,k−1 )+ ∑ ja ij g(x i,k−1 , x j,k−1 ))}],and 〈· · · 〉 represents the average over the ensemble of networks. J is thefunctional Jacobian term,⎛⎞J = exp ⎝− ∆t ∑ ∂(f i (x ik ) + a ij g(x ik , x jk ))⎠ .2∂x ikijkIchinomiya considered such a form of the network model in which{ 1 with probability pij ,a ij =0 with probability 1 − p ij .


<strong>Geom</strong>etrical Path Integrals and Their Applications 1039Note that p ij can be a function of variables such as i or j. For example, inthe 1D chain model, p ij is 1 if |i − j| = 1, else it is 0. The average over allnetworks can be expressed as〈 ⎡⎤exp ⎣ ∑ 〉∑i∆t¯x ik a ij g(x i,k−1 , x j,k−1 ) ⎦ =ikj[} ]∏p ij exp i∆t¯x ik g(x i,k−1 , x j,k−1 ) + 1 − p ij ,so we getij{ ∑k〈e −S 〉 = exp(−S 0 ) ∏ ij[ { ∑p ij expki∆t¯x ik g(x i,k−1 , x j,k−1 )}+ 1 − p ij],whereS 0 = ∑ ikσ 2 ∆t¯x 2 ik + i¯x ik {x ik − x i,k−1 − ∆tf i (x i,k−1 )}.2This expression can be applied to the dynamics of any complex networkmodel. [Ichinomiya (2005)] applied this model to analysis of the Kuramototransition in random sparse networks.6.3.8 Application: Dissipative Quantum Brain ModelThe conservative brain model was originally formulated within the frameworkof the quantum field theory (QFT) by [Ricciardi and Umezawa (1967)]and subsequently developed in [Stuart et al. (1978); Stuart et al. (1979);Jibu and Yasue (1995); Jibu et al (1996)]. The conservative brain model hasbeen recently extended to the dissipative quantum dynamics in the work ofG. Vitiello and collaborators [Vitiello (1995); Alfinito and Vitiello (2000);Pessa and Vitiello (1999); Vitiello (2001); Pessa and Vitiello (2003);Pessa and Vitiello (2004)].The canonical quantization procedure of a dissipative system requiresto include in the formalism also the system representing the environment(usually the heat bath) in which the system is embedded. One possible wayto do that is to depict the environment as the time–reversal image of thesystem [Celeghini et al. (1992)]: the environment is thus described as thedouble of the system in the time–reversed dynamics (the system image inthe mirror of time).Within the framework of dissipative QFT, the brain system is describedin terms of an infinite collection of damped harmonic oscillators A κ (the


1040 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsimplest prototype of a dissipative system) representing the DWQ [Vitiello(1995)]. Now, the collection of damped harmonic oscillators is ruled by theHamiltonian [Vitiello (1995); Celeghini et al. (1992)]H = H 0 + H I ,withH 0 = Ω κ (A † κA κ − Æ κÃκ), H I = iΓ κ (A † κÆ κ − A κ à κ ),where Ω κ is the frequency and Γ κ is the damping constant. The Ãκ modesare the ‘time–reversed mirror image’ (i.e., the ‘mirror modes’) of the A κmodes. They are the doubled modes, representing the environment modes,in such a way that κ generically labels their degrees of freedom. In particular,we consider the damped harmonic oscillator (DHO)mẍ + γẋ + κx = 0, (6.63)as a simple prototype for dissipative systems (with intention that thus getresults also apply to more general systems). The damped oscillator (6.63) isa non–Hamiltonian system and therefore the customary canonical quantizationprocedure cannot be followed. However, one can face the problem byresorting to well known tools such as the density matrix ρ and the Wignerfunction W .Let us start with the special case of a conservative particle in the absenceof friction γ, with the standard Hamiltonian, H = −(∂ x ) 2 /2m + V (x).Recall (from the previous subsection) that the density matrix equationof motion, i.e., quantum Liouville equation, is given byThe density matrix function ρ is defined byi˙ρ = [H, ρ]. (6.64)〈x + 1 2 y|ρ(t)|x − 1 2 y〉 = ψ∗ (x + 1 2 y, t)ψ(x − 1 y, t) ≡ W (x, y, t),2with the associated standard expression for the Wigner function (see [Feynmanand Hibbs (1965)]),W (p, x, t) = 1 ∫W (x, y, t) e (−i py ) dy.2πNow, in the coordinate x−representation, by introducing the notationx ± = x ± 1 y, (6.65)2


<strong>Geom</strong>etrical Path Integrals and Their Applications 1041the Liouville equation (6.64) can be expanded asi ∂ t 〈x + |ρ(t)|x − 〉 = (6.66)]}{−[∂ 2x 2 2m +− ∂x 2 −+ [V (x + ) − V (x − )] 〈x + |ρ(t)|x − 〉,while the Wigner function W (p, x, t) is now given byi ∂ t W (x, y, t) = H o W (x, y, t),withH o = 1 m p xp y + V (x + 1 2 y) − V (x − 1 y), (6.67)2and p x = −i∂ x , p y = −i∂ y .The new Hamiltonian H o (6.67) may be get from the corresponding LagrangianL o = mẋẏ − V (x + 1 2 y) + V (x − 1 y). (6.68)2In this way, Vitiello concluded that the density matrix and the Wignerfunction formalism required, even in the conservative case (with zero mechanicalresistance γ), the introduction of a ‘doubled’ set of coordinates,x ± , or, alternatively, x and y. One may understand this as related to theintroduction of the ‘couple’ of indices necessary to label the density matrixelements (6.66).Let us now consider the case of the particle interacting with a thermalbath at temperature T . Let f denote the random force on the particle atthe position x due to the bath. The interaction Hamiltonian between thebath and the particle is written asH int = −fx. (6.69)Now, in the Feynman–Vernon formalism (see [Feynman (1972)]), theeffective action A[x, y] for the particle is given byA[x, y] =∫ tfwith L o defined by (6.68) ande i I[x,y] = 〈(e − i R t ft it iL o (ẋ, ẏ, x, y) dt + I[x, y],f(t)x −(t)dt ) − (e i R t ft if(t)x +(t)dt ) + 〉, (6.70)where the symbol 〈.〉 denotes the average with respect to the thermal bath;‘(.) + ’ and ‘(.) − ’ denote time ordering and anti–time ordering, respectively;the coordinates x ± are defined as in (6.65). If the interaction between the


1042 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbath and the coordinate x (6.69) were turned off, then the operator f ofthe bath would develop in time according to f(t) = e iHγt/ fe −iHγt/ , whereH γ is the Hamiltonian of the isolated bath (decoupled from the coordinatex). f(t) is then the force operator of the bath to be used in (6.70).The interaction I[x, y] between the bath and the particle has been evaluatedin [Srivastava et al. (1995)] for a linear passive damping due tothermal bath by following Feynman–Vernon and Schwinger [Feynman andHibbs (1965)]. The final result from [Srivastava et al. (1995)] is:I[x, y] = 1 2+ i2∫ tfdt [x(t)F retyt i∫ tf∫ tft it i(t) + y(t)F adv (t)]dtds N(t − s)y(t)y(s),where the retarded force on y, Fyret , and the advanced force on x, Fxadv , aregiven in terms of the retarded and advanced Green functions G ret (t − s)and G adv (t − s) byF rety (t) =∫ tft ids G ret (t − s)y(s), F advx (t) =∫ tft ixds G adv (t − s)x(s),respectively. In (6.71), N(t − s) is the quantum noise in the fluctuatingrandom force given by: N(t − s) = 1 2〈f(t)f(s) + f(s)f(t)〉.The real and the imaginary part of the action are given respectively byRe (A[x, y]) =∫ tfL = mẋẏ −[V (x + 1 2 y) − V (x − 1 ]2 y) + 1 [xFrety2t iL dt, (6.71)+ yFxadv ], (6.72)and Im (A[x, y]) = 1 ∫ tf∫ tfN(t − s)y(t)y(s) dtds. (6.73)2 t i t iEquations (6.71–6.73), are exact results for linear passive damping dueto the bath. They show that in the classical limit ‘ → 0’ nonzero y yields an‘unlikely process’ in view of the large imaginary part of the action implicitin (6.73). Nonzero y, indeed, may lead to a negative real exponent in theevolution operator, which in the limit → 0 may produce a negligiblecontribution to the probability amplitude. On the contrary, at quantumlevel nonzero y accounts for quantum noise effects in the fluctuating random


<strong>Geom</strong>etrical Path Integrals and Their Applications 1043force in the system–environment coupling arising from the imaginary partof the action (see [Srivastava et al. (1995)]).When in (6.72) we useL(ẋ, ẏ, x, y) = mẋẏ − VF rety = γẏ and F advx = −γẋ we get,(x + 1 )2 y + V(x − 1 )2 y + γ (xẏ − yẋ). (6.74)2By usingV(x ± 1 )2 y = 1 2 κ(x ± 1 2 y)2in (6.74), the DHO equation (6.63) and its complementary equation for they coordinatemÿ − γẏ + κy = 0. (6.75)are derived. The y−oscillator is the time–reversed image of the x−oscillator(6.63). From the manifolds of solutions to equations (6.63) and (6.75), wecould choose those for which the y coordinate is constrained to be zero,they simplify tomẍ + γẋ + κx = 0, y = 0.Thus we get the classical damped oscillator equation from a Lagrangiantheory at the expense of introducing an ‘extra’ coordinate y, later constrainedto vanish. Note that the constraint y(t) = 0 is not in violation ofthe equations of motion since it is a true solution to (6.63) and (6.75).6.3.9 Application: Cerebellum as a Neural Path–IntegralRecall that human motion is naturally driven by synergistic action of morethan 600 skeletal muscles. While the muscles generate driving torques inthe moving joints, subcortical neural system performs both local and global(loco)motion control: first reflexly controlling contractions of individualmuscles, and then orchestrating all the muscles into synergetic actions inorder to produce efficient movements. While the local reflex control ofindividual muscles is performed on the spinal control level, the global integrationof all the muscles into coordinated movements is performed withinthe cerebellum.


1044 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAll hierarchical subcortical neuro–muscular physiology, from the bottomlevel of a single muscle fiber, to the top level of cerebellar muscular synergy,acts as a temporal < out|in > reaction, in such a way that the higher levelacts as a command/control space for the lower level, itself representing anabstract image of the lower one:(1) At the muscular level, we have excitation–contraction dynamics [Hatze(1977a); Hatze (1978); Hatze (1977b)], in which < out|in > is given bythe following sequence of nonlinear diffusion processes: neural-actionpotential synaptic-potentialmuscular-action-potentialexcitationcontraction-coupling muscle-tension-generating [<strong>Ivancevic</strong> (1991);<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)]. Its purpose is the generation of muscularforces, to be transferred into driving torques within the jointanatomical geometry.(2) At the spinal level, < out|in > is given by autogenetic–reflex stimulus–response control [Houk (1979)]. Here we have a neural image of allindividual muscles. The main purpose of the spinal control level is togive both positive and negative feedbacks to stabilize generated muscularforces within the ‘homeostatic’ (or, more appropriately, ‘homeokinetic’)limits. The individual muscular actions are combined intoflexor–extensor (or agonist–antagonist) pairs, mutually controlling eachother. This is the mechanism of reciprocal innervation of agonists andinhibition of antagonists. It has a purely mechanical purpose to formthe so–called equivalent muscular actuators (EMAs), which would generatedriving torques T i (t) for all movable joints.(3) At the cerebellar level, < out|in > is given by sensory–motor integration[Houk et al. (1996)]. Here we have an abstracted image of allautogenetic reflexes. The main purpose of the cerebellar control levelis integration and fine tuning of the action of all active EMAs intoa synchronized movement, by supervising the individual autogeneticreflex circuits. At the same time, to be able to perform in new andunknown conditions, the cerebellum is continuously adapting its ownneural circuitry by unsupervised (self–organizing) learning. Its actionis subconscious and automatic, both in humans and in animals.Naturally, we can ask the question: Can we assign a single < out|in >measure to all these neuro–muscular stimulus–response reactions? We thinkthat we can do it; so in this Letter, we propose the concept of adaptivesensory–motor transition amplitude as a unique measure for this temporal


<strong>Geom</strong>etrical Path Integrals and Their Applications 1045< out|in > relation. Conceptually, this < out|in > −amplitude can beformulated as the ‘neural path integral’:∫< out|in >≡ 〈motor|sensory〉 =amplitudeD[w, x] e i S[x] . (6.76)Here, the integral is taken over all activated (or, ‘fired’) neural pathwaysx i = x i (t) of the cerebellum, connecting its input sensory−state with itsoutput motor−state, symbolically described by adaptive neural measureD[w, x], defined by the weighted product (of discrete time steps)n∏D[w, x] = lim w i (t) dx i (t),n−→∞in which the synaptic weights w i = w i (t), included in all active neuralpathways x i = x i (t), are updated by the unsupervised Hebbian–like learningrule 6.7, namelyt=1w i (t + 1) = w i (t) + σ η (wi d(t) − w i a(t)), (6.77)where σ = σ(t), η = η(t) represent local neural signal and noise amplitudes,respectively, while superscripts d and a denote desired and achievedneural states, respectively. Theoretically, equations (6.76–6.77) define an∞−dimensional neural network. Practically, in a computer simulation wecan use 10 7 ≤ n ≤ 10 8 , roughly corresponding to the number of neurons inthe cerebellum.The exponent term S[x] in equation (6.76) represents the autogenetic–reflex action, describing reflexly–induced motion of all active EMAs, fromtheir initial stimulus−state to their final response−state, along the familyof extremal (i.e., Euler–Lagrangian) paths x i min (t). (S[x] is properlyderived in (6.80–6.81) below.)6.3.9.1 Spinal Autogenetic Reflex ControlRecall (from Introduction) that at the spinal control level we have the autogeneticreflex motor servo [Houk (1979)], providing the local, reflex feedbackloops for individual muscular contractions. A voluntary contraction forceF of human skeletal muscle is reflexly excited (positive feedback +F −1 )by the responses of its spindle receptors to stretch and is reflexly inhibited(negative feedback -F −1 ) by the responses of its Golgi tendon organs


1046 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionto contraction. Stretch and unloading reflexes are mediated by combinedactions of several autogenetic neural pathways, forming the motor servo.In other words, branches of the afferent fibers also synapse with with interneuronsthat inhibit motor neurons controlling the antagonistic muscles– reciprocal inhibition. Consequently, the stretch stimulus causes the antagoniststo relax so that they cannot resists the shortening of the stretchedmuscle caused by the main reflex arc. Similarly, firing of the Golgi tendonreceptors causes inhibition of the muscle contracting too strong andsimultaneous reciprocal activation of its antagonist. Both mechanisms ofreciprocal inhibition and activation performed by the autogenetic circuits+F −1 and -F −1 , serve to generate the well–tuned EMA–driving torquesT i .Now, once we have properly defined the symplectic musculo–skeletal dynamics[<strong>Ivancevic</strong> (2004)] on the biomechanical (momentum) phase–spacemanifold T ∗ M N , we can proceed in formalizing its hierarchical subcorticalneural control. By introducing the coupling Hamiltonians H m = H m (q, p),selectively corresponding only to the M ≤ N active joints, we define theaffine Hamiltonian control function H aff : T ∗ M N → R, in local canonicalcoordinates on T ∗ M N given by (adapted from [Nijmeijer and van derSchaft (1990)] for the biomechanical purpose)H aff (q, p) = H 0 (q, p) − H m (q, p) T m , (m = 1, . . . , M ≤ N), (6.78)where T m = T m (t, q, p) are affine feedback torque one–forms, different fromthe initial driving torques T i acting in all the joints. Using the affine Hamiltonianfunction (6.78), we get the affine Hamiltonian servo–system [<strong>Ivancevic</strong>(2004)],˙q i = ∂H 0(q, p)∂p i− ∂Hm (q, p)∂p iT m , (6.79)ṗ i = − ∂H 0(q, p)∂q i + ∂Hm (q, p)∂q i T m ,q i (0) = q i 0, p i (0) = p 0 i , (i = 1, . . . , N; m = 1, . . . , M ≤ N).The affine Hamiltonian control system (6.79) gives our formal descriptionfor the autogenetic spinal motor–servo for all M ≤ N activated (i.e., working)EMAs.


<strong>Geom</strong>etrical Path Integrals and Their Applications 10476.3.9.2 Cerebellum – the ComparatorHaving, thus, defined the spinal reflex control level, we proceed to model thetop subcortical commander/controller, the cerebellum. It is a brain regionanatomically located at the bottom rear of the head (the hindbrain), directlyabove the brainstem, which is important for a number of subconsciousand automatic motor functions, including motor learning. It processes informationreceived from the motor cortex, as well as from proprioceptorsand visual and equilibrium pathways, and gives ‘instructions’ to the motorcortex and other subcortical motor centers (like the basal nuclei), whichresult in proper balance and posture, as well as smooth, coordinated skeletalmovements, like walking, running, jumping, driving, typing, playing thepiano, etc. Patients with cerebellar dysfunction have problems with precisemovements, such as walking and balance, and hand and arm movements.The cerebellum looks similar in all animals, from fish to mice to humans.This has been taken as evidence that it performs a common function, suchas regulating motor learning and the timing of movements, in all animals.Studies of simple forms of motor learning in the vestibulo–ocular reflexand eye–blink conditioning are demonstrating that timing and amplitudeof learned movements are encoded by the cerebellum.The cerebellum is responsible for coordinating precisely timed activity by integrating motor output with ongoing sensory feedback.It receives extensive projections from sensory–motor areas of thecortex and the periphery and directs it back to premotor and motor cortex[Ghez (1990); Ghez (1991)]. This suggests a role in sensory–motorintegration and the timing and execution of human movements. The cerebellumstores patterns of motor control for frequently performed movements,and therefore, its circuits are changed by experience and training.It was termed the adjustable pattern generator in the work of J. Houkand collaborators [Houk et al. (1996)]. Also, it has become the inspiring‘brain–model’ in the recent robotic research [Schaal and Atkeson (1998);Schaal (1998)].Comparing the number of its neurons (10 7 − 10 8 ), to the size of conventionalneural networks, suggests that artificial neural nets cannot satisfactorilymodel the function of this sophisticated ‘super–bio–computer’, asits dimensionality is virtually infinite. Despite a lot of research dedicatedto its structure and function (see [Houk et al. (1996)] and references therecited), the real nature of the cerebellum still remains a ‘mystery’.The main function of the cerebellum as a motor controller is depicted


1048 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 6.5 Schematic < out|in > organization of the primary cerebellar circuit. In essence,excitatory inputs, conveyed by collateral axons of Mossy and Climbing fibers activatedirectly neurones in the Deep cerebellar nuclei. The activity of these latter is alsomodulated by the inhibitory action of the cerebellar cortex, mediated by the Purkinjecells.Fig. 6.6The cerebellum as a motor controller.in Figure 6.6. A coordinated movement is easy to recognize, but we knowlittle about how it is achieved. In search of the neural basis of coordination,a model of spinocerebellar interactions was recently presented in [Apps andGarwicz (2005)], in which the structural and functional organizing principleis a division of the cerebellum into discrete micro–complexes. Each micro–complex is the recipient of a specific motor error signal - that is, a signalthat conveys information about an inappropriate movement. These signalsare encoded by spinal reflex circuits and conveyed to the cerebellar cortex


<strong>Geom</strong>etrical Path Integrals and Their Applications 1049through climbing fibre afferents. This organization reveals salient featuresof cerebellar information processing, but also highlights the importance ofsystems level analysis for a fuller understanding of the neural mechanismsthat underlie behavior.6.3.9.3 Hamiltonian Action and Neural Path IntegralHere, we propose a quantum–like adaptive control approach to modellingthe ‘cerebellar mystery’. Corresponding to the affine Hamiltonian controlfunction (6.78) we define the affine Hamiltonian control action,S aff [q, p] =∫ toutt indτ [ p i ˙q i − H aff (q, p) ] . (6.80)From the affine Hamiltonian action (6.80) we further derive the associatedexpression for the neural phase–space path integral (in normal units),representing the cerebellar sensory–motor amplitude < out|in >,∫〉=〈qiout , p outi|q i in, p iniwithD[w, q, p] e i S aff [q,p](6.81)∫{ ∫ tout= D[w, q, p] exp i dτ [ p i ˙q i − H aff (q, p) ]} ,t in∫∫∏ nw i (τ)dp i (τ)dq i (τ)D[w, q, p] =,2πwhere w i = w i (t) denote the cerebellar synaptic weights positioned along itsneural pathways, being continuously updated using the Hebbian–like self–organizing learning rule (6.77). Given the transition amplitude < out|in >(6.81), the cerebellar sensory–motor transition probability is defined as itsabsolute square, | < out|in > | 2 .In (6.81), qin i = qi in (t), qi out = qout(t); i p ini = p ini (t), pout i =p outi (t); t in ≤ t ≤ t out , for all discrete time steps, t = 1, ..., n −→∞, and we are allowing for the affine Hamiltonian H aff (q, p) to dependupon all the (M ≤ N) EMA–angles and angular momenta collectively.Here, we actually systematically took a discretized differentialtime limit of the form t σ − t σ−1 ≡ dτ (both σ and τ denote discretetime steps) and wrote (qi σ −qi σ−1 )(t σ−t σ−1)≡ ˙q i . For technical details regardingthe path integral calculations on Riemannian and symplectic manifolds(including the standard regularization procedures), see [Klauder (1997);Klauder (2000)].τ=1


1050 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNow, motor learning occurring in the cerebellum can be observed usingfunctional MR imaging, showing changes in the cerebellar action potential,related to the motor tasks (see, e.g., [Mascalchi et. al. (2002)]). To accountfor these electro–physiological currents, we need to add the source termJ i (t)q i (t) to the affine Hamiltonian action (6.80), (the current J i = J i (t)acts as a source J i A i of the cerebellar electrical potential A i = A i (t)),S aff [q, p, J] =∫ toutt indτ [ p i ˙q i − H aff (q, p) + J i q i] ,which, subsequently gives the cerebellar path integral with the action potentialsource, coming either from the motor cortex or from other subcorticalareas.Note that the standard Wick rotation: t ↦→ it (see [Klauder (1997);Klauder (2000)]), makes all our path integrals real, i.e.,∫∫D[w, q, p] e i S aff [q,p]W−−−→ick D[w, q, p] e − S aff [q,p] ,while their subsequent discretization gives the standard thermodynamicpartition functions,Z = ∑ je −wjEj /T , (6.82)where E j is the energy eigenvalue corresponding to the affine HamiltonianH aff (q, p), T is the temperature–like environmental control parameter, andthe sum runs over all energy eigenstates (labelled by the index j). From(6.82), we can further calculate all statistical and thermodynamic systemproperties (see [Feynman (1972)]), as for example, transition entropy S =k B ln Z, etc.6.3.10 Path Integrals via Jets: Perturbative QuantumFieldsRecall that an elegant way to make geometrical path integrals rigorous is toformulate them using the jet formalism. In this way the covariant Hamiltonianfield systems were presented in [Bashkirov and Sardanashvily (2004)].In this subsection we give a brief review of this perturbative quantum fieldmodel.Let us quantize a Lagrangian system with the Lagrangian L N (5.316)on the constraint manifold N L (5.314). In the framework of a perturbative


<strong>Geom</strong>etrical Path Integrals and Their Applications 1051quantum field theory, we should assume that X = R n and Y → X is a trivialaffine bundle. It follows that both the original coordinates (x α , y i , p α i )and the adapted coordinates (x α , y i , p a , p A ) on the Legendre bundle Π areglobal. Passing to field theory on an Euclidean space R n , we also assumethat the matrix a in the Lagrangian L (5.312) is positive–definite, i.e.,a AA > 0.Let us start from a Lagrangian (5.316) without gauge symmetries. Sincethe Lagrangian constraint space N L can be equipped with the adapted coordinatesp A , the generating functional of Euclidean Green functions of theLagrangian system in question reads [Bashkirov and Sardanashvily (2004)]∫ ∫Z = N −1 exp{ (L N + 1 2 Tr(ln σ 0)+iJ i y i +iJ A p A )ω} ∏ [dp A (x)][dy(x)],x(6.83)where L N is given by the expression (5.322) and σ 0 is the square matrixσ AB0 = M −1αAiM −1µBjσ 0ijαµ = δ AB (a AA ) −1 .The generating functional (6.83) a Gaussian integral of variables p A (x). Itsintegration with respect to p A (x) under the condition J A = 0 restarts thegenerating functional∫ ∫Z = N −1 exp{ (L + iJ i y i )ω} ∏ [dy(x)], (6.84)xof the original Lagrangian field system on Y with the Lagrangian (5.312).However, the generating functional (6.83) cannot be rewritten with respectto the original variables p µ i , unless a is a nondegenerate matrix function.In order to overcome this difficulty, let us consider a Lagrangian systemon the whole Legendre manifold Π with the Lagrangian L Π (5.319). Sincethis Lagrangian is constant along the fibres of the vector bundle Π → N L ,an integration of the generating functional of this field model with respect tovariables p a (x) should be finite. One can choose the generating functionalin the form [Bashkirov and Sardanashvily (2004)]∫ ∫Z = N −1 exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j (6.85)+ 1 2 Tr(ln σ) + iJ iy i + iJ i µp µ i )ω} ∏ x[dp(x)][dy(x)].Its integration with respect to momenta p α i (x) restarts the generating functional(6.84) of the original Lagrangian system on Y . In order to get the


1052 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongenerating functional (6.85), one can follow a procedure of quantizationof gauge–invariant Lagrangian systems. In the case of the Lagrangian L Π(5.319), this procedure is rather trivial, since the space of momenta variablesp a (x) coincides with the translation subgroup of the gauge groupAut Ker σ 0 .Now let us suppose that the Lagrangian L N (5.316) and, consequently,the Lagrangian L Π (5.319) are invariant under some gauge group G X ofvertical automorphisms of the fibre bundle Y → X (and the induced automorphismsof Π → X) which acts freely on the space of sections ofY → X. Its infinitesimal generators are represented by vertical vector–fields u = u i (x µ , y j )∂ i on Y → X which induce the vector–fieldsu = u i ∂ i − ∂ j u i p α i ∂ j α + d α u i ∂ α i , d α = ∂ α + y i α∂ i , (6.86)on Π × J 1 (X, Y ). Let us also assume that G X is indexed by m parameterfunctions ξ r (x) such that u = u i (x α , y j , ξ r )∂ i , whereu i (x α , y j , ξ r ) = u i r(x α , y j )ξ r + u iµr (x α , y j )∂ µ ξ r (6.87)are linear first–order differential operators on the space of parameters ξ r (x).The vector–fields u(ξ r ) must satisfy the commutation relations[u(ξ q ), u(ξ ′p )] = u(c r pqξ ′p ξ q ),where c r pq are structure constants. The Lagrangian L Π (5.319) is invariantunder the above gauge transformations iff its Lie derivative L u L Π alongvector–fields (6.86) vanishes, i.e.,(u i ∂ i − ∂ j u i p α i ∂ j α + d α u i ∂ α i )L Π = 0. (6.88)Since the operator L u is linear in momenta p µ i , the condition (6.88) fallsinto the independent conditions(u k ∂ k − ∂ j u k p ν k∂ j ν + d ν u j ∂ ν j )(p α i F i α) = 0, (6.89)(u k ∂ k − ∂ j u k p ν k∂ν)(σ j ij0 λµ pα i p µ j ) = 0, (6.90)u i ∂ i c ′ = 0. (6.91)It follows that the Lagrangian L Π is gauge–invariant iff its three summandsare separately gauge–invariant.Note that, if the Lagrangian L Π on Π is gauge–invariant, the originalLagrangian L (5.312) is also invariant under the same gauge transforma-


<strong>Geom</strong>etrical Path Integrals and Their Applications 1053tions. Indeed, one gets at once from the condition (6.89) thatu(F i µ) = ∂ j u i F j µ, (6.92)i.e., the quantity F is transformed as the dual of momenta p. Then thecondition (6.90) shows that the quantity σ 0 p is transformed by the samelaw as F. It follows that the term aFF in the Lagrangian L (5.312) istransformed exactly as a(σ 0 p)(σ 0 p) = σ 0 pp, i.e., is gauge–invariant. Thenthis Lagrangian is gauge–invariant due to the equality (6.91).Since S i α = y i α − F i α, one can derive from the formula (6.92) the transformationlaw of S,u(S i µ) = d µ u i − ∂ j u i F j µ = d µ u i − ∂ j u i (y j µ − S j µ) = ∂ µ u i + ∂ j u i S j µ. (6.93)This expression shows that the gauge group G X acts freely on the spaceof sections S(x) of the fibre bundle Ker ̂L → Y in the splitting (5.309).Let the number m of parameters of the gauge group G X do not exceed thefibre dimension of Ker ̂L → Y . Then some combinations b r µi Si µ of Sµ i canbe used as the gauge conditionb rµ i Si µ(x) − α r (x) = 0,similar to the generalized Lorentz gauge in Yang–Mills gauge theory.Now we turn to path–integral quantization of a Lagrangian systemwith the gauge–invariant Lagrangian L Π (5.319). In accordance with thewell–known quantization procedure, let us modify the generating functional(6.85) as follows [Bashkirov and Sardanashvily (2004)]Z = N −1 ∫∫exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j+ 1 2 Tr(ln σ) − 1 2 h rsα r α s + iJ i y i + iJµp i µ i )ω} (6.94)∆ ∏ × r δ(b rµ i Si µ(x) − α r (x))[dα(x)][dp(x)][dy(x)]x∫ ∫= N ′−1 exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j + 1 Tr(ln σ)2− 1 2 h rsb rµ i bsα j SµS i α j + iJ i y i + iJµp i µ i )ω}∆ ∏ [dp(x)][dy(x)],x∫ ∫where exp{ (− 1 2 h rsα r α s )ω} ∏ [dα(x)]x


1054 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis a Gaussian integral, while the factor ∆ is defined by the condition∫ ∏∆x× r δ(u(ξ)(b rµ i Si µ))[dξ(x)] = 1.We have the linear second–order differential operatorM r s ξ s = u(ξ)(b rµ i Si µ(x)) = b rµ i (∂ µu i (ξ) + ∂ j u i (ξ)S j µ) (6.95)on the parameter functions ξ(x), and get ∆ = det M. Then the generatingfunctional (6.95) takes the form∫ ∫Z = N ′−1 exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j + 1 2 Tr ln σ − 1 2 h rsb rµ i bsα j SµS i αj− c r M r s c s + iJ i y i + iJ i µp µ i )ω} ∏ x[dc][dc][dp(x)][dy(x)], (6.96)where c r , c s are odd ghost fields. Integrating Z (6.96) with respect tomomenta under the condition Jµ i = 0, we come to the generating functional∫ ∫Z = N ′−1 exp{ (L− 1 2 h rsb rµ i bsα j SµS i α−c j r Ms r c s +iJ i y i )ω} ∏ [dc][dc][dy(x)]x(6.97)of the original field model on Y with the gauge–invariant Lagrangian L(5.312).Note that the LagrangianL ′ = L − 1 2 h rsb rµ i bsα j S i µS j α − c r M r s c s (6.98)fails to be gauge–invariant, but it admits the so–called BRST symmetry 10whose odd operator readsϑ = u i (x µ , y i , c s )∂ i + d α u i (x µ , y i , c s )∂iα + v r (x µ , y i , yµ) i ∂ (6.99)∂c r+ v r (x µ , y i , c s ) ∂∂c r + d αv r (x µ , y i , c s ) ∂∂c r + d µ d α v r (x µ , y i , c s ∂)α∂c r ,µλd α = ∂ α + yα∂ i i + yλµ∂ i µ i + ∂cr α∂c r + ∂cr λµ .10 Recall that the BRST formalism is a method of implementing first class constraints.The letters BRST stand for Becchi, Rouet, Stora, and (independently) Tyutin whodiscovered this formalism. It is a rigorous method to deal with quantum theories withgauge invariance.∂c r µ


<strong>Geom</strong>etrical Path Integrals and Their Applications 1055Its components u i (x µ , y i , c s ) are given by the expression (6.87) where parameterfunctions ξ r (x) are replaced with the ghosts c r . The componentsv r and v r of the BRST operator ϑ can be derived from the conditionϑ(L ′ ) = −h rs M r q b sα j S j αc q − v r M r q c q + c r ϑ(ϑ(b rα j S j α)) = 0of the BRST invariance of L ′ . This condition falls into the two independentrelationsh rs M r q b sα j S j α + v r M r q = 0,ϑ(c q )(ϑ(c p )(b rα j S j α)) = u(c p )(u(c q )(b rα j S j α)) + u(v r )(b rα j S j α)= u( 1 2 cr pqc p c q + v r )(b rα j S j α) = 0.Hence, we get: v r = −h rs b sα j Sj α, and v r = − 1 2 cr pqc p c q .6.4 Sum over <strong>Geom</strong>etries and TopologiesRecall that the term quantum gravity (or quantum geometrodynamics, orquantum geometry), is usually understood as a consistent fundamentalquantum description of gravitational space–time geometry whose classicallimit is Einstein’s general relativity. Among the possible ramificationsof such a theory are a model for the structure of space–time nearthe Planck scale, a consistent calculational scheme to calculate gravitationaleffects at all energies, a description of quantum geometry nearspace–time singularities and a non–perturbative quantum description of4D black holes. It might also help us in understanding cosmologicalissues about the beginning and end of the universe, i.e., the so–called‘big bang’ and ‘big crunch’ (see e.g., [Penrose (1989); Penrose (1994);Penrose (1997)]).From what we know about the quantum dynamics of other fundamentalinteractions it seems eminently plausible that also the gravitational excitationsshould at very short scales be governed by quantum laws. Now,conventional perturbative path integral expansions of gravity, as well as perturbativeexpansion in the string coupling in the case of unified approaches,both have difficulty in finding any direct or indirect evidence for quantumgravitational effects, be they experimental or observational, which couldgive a feedback for model building. The outstanding problems mentionedabove require a non–perturbative treatment; it is not sufficient to know thefirst few terms of a perturbation series. The real goal is to search for a


1056 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionnon–perturbative definition of such a theory, where the initial input of anyfixed ‘background metric’ is inessential (or even undesirable), and where‘space–time’ is determined dynamically. Whether or not such an approachnecessarily requires the inclusion of higher dimensions and fundamental supersymmetryis currently unknown (see [Ambjørn and Kristjansen (1993);Ambjørn and Loll (1998); Ambjørn et. al. (2000a); Ambjørn et. al. (2000b);Ambjørn et. al. (2001a); Ambjørn et. al. (2001b); Ambjørn et. al. (2001c);Dasgupta and Loll (2001)]).Such a non–perturbative viewpoint is very much in line with howone proceeds in classical geometrodynamics, where a metric space–time(M, g µν ) (+ matter) emerges only as a solution to the familiar EinsteinequationG µν [g] ≡ R µν [g] − 1 2 g µνR[g] = −8πT µν [Φ], (6.100)which define the classical dynamics of fields Φ = Φ µν on the space M(M),the space of all metrics g = g µν on a given smooth manifold M. Theanalogous question we want to address in the quantum theory is: Canwe get ‘quantum space–time’ as a solution to a set of non–perturbativequantum equations of motion on a suitable quantum analogue of M(M) orrather, of the space of geometries, <strong>Geom</strong>(M) = M(M)/<strong>Diff</strong>(M)?Now, this is not a completely straightforward task. Whichever waywe want to proceed non–perturbatively, if we give up the privileged roleof a flat, Minkowskian background space–time on which the quantizationis to take place, we also have to abandon the central role usually playedby the Poincaré group, and with it most standard quantum field–theoretictools for regularization and renormalization. If one works in a continuummetric formulation of gravity, the symmetry group of the Einstein–Hilbertaction is instead the group <strong>Diff</strong>(M) of diffeomorphisms on M, which interms of local charts are the smooth invertible coordinate transformationsx µ ↦→ y µ (x µ ).In the following, we will describe a non–perturbative path integral approachto quantum gravity, defined on the space of all geometries, withoutdistinguishing any background metric structure [Loll (2001)]. This is closelyrelated in spirit with the canonical approach of loop quantum gravity [Rovelli(1998)] and its more recent incarnations using so–called spin networks(see, e.g., [Oriti (2001)]). ‘Non–perturbative’ here means in a covariantcontext that the path sum or integral will have to be performed explicitly,and not just evaluated around its stationary points, which can only be


<strong>Geom</strong>etrical Path Integrals and Their Applications 1057achieved in an appropriate regularization. The method we will employ usesa discrete lattice regularization as an intermediate step in the constructionof the quantum theory.6.4.1 Simplicial Quantum <strong>Geom</strong>etryIn this section we will explain how one may construct a theory of quantumgravity from a non–perturbative path integral, using the method ofLorentzian dynamical triangulations. The method is minimal in the senseof employing standard tools from quantum field theory and the theory ofcritical phenomena and adapting them to the case of generally covariantsystems, without invoking any symmetries beyond those of the classical theory.At an intermediate stage of the construction, we use a regularizationin terms of simplicial Regge geometries, that is, piecewise linear manifolds.In this approach, ‘computing the path integral’ amounts to a conceptuallysimple and geometrically transparent ‘counting of geometries’, with additionalweight factors which are determined by the EH action. This is donefirst of all at a regularized level. Subsequently, one searches for interestingcontinuum limits of these discrete models which are possible candidates fortheories of quantum gravity, a step that will always involve a renormalization.From the point of view of statistical mechanics, one may think ofLorentzian dynamical triangulations as a new class of statistical models ofLorentzian random surfaces in various dimensions, whose building blocksare flat simplices which carry a ‘time arrow’, and whose dynamics is entirelygoverned by their intrinsic geometric properties.Before describing the details of the construction, it may be helpful torecall the path integral representation for a 1D non–relativistic particle (seeprevious section). The time evolution of the particle’s wave function ψ maybe described by the integral equation (6.3) above, where the propagator,or the Feynman kernel G, is defined through a limiting procedure (6.4).The time interval t ′′ − t ′ has been discretized into N steps of length ɛ =(t ′′ − t ′ )/N, and the r.h.s. of (6.4) represents an integral over all piecewiselinear paths x(t) of a ‘virtual’ particle propagating from x ′ to x ′′ , illustratedin Figure 6.4 above.The prefactor A −N is a normalization and L denotes the Lagrangianfunction of the particle. Knowing the propagator G is tantamount to havingsolved the quantum dynamics. This is the simplest instance of a path


1058 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 6.7 The time–honored way of illustrating the gravitational path integral as thepropagator from an initial to a final spatial boundary geometry (see text for explanation).integral, and is often written schematically as∫G(x ′ , t ′ ; x ′′ , t ′′ ) = Σ, D[x(t)] e iS[x(t)] , (6.101)where D[x(t)] is a functional measure on the ‘space of all paths’, and theexponential weight depends on the classical action S[x(t)] of a path. Recallalso that this procedure can be defined in a mathematically clean way ifwe Wick–rotate the time variable t to imaginary values t ↦→ τ = it, therebymaking all integrals real [Reed and Simon (1975)].Can a similar strategy work for the case of Einstein geometrodynamics?As an analogue of the particle’s position we can take the geometry [g ij (x)](i.e., an equivalence class of spatial metrics) of a constant–time slice. Canone then define a gravitational propagatorG([g ′ ij], [g ′′ij]) =∫Σ <strong>Geom</strong>(M) D[g µν ] e iSEH [g µν](6.102)from an initial geometry [g ′ ] to a final geometry [g ′′ ] (Figure 6.7) as alimit of some discrete construction analogous to that of the non-relativisticparticle (6.4)? And crucially, what would be a suitable class of ‘paths’, thatis, space–times [g µν ] to sum over? ∫Now, to be able to perform the integration ΣD[g µν ] in a meaningfulway, the strategy we will be following starts from a regularized version ofthe space <strong>Geom</strong>(M) of all geometries. A regularized path integral G(a) canbe defined which depends on an ultraviolet cutoff a and is convergent in anon–trivial region of the space of coupling constants. Taking the continuumlimit corresponds to letting a → 0. The resulting continuum theory – if itcan be shown to exist – is then investigated with regard to its geometric


<strong>Geom</strong>etrical Path Integrals and Their Applications 1059properties and in particular its semiclassical limit.6.4.2 Discrete Gravitational Path IntegralsTrying to construct non–perturbative path integrals for gravity from sumsover discretized geometries, using approach of Lorentzian dynamical triangulations,is not a new idea. Inspired by the successes of lattice gaugetheory, attempts to describe quantum gravity by similar methods have beenpopular on and off since the late 70’s. Initially the emphasis was on gauge–theoretic, first–order formulations of gravity, usually based on (compactifiedversions of) the Lorentz group, followed in the 80’s by ‘quantum Reggecalculus’, an attempt to represent the gravitational path integral as an integralover certain piecewise linear geometries (see [Williams (1997)] andreferences therein), which had first made an appearance in approximatedescriptions of classical solutions of the Einstein equations. A variant ofthis approach by the name of ‘dynamical triangulation(s)’ attracted a lotof interest during the 90’s, partly because it had proved a powerful tool indescribing 2D quantum gravity (see the textbook [Ambjørn et. al. (1997)]and lecture notes [Ambjørn et. al. (2000a)] for more details).The problem is that none of these attempts have so far come up withconvincing evidence for the existence of an underlying continuum theoryof 4D quantum gravity. This conclusion is drawn largely on the basis ofnumerical simulations, so it is by no means water–tight, although one canmake an argument that the ‘symptoms’ of failure are related in the variousapproaches [Loll (1998)]. What goes wrong generically seems to be adominance in the continuum limit of highly degenerate geometries, whoseprecise form depends on the approach chosen. One would expect that non–smooth geometries play a decisive role, in the same way as it can be shownin the particle case that the support of the measure in the continuum limitis on a set of nowhere differentiable paths. However, what seems to happenin the case of the path integral for 4–geometries is that the structures getare too wild, in the sense of not generating, even at coarse–grained scales,an effective geometry whose dimension is anywhere near four.The schematic phase diagram of Euclidean dynamical triangulationsshown in Figure 6.8 gives an example of what can happen. The pictureturns out to be essentially the same in both three and four dimensions: themodel possesses infinite-volume limits everywhere along the critical linek3 crit (k 0 ), which fixes the bare cosmological constant as a function of theinverse Newton constant k 0 ∼ G −1N. Along this line, there is a critical point


1060 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionk0 crit (which we now know to be of first–order in d = 3, 4) below whichgeometries generically have a very large effective or Hausdorff dimension. 11Above k0 crit we find the opposite phenomenon of ‘polymerization’: a typicalelement contributing to the state sum is a thin branched polymer, with oneor more dimensions ‘curled up’ such that its effective dimension is aroundtwo.Fig. 6.8 The phase diagram of 3D and 4D Euclidean dynamical triangulations (see textfor explanation).This problem has to do with the fact that the gravitational action isunbounded below, causing potential havoc in Euclidean versions of thepath integral. Namely, what all the above-mentioned approaches have incommon is that they work from the outset with Euclidean geometries, andassociated Boltzmann-type weights exp(-S eu ) in the path integral. In otherwords, they integrate over ‘space–times’ which know nothing about time,light cones and causality. This is done mainly for technical reasons, sinceit is difficult to set up simulations with complex weights and since untilrecently a suitable Wick rotation was not known.‘Lorentzian dynamical triangulations’, first proposed in [Ambjørn andLoll (1998)] and further elaborated in [Ambjørn et. al. (2000b); Ambjørnet. al. (2001a)] tries to establish a logical connection between the fact thatnon–perturbative path integrals were constructed for Euclidean instead ofLorentzian geometries and their apparent failure to lead to an interestingcontinuum theory.11 In terms of geometry, this means that there are a few vertices at which the entirespace–time ‘condenses’ in the sense that almost every other vertex in the simplicialspace–time is about one link-distance away from them.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1061Fig. 6.9 Positive (a) and negative (b) space–like deficit angles δ (adapted from [Loll(2001); Loll (1998)]).6.4.3 Regge CalculusThe use of simplicial methods in general relativity goes back to the pioneeringwork of Regge [Regge (1961)]. In classical applications one tries toapproximate a classical space–time geometry by a triangulation, that is, apiecewise linear space get by gluing together flat simplicial building blocks,which in dimension d are dD generalizations of triangles. By ‘flat’ we meanthat they are isometric to a subspace of dD Euclidean or Minkowski space.We will only be interested in gluings leading to genuine manifolds, whichtherefore look locally like an R d . A nice feature of such simplicial manifoldsis that their geometric properties are completely described by the discreteset {li 2 } of the squared lengths of their edges. Note that this amounts toa description of geometry without the use of coordinates. There is nothingto prevent us from re–introducing coordinate patches covering the piecewiselinear manifold, for example, on each individual simplex, with suitabletransition functions between patches. In such a coordinate system the metrictensor will then assume a definite form. However, for the purposes offormulating the path integral we will not be interested in doing this, butrather work with the edge lengths, which constitute a direct, regularizedparametrization of the space <strong>Geom</strong>(M) of geometries.How precisely is the intrinsic geometry of a simplicial space, most importantly,its curvature, encoded in its edge lengths? A useful example tokeep in mind is the case of dimension two, which can easily be visualized.A 2D piecewise linear space is a triangulation, and its scalar curvatureR(x) coincides with the Gaussian curvature (see section 3.10.1.3 above).One way of measuring this curvature is by parallel–transporting a vectoraround closed curves in the manifold. In our piecewise–flat manifold such avector will always return to its original orientation unless it has surroundedlattice vertices v at which the surrounding angles did not add up to 2π, but


1062 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction∑i⊃v α i = 2π − δ, for δ ≠ 0, see Figure 6.9. The so–called deficit angle δ isprecisely the rotation angle picked up by the vector and is a direct measurefor the scalar curvature at the vertex. The operational description to getthe scalar curvature in higher dimensions is very similar, one basically hasto sum in each point over the Gaussian curvatures of all 2D submanifolds.This explains why in Regge calculus the curvature part of the EH actionis given by a sum over building blocks of dimension (d − 2) which are theobjects dual to those local 2D submanifolds. More precisely, the continuumcurvature and volume terms of the action become12∫d d x √ | det g| (d) R −→ ∑ Vol(i th (d − 2)−simplex) δ i (6.103)Ri∈R∫d d x √ | det g| −→ ∑ Vol(i th d−simplex) (6.104)Ri∈Rin the simplicial discretization. It is then a simple exercise in trigonometryto express the volumes and angles appearing in these formulas as functionsof the edge lengths l i , both in the Euclidean and the Minkowskian case.The approach of dynamical triangulations uses a certain class of suchsimplicial space–times as an explicit, regularized realization of the space<strong>Geom</strong>(M). For a given volume N d , this class consists of all gluings ofmanifold–type of a set of N d simplicial building blocks of top–dimension dwhose edge lengths are restricted to take either one or one out of two values.In the Euclidean case we set li2 = a2 for all i, and in the Lorentzian casewe allow for both space- and time–like links with li 2 ∈ {−a2 , a 2 }, where thegeodesic distance a serves as a short-distance cutoff, which will be takento zero later. Coming from the classical theory this may seem a grave restrictionat first, but this is indeed not the case. Firstly, keep in mind thatfor the purposes of the quantum theory we want to sample the space ofgeometries ‘ergodically’ at a coarse-grained scale of order a. This shouldbe contrasted with the classical theory where the objective is usually to approximatea given, fixed space–time to within a length scale a. In the lattercase one typically requires a much finer topology on the space of metrics orgeometries. It is also straightforward to see that no local curvature degreesof freedom are suppressed by fixing the edge lengths; deficit angles in alldirections are still present, although they take on only a discretized set ofvalues. In this sense, in dynamical triangulations all geometry is in thegluing of the fundamental building blocks. This is dual to how quantumRegge calculus is set up, where one usually fixes a triangulation T and then


<strong>Geom</strong>etrical Path Integrals and Their Applications 1063‘scans’ the space of geometries by letting the l i ’s run continuously over allvalues compatible with the triangular inequalities.In a nutshell, Lorentzian dynamical triangulations give a definite meaningto the ‘integral over geometries’, namely, as a sum over inequivalentLorentzian gluings T over any number N d of d−simplices,∫Σ <strong>Geom</strong>(M) D[g µν ] e iS[gµν]LDT−→∑ T ∈T1C Te iSReg (T ) , (6.105)where the symmetry factor C T = |Aut(T )| on the r.h.s. is the order ofthe automorphism group of the triangulation, consisting of all maps of Tonto itself which preserve the connectivity of the simplicial lattice. We willspecify below what precise class T of triangulations should appear in thesummation.It follows from the above that in this formulation all curvatures andvolumes contributing to the Regge simplicial action come in discrete units.This can be illustrated by the case of a 2D triangulation with Euclideansignature, which according to the prescription of dynamical triangulationsconsists of equilateral triangles with squared edge lengths +a 2 . All interiorangles of such a triangle are equal to π/3, which implies that the deficitangle at any vertex v can take the values 2π − k v π/3, where k v is thenumber of triangles meeting at v. As a consequence, the Einstein–Reggeaction S Reg assumes the simple formS Reg (T ) = κ d−2 N d−2 − κ d N d , (6.106)where the coupling constants κ i = κ i (λ, G N ) are simple functions of thebare cosmological and Newton constants in d dimensions. Substituting thisinto the path sum in (6.105) leads toZ(κ d−2 , κ d ) = ∑ N de −iκ dN d∑N d−2e iκ d−2N d−2∑T | Nd ,N d−21C T, (6.107)The point of taking separate sums over the numbers of d− and (d −2)−simplices in (6.107) is to make explicit that ‘doing the sum’ is tantamountto the combinatorial problem of counting triangulations of a givenvolume and number of simplices of codimension 2 (corresponding to thelast summation in (6.107)). 12 It turns out that at least in two space–timedimensions the counting of geometries can be done completely explicitly,12 The symmetry factor C T is almost always equal to 1 for large triangulations.


1064 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 6.10Two types of Minkowskian 4–simplices in 4D (see text for explanation).turning both Lorentzian and Euclidean quantum gravity into exactly solublestatistical models.6.4.4 Lorentzian Path IntegralNow, the simplicial building blocks of the models are taken to be piecesof Minkowski space, and their edges have squared lengths +a 2 or -a 2 . Forexample, the two types of 4–simplices that are used in Lorentzian dynamicaltriangulations in dimension four are shown in Figure 6.10. The first of themhas four time–like and six space–like links (and therefore contains 4 time–like and 1 space–like tetrahedron), whereas the second one has six time–likeand four space–like links (and contains 5 time–like tetrahedra). Since bothare subspaces of flat space with signature (− + ++), they possess well–defined light–cone structures everywhere [Loll (2001); Loll (1998)].In general, gluings between pairs of d−simplices are only possible whenthe metric properties of their (d − 1)−faces match. Having local light conesimplies causal relations between pairs of points in local neighborhoods.Creating closed time–like curves will be avoided by requiring that all space–times contributing to the path sum possess a global ‘time’ function t. Interms of the triangulation this means that the d−simplices are arrangedsuch that their space–like links all lie in slices of constant integer t, andtheir time–like links interpolate between adjacent spatial slices t and t +1. Moreover, with respect to this time, we will not allow for any spatialtopology changes 13 .This latter condition is always satisfied in classical applications, where‘trouser points’ like the one depicted in Figure 6.11 are ruled out by therequirement of having a non–degenerate Lorentzian metric defined everywhereon M (it is geometrically obvious that the light cone and hence g µνmust degenerate in at least one point along the ‘crotch’). Another way ofthinking about such configurations (and their time–reversed counterparts)13 Note that if we were in the continuum and had introduced coordinates on space–time,such a statement would actually be diffeomorphism–invariant.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1065Fig. 6.11 At a branching point associated with a spatial topology change, light–conesget ‘squeezed’.is that the causal past (future) of an observer changes discontinuously asher world–line passes near the singular point (see [Dowker (2002)] and referencestherein for related discussions about the issue of topology changein quantum gravity).There is no a priori reason in the quantum theory to not relax someof these classical causality constraints. After all, as we stressed right atthe outset, path integral histories are not in general classical solutions, norcan we attribute any other direct physical meaning to them individually.It might well be that one can construct models whose path integral configurationsviolate causality in this strict sense, but where this notion issomehow recovered in the resulting continuum theory. What the approachof Lorentzian dynamical triangulations has demonstrated is that imposingcausality constraints will in general lead to a different continuum theory.This is in contrast with the intuition one may have that ‘including a fewisolated singular points will not make any difference’. On the contrary,tampering with causality in this way is not innocent at all, as was alreadyanticipated by Teitelboim many years ago [Teitelboim (1983)].We want to point out that one cannot conclude from the above that spatialtopology changes or even fluctuations in the space–time topology cannotbe treated in the formulation of dynamical triangulations. However, if oneinsists on including geometries of variable topology in a Lorentzian discretecontext, one has to come up with a prescription of how to weigh these singularpoints in the path integral, both before and after the Wick rotation[Dasgupta (2002)]. Maybe this can be done along the lines suggested in[Louko and Sorkin (1997)]; this is clearly an interesting issue for furtherresearch.Having said this, we next have to address the question of the Wickrotation, in other words, of how to get rid of the factor of i in the exponent of


1066 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(6.107). Without it, this expression is an infinite sum (since the volume canbecome arbitrarily large) of complex terms whose convergence propertieswill be very difficult to establish. In this situation, a Wick rotation is simplya technical tool which – in the best of all worlds – enables us to performthe state sum and determine its continuum limit. The end result will haveto be Wick–rotated back to Lorentzian signature.Fortunately, Lorentzian dynamical triangulations come with a naturalnotion of Wick rotation, and the strategy we just outlined can be carriedout explicitly in two space–time dimensions, leading to a unitary theory.In higher dimensions we do not yet have sufficient analytical control of thecontinuum theories to make specific statements about the inverse Wickrotation. Since we use the Wick rotation at an intermediate step, one canask whether other Wick rotations would lead to the same result. Currentlythis is a somewhat academic question, since it is in practice difficult to findsuch alternatives. In fact, it is quite miraculous we have found a singleprescription for Wick–rotating in our regularized setting, and it does notseem to have a direct continuum analogue (for more comments on this issue,see [Dasgupta and Loll (2001); Dasgupta (2002)]).Our Wick rotation W in any dimension is an injective map fromLorentzian– to Euclidean–signature simplicial space–times. Using the notationT for a simplicial manifold together with length assignments ls 2 andlt 2 to its space– and time–like links, it is defined byT lor = (T, {ls 2 = a 2 , lt 2 = −a 2 W}) ↦−→ T eu = (T, {ls 2 = a 2 , lt 2 = a 2 }).(6.108)Note that we have not touched the connectivity of the simplicial manifoldT , but only its metric properties, by mapping all time–like links of T intospace–like ones, resulting in a Euclidean ‘space–time’ of equilateral buildingblocks. It can be shown [Ambjørn et. al. (2001a)] that at the level of thecorresponding weight factors in the path integral this Wick rotation 14 hasprecisely the desired effect of rotating to the exponentiated Regge action ofthe ‘Euclideanized’ geometry,e iS(T lor )W↦−→ e−S(T eu) . (6.109)14 To get a genuine Wick rotation and not just a discrete map, one introduces a complexparameter α in l 2 t = −αa2 . The proper prescription leading to (6.109) is then an analyticcontinuation of α from 1 to -1 through the lower–half complex plane.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1067The Euclideanized path sum after the Wick rotation has the formZ eu (κ d−2 , κ d ) = ∑ T1C Te −κ dN d (T )+κ d−2 N d−2 (T )= ∑ N de −κ dN d∑T | Nd1C Te κ d−2N d−2 (T )= ∑ N de −κ dN de κcrit d (κ d−2)N d× subleading(N d ). (6.110)In the last equality we have used that the number of Lorentzian triangulationsof discrete volume N d to leading order scales exponentially with N d forlarge volumes. This can be shown explicitly in space–time dimension 2 and3. For d = 4, there is strong (numerical) evidence for such an exponentialbound for Euclidean triangulations, from which the desired result for theLorentzian case follows (since W maps to a strict subset of all Euclideansimplicial manifolds).From the functional form of the last line of (6.110) one can immediatelyread off some qualitative features of the phase diagram, an example of whichappeared already earlier in Figure 6.8. Namely, the sum over geometriesZ eu converges for values κ d > κ critdof the bare cosmological constant, anddiverges (ie. is not defined) below this critical line. Generically, for allmodels of dynamical triangulations the infinite–volume limit is attained byapproaching the critical line κ critd(κ d−2 ) from above, ie. from inside theregion of convergence of Z eu . In the process of taking N d → ∞ and thecutoff a → 0, one gets a renormalized cosmological constant Λ through(κ d − κ critd ) = a µ Λ + O(a µ+1 ). (6.111)If the scaling is canonical (which means that the dimensionality of the renormalizedcoupling constant is the one expected from the classical theory),the exponent is given by µ = d. Note that this construction requires a positivebare cosmological constant in order to make the state sum converge.Moreover, by virtue of relation (6.111) also the renormalized cosmologicalconstant must be positive. Other than that, its numerical value is not determinedby this argument, but by comparing observables of the theory whichdepend on Λ with actual physical measurements. 15 Another interestingobservation is that the inclusion of a sum over topologies in the discretized15 The non–negativity of the renormalized cosmological coupling may be taken as afirst ‘prediction’ of our construction, which in the physical case of four dimensions isindeed in agreement with current observations.


1068 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsum (6.110) would lead to a super–exponential growth of at least ∝ N d ! ofthe number of triangulations with the volume N d . Such a divergence of thepath integral cannot be compensated by an additive renormalization of thecosmological constant of the kind outlined above.There are ways in which one can sum divergent series of this type, forexample, by performing a Borel sum. The problem with these stems fromthe fact that two different functions can share the same asymptotic expansion.Therefore, the series in itself is not sufficient to define the underlyingtheory uniquely. The non–uniqueness arises because of non–perturbativecontributions to the path integral which are not represented in the perturbativeexpansion. 16 In order to fix these uniquely, an independent, non–perturbative definition of the theory is necessary. Unfortunately, for dynamicallytriangulated models of quantum gravity, no such definitions havebeen found so far. In the context of 2D (Euclidean) quantum gravity thisdifficulty is known as the ‘absence of a physically motivated double-scalinglimit’ [Ambjørn and Kristjansen (1993)].Lastly, getting an interesting continuum limit may or may not require anadditional fine–tuning of the inverse gravitational coupling κ d−2 , dependingon the dimension d. In four dimensions, one would expect to find a secondordertransition along the critical line, corresponding to local gravitonicexcitations. The situation in d = 3 is less clear, but results get so farindicate that no fine–tuning of Newton’s constant is necessary [Ambjørnet. al. (2001b); Ambjørn et. al. (2001c)].Before delving into the details, let us summarize briefly the results thathave been get so far in the approach of Lorentzian dynamical triangulations.At the regularized level, that is, in the presence of a finite cutoff a for theedge lengths and an infrared cutoff for large space–time volume, they arewell–defined statistical models of Lorentzian random geometries in d =2, 3, 4. In particular, they obey a suitable notion of reflection-positivityand possess self–adjoint Hamiltonians.The crucial questions are then to what extent the underlying combinatorialproblems of counting all dD geometries with certain causal propertiescan be solved, whether continuum theories with non–trivial dynamics existand how their bare coupling constants get renormalized in the process.What we know about Lorentzian dynamical triangulations so far is thatthey lead to continuum theories of quantum gravity in dimension 2 and 3.In d = 2, there is a complete analytic solution, which is distinct from the16 A field–theoretic example would be instantons and renormalons in QCD.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1069continuum theory produced by Euclidean dynamical triangulations. Alsothe matter–coupled model has been studied. In d = 3, there are numericaland partial analytical results which show that both a continuum theoryexists and that it again differs from its Euclidean counterpart. Work ona more complete analytic solution which would give details about the geometricproperties of the quantum theory is under way. In d = 4, the firstnumerical simulations are currently being set up. The challenge here is todo this for sufficiently large lattices, to be able to perform meaningful measurements.So far, we cannot make any statements about the existence andproperties of a continuum theory in this physically most interesting case.6.4.5 Application: Topological Phase Transitions andHamiltonian Chaos6.4.5.1 Phase Transitions in Hamiltonian SystemsRecall that phase transitions (PTs) are phenomena which bring about qualitativephysical changes at the macroscopic level in presence of the samemicroscopic forces acting among the constituents of a system. Their mathematicaldescription requires to translate into quantitative terms the mentionedqualitative changes. The standard way of doing this is to considerhow the values of thermodynamic observables, get in laboratory experiments,vary with temperature, or volume, or an external field, and thento associate the experimentally observed discontinuities at a PT to the appearanceof some kind of singularity entailing a loss of analyticity. Despitethe smoothness of the statistical measures, after the Yang–Lee Theorem[Yang and Lee (1952)] we know that in the N → ∞ limit non–analytic behaviorsof thermodynamic functions are possible whenever the analyticityradius in the complex fugacity plane shrinks to zero, because this entailsthe loss of uniform convergence in N (number of degrees of freedom) ofany sequence of real–valued thermodynamic functions, and all this dependson the distribution of the zeros of the grand canonical partition function.Also the other developments of the rigorous theory of PTs [Georgii (1988);Ruelle (1978)], identify PTs with the loss of analyticity.In this subsection we will address a recently proposed geometric approachto thermodynamic phase transitions (see [Caiani et al. (1997);Franzosi et al. (1999); Franzosi et al. (2000); Franzosi and Pettini (2004)]).Given any Hamiltonian system, the configuration space can be equippedwith a metric, in order to get a Riemannian geometrization of the dynam-


1070 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionics. At the beginning, several numerical and analytical studies of a varietyof models showed that the fluctuation of the curvature becomes singular atthe transition point. Then the following conjecture was proposed in [Caianiet al. (1997)]: The phase transition is determined by a change in thetopology of the configuration space, and the loss of analyticity in the thermodynamicobservables is nothing but a consequence of such topologicalchange. The latter conjecture is also known as the topological hypothesis.The topological hypothesis states that suitable topology changes ofequipotential submanifolds of the Hamiltonian system’s configuration manifoldcan entail thermodynamic phase transitions [Franzosi et al. (2000)].The authors of the topological hypothesis gave both a theoretical argumentand numerical demonstration in case of 2D lattice ϕ 4 model. They consideredclassical many–particle (or many–subsystem) systems described bystandard mechanical HamiltoniansH(p, q) =N∑i=1p 2 i+ V (q), (6.112)2mwhere the coordinates q i = q i (t) and momenta p i = p i (t), (i = 1, ..., N),have continuous values and the system’s potential energy V (q) is boundedbelow.Now, assuming a large number of subsystems N, the statistical behaviorof physical systems described by Hamiltonians of the type (6.112) is usuallyencompassed, in the system’s canonical ensemble, by the partition functionin the system’s phase–space∫Z N (β) ==N∏i=1( ) N ∫ πdp i dq i e −βH(p,q) 2 ∏ N=dq i −βV (q)eβ( πβ) N2 ∫ ∞0dv e −βv ∫M vi=1dσ‖∇V ‖ , (6.113)where the last term is written using a co–area formula [Federer (1969)], andv labels the equipotential hypersurfaces M v of the system’s configurationmanifold M,M v = {(q 1 , . . . , q N ) ∈ R N |V (q 1 , . . . , q N ) = v}. (6.114)Equation (6.113) shows that for Hamiltonians (3.34) the relevant statistical


<strong>Geom</strong>etrical Path Integrals and Their Applications 1071information is contained in the canonical configurational partition function∫ZN C =N∏i=1dq i exp[−βV (q)].Therefore, partition function ZNC is decomposed – in the last term ofequation∫(6.113) – into an infinite summation of geometric integrals,M vdσ /‖∇V ‖, defined on the {M v } v∈R . Once the microscopic interactionpotential V (q) is given, the configuration space of the system is automaticallyfoliated into the family {M v } v∈R of these equipotential hypersurfaces.Now, from standard statistical mechanical arguments we know that, atany given value of the inverse temperature β, the larger the number Nof particles the closer to M v ≡ M uβ are the microstates that significantlycontribute to the averages – computed through Z N (β) – of thermodynamicobservables. The hypersurface M uβ is the one associated with the averagepotential energy computed at a given β,u β = (Z C N ) −1 ∫N∏i=1dq i V (q) exp[−βV (q)].Thus, at any β, if N is very large the effective support of the canonicalmeasure shrinks very close to a single M v = M uβ .Explicitly, the topological hypothesis reads: the basic origin of a phasetransition lies in a suitable topology change of the {M v }, occurring at somev c . This topology change induces the singular behavior of the thermodynamicobservables at a phase transition. By change of topology we meanthat {M v } vvc . In other words,canonical measure should ‘feel’ a big and sudden change of the topology ofthe equipotential hypersurfaces of its underlying support, the consequencebeing the appearance of the typical signals of a phase transition.This point of view has the interesting consequence that – also at finiteN – in principle different mathematical objects, i.e., manifolds of differentcohomology type, could be associated to different thermodynamical phases,whereas from the point of view of measure theory [Yang and Lee (1952)] theonly mathematical property available to signal the appearance of a phasetransition is the loss of analyticity of the grand–canonical and canonicalaverages, a fact which is compatible with analytic statistical measures onlyin the mathematical N → ∞ limit.As it is conjectured that the counterpart of a phase transition is a breakingof diffeomorphicity among the surfaces M v , it is appropriate to choose


1072 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona diffeomorphism invariant to probe if and how the topology of the M vchanges as a function of v. This is a very challenging task because we haveto deal with high dimensional manifolds. Fortunately a topological invariantexists whose computation is feasible, yet demands a big effort. Recall(from subsection 3.10.1 above) that this is the Euler characteristic, a diffeomorphisminvariant of the system’s configuration manifold, expressingits fundamental topological information.6.4.5.2 <strong>Geom</strong>etry of the Largest Lyapunov ExponentNow, the topological hypothesis has recently been promoted into a topologicalTheorem [Franzosi and Pettini (2004)]. The new Theorem says thatnon–analyticity is the ‘shadow’ of a more fundamental phenomenon occurringin the system’s configuration manifold: a topology change within thefamily of equipotential hypersurfaces (6.114). This topological approach toPTs stems from the numerical study of the Hamiltonian dynamical counterpartof phase transitions, and precisely from the observation of discontinuousor cuspy patterns, displayed by the largest Lyapunov exponent atthe transition energy (or temperature).Recall that the Lyapunov exponents measure the strength of dynamicalchaos and cannot be measured in laboratory experiments, at variancewith thermodynamic observables, thus, being genuine dynamical observablesthey are only measurable in numerical simulations of the microscopicdynamics. To get a hold of the reason why the largest Lyapunov exponentλ 1 should probe configuration space topology, let us first remember that forstandard Hamiltonian systems, λ 1 is computed by solving the tangent dynamicsequation for Hamiltonian systems (see Jacobi equation of geodesicdeviation (3.133)),( ∂ 2 V¨ξi +∂q i ∂q)q(t)j ξ j = 0, (6.115)which, for the nonlinear Hamiltonian system˙q 1 = p 1 , ṗ 1 = −∂ q 1V,... ...˙q N = p N , ṗ N = −∂ q N V,


<strong>Geom</strong>etrical Path Integrals and Their Applications 1073expands into linearized Hamiltonian dynamicsN∑( ∂˙ξ 2 V1 = ξ N+1 , ˙ξN+1 = −ξ∂q 1 ∂q j ,j)q(t)j=1... ... (6.116)˙ξ n = ξ 2N ,N∑( ∂ 2 V˙ξ2N = −ξ∂q N ∂q j .j)q(t)Using (6.115) we can get the analytical expression for the largest Lyapunovexponent[1 ξ 2 1(t) + · · · + ξ 2 N(t) + ˙ξ 2 1(t) + · · · + ˙ξ] 2 1/2N(t)λ 1 = limt→∞ t log [ξ 2 1(0) + · · · + ξ 2 N(0) + ˙ξ 2 1(0) + · · · + ˙ξ] 2 1/2. (6.117)N (0)If there are critical points of V in configuration space, that is pointsq c = [q 1 , . . . , q N ] such that ∇V (q)| q=qc= 0, according to the Morse lemma(see e.g., [Hirsch (1976)]), in the neighborhood of any critical point q c therealways exists a coordinate system ˜q(t) = [q 1 (t), . . . , q N (t)] for whichV (˜q) = V (q c ) − ( q 1) 2− · · · −(qk ) 2+(qk+1 ) 2+ · · · +(qN ) 2, (6.118)where k is the index of the critical point, i.e., the number of negativeeigenvalues of the Hessian of V . In the neighborhood of a critical point,equation (6.118) yieldsj=1∂ 2 V/∂q i ∂q j = ±δ ij ,which, substituted into equation (6.115), gives k unstable directions whichcontribute to the exponential growth of the norm of the tangent vectorξ = ξ(t). This means that the strength of dynamical chaos, measured by thelargest Lyapunov exponent λ 1 , is affected by the existence of critical pointsof V . In particular, let us consider the possibility of a sudden variation,with the potential energy v, of the number of critical points (or of theirindexes) in configuration space at some value v c , it is then reasonable toexpect that the pattern of λ 1 (v) – as well as that of λ 1 (E) since v = v(E)– will be consequently affected, thus displaying jumps or cusps or othersingular patterns at v c .On the other hand, recall that Morse theory teaches us that the existenceof critical points of V is associated with topology changes of thehypersurfaces {M v } v∈R , provided that V is a good Morse function (that is:


1074 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionbounded below, with no vanishing eigenvalues of its Hessian matrix). Thusthe existence of critical points of the potential V makes possible a conceptuallink between dynamics and configuration space topology, which, on thebasis of both direct and indirect evidence for a few particular models, hasbeen formulated as a topological hypothesis about the relevance of topologyfor PTs phenomena (see [Franzosi et al. (2000); Franzosi and Pettini (2004);Grinza and Mossa (2004)]).Here we give two simple examples of standard Hamiltonian systemsof the form (6.112), namely Peyrard–Bishop system and mean–field XYmodel.Peyrard–Bishop Hamiltonian SystemThe Peyrard–Bishop system [Peyrard and Bishop (1989)] 17 exhibits asecond–order phase transition. It is defined by the following potential energyN∑[ ]KV (q) =2 (qi+1 − q i ) 2 + D(e −aqi − 1) 2 + Dhaq i , (6.119)i=1which represents the energy of a string of N base pairs of reduced mass m.Each hydrogen bond is characterized by the stretching q i and its conjugatemomentum p i = m ˙q i . The elastic transverse force between neighboringpairs is tuned by the constant K, while the energy D and the inverselength a determine, respectively, the plateau and the narrowness of the on–site potential well that mimics the interaction between bases in each pair. Itis understood that K, D, and a are all positive parameters. The transverse,external stress h ≥ 0 is a computational tool useful in the evaluation of thesusceptibility. Our interest in it lies in the fact that a phase transition canoccur only when h = 0. We assume periodic boundary conditions.The transfer operator technique [Dauxois et al. (2002)] maps the problemof computing the classical partition function into the easier task ofevaluating the lowest energy eigenvalues of a ‘quantum’ mechanical Morseoscillator (no real quantum mechanics is involved, since the temperatureplays the role of ). One can then observe that, as the temperature increases,the number of levels belonging to the discrete spectrum decreases,until for some critical temperature T c = 2 √ 2KD/(ak B ) only the continuousspectrum survives. This passage from a localized ground state to an17 The Peyrard–Bishop system has been proposed as a simple model for describing theDNA thermally induced denaturation [Grinza and Mossa (2004)].


<strong>Geom</strong>etrical Path Integrals and Their Applications 1075unnormalizable one corresponds to the second–order phase transition of thestatistical model. Various critical exponents can be analytically computedand all applicable scaling laws can be checked. The simplicity of this modelpermits an analytical computation of the largest Lyapunov exponent byexploiting the geometric method proposed in [Caiani et al. (1997)].Mean–Field XY Hamiltonian SystemThe mean–field XY model describes a system of N equally coupled planarclassical rotators (see [Antoni and Ruffo (1995); Casetti et al. (1999)]). Itis defined by a Hamiltonian of the class (6.112) where the potential energyisV (ϕ) =J2NN∑ [1 − cos(ϕi − ϕ j ) ] N∑− h cos ϕ i . (6.120)i,j=1Here ϕ i ∈ [0, 2π] is the rotation angle of the ith rotator and h is an externalfield. Defining at each site i a classical spin vector s i = (cos ϕ i , sin ϕ i )the model describes a planar (XY) Heisenberg system with interactions ofequal strength among all the spins. We consider only the ferromagnetic caseJ > 0; for the sake of simplicity, we set J = 1. The equilibrium statisticalmechanics of this system is exactly described, in the thermodynamic limit,by the mean–field theory [Antoni and Ruffo (1995)]. In the limit h → 0, thesystem has a continuous phase transition, with classical critical exponents,at T c = 1/2, or ε c = 3/4, where ε = E/N is the energy per particle.The Lyapunov exponent λ 1 of this system is extremely sensitive to thephase transition. According to reported numerical simulations (see [Casettiet al. (1999)]), λ 1 (ε) is positive for 0 < ε < ε c , shows a sharp maximumimmediately below the critical energy, and drops to zero at ε c in the thermodynamiclimit, where it remains zero in the whole region ε > ε c , whichcorresponds to the thermodynamic disordered phase. In fact in this phasethe system is integrable, reducing to an assembly of uncoupled rotators.i=16.4.5.3 Euler Characteristics of Hamiltonian SystemsRecall that Euler characteristic χ is a number that is a characterizationof the various classes of geometric figures based only on the topologicalrelationship between the numbers of vertices V , edges E, and faces F , of ageometric Figure. This number, χ = F − E + V, is the same for all figuresthe boundaries of which are composed of the same number of connected


1076 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionpieces. Therefore, the Euler characteristic is a topological invariant, i.e.,any two geometric figures that are homeomorphic to each other have thesame Euler characteristic.More specifically, a standard way to analyze a geometric Figure is tofragment it into other more familiar objects and then to examine how thesepieces fit together. Take for example a surface M in the Euclidean 3D space.Slice M into pieces that are curved triangles (this is called a triangulation ofthe surface). Then count the number F of faces of the triangles, the numberE of edges, and the number V of vertices on the tesselated surface. Now,no matter how we triangulate a compact surface Σ, its Euler characteristic,χ(Σ) = F − E + V , will always equal a constant which is characteristic ofthe surface and which is invariant under diffeomorphisms φ : Σ → Σ ′ .At higher dimensions this can be again defined by using higher dimensionalgeneralizations of triangles (simplexes) and by defining the Eulercharacteristic χ(M) of the nD manifold M to be the alternating sum:{number of points} − {number of 2-simplices} +{number of 3-simplices} − {number of 4-simplices} + ...n∑i.e., χ(M) = (−1) k (number of faces of dimension k).k=0and then define the Euler characteristic of a manifold as the Euler characteristicof any simplicial complex homeomorphic to it. With this definition,circles and squares have Euler characteristic 0 and solid balls have Eulercharacteristic 1.The Euler characteristic χ of a manifold is closely related to its genusg as χ = 2 − 2g. 18Recall that a more standard topological definition of χ(M) isχ(M) =n∑(−1) k b k (M), (6.121)k=0where b k are the kth Betti numbers of M.18 Recall that the genus of a topological space such as a surface is a topologicallyinvariant property defined as the largest number of nonintersecting simple closed curvesthat can be drawn on the surface without separating it, i.e., an integer representing themaximum number of cuts that can be made through it without rendering it disconnected.This is roughly equivalent to the number of holes in it, or handles on it. For instance: apoint, line, and a sphere all have genus 0; a torus has genus 1, as does a coffee cup as asolid object (solid torus), a Möbius strip, and the symbol 0; the symbols 8 and B havegenus 2; etc.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1077In general, it would be hopeless to try to practically calculate χ(M)from (6.121) in the case of non–trivial physical models at large dimension.Fortunately, there is a possibility given by the Gauss–Bonnet formula, thatrelates χ(M) with the total Gauss–Kronecker curvature of the manifold,(compare with (3.126) and (3.135))∫χ(M) = γ K G dσ, (6.122)which is valid for even dimensional hypersurfaces of Euclidean spaces R N[here dim(M) = n ≡ N − 1], and where:Mγ = 2/ Vol(S n 1 )is twice the inverse of the volume of an n−dimensional sphere of unit radiusS n 1 ; K G is the Gauss–Kronecker curvature of the manifold;dσ = √ det(g) dx 1 dx 2 · · · dx nis the invariant volume measure of M and g is its Riemannian metric (inducedfrom R N ). Let us briefly sketch the meaning and definition of theGauss–Kronecker curvature. The study of the way in which an n−surfaceM curves around in R N is measured by the way the normal directionchanges as we move from point to point on the surface. The rate of changeof the normal direction ξ at a point x ∈ M in direction v is described bythe shape operatorL x (v) = −L v ξ = [v, ξ],where v is a tangent vector at x and L v is the Lie derivative, henceL x (v) = −(∇ξ 1 · v, . . . , ∇ξ n+1 · v);gradients and vectors are represented in R N . As L x is an operator ofthe tangent space at x into itself, there are n independent eigenvaluesκ 1 (x), . . . , κ n (x) which are called the principal curvatures of M at x [Thorpe(1979)]. Their product is the Gauss–Kronecker curvature:K G (x) =n∏κ i (x) = det(L x ).i=1Alternatively, recall that according to the Morse theory, it is possibleto understand the topology of a given manifold by studying the regularcritical points of a smooth Morse function defined on it. In our case, the


1078 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionmanifold M is the configuration space R N and the natural choice for theMorse function is the potential V (q). Hence, one is lead to define the familyM v (6.114) of submanifolds of M.A full characterization of the topological properties of M v generallyrequires the critical points of V (q), which means solving the equations∂ q iV = 0, (i = 1, . . . , N). (6.123)Moreover, one has to calculate the indexes of all the critical points, thatis the number of negative eigenvalues of the Hessian ∂ 2 V/(∂q i ∂q j ). Thenthe Euler characteristic χ(M v ) can be computed by means of the formulaχ(M v ) =N∑(−1) k µ k (M v ), (6.124)k=0where µ k (M v ) is the total number of critical points of V (q) on M v whichhave index k, i.e., the so–called Morse numbers of a manifold M, whichhappen to be upper bounds of the Betti numbers,b k (M) ≤ µ k (M) (k = 0, . . . , n). (6.125)Among all the Morse functions on a manifold M, there is a special class,called perfect Morse functions, for which the Morse inequalities (6.125) holdas equalities. Perfect Morse functions characterize completely the topologyof a manifold.Now, we continue with our two examples started before.Peyrard–Bishop System. If applied to any generic model, calculationof (6.124) turns out to be quite formidable, but the exceptional simplicity ofthe Peyrard–Bishop model (6.119) makes it possible to carry on completelythe topological analysis without invoking equation (6.124).For the potential in exam, equation (6.123) results in the nonlinearsystemaR (qi+1 − 2q i + q i−1 ) = h − 2(e −2aqi − e −aqi ),where R = Da 2 /K is a dimensionless ratio.particular solution is given byIt is easy to verify that aq i = − 1 a ln 1 + √ 1 + 2h, (i = 1, . . . , N).2


<strong>Geom</strong>etrical Path Integrals and Their Applications 1079The corresponding minimum of potential energy is( √1 + h − 1 + 2hV min = ND− h ln 1 + √ )1 + 2h.22Mean–Field XY Model. In the case of the mean–field XY model(6.120) it is possible to show analytically that a topological change in theconfiguration space exists and that it can be related to the thermodynamicphase transition. Consider again the family M v of submanifolds of theconfiguration space defined in (6.114); now the potential energy per degreeof freedom is that of the mean–field XY model, i.e.,V(ϕ) = V (ϕ)N = J2N 2N∑i,j=1[1 − cos(ϕi − ϕ j ) ] − hN∑cos ϕ i ,where ϕ i ∈ [0, 2π]. Such a function can be considered a Morse function onM, so that, according to Morse theory, all these manifolds have the sametopology until a critical level V −1 (v c ) is crossed, where the topology of M vchanges.A change in the topology of M v can only occur when v passes througha critical value of V. Thus in order to detect topological changes in M v wehave to find the critical values of V, which means solving the equationsi=1∂ϕ i V(ϕ) = 0, (i = 1, . . . , N). (6.126)For a general potential energy function V, the solution of (6.126) would bea formidable task, but in the case of the mean–field XY model, the mean–field character of the interaction greatly simplifies the analysis, allowing ananalytical treatment of (6.126); moreover, a projection of the configurationspace onto a 2D plane is possible [Casetti et al. (1999); Casetti et al.(2003)].6.4.6 Application: Force–Field PsychodynamicsIn this section, which is written in the fashion of the above quantum brainmodelling (see subsection 6.3.8 above), we present the top level of naturalbiodynamics, using geometrical generalization of the Feynman path integral.To formulate the basics of force–field psychodynamics, we use the action–amplitude picture of the BODY ⇄ MIND adjunction:


1080 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction↓ Deterministic (causal) world of Human BODY↓Action : S[q n ] =∫ toutt in(E k − E p + W rk + Src ± ) dt− − − − − − − − − − −∫− − − − − − − −Amplitude : 〈out|in〉 = Σ D[w n q n ] e iS[qn ]↑Probabilistic (fuzzy) world of Human MIND ↑In the action integral, E k , E p , W rk and Src ± denote the kinetic end potentialenergies, work done by dissipative/driving forces and other energysources/sinks, respectively. In the amplitude integral, the peculiar sign∫Σ denotes integration along smooth paths and summation along discreteMarkov chains; i is the imaginary unit, w n are synaptic–like weights, whileD is the Feynman path differential (defined below) calculated along theconfiguration trajectories q n . The action S[q n ], through the least actionprinciple δS = 0, leads to all biodynamic equations considered so far (ingeneralized Lagrangian and Hamiltonian form). At the ∫same time, theaction S[q n ] figures in the exponent of the path integral Σ, defining theprobability transition amplitude 〈out|in〉. In this way, the whole body dynamicsis incorporated in the mind dynamics. This adaptive path integralrepresents an infinite–dimensional neural network, suggesting an infinitecapacity of human brain/mind.For a long time the cortical systems for language and actions were believedto be independent modules. However, according to the recent researchof [Pulvermüller (2005)], as these systems are reciprocally connectedwith each other, information about language and actions might interact indistributed neuronal assemblies. A critical case is that of action words thatare semantically related to different parts of the body (e.g. ‘pick’, ‘kick’,‘lick’,...). The author suggests that the comprehension of these words mightspecifically, rapidly and automatically activate the motor system in a somatotopicmanner, and that their comprehension rely on activity in theaction system.6.4.6.1 Motivational Cognition in the Life Space FoamApplications of nonlinear dynamical systems (NDS) theory in psychologyhave been encouraging, if not universally productive/effective [Met-


<strong>Geom</strong>etrical Path Integrals and Their Applications 1081zger (1997)]. Its historical antecedents can be traced back to Piaget’s[Piaget et. al. (1992)] and Vygotsky’s [Vygotsky (1982)] interpretationsof the dynamic relations between action and thought, Lewinian theoryof social dynamics and cognitive–affective development [Lewin (1951);Gold (1999)], and Bernstein’s [Bernstein (1947)] theory of self–adjusting,goal–driven motor action.Now, both the original Lewinian force–field theory in psychology (see[Lewin (1951); Gold (1999)]) and modern decision–field dynamics (see[Busemeyer and Townsend (1993); Roe et al. (2001); Busemeyer andDiederich (2002)]) are based on the classical Lewinian concept of an individual’slife space. 19 As a topological construct, Lewinian life space representsa person’s psychological environment that contains regions separated by dynamicalpermeable boundaries. As a field construct, on the other hand, thelife space is not empty: each of its regions is characterized by valence (rangingfrom positive or negative and resulting from an interaction between theperson’s needs and the dynamics of their environment). Need is an energyconstruct, according to Lewin. It creates tension in the person, which, incombination with other tensions, initiates and sustains behavior. Needsvary from the most primitive urges to the most idiosyncratic intentionsand can be both internally generated (e.g., thirst or hunger) and stimulus–induced (e.g., an urge to buy something in response to a TV advertisement).Valences are, in essence, personal values dynamically derived from the person’sneeds and attached to various regions in their life space. As a field, thelife space generates forces pulling the person towards positively–valencedregions and pushing them away from regions with negative valence. Lewin’sterm for these forces is vectors. Combinations of multiple vectors in the lifespace cause the person to move from one region towards another. Thismovement is termed locomotion and it may range from overt behavior tocognitive shifts (e.g., between alternatives in a decision–making process).Locomotion normally results in crossing the boundaries between regions.When their permeability is degraded, these boundaries become barriersthat restrain locomotion. Life space model, thus, offers a meta–theoreticallanguage to describe a wide range of behaviors, from goal–directed actionto intrapersonal conflicts and multi–alternative decision–making.In order to formalize the Lewinian life–space concept, a set of actionprinciples need to be associated to Lewinian force–fields, (loco)motion19 The work presented in this subsection has been developed in collaboration with Dr.Eugene Aidman, Senior Research Scientist, Human Systems Integration, Land OperationsDivision, Defence Science & Technology Organisation, Australia.


1082 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionpaths (representing mental abstractions of biomechanical paths [<strong>Ivancevic</strong>(2004)]) and life space geometry. As an extension of the Lewinian concept,in this paper we introduce a new concept of life–space foam (LSF, seeFigure 6.12). According to this new concept, Lewin’s life space can be representedas a geometrical functor with globally smooth macro–dynamics,which is at the same time underpinned by wildly fluctuating, non–smooth,local micro–dynamics, describable by Feynman’s: (i) sum–over–histories∫∫Σ fields , andΣ paths , (ii) sum–over–fields∫(iii) sum–over–geometries Σ geom .LSF is thus a two–level geometrodynamical functor, representing thesetwo distinct types of dynamics within the Lewinian life space. At itsmacroscopic spatio–temporal level, LSF appears as a ‘nice & smooth’ geometricalfunctor with globally predictable dynamics – formally, a smoothn−dimensional manifold M with local Riemannian metrics g ij (x), smoothforce–fields and smooth (loco)motion paths, as conceptualized in theLewinian theory. To model the global and smooth macro–level LSF–paths,fields and geometry, we use the general physics–like principle of the leastaction.Now, the apparent smoothness of the macro–level LSF is achieved bythe existence of another level underneath it. This micro–level LSF is actuallya collection of wildly fluctuating force–fields, (loco)motion paths,curved regional geometries and topologies with holes. The micro–level LSFis proposed as an extension of the Lewinian concept: it is characterizedby uncertainties and fluctuations, enabled by microscopic time–level, microscopictransition paths, microscopic force–fields, local geometries andvarying topologies with holes. To model these fluctuating microscopicLSF–structures, we use three instances of adaptive path integral, defininga multi–phase and multi–path (also multi–field and multi–geometry)transition process from intention to the goal–driven action.We use the new LSF concept to develop modelling framework for motivationaldynamics (MD) and induced cognitive dynamics (CD).According to Heckhausen (see [Heckhausen (1977)]), motivation canbe thought of as a process of energizing and directing the action. Theprocess of energizing can be represented by Lewin’s force–field analysis andVygotsky’s motive formation (see [Vygotsky (1982); Aidman and Leontiev(1991)]), while the process of directing can be represented by hierarchicalaction control (see [Bernstein (1947); Bernstein (1935); Kuhl (1985)]).Motivation processes both precede and coincide with every goal–directed


<strong>Geom</strong>etrical Path Integrals and Their Applications 1083Fig. 6.12 Diagram of the life space foam: Lewinian life space with an adaptive pathintegral acting inside it and generating microscopic fluctuation dynamics.action. Usually these motivation processes include the sequence of thefollowing four feedforward phases [Vygotsky (1982); Aidman and Leontiev(1991)]: (*)(1) Intention Formation F, including: decision making, commitmentbuilding, etc.(2) Action Initiation I, including: handling conflict of motives, resistanceto alternatives, etc.(3) Maintaining the Action M, including: resistance to fatigue, distractions,etc.(4) Termination T , including parking and avoiding addiction, i.e., stayingin control.With each of the phases {F, I, M, T } in (*), we can associate a transitionpropagator – an ensemble of (possibly crossing) feedforward paths propagatingthrough the ‘wood of obstacles’ (including topological holes in theLSF, see Figure 6.13), so that the complete transition functor T A is a productof propagators (as well as sum over paths). All the phases–propagatorsare controlled by a unique Monitor feedback process.In this subsection we propose an adaptive path integral formulationfor the motivational–transition functor T A. In essence, we sum/integrateover different paths and make a product (composition) of different phases–propagators. Recall that this is the most general description of the generalMarkov stochastic process.We will also attempt to demonstrate the utility of the same LSF–formalisms in representing cognitive functions, such as memory, learningand decision making. For example, in the classical Stimulus encoding−→ Search −→ Decision −→ Response sequence [Sternberg (1969);


1084 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 6.13 Transition–propagator corresponding to each of the motivational phases{F, I, M, T }, consisting of an ensemble of feedforward paths propagating through the‘wood of obstacles’. The paths affected by driving and restraining force–fields, as well asby the local LSF–geometry. Transition goes from Intention, occurring at a sample timeinstant t 0 , to Action, occurring at some later time t 1 . Each propagator is controlled byits own Monitor feedback. All together they form the transition functor T A.Ashcraft (1994)], the environmental input–triggered sensory memory andworking memory (WM) can be interpreted as operating at the micro–levelforce–field under the executive control of the Monitor feedback, whereassearch can be formalized as a control mechanism guiding retrieval from thelong–term memory (LTM, itself shaped by learning) and filtering materialrelevant to decision making into the WM. The essential measure of thesemental processes, the processing speed (essentially determined by Sternberg’sreaction–time) can be represented by our (loco)motion speed ẋ.Six Faces of the Life Space FoamThe LSF has three forms of appearance: paths + field + geometries,acting on both macro–level and micro–level, which is six modes in total. Inthis section, we develop three least action principles for the macro–LSF–level and three adaptive path integrals for the micro–LSF–level. Whiledeveloping our psycho–physical formalism, we will address the behavioralissues of motivational fatigue, learning, memory and decision making.General FormalismAt both macro– and micro–levels, the total LSF represents a union of


<strong>Geom</strong>etrical Path Integrals and Their Applications 1085transition paths, force–fields and geometries, formally written as⋃ ⋃LSF total := LSF paths LSFfields LSFgeom (6.127)∫ ∫ ∫≡ Σ paths + Σ fields + Σ geom .Corresponding to each of the three LSF–subspaces in (6.127) we formulate:(1) The least action principle, to model deterministic and predictive,macro–level MD & CD, giving a unique, global, causal and smoothpath–field–geometry on the macroscopic spatio–temporal level; and(2) Associated adaptive path integral to model uncertain, fluctuating andprobabilistic, micro–level MD & CD, as an ensemble of local paths–fields–geometries on the microscopic spatio–temporal level, to whichthe global macro–level MD & CD represents both time and ensembleaverage (which are equal according to the ergodic hypothesis).In the proposed formalism, transition paths x i (t) are affected by the force–fields ϕ k (t), which are themselves affected by geometry with metric g ij .Global Macro–Level of LSF total . In general, at the macroscopicLSF–level we first formulate the total action S[Φ], the central quantity inour formalism that has psycho–physical dimensions of Energy × T ime =Effort, with immediate cognitive and motivational applications: thegreater the action – the higher the speed of cognitive processes and the lowerthe macroscopic fatigue (which includes all sources of physical, cognitiveand emotional fatigue that influence motivational dynamics). The actionS[Φ] depends on macroscopic paths, fields and geometries, commonly denotedby an abstract field symbol Φ i . The action S[Φ] is formally definedas a temporal integral from the initial time instant t ini to the final timeinstant t fin ,S[Φ] =∫ tfinwith Lagrangian density given by∫L[Φ] = d n x L(Φ i , ∂ x j Φ i ),t iniL[Φ] dt, (6.128)where the integral is taken over all n coordinates x j = x j (t) of the LSF,and ∂ x j Φ i are time and space partial derivatives of the Φ i −variables overcoordinates.


1086 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSecond, we formulate the least action principle as a minimal variationδ of the action S[Φ]δS[Φ] = 0, (6.129)which, using techniques from the calculus of variations gives, in the form ofthe so–called Euler–Lagrangian equations, a shortest (loco)motion path, anextreme force–field, and a life–space geometry of minimal curvature (andwithout holes). In this way, we effectively derive a unique globally smoothtransition functorT A : INT ENT ION tini ⇛ ACT ION tfin , (6.130)performed at a macroscopic (global) time–level from some initial time t inito the final time t fin .In this way, we get macro–objects in the global LSF: a single pathdescribed Newtonian–like equation of motion, a single force–field describedby Maxwellian–like field equations, and a single obstacle–free Riemanniangeometry (with global topology without holes).For example, recall that in the period 1945–1949 J. Wheeler and R.Feynman developed their action-at-a-distance electrodynamics [Wheelerand Feynman (1949)], in complete experimental agreement with the classicalMaxwell’s electromagnetic theory, but at the same time avoiding thecomplications of divergent self–interaction of the Maxwell’s theory as wellas eliminating its infinite number of field degrees of freedom. In Wheeler–Feynman view, “Matter consists of electrically charged particles,” so theyfound a form for the action directly involving the motions of the chargesonly, which upon variation would give the Newtonian–like equations of motionof these charges. Here is the expression for this action in the flatspace–time, which is in the core of quantum electrodynamics:S[x; t i , t j ] = 1 ∫2 m i (ẋ i µ) 2 dt i + 1 ∫ ∫2 e ie j δ(Iij) 2 ẋ i µ(t i )ẋ j µ(t j ) dt i dt jwith (6.131)I 2 ij = [ x i µ(t i ) − x j µ(t j ) ] [ x i µ(t i ) − x j µ(t j ) ] ,where x i µ = x i µ(t i ) is the four–vector position of the ith particle as a functionof the proper time t i , while ẋ i µ(t i ) = dx i µ/dt i is the velocity four–vector.The first term in the action (6.131) is the ordinary mechanical action inEuclidean space, while the second term defines the electrical interaction of


<strong>Geom</strong>etrical Path Integrals and Their Applications 1087the charges, representing the Maxwell–like field (it is summed over eachpair of charges; the factor 1 2is to count each pair once, while the term i = jis omitted to avoid self–action; the interaction is a double integral over adelta function of the square of space–time interval I 2 between two pointson the paths; thus, interaction occurs only when this interval vanishes, thatis, along light cones [Wheeler and Feynman (1949)]).Now, from the point of view of Lewinian geometrical force–fields and(loco)motion paths, we can give the following life–space interpretation tothe Wheeler–Feynman action (6.131). The mechanical–like locomotionterm occurring at the single time t, needs a covariant generalization fromthe flat 4D Euclidean space to the nD smooth Riemannian manifold, so itbecomes (see e.g., [<strong>Ivancevic</strong> (2004)])S[x] = 1 2∫ tfint inig ij ẋ i ẋ j dt,where g ij is the Riemannian metric tensor that generates the total ‘kineticenergy’ of (loco)motions in the life space.The second term in (6.131) gives the sophisticated definition of Lewinianforce–fields that drive the psychological (loco)motions, if we interpret electricalcharges e i occurring at different times t i as motivational charges –needs.Local Micro–Level of LSF total . After having properly definedmacro–level MD & CD, with a unique transition map F (including a uniquemotion path, driving field and smooth geometry), we move down to the microscopicLSF–level of rapidly fluctuating MD & CD, where we cannot definea unique and smooth path–field–geometry. The most we can do at thislevel of fluctuating uncertainty, is to formulate an adaptive path integral andcalculate overall probability amplitudes for ensembles of local transitionsfrom one LSF–point to the neighboring one. This probabilistic transitionmicro–dynamics functor is defined by a multi–path (field and geometry,respectively) and multi–phase transition amplitude 〈Action|Intention〉 ofcorresponding to the globally–smooth transition map (6.130). This absolutesquare of this probability amplitude gives the transition probability ofoccurring the final state of Action given the initial state of Intention,P (Action|Intention) = |〈Action|Intention〉| 2 .The total transition amplitude from the state of Intention to the state of


1088 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAction is defined on LSF totalT A ≡ 〈Action|Intention〉 total : INT ENT ION t0 ⇛ ACT ION t1 , (6.132)given by adaptive generalization of the Feynman’s path integral [Feynmanand Hibbs (1965); Feynman (1972); Feynman (1998)]. The transition map(6.132) calculates the overall probability amplitude along a multitude ofwildly fluctuating paths, fields and geometries, performing the microscopictransition from the micro–state INT ENT ION t0 occurring at initial micro–time instant t 0 to the micro–state ACT ION t1 at some later micro–timeinstant t 1 , such that all micro–time instants fit inside the global transitioninterval t 0 , t 1 , ..., t s ∈ [t ini , t fin ]. It is symbolically written as∫〈Action|Intention〉 total := Σ D[wΦ] e iS[Φ] , (6.133)where the Lebesgue integration is performed over all continuous Φ i con =paths + field + geometries, while summation is performed over all discreteprocesses and regional topologies Φ j dis). The symbolic differential D[wΦ]in the general path integral (6.133), represents an adaptive path measure,defined as a weighted productN∏D[wΦ] = lim w s dΦ i s, (i = 1, ..., n = con + dis), (6.134)N−→∞s=1which is in practice satisfied with a large N corresponding to infinitesimaltemporal division of the four motivational phases (*). Technically, thepath integral (6.133) calculates the amplitude for the transition functorT A : Intention ⇛ Action.In the exponent of the path integral (6.133) we have the action S[Φ]and the imaginary unit i = √ −1 (i can be converted into the real number-1 using the so–called Wick rotation, see next subsection).In this way, we get a range of micro–objects in the local LSF at theshort time–level: ensembles of rapidly fluctuating, noisy and crossing paths,force–fields, local geometries with obstacles and topologies with holes. However,by averaging process, both in time and along ensembles of paths, fieldsand geometries, we recover the corresponding global MD & CD variables.Infinite–Dimensional Neural Network. The adaptive path integral(6.133) incorporates the local learning process according to the standardformula: New V alue = Old V alue + Innovation. The general weightsw s = w s (t) in (6.134) are updated by the MONIT OR feedback during the


<strong>Geom</strong>etrical Path Integrals and Their Applications 1089transition process, according to one of the two standard neural learningschemes, in which the micro–time level is traversed in discrete steps, i.e., ift = t 0 , t 1 , ..., t s then t + 1 = t 1 , t 2 , ..., t s+1 :(1) A self–organized, unsupervised (e.g., Hebbian–like [Hebb (1949)]) learningrule:w s (t + 1) = w s (t) + σ η (wd s(t) − w a s (t)), (6.135)where σ = σ(t), η = η(t) denote signal and noise, respectively, whilesuperscripts d and a denote desired and achieved micro–states, respectively;or(2) A certain form of a supervised gradient descent learning:w s (t + 1) = w s (t) − η∇J(t), (6.136)where η is a small constant, called the step size, or the learning rateand ∇J(n) denotes the gradient of the ‘performance hyper–surface’ atthe t−th iteration.Both Hebbian and supervised learning are used for the local decision makingprocess (see below) occurring at the intention formation faze F.In this way, local micro–level of LSF total represents an infinite–dimensional neural network. In the cognitive psychology framework, ouradaptive path integral (6.133) can be interpreted as semantic integration(see [Bransford and Franks (1971); Ashcraft (1994)]).Motion and Decision Making in LSF pathsOn the macro–level in the subspace LSF paths we have the (loco)motionaction principleδS[x] = 0,with the Newtonian–like action S[x] given byS[x] =∫ tfint inidt [ 1 2 g ij ẋ i ẋ j + ϕ i (x i )], (6.137)where overdot denotes time derivative, so that ẋ i represents processingspeed, or (loco)motion velocity vector. The first bracket term in (6.137)represents the kinetic energy T ,T = 1 2 g ij ẋ i ẋ j ,


1090 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiongenerated by the Riemannian metric tensor g ij , while the second bracketterm, ϕ i (x i ), denotes the family of potential force–fields, driving the(loco)mo-tions x i = x i (t) (the strengths of the fields ϕ i (x i ) depend ontheir positions x i in LSF, see LSF fields below). The corresponding Euler–Lagrangian equation gives the Newtonian–like equation of motionddt T ẋ i − T x i = −ϕi xi, (6.138)(subscripts denote the partial derivatives), which can be put into the standardLagrangian formddt L ẋ i = L x i, with L = T − ϕi (x i ).In the next subsection we use the micro–level implications of the actionS[x] as given by (6.137), for dynamical descriptions of the local decision–making process.On the micro–level in the subspace LSF paths , instead of a single pathdefined by the Newtonian–like equation of motion (6.138), we have an ensembleof fluctuating and crossing paths with weighted probabilities (of theunit total sum). This ensemble of micro–paths is defined by the simplestinstance of our adaptive path integral (6.133), similar to the Feynman’soriginal sum over histories,∫〈Action|Intention〉 paths = Σ D[wx] e iS[x] , (6.139)where D[wx] is a functional measure on the space of all weighted paths,and the exponential depends on the action S[x] given by (6.137). Thisprocedure can be redefined in a mathematically cleaner way if we Wick–rotate the time variable t to imaginary values t ↦→ τ = it, thereby makingall integrals real:∫ ∫Σ D[wx] e iS[x] W ick✲ Σ D[wx] e −S[x] . (6.140)Discretization of (6.140) gives the thermodynamic–like partition functionZ = ∑ je −wjEj /T , (6.141)where E j is the motion energy eigenvalue (reflecting each possible motivationalenergetic state), T is the temperature–like environmental controlparameter, and the sum runs over all motion energy eigenstates (labelled


<strong>Geom</strong>etrical Path Integrals and Their Applications 1091by the index j). From (6.141), we can further calculate all thermodynamic–like and statistical properties of MD & CD (see e.g., [Feynman (1972)]), asfor example, transition entropy S = k B ln Z, etc.From cognitive perspective, our adaptive path integral (6.139) calculatesall (alternative) pathways of information flow during the transitionIntention −→ Action.In the language of transition–propagators, the integral over histories(6.139) can be decomposed into the product of propagators (i.e., Fredholmkernels or Green functions) corresponding to the cascade of the four motivationalphases (*)∫〈Action|Intention〉 paths = Σ dx F dx I dx M dx T K(F, I)K(I, M)K(M, T ),(6.142)satisfying the Schrödinger–like equation (see e.g., [Dirac (1982)])i ∂ t 〈Action|Intention〉 paths = H Action 〈Action|Intention〉 paths , (6.143)where H Action represents the Hamiltonian (total energy) function availableat the state of Action. Here our ‘golden rule’ is: the higher the H Action ,the lower the microscopic fatigue.In the connectionist language, our propagator expressions (6.142–6.143)represent activation dynamics, to which our Monitor process gives a kind ofbackpropagation feedback, a version of the basic supervised learning (6.136).Mechanisms of Decision–Making under Uncertainty. The basicquestion about our local decision making process, occurring underuncertainty at the intention formation faze F, is: Which alternative tochoose? (see [Roe et al. (2001); Grossberg (1982); Grossberg (1999);Grossberg (1988); Ashcraft (1994)]). In our path–integral language thisreads: Which path (alternative) should be given the highest probabilityweight w? Naturally, this problem is iteratively solved by the learning process(6.135–6.136), controlled by the MONIT OR feedback, which we termalgorithmic approach.In addition, here we analyze qualitative mechanics of the local decisionmaking process under uncertainty, as a heuristic approach. This qualitativeanalysis is based on the micro–level interpretation of the Newtonian–likeaction S[x], given by (6.137) and figuring both processing speed ẋ andLTM (i.e., the force–field ϕ(x), see next subsection). Here we considerthree different cases:


1092 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction(1) If the potential ϕ(x) is not very dependent upon position x(t), thenthe more direct paths contribute the most, as longer paths, with highermean square velocities [ẋ(t)] 2 make the exponent more negative (afterWick rotation (6.140)).(2) On the other hand, suppose that ϕ(x) does indeed depend on positionx. For simplicity, let the potential increase for the larger values of x.Then a direct path does not necessarily give the largest contribution tothe overall transition probability, because the integrated value of thepotential is higher than over another paths.(3) Finally, consider a path that deviates widely from the direct path. Thenϕ(x) decreases over that path, but at the same time the velocity ẋincreases. In this case, we expect that the increased velocity ẋ wouldmore than compensate for the decreased potential over the path.Therefore, the most important path (i.e., the path with the highest weightw) would be one for which any smaller integrated value of the surroundingfield potential ϕ(x) is more than compensated for by an increase in kinetic–like energy m 2 ẋ2 . In principle, this is neither the most direct path, nor thelongest path, but rather a middle way between the two. Formally, it is thepath along which the average Lagrangian is minimal,< m 2 ẋ2 + ϕ(x) > ✲ min, (6.144)i.e., the path that requires minimal memory (both LTM and WM, seeLSF fields below) and processing speed. This mechanical result is consistentwith the ‘filter theory’ of selective attention [Broadbent (1958)], proposedin an attempt to explain a range of the existing experimental results. Thistheory postulates a low level filter that allows only a limited number ofpercepts to reach the brain at any time. In this theory, the importance ofconscious, directed attention is minimized. The type of attention involvinglow level filtering corresponds to the concept of early selection [Broadbent(1958)].Although we termed this ‘heuristic approach’ in the sense that we caninstantly feel both the processing speed ẋ and the LTM field ϕ(x) involved,there is clearly a psycho–physical rule in the background, namely the averagingminimum relation (6.144).From the decision making point of view, all possible paths (alternatives)represent the consequences of decision making. They are, by default,short–term consequences, as they are modelled in the micro–time–level.However, the path integral formalism allows calculation of the long–term


<strong>Geom</strong>etrical Path Integrals and Their Applications 1093consequences, just by extending the integration time, t fin −→ ∞. Besides,this averaging decision mechanics – choosing the optimal path – actuallyperforms the ‘averaging lift’ in the LSF: from micro– to the macro–level.Force–Fields and Memory in LSF fieldsAt the macro–level in the subspace LSF fields we formulate the force–field action principleδS[ϕ] = 0, (6.145)with the action S[ϕ] dependent on Lewinian force–fields ϕ i = ϕ i (x) (i =1, ..., N), defined as a temporal integralS[ϕ] =∫ tfinwith Lagrangian density given by∫L[ϕ] = d n x L(ϕ i , ∂ x j ϕ i ),t iniL[ϕ] dt, (6.146)where the integral is taken over all n coordinates x j = x j (t) of the LSF,and ∂ x j ϕ i are partial derivatives of the field variables over coordinates.On the micro–level in the subspace LSF fields we have the Feynman–type sum over fields ϕ i (i = 1, ..., N) given by the adaptive path integral∫∫〈Action|Intention〉 fields = Σ D[wϕ] e iS[ϕ] W ick✲ Σ D[wϕ] e −S[ϕ] ,(6.147)with action S[ϕ] given by temporal integral (6.146). (Choosing specialforms of the force–field action S[ϕ] in (6.147) defines micro–level MD & CD,in the LSF fields space, that is similar to standard quantum–field equations,see e.g., [Ramond (1990)].) The corresponding partition function has theform similar to (6.141), but with field energy levels.Regarding topology of the force fields, we have in place n−categoricalLagrangian–field structure on the Riemannian LSF manifold M,Φ i : [0, 1] → M, Φ i : Φ i 0 ↦→ Φ i 1,generalized from the recursive homotopy dynamics (3.205) above, using( )ddt f ẋ i = f ✲ ∂Lx i ∂µ∂ µ Φ i = ∂L∂Φ i ,with [x 0 , x 1 ] ✲ [Φi0 , Φ i 1].


1094 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRelationship between Memory and Force–Fields. As alreadymentioned, the subspace LSF fields is related to our memory storage[Ashcraft (1994)]. Its global macro–level represents the long–term memory(LTM), defined by the least action principle (6.145), related tocognitive economy in the model of semantic memory [Ratcliff (1978);Collins and Quillian (1969)]. Its local micro–level represents working memory(WM), a limited–capacity ‘bottleneck’ defined by the adaptive pathintegral (6.147). According to our formalism, each of Miller’s 7 ± 2 units[Miller (1956)] of the local WM are adaptively stored and averaged to givethe global LTM capacity (similar to the physical notion of potential). Thisaveraging memory lift, from WM to LTM represents retroactive interference,while the opposite direction, given by the path integral (6.147) itself,represents proactive interference. Both retroactive and proactive interferencesare examples of the impact of cognitive contexts on memory. Motivationalcontexts can exert their influence, too. For example, a reduction intask–related recall following the completion of the task is one of the clearestexamples of force–field influences on memory: the amount of detailsremembered of a task declines as the force–field tension to complete thetask is reduced by actually completing it.Once defined, the global LTM potential ϕ = ϕ(x) is then affecting thelocomotion transition paths through the path action principle (6.137), aswell as general learning (6.135–6.136) and decision making process (6.144).On the other hand, the two levels of LSF fields fit nicely into the two levelsof processing framework, as presented by [Craik and Lockhart (1972)],as an alternative to theories of separate stages for sensory, working andlong–term memory. According to the levels of processing framework, stimulusinformation is processed at multiple levels simultaneously dependingupon its characteristics. In this framework, our macro–level memory field,defined by the fields action principle (6.145), corresponds to the shallowmemory, while our micro–level memory field, defined by the adaptive pathintegral (6.147), corresponds to the deep memory.<strong>Geom</strong>etries, Topologies and Noise in LSF geomOn the macro–level in the subspace LSF geom representing ann−dimensional smooth manifold M with the global Riemannian metrictensor g ij , we formulate the geometrical action principleδS[g ij ] = 0,


<strong>Geom</strong>etrical Path Integrals and Their Applications 1095where S = S[g ij ] is the n−dimensional geodesic action on M,∫ √S[g ij ] = d n x g ij dx i dx j . (6.148)The corresponding Euler–Lagrangian equation gives the geodesic equationof the shortest path in the manifold M,ẍ i + Γ i jk ẋ j ẋ k = 0,where the symbol Γ i jkdenotes the so–called affine connection which is thesource of curvature, which is geometrical description for noise (see [Ingber(1997); Ingber (1998)]). The higher the local curvatures of the LSF–manifold M, the greater the noise in the life space. This noise is the sourceof our micro–level fluctuations. It can be internal or external; in both casesit curves our micro–LSF.Otherwise, if instead we choose an n−dimensional Hilbert–like action(see [Misner et al. (1973)]),∫ √S[g ij ] = d n x det |g ij |R, (6.149)where R is the scalar curvature (derived from Γ i jk), we get then−dimensional Einstein–like equation: G ij = 8πT ij , where G ij is theEinstein–like tensor representing geometry of the LSF manifold M (G ij isthe trace–reversed Ricci tensor R ij , which is itself the trace of the Riemanncurvature tensor of the manifold M), while T ij is the n−dimensional stress–energy–momentum tensor. This equation explicitly states that psycho–physics of the LSF is proportional to its geometry. T ij is important quantity,representing motivational energy, geometry–imposed stress and momentumof (loco)motion. As before, we have our ‘golden rule’: the greaterthe T ij −components, the higher the speed of cognitive processes and thelower the macroscopic fatigue.The choice between the geodesic action (6.148) and the Hilbert action(6.149) depends on our interpretation of time. If time is not included in theLSF manifold M (non–relativistic approach) then we choose the geodesicaction. If time is included in the LSF manifold M (making it a relativistic–like n−dimensional space–time) then the Hilbert action is preferred. Thefirst approach is more related to the information processing and the workingmemory. The later, space–time approach can be related to the long–termmemory: we usually recall events closely associated with the times of theirhappening.


1096 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOn the micro–level in the subspace LSF geom we have the adaptive sumover geometries, represented by the path integral over all local (regional)Riemannian metrics g ij = g ij (x) varying from point to point on M (modulodiffeomorphisms),〈Action|Intention〉 geom =∫Σ D[wg ij ] e iS[gij]W ick✲∫Σ D[wg ij ] e −S[gij] ,(6.150)where D[g ij ] is diffeomorphism equivalence class of g ij (x) ∈ M.To include the topological structure (e.g., a number of holes) in M, wecan extend (6.150) as∫〈Action|Intention〉 geom/top = ∑Σ D[wg ij ] e iS[gij] , (6.151)topol.where the topological sum is taken over all connectedness–components ofM determined by the Euler characteristic χ of M. This type of integraldefines the theory of fluctuating geometries, a propagator between(n − 1)−dimensional boundaries of the n−dimensional manifold M. Onehas to contribute a meaning to the integration over geometries. A keyingredient in doing so is to approximate (using simplicial approximationand Regge calculus [Misner et al. (1973)]) in a natural way the smoothstructures of the manifold M by piecewise linear structures (mostly usingtopological simplices ∆). In this way, after the Wick–rotation (6.140), theintegral (6.150–6.151) becomes a simple statistical system, given by partitionfunction Z = ∑ ∆ 1C ∆e −S∆ , where the summation is over all triangulations∆ of the manifold M, while C T is the order of the automorphismgroup of the performed triangulation.Micro–Level <strong>Geom</strong>etry: the source of noise and stress in LSF.The subspace LSF geom is the source of noise, fluctuations and obstacles,as well as psycho–physical stress. Its micro–level is adaptive, reflectingthe human ability to efficiently act within the noisy environment and underthe stress conditions. By averaging it produces smooth geometry ofcertain curvature, which is at the same time the smooth psycho–physics.This macro–level geometry directly affects the memory fields and indirectlyaffects the (loco)motion transition paths.The Mental Force Law. As an effective summary of thissection, we state that the psychodynamic transition functor T A :INT ENT ION tini ⇛ ACT ION tfin , defined by the generic path integral


<strong>Geom</strong>etrical Path Integrals and Their Applications 1097(6.133), can be interpreted as a mental force law, analogous to our musculo–skeletal covariant force law, F i = mg ij a j , and its associated covariant forcefunctor F ∗ : T T ∗ M → T T M [<strong>Ivancevic</strong> and <strong>Ivancevic</strong> (2006)].6.5 Application: Witten’s TQFT, SW–Monopoles andStringsIn this section we review three parts of modern physics associated to thename of the Fields Medalist Edward Witten: topological quantum field theory,Seiberg–Witten monopole theory and open superstring theory. Theyare all extensions of a general quantum field theory (QFT).Recall that any QFT deals with smooth maps γ : Σ → M of Riemannianmanifolds Σ and M such that the dimension of Σ is the dimension of thetheory. On the set Map(Σ, M) of all smooth maps γ = γ(φ), we also havedefined an action function S[φ] of the field variables φ. A non–relativisticQFT studies real–valued (Euclidean) path integrals of the form∫V (φ)D[φ] e −S[φ]/ħ , (6.152)Map(Σ,M)where D[φ] represents some measure on the space of paths, is the Planckconstant and V : Map(Σ, M) → R is an insertion function. The numbere −S[φ]/ħ should be interpreted as the probability amplitude of the contributionof the map γ : Σ → M to the path integral. The associated integral∫Z E =dφ e −S[φ]/ħ ,Map(Σ,M)is the partition function of the theory. In a relativistic QFT, the space Σhas a Lorentzian metric of signature (−, +, ..., +). The first coordinate isreserved for time, the rest are for space. In this case, the real–valued pathintegral (6.152) is replaced with the complex–valued path integral∫Z M =V (φ)D[φ] e iS[φ]/ħ .Map(Σ,M)6.5.1 Topological Quantum Field TheoryBefore we come to (super)strings, we give a brief on topological quantumfield theory (TQFT), as developed by Ed Witten, from his original pathintegral point of view (see [Witten (1988a); Labastida and Lozano (1998)]).


1098 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionTQFT originated in 1982, when Witten rewrote classical Morse theory (seesection 3.10.5.1 above, as well as section 3.13.5.2 below) in Dick Feynman’slanguage of quantum field theory [Witten (1982)]. Witten’s argumentsmade use of Feynman’s path integrals and consequently, at first, they wereregarded as mathematically non–rigorous. However, a few years later, A.Floer reformulated a rigorous Morse–Witten theory [Floer (1987)] (thatwon a Fields medal for Witten). This trend in which some mathematicalstructure is first constructed by quantum field theory methods and thenreformulated in a rigorous mathematical ground constitutes one of the tendenciesin modern physics.In TQFT our basic topological space is an nD Riemannian manifoldM with a metric g µν . Let us consider on it a set of fields {φ i }, and letS[φ i ] be a real functional of these fields which is regarded as the action ofthe theory. We consider ‘operators’, O α (φ i ), which are in general arbitraryfunctionals of the fields. In TQFT these functionals are real functionalslabelled by some set of indices α carrying topological or group–theoreticaldata. The vacuum expectation value (VEV) of a product of these operatorsis defined as∫〈O α1 O α2 · · · O αp 〉 = [Dφ i ]O α1 (φ i )O α2 (φ i ) · · · O αp (φ i ) exp (−S[φ i ]) .A quantum field theory is considered topological if the following relation issatisfied:δδg µν 〈O α 1O α2 · · · O αp 〉 = 0, (6.153)i.e., if the VEV of some set of selected operators is independent of the metricg µν on M. If such is the case those operators are called ‘observables’.There are two ways to guarantee, at least formally, that condition(6.153) is satisfied. The first one corresponds to the situation in whichboth, the action S[φ i ], as well as the operators O αi are metric independent.These TQFTs are called of Schwarz type. The most important representativeis Chern–Simons gauge theory. The second one corresponds to thecase in which there exist a symmetry, whose infinitesimal form is denotedby δ, satisfying the following properties:δO αi = 0, T µν = δG µν , (6.154)


<strong>Geom</strong>etrical Path Integrals and Their Applications 1099where T µν is the SEM–tensor of the theory, i.e.,T µν (φ i ) =δδg µν S[φ i]. (6.155)The fact that δ in (6.154) is a symmetry of the theory implies thatthe transformations δφ i of the fields are such that both δA[φ i ] = 0 andδO αi (φ i ) = 0. Conditions (6.154) lead, at least formally, to the followingrelation for VEVs:∫δδg µν 〈O α 1O α2 · · · O αp 〉 = − [Dφ i ]O α1 (φ i )O α2 (φ i ) · · · O αp (φ i )T µν e −S[φ i ]∫= − [Dφ i ]δ ( O α1 (φ i )O α2 (φ i ) · · · O αp (φ i )G µν exp (−S[φ i ]) ) = 0, (6.156)which implies that the quantum field theory can be regarded as topological.This second type of TQFTs are called of Witten type. One of its mainrepresentatives is the theory related to Donaldson invariants, which is atwisted version of N = 2 supersymmetric Yang–Mills gauge theory. Itis important to remark that the symmetry δ must be a scalar symmetry,i.e., that its symmetry parameter must be a scalar. The reason is that,being a global symmetry, this parameter must be covariantly constant andfor arbitrary manifolds this property, if it is satisfied at all, implies strongrestrictions unless the parameter is a scalar.Most of the TQFTs of cohomological type satisfy the relation:S[φ i ] = δΛ(φ i ), (6.157)for some functional Λ(φ i ). This has far–reaching consequences, for it meansthat the topological observables of the theory, in particular the partitionfunction, (path integral) itself are independent of the value of the couplingconstant. Indeed, let us consider for example the VEV:∫〈O α1 O α2 · · · O αp 〉 =[Dφ i ]O α1 (φ i )O α2 (φ i ) · · · O αp (φ i ) e − 1g 2 S[φ i ] . (6.158)Under a change in the coupling constant, 1/g 2 → 1/g 2 −∆, one has (assumingthat the observables do not depend on the coupling), up to first–orderin ∆:∫+ ∆ [Dφ i ]δ〈O α1 O α2 · · · O αp 〉 −→ 〈O α1 O α2 · · · O αp 〉[O α1 (φ i )O α2 (φ i ) · · · O αp (φ i )Λ(φ i ) exp(− 1 )]g 2 S[φ i]= 〈O α1 O α2 · · · O αp 〉. (6.159)


1100 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionHence, observables can be computed either in the weak coupling limit, g →0, or in the strong coupling limit, g → ∞.So far we have presented a rather general definition of TQFT and madea series of elementary remarks. Now we will analyze some aspects of itsstructure. We begin pointing out that given a theory in which (6.154) holdsone can build correlators which correspond to topological invariants (in thesense that they are invariant under deformations of the metric g µν ) justby considering the operators of the theory which are invariant under thesymmetry. We will call these operators observables. In virtue of (6.156),if one of these operators can be written as a symmetry transformation ofanother operator, its presence in a correlation function will make it vanish.Thus we may identify operators satisfying (6.154) which differ by an operatorwhich corresponds to a symmetry transformation of another operator.Let us denote the set of the resulting classes by {Φ}. By restricting theanalysis to the appropriate set of operators, one has that in fact,δ 2 = 0. (6.160)Property (6.160) has consequences on the features of TQFT. First, thesymmetry must be odd which implies the presence in the theory of commutingand anticommuting fields. For example, the tensor G µν in (6.154) mustbe anticommuting. This is the first appearance of an odd non–spinorialfield in TQFT. Those kinds of objects are standard features of cohomologicalTQFTs. Second, if we denote by Q the operator which implements thissymmetry, the observables of the theory can be described as the cohomologyclasses of Q:{Φ} = Ker QIm Q , Q2 = 0. (6.161)Equation (6.154) means that in addition to the Poincaré group thetheory possesses a symmetry generated by an odd version of the Poincarégroup. The corresponding odd generators are constructed out of the tensorG µν in much the same way as the ordinary Poincaré generators are builtout of T µν . For example, if P µ represents the ordinary momentum operator,there exists a corresponding odd one G µ such thatP µ = {Q, G µ }. (6.162)Now, let us discuss the structure of the Hilbert space of the theory invirtue of the symmetries that we have just described. The states of thisspace must correspond to representations of the algebra generated by the


<strong>Geom</strong>etrical Path Integrals and Their Applications 1101operators in the Poincaré groups and by Q. Furthermore, as follows fromour analysis of operators leading to (6.161), if one is interested only in states|Ψ〉 leading to topological invariants one must consider states which satisfyQ|Ψ〉 = 0, (6.163)and two states which differ by a Q−exact state must be identified. Theodd Poincaré group can be used to generate descendant states out of astate satisfying (6.163). The operators G µ act non–trivially on the statesand in fact, out of a state satisfying (6.163) we can build additional statesusing this generator. The simplest case consists ofwhere γ 1 is a 1–cycle.satisfies (6.163):∫γ 1G µ |Ψ〉,One can verify using (6.154) that this new state∫∫∫Q G µ |Ψ〉 =γ 1{Q, G µ }|Ψ〉 =γ 1P µ |Ψ〉 = 0.γ 1Similarly, one may construct other invariants tensoring n operators G µ andintegrating over n−cycles γ n :∫γ nG µ1 G µ2 ...G µn |Ψ〉. (6.164)Notice that since the operator G µ is odd and its algebra is Poincaré–like theintegrand in this expression is an exterior differential n−form. These statesalso satisfy condition (6.163). Therefore, starting from a state |Ψ〉 ∈ ker Qwe have built a set of partners or descendants giving rise to a topologicalmultiplet. The members of a multiplet have well defined ghost number.If one assigns ghost number -1 to the operator G µ the state in (6.164)has ghost number -n plus the ghost number of |Ψ〉. Now, n is boundedby the dimension of the manifold X. Among the states constructed inthis way there may be many which are related via another state which isQ−exact, i.e., which can be written as Q acting on some other state. Letus try to single out representatives at each level of ghost number in a giventopological multiplet.Consider an (n − 1)−cycle which is the boundary of an nD surface,


1102 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionγ n−1 = ∂S n . If one builds a state taking such a cycle one finds (P µ = −i∂ µ ),∫∫G µ1 G µ2 ...G µn−1 |Ψ〉 = i P [µ1 G µ2 G µ3 ...G µn ]|Ψ〉 (6.165)γ n−1 S n∫= iQ G µ1 G µ2 ...G µn |Ψ〉,S ni.e., it is Q−exact. The square–bracketed subscripts in (6.165) denote thatall indices between them must by antisymmetrized. In (6.165) use hasbeen made of (6.162). This result tells us that the representatives we arelooking for are built out of the homology cycles of the manifold X. Given amanifold X, the homology cycles are equivalence classes among cycles, theequivalence relation being that two n−cycles are equivalent if they differby a cycle which is the boundary of an n + 1 surface. Thus, knowledgeon the homology of the manifold on which the TQFT is defined allows usto classify the representatives among the operators (6.164). Let us assumethat X has dimension d and that its homology cycles are γ in, (i n = 1, ..., d n ,n = 0, ..., d), where d n is the dimension of the n−homology group, and dthe dimension of X. Then, the non–trivial partners or descendants of agiven |Ψ〉 highest–ghost–number state are labelled in the following way:∫G µ1 G µ2 ...G µn |Ψ〉, (i n = 1, ..., d n , n = 0, ..., d).γ inA similar construction to the one just described can be made for fields.Starting with a field φ(x) which satisfies,[Q, φ(x)] = 0, (6.166)one can construct other fields using the operators G µ . These fields, whichwe call partners are antisymmetric tensors defined as,φ (n)µ 1 µ 2 ...µ n(x) = 1 n! [G µ 1, [G µ2 ...[G µn , φ(x)}...}}, (n = 1, ..., d).Using (6.162) and (6.166) one finds that these fields satisfy the so–calledtopological descent equations:dφ (n) = i[Q, φ (n+1) },where the subindices of the forms have been suppressed for simplicity, andthe highest–ghost–number field φ(x) has been denoted as φ (0) (x). Theseequations enclose all the relevant properties of the observables which are


<strong>Geom</strong>etrical Path Integrals and Their Applications 1103constructed out of them. They constitute a very useful tool to build theobservables of the theory.6.5.2 Seiberg–Witten Theory and TQFTRecall that the field of low–dimensional geometry and topology [Atiyah(1988b)] has undergone a dramatic phase of progress in the last decade ofthe 20th Century, prompted, to a large extend, by new ideas and discoveriesin mathematical physics. The discovery of quantum groups [Drinfeld(1986)] in the study of the Yang–Baxter equation [Baxter (1982)] has reshapedthe theory of knots and links [Jones (1985); Reshetikhin and Turaev(1991); Zhang et. al. (1991)]; the study of conformal field theory andquantum Chern–Simons theory [Witten (1989)] in physics had a profoundimpact on the theory of 3–manifolds; and most importantly, investigationsof the classical Yang–Mills (YM) theory led to the creation of the Donaldsontheory of 4–manifolds [Freed and Uhlenbeck (1984); Donaldson (1987)].Witten [Witten (1994)] discovered a new set of invariants of 4–manifoldsin the study of the Seiberg–Witten (SW) monopole equations, which havetheir origin in supersymmetric gauge theory. The SW theory, while closelyrelated to Donaldson theory, is much easier to handle. Using SW theory,proofs of many theorems in Donaldson theory have been simplified,and several important new results have also been obtained [Taubes (1990);Taubes (1994)].In [Zhang et. al. (1995)] a topological quantum field theory was introducedwhich reproduces the SW invariants of 4–manifolds. A geometricalinterpretation of the 3D quantum field theory was also given.6.5.2.1 SW Invariants and Monopole EquationsRecall that the SW monopole equations are classical field theoretical equationsinvolving a U(1) gauge field and a complex Weyl spinor on a 4Dmanifold. Let X denote the 4–manifold, which is assumed to be orientedand closed. If X is spin, there exist positive and negative spin bundles S ±of rank two. Introduce a complex line bundle L → X. Let A be a connectionon L and M be a section of the product bundle S + ⊗ L. Recall thatthe SW monopole equations readF + kl = − i 2 ¯MΓ kl M, D A M = 0, (6.167)


1104 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere D A is the twisted Dirac operator, Γ ij = 1 2 [γ i, γ j ], and F + representsthe self–dual part of the curvature of L with connection A.If X is not a spin manifold, then spin bundles do not exist. However,it is always possible to introduce the so called Spin c bundles S ± ⊗ L,with L 2 being a line bundle. Then in this more general setting, the SWmonopoles equations look formally the same as (6.167), but the M shouldbe interpreted as a section of the the Spin C bundle S + ⊗ L.Denote by M the moduli space of solutions of the SW monopole equationsup to gauge transformations. Generically, this space is a manifold.Its virtual dimension is equal to the number of solutions of the followingequations(dψ) + kl + i 2( ¯MΓkl N + ¯NΓ kl M ) = 0, D A N + ψM = 0,∇ k ψ k + i (NM − MN) = 0, (6.168)2where A and M are a given solution of (6.167), ψ ∈ Ω 1 (X) is a oneform, (dψ) + ∈ Ω 2,+ (X) is the self dual part of the two form dψ, andN ∈ S + ⊗ L. The first two of the equations in (6.168) are the linearizationof the monopole equations (6.167), while the last one is a gauge fixingcondition. Though with a rather unusual form, it arises naturally from thedual operator governing gauge transformationsLetC : Ω 0 (X) → Ω 1 (X) ⊕ (S + ⊗ L),φ ↦→ (−dφ, iφM).T : Ω 1 (X) ⊕ (S + ⊗ L) → Ω 0 (X) ⊕ Ω 2,+ (X) ⊕ (S − ⊗ L),be the operator governing equation (6.168), namely, the operator whichallows us to rewrite (6.168) as T (ψ, N) = 0. Then T is an elliptic operator,the index Ind(T ) of which yields the virtual dimension of M. Astraightforward application of the Atiyah–Singer index Theorem gives2χ(X) + 3σ(X)Ind(T ) = − + c 1 (L) 2 ,4where χ(X) is the Euler character of X, σ(X) its signature index and c 1 (L) 2is the square of the first Chern class of L evaluated on X in the standardway.When Ind(T ) equals zero, the moduli space generically consists of afinite number of points, M = {p t : t = 1, 2, ..., I}. Let ɛ t denote the signof the determinant of the operator T at p t , which can be defined with


<strong>Geom</strong>etrical Path Integrals and Their Applications 1105mathematical rigor. Then the SW invariant of the 4–manifold X is definedby ∑ I1 ɛ t.The fact that this is indeed an invariant(i.e., independent of the metric)of X is not very difficult to prove, and we refer to [Witten (1994)] fordetails. As a matter of fact, the number of solutions of a system of equationsweighted by the sign of the operator governing the equations(i.e., the analogof T ) is a topological invariant in general [Witten (1994)]. This point of viewhas been extensively explored by Vafa and Witten [Vafa and Witten (1994)]within the framework of topological quantum field theory in connection withthe so called S duality. Here we wish to explore the SW invariants followinga similar line as that taken in [Witten (1988a); Vafa and Witten (1994)].6.5.2.2 Topological LagrangianIntroduce a Lie super–algebra with an odd generator Q and two even generatorsU and δ obeying the following (anti)commutation relations [Zhanget. al. (1995)][U, Q] = Q, [Q, Q] = 2δ, [Q, δ] = 0. (6.169)We will call U the ghost number operator, and Q the BRST–operator.Let A be a connection of L and M ∈ S + ⊗ L. We define the action ofthe super–algebra on these fields by requiring that δ coincide with a gaugetransformation with a gauge parameter φ ∈ Ω 0 (X). The field multipletsassociated with A and M furnishing representations of the super–algebraare (A, ψ, φ), and (M, N), where ψ ∈ Ω 1 (X), φ ∈ Ω 0 (X), and N is a sectionof S + ⊗L. They transform under the action of the super–algebra accordingto[Q, A i ] = ψ i , [Q, M] = N,[Q, ψ i ] = −∂ i φ, [Q, N] = iφM, [Q, φ] = 0.We assume that both A and M have ghost number 0, and thus will beregarded as bosonic fields when we study their quantum field theory. Theghost numbers of other fields can be read off the above transformation rules.We have that ψ and N are of ghost number 1, thus are fermionic, and φ is ofghost number 2 and bosonic. Note that the multiplet (A, ψ, φ) is what onewould get in the topological field theory for Donaldson invariants exceptthat our gauge group is U(1), while the existence of M and N is a newfeature. Also note that both M and ψ have the wrong statistics.


1106 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn order to construct a quantum field theory which will reproduce theSW invariants as correlation functions, anti–ghosts and Lagrangian multipliersare also required. We introduce the anti–ghost multiplet (λ, η)∈ Ω 0 (X), such that[U, λ] = −2λ, [Q, λ] = η, [Q, η] = 0,and the Lagrangian multipliers (χ, H) ∈ Ω 2,+ (X), and (µ, ν) ∈ S − ⊗L suchthat[U, χ] = −χ, [Q, χ] = H, [Q, H] = 0;[U, µ] = −µ, [Q, µ] = ν, [Q, ν] = iφµ.With the given fields, we construct the following functional which hasghost number -1:∫V ={[∇ k ψ k + i (X2 (NM − MN)]λ − χkl H kl − F + kl − i )2 ¯MΓ kl M}− ¯µ (ν − iD A M) − (ν − iD A M)µ , (6.170)where the indices of the tensorial fields are raised and lowered by a givenmetric g on X, and the integration measure is the standard √ gd 4 x. Also,M and ¯µ etc. represent the Hermitian conjugate of the spinorial fields. Ina formal language, M ∈ S + ⊗ L −1 and ¯µ, ¯ν, D A M ∈ S − ⊗ L −1 . Followingthe standard procedure in constructing topological quantum field theory,we take the classical action of our theory to be [Zhang et. al. (1995)]:S = [Q, V ], which has ghost number 0. One can easily show that S isalso BRST invariant, i.e., [Q, S] = 0, thus it is invariant under the fullsuper–algebra (6.169).The bosonic Lagrangian multiplier fields H and ν do not have any dynamics,and so can be eliminated from the action by using their equationsof motionH kl = 1 2(F + kl + i )2 ¯MΓ kl M , ν = 1 2 iD AM. (6.171)Then we arrive at the following expression for the action [Zhang et. al.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1107(1995)]∫S =X{[−∆φ + MMφ − iNN]λ − [∇ k ψ k + i (NM − MN)]η + 2iφ¯µµ2+ (iD A N − γ.ψM)µ − ¯µ (iD A N − γ.ψM)[ (− χ kl ∇ k ψ l − ∇ l ψ k) + i (+ ¯MΓkl N +2¯NΓ kl M )]} + S 0 , (6.172)where S 0 is given by∫S 0 =X{ 14 |F + + i 2 ¯MΓM| 2 + 1 2 |D AM| 2 }.It is interesting to observe that S 0 is nonnegative, and vanishes if and onlyif A and M satisfy the SW monopole equations. As pointed out in [Witten(1994)], S 0 can be rewritten as∫ { 1S 0 =4 |F + | 2 + 1 4 |M|4 + 1 8 R|M|2 + g ij D i MD j M},Xwhere R is the scalar curvature of X associated with the metric g. If Ris nonnegative over the entire X, then the only square integrable solutionof the monopole equations (6.167) is A is a anti-self-dual connection andM = 0.6.5.2.3 Quantum Field TheoryWe will now investigate the quantum field theory defined by the classicalaction (6.172) with the path integral method. Let F collectively denote allthe fields. The partition function of the theory is defined by [Zhang et. al.(1995)]∫Z = DF exp(− 1 e 2 S),where e ∈ R is the coupling constant. The integration measure DF isdefined on the space of all the fields. However, since S is invariant underthe gauge transformations, we assume the integration over the gauge fieldto be performed over the gauge orbits of A. In other words, we fix a gaugefor the A field using, say, a Faddeev–Popov procedure. This can be carriedout in the standard manner, thus there is no need for us to spell out thedetails here. The integration measure DF can be shown to be invariantunder the super charge Q. Also, it does not explicitly involve the metric gof X.


1108 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionbyLet W be any operator in the theory. Its correlation function is defined∫Z[W ] = DF exp(− 1 e 2 S)W.It follows from the Q invariance of both the action S and the path integrationmeasure that for any operator W ,∫Z[[Q, W ]] = DF exp(− 1 S)[Q, W ] = 0.e2 For the purpose of constructing topological invariants of the 4–manifoldX, we are particularly interested in operators W which are BRST–closed,[Q, W ] = 0, (6.173)but not BRST–exact, i.e., can not be expressed as the (anti)–commutatorsof Q with other operators. For such a W , if its variation with respect tothe metric g is BRST exact,δ g W = [Q, W ′ ], (6.174)then its correlation function Z[W ] is a topological invariant of X (by thatwe really mean that it does not depend on the metric g):∫δ g Z[W ] = DF exp(− 1 e 2 S)[Q, W ′ − 1 e 2 δ gV.W ] = 0.In particular, the partition function Z itself is a topological invariant.Another important property of the partition function is that it does notdepend on the coupling constant e:∫∂Z∂e 2 =DF 1 e 4 exp(− 1 S)[Q, V ] = 0.e2 Therefore, Z can be computed exactly in the limit when the coupling constantgoes to zero. Such a computation can be carried out in the standardway: Let A o , M o be a solution of the equations of motion of A and M arisingfrom the action S. We expand the fields A and M around this classicalconfiguration,A = A o + ea,M = M o + em,where a and m are the quantum fluctuations of A and M respectively. Allthe other fields do not acquire background components, thus are purely


<strong>Geom</strong>etrical Path Integrals and Their Applications 1109quantum mechanical. We scale them by the coupling constant e, by settingN to eN, φ to eφ etc.. To the order o(1) in e 2 , we haveZ = ∑ exp(− 1 ∫e 2 S(p) cl) DF ′ exp(−S q (p) ),pwhere S q(p) is the quadratic part of the action in the quantum fields anddepends on the gauge orbit of the classical configuration A o , M o , which welabel by p. Explicitly [Zhang et. al. (1995)],S (p)q =∫X{[−∆φ + M o M o φ − iNN]λ − [∇ k ψ k + i 2 (NMo − M o N)]η + 2iφ¯µµ+ (iD A oN − γ.ψM o )µ − ¯µ (iD A oN − γ.ψM o )[ (− χ kl ∇ k ψ l − ∇ l ψ k) + i (+ ¯M o Γ kl N +2¯NΓ kl M o)]+ 1 4 |f + + i 2 ( ¯mΓM o + ¯M o Γm)| 2 + 1 }2 |iD A om + γ.aM o | 2 ,with f + the self–dual part of f = da. The classical part of the action isgiven by S (p)cl= S 0 | A=A o ,M=M o.The integration measure DF ′ has exactlythe same form as DF but with A replaced by a, and M by m, ¯M by ¯mrespectively. Needless to say, the summation over p runs through all gaugeclasses of classical configurations.Let us now examine further features of our quantum field theory. Agauge class of classical configurations may give a non–zero contributionto the partition function in the limit e 2 → 0 only if S (p)clvanishes, andthis happens if and only if A o and M o satisfy (6.167). Therefore, the SWmonopole equations are recovered from the quantum field theory.The equations of motion of the fields ψ and N in the semi–classical approximationcan be easily derived from the quadratic action S q(p) , solutionsof which are the zero modes of the quantum fields ψ and N. The equationsof motion read(dψ) + kl + i 2( ¯M o Γ kl N + ¯NΓ kl M o) = 0, D A oN + γ.ψM 0 = 0,∇ k ψ k + i (NM − MN) = 0. (6.175)2Note that they are exactly the same equations which we have alreadydiscussed in (6.168). The first two equations are the linearization of themonopole equations, while the last is a ‘gauge fixing condition’ for ψ. The


1110 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiondimension of the space of solutions of these equations is the virtual dimensionof the moduli space M. Thus, within the context of our quantum fieldtheoretical model, the virtual dimension of M is identified with the numberof the zero modes of the quantum fields ψ and N.For simplicity we assume that there are no zero modes of ψ and N, i.e.,the moduli space is zero–dimensional. Then no zero modes exist for theother two fermionic fields χ and µ. To compute the partition function inthis case, we first observe that the quadratic action S q(p) is invariant underthe supersymmetry obtained by expanding Q to first order in the quantumfields around the monopole solution A o , M o (equations of motion for thenonpropagating fields H and ν should also be used.). This supersymmetrytransforms the set of 8 real bosonic fields (each complex field is counted astwo real ones; the a i contribute 2 upon gauge fixing.) and the set of 16fermionic fields to each other. Thus at a given monopole background weget [Zhang et. al. (1995)]∫DF ′ exp(−S (p)q ) = Pfaff(∇ F)|Pfaff(∇ F )| = ɛ(p) ,where ɛ (p) is +1 or -1. In the above equation, ∇ F is the skew symmetricfirst order differential operator defining the fermionic ( part ) of the actionS q(p) , which can be read off from S q (p)0 Tto be ∇ F =−T ∗ . Therefore,0ɛ (p) is the sign of the determinant of the elliptic operator T at the monopolebackground A o , M o , and the partition function Z = ∑ p ɛ(p) coincides withthe SW invariant of the 4–manifold X.When the dimension of the moduli space M is greater than zero, thepartition function Z vanishes identically, due to integration over zero modesof the fermionic fields. In order to get any non trivial topological invariantsfor the underlying manifold X, we need to examine correlations functions ofoperators satisfying equations (6.173) and (6.174). A class of such operatorscan be constructed following the standard procedure [Witten (1994)]. Wedefine the following set of operators


<strong>Geom</strong>etrical Path Integrals and Their Applications 1111W k,0 = φkk! , W k,1 = ψW k−1,0 ,W k,2 = FW k−1,0 − 1 2 ψ ∧ ψW k−2,0, (6.176)W k,3 = F ∧ ψW k−2,0 − 1 3! ψ ∧ ψ ∧ ψW k−3,0,W k,4 = 1 2 F ∧ F W k−2,0 − 1 2 F ∧ ψ ∧ ψW k−3,0 − 1 4! ψ ∧ ψ ∧ ψ ∧ ψW k−4,0.These operators are clearly independent of the metric g of X. Although theyare not BRST invariant except for W k,0 , they obey the following equations[Zhang et. al. (1995)]dW k,0 = −[Q, W k,1 ], dW k,1 = [Q, W k,2 ],dW k,2 = −[Q, W k,3 ], dW k,3 = [Q, W k,4 ], dW k,4 = 0,which allow us to construct BRST invariant operators from the the W ’sin the following way: Let X i , i = 1, 2, 3, X 4 = X, be compact manifoldswithout boundary embedded in X. We assume that these submanifolds arehomologically nontrivial. Define∫Ô k,0 = W k,0 , Ô k,i = W k,i , (i = 1, 2, 3, 4). (6.177)X iAs we have already pointed out, Ô k,0 is BRST invariant. It follows fromthe descendent equations that[Q, Ôk,i] =∫[Q, W k,i ] =X i∫dW k,i−1 = 0.X iTherefore the operators Ô indeed have the properties (6.173) and (6.174).Also, for the boundary ∂K of an i + 1D manifold K embedded in X, wehave∫ ∫∫W k,i = dW k,i = [Q, W k,i+1 ],∂KKKis BRST trivial. The correlation function of ∫ ∂K W k,i with any BRSTinvariant operator is identically zero. This in particular shows that the Ô’sonly depend on the homological classes of the submanifolds X i .


1112 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction6.5.2.4 Dimensional Reduction and 3D Field TheoryIn this subsection we dimensionally reduce the quantum field theoreticalmodel for the SW invariant from 4D to 3D, thus to get a new topologicalquantum field theory defined on 3−manifolds. Its partition function yields a3−manifold invariant, which can be regarded as the SW version of Casson’sinvariant [Akbulut and McCarthy (1990); Taubes (1994)].We take the 4–manifold X to be of the form Y × [0, 1] with Y being acompact 3−manifold without boundary. The metric on X will be taken tobe(ds) 2 = (dt) 2 + g ij (x)dx i dx j ,where the ‘time’ t−independent g(x) is the Riemannian metric on Y . Weassume that Y admits a spin structure which is compatible with the Spin cstructure of X, i.e., if we think of Y as embedded in X, then this embeddinginduces maps from the Spin c bundles S ± ⊗ L of X to ˜S ⊗ L, where ˜S is aspin bundle and L is a line bundle over Y .To perform the dimensional reduction, we impose the condition that allfields are t in dependent. This leads to the following action [Zhang et. al.(1995)]∫ { √gdS =3 x [−∆φ + MMφ − iNN]λ − [∇ k ψ k + i (NM − MN)]η + 2iφ¯µµ2+ [i(D A + b)N − (σ.ψ − τ)M]µ − ¯µ [i(D A + b)N − (σ.ψ − τ)M]− 2χ k [ −∂ k τ + ∗(∇ψ) k − ¯Mσ k N − ¯Nσ k M ]+ 1 4 | ∗ F − ∂b − ¯MσM| 2 + 1 2 |(D A + b)M| 2 }, (6.178)where the k is a 3D index, and σ k are the Pauli matrices. The fieldsb, τ ∈ Ω 0 (Y ) respectively arose from A 0 and ψ 0 of the 4D theory, while themeanings of the other fields are clear. The BRST symmetry in 4D carriesover to the 3D theory. The BRST transformations rules for (A i , ψ i , φ),i = 1, 2, 3, (M, N), and (λ, η) are the same as before, but for the otherfields, we have[Q, b] = τ, [Q, τ] = 0,[Q, χ k ] = 1 (∗Fk − ∂ k b −2¯Mσ k M ) ,[Q, µ] = 1 2 i(D A + b)M.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1113The action S is cohomological in the sense that S = [Q, V 3 ], withV 3 being the dimensionally reduced version of V defined by (6.170), and[Q, S] = 0. Thus it gives rise to a topological field theory upon quantization.The partition function of the theory∫Z = DF exp(− 1 e 2 S),can be computed exactly in the limit e 2 → 0, as it is coupling constantindependent. We have, as before,Z = ∑ exp(− 1 ∫e 2 S(p) cl) DF ′ exp(−S q (p) ),pwhere S q(p) is the quadratic part of S expanded around a classical configurationwith the classical parts for the fields A, M, b being A o , M o , b o , whilethose for all the other fields being zero. The classical action S (p)clis givenby∫ {S (p) 1cl=Y 4 | ∗ F o − db o − ¯M o σM o | 2 + 1 }2 |(D A o + bo )M o | 2 ,which can be rewritten as [Zhang et. al. (1995)]∫ {S (p) 1cl=4 | ∗ F o − ¯M o σM o | 2 + 1 2 |D A oMo | 2 + 1 2 |dbo | 2 + 1 }2 |bo M o | 2 .YIn order for the classical configuration to have non–vanishing contributionsto the partition function, all the terms in S (p)clshould vanish separately.Therefore,∗ F o − ¯M o σM o = 0, D A oM o = 0, b o = 0, (6.179)where the last condition requires some explanation. When we have a trivialsolution of the equations (6.179), it can be replaced by the less stringentcondition db o = 0. However, in a more rigorous treatment of the problem athand, we in general perturb the equations (6.179), then the trivial solutiondoes not arise.Let us define an operator˜T : Ω 0 (Y ) ⊕ Ω 1 (Y ) ⊕ ( ˜S ⊗ L) → Ω 0 (Y ) ⊕ Ω 1 (Y ) ⊕ ( ˜S ⊗ L),(τ, ψ, N) ↦→ (−d ∗ ψ + i (NM − MN),2 ∗(dψ) − dτ − ¯NσM − MσN,iD A N − (σ.ψ − τ)M), (6.180)


1114 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere the complex bundle ˜S ⊗ L should be regarded as a real one withtwice the rank. This operator is self–adjoint, and is also obviously elliptic.We will assume that it is Fredholm as well. In terms of ˜T , the equationsof motion of the fields χ i and µ can be expressed as [Zhang et. al. (1995)]˜T (p) (τ, ψ, N) = 0, where ˜T (p) is the opeartor ˜T with the background fields(A o , M o ) belonging to the gauge class p of classical configurations .When the kernel of ˜T is zero, the partition function Z does not vanishidentically. An easy computation leads to Z = ∑ p ɛ(p) , where the sum isover all gauge inequivalent solutions of (6.179), and ɛ (p) is the sign of thedeterminant of ˜T (p) .A rigorous definition of the sign of the det( ˜T ) can be devised. However,if we are to compute only the absolute value of Z, then it is sufficient toknow the sign of det( ˜T ) relative to a fixed gauge class of classical configurations.This can be achieved using the mod − 2 spectral flow of a familyof Fredholm operators ˜T t along a path of solutions of (6.179). More explicitly,let (A o , M o ) belong to the gauge class of classical configurationsp, and (Ão , ˜M o ) in ˜p. We consider the solution of the SW equation onX = Y × [0, 1] with A 0 = 0 and also satisfying the following conditions(A, M)| t=0 = (A o , M o ), (A, M)| t=1 = (Ão , ˜M o ).Using this solution in ˜T results in a family of Fredholm operators, whichhas zero kernels at t = 0 and 1. The spectral flow of ˜Tt , denoted byq(p, ˜p), is defined to be the number of eigenvalues which cross zero witha positive slope minus the number which cross zero with a negative slope.This number is a well defined quantity, and is given by the index of theoperator ∂ ∂t − ˜T t . In terms of the spectral flow, we have [Zhang et. al.(1995)]det( ˜T (p) )det( ˜T (˜p) ) = (−1)q(p,˜p) .Equations (6.179) can be derived from the functionalS c−s = 1 ∫∫√A ∧ F + i gd 3 xMD A M.2Y(It is interesting to observe that this is almost the standard Lagrangianof a U(1) Chern–Simons theory coupled to spinors, except that we havetaken M to have bosonic statistics.) S c−s is gauge invariant modulo aconstant arising from the Chern–Simons term upon a gauge transformation.Y


<strong>Geom</strong>etrical Path Integrals and Their Applications 1115Therefore, ( δSc−sδA, δSc−sδ ¯M ) defines a vector field on the quotient space of allU(1) connections A tensored with the ˜S × L sections by the U(1) gaugegroup G, i.e., W = (A × ( ˜S ⊗ L))/G. Solutions of (6.179) are zeros ofthis vector field, and ˜T (p) is the Hessian at the point p ∈ W. Thus thepartition Z is nothing else but the Euler character of W. This geometricalinterpretation will be spelt out more explicitly in the next subsection byre–interpreting the theory using the Mathai–Quillen formula [Mathai andQuillen (1986)].6.5.2.5 <strong>Geom</strong>etrical InterpretationTo elucidate the geometric meaning of the 3D theory obtained in the lastsection, we now cast it into the framework of Atiyah and Jeffrey [Atiyahand Jeffrey (1990)]. Let us briefly recall the geometric set up of the Mathai–Quillen formula as reformulated in reference [Atiyah and Jeffrey (1990)].Let P be a Riemannian manifold of dimension 2m + dim G, and G be acompact Lie group acting on P by isometries. Then P → P/G is a principlebundle. Let V be a 2m dimensional real vector space, which furnishes arepresentation G → SO(2m). Form the associated vector bundle P × G V .Now the Thom form of P × G V can be expressed [Zhang et. al. (1995)]∫U = exp(−x2 )(2π) dim G π m{ iχφχexp + iχdx − i〈δν, λ〉4− 〈φ, Rλ〉 +〈ν, η〉} DηDχDφDλ, (6.181)where x = (x 1 , ..., x 2m ) is the coordinates of V , φ and λ are bosonic variablesin the Lie algebra g of G, and η and χ are Grassmannian variablesvalued in the Lie algebra and the tangent space of the fiber respectively. Inthe above equation, C maps any η ∈ g to the element of the vertical part ofT P generated by η; ν is the g - valued one form on P defined by 〈ν(α), η〉= 〈α, C(η)〉, for all vector fields α; and R = C ∗ C. Also, δ is the exteriorderivative on P .Now we choose a G invariant map s : P → V , and pull back the Thomform U. Then the top form on P in s ∗ U is the Euler class. If {δp} forms abasis of the cotangent space of P (note that ν and δs are one forms on P ),we replace it by a set of Grassmannian variables {ψ} in s ∗ U, then intergrate


1116 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthem away. We arrive atΥ =∫1(2π) dim G π m{exp −|s| 2 + iχφχ + iχδs − i〈δν, λ〉4− 〈φ, Rλ〉 +〈ψ, Cη〉} DηDχDφDλDψ, (6.182)the precise relationship of which with the Euler character of P × G V is∫Υ = Vol(G)χ(P × G ).PIt is rather obvious that the action S defined by (6.172) for the 4Dtheory can be interpreted as the exponent in the integrand of (6.182), ifwe identify P with A × Γ(W + ), and V with Ω 2,+ (X) × Γ(W − ), and sets = (F + + i ¯MΓM, 2D A M). Here A is the space of all U(1) connections ofdet(W + ), and Γ(W ± ) are the sections of S ± ⊗ L respectively.For the 3D theory, we wish to show that the partition function yieldsthe Euler number of W. However, the tangent bundle of W cannot beregarded as an associated bundle with the principal bundle, for which forthe formulae (6.181) or (6.182) can readily apply, some further work isrequired.Let P be the principal bundle over P/G, V , V ′ be two orthogonalrepresentions of G. Suppose there is an embedding from P × G V ′ to P × G Vvia a G−map γ(p) : V ′ → V for p ∈ P . Denote the resulting quotientbundle as E. In order to derive the Thom class for E, one needs to choosea section of E, or equivalently, a G−map s : P → V such that s(p) ∈(Imγ(p)) ⊥ . Then the Euler class of E can be expressed as π ∗ ρ ∗ U, where Uis the Thom class of P × G V , ρ is a G−map: P × V ′ → P × V defined byρ(p, τ) = (p, γ(p)τ + s(p)),and π ∗ is the integration along the fiber for the projection π : P × V ′ →P/G. Explicitly, [Zhang et. al. (1995)]∫π ∗ ρ ∗ (U) = exp { −|γ(p)τ + s(p)| 2 + iχφχ + iχδ(γ(p)τ + s(p))− i〈δν, λ〉 − 〈φ, Rλ〉 + 〈ν, Cη〉 } DχDφDτDηDλ. (6.183)Consider the exact sequence0 −→ (A × Γ(W )) × G Ω 0 (Y )j−→ (A × Γ(W )) × G (Ω 1 (Y ) × Γ(W )),


<strong>Geom</strong>etrical Path Integrals and Their Applications 1117where j (A,M) : b ↦→ (−db, bM) (assuming that M ≠ 0). Then the tangentbundle of A × G Γ(W ) can be Regarded as the quotient bundle(A × Γ(W )) × G (Ω 1 (Y ) × Γ(W ))/Im(j).We define a vector field on A × G Γ(W ) bys(A, M) = (∗F A − ¯MσM, D A M),which lies in Im(j) ⊥ :∫∫(∗F A − ¯MσM) ∧ ∗(−db) +YY√ gd 3 x〈D A M, bM〉 = 0, (6.184)where we have used the short hand notation 〈M 1 , M 2 〉 = 1 2 (M 1M 2 +M 2 M 1 ).Formally applying the formula (6.183) to the present infinite–dimensional situation, we get the Euler class π ∗ ρ ∗ (U) for the tangent bundleT (A × G Γ(W )), where ρ is the G−invariant map ρ is defined byρ : Ω 0 (Y ) −→ Ω 1 (Y )×Γ(W ), ρ(b) = (−db+∗F A − ¯MσM, (D A +b)M),π is the projection (A × Γ(W )) × G Ω 0 (Y ) −→ A × G Γ(W ), and π ∗ signifiesthe integration along the fiber. Also U is the Thom form of the bundle(A × Γ(W )) × G (Ω 1 (Y ) × Γ(W )) −→ A × G Γ(W ).To get a concrete feel about U, we need to explain the geometry of thisbundle. The metric on Y and the Hermitian metric 〈. , .〉 on Γ(W ) naturallydefine a connection. The Maurer–Cartan connection on A −→ A/G is flatwhile the Hermitian connection on has the curvature iφµ ∧ ¯µ. This givesthe expression of term i(χ, µ)φ(χ, µ) in (6.182) in our case.In our infinite–dimensional setting, the map C is given byC : Ω 0 (Y ) −→ T (A,M) (A × Γ(W )), C(η) = (−dη, iηM),and its dual is given byC ∗ : Ω 1 (Y ) × Γ(W ) −→ Ω 0 (Y ), C ∗ (ψ, N) = −d ∗ ψ + 〈N, iM〉.The one form 〈ν, η〉 on A × Γ(W ) takes the value〈(ψ, N), Cη〉 = 〈−d ∗ ψ, η〉 + 〈N, iM〉η


1118 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionon the vector field (ψ, N). We also easily get R(λ) = −∆λ + 〈M, M〉λ,where ∆ = d ∗ d. The 〈δν, λ〉 is a two form on A × Γ(W ) whose value on(ψ 1 , N 1 ), (ψ 2 , N 2 ) is -〈N 1 , N 2 〉λ.Combining all the information together, we arrive at the following formula,∫ {π ∗ ρ ∗ (U) = exp − 1 2 |ρ|2 + i(χ, µ)δρ + 2iφµ¯µ+ 〈∆φ, λ〉 − φλ〈M, M〉 + i〈N, N〉λ+ 〈ν, η〉} DχDφDλDηDb. (6.185)Note that the 1–form i(χ, µ)δρ on A × Γ(W ) × Ω 0 (Y ) contacted with thevector field (φ, N, b) leads to2χ k [ −∂ k τ + ∗(∇ψ) k − ¯Mσ k N − ¯Nσ k M ] +2〈µ, [i(D A + b)N − (σ.ψ − τ)M]〉;and the relation (6.184) gives |ρ| 2 = |∗F− ¯MσM| 2 +|db| 2 +|D A M| 2 +b 2 |M| 2 .Finally we get the Euler character∫π ∗ ρ ∗ (U) = exp(−S)DχDφDλDηDb, (6.186)where S is the action (6.178) of the 3D theory defined on the manifold Y .Integrating (6.186) over A × G Γ(W ) leads to the Euler number∑ɛ (A,M) ,[(A,M)]:s(A,M)=0which coincides with the partition function Z of our 3D theory.6.5.3 TQFTs Associated with SW–MonopolesRecall that TQFTs are often used to study topological nature of manifolds.In particular, 3D and 4D TQFTs are well developed. The mostwell–known 3D TQFT would be the Chern–Simons theory, whose partionfunction gives Ray–Singer torsion of 3–manifolds and the other topologicalinvariants can be obtained as gauge invariant observables i.e., Wilsonloops. The correlation functions can be identified with knot or link invariantse.g., Jones polynomal or its generalizations. On the other hand,in 4D, a twisted N = 2 supersymmetric YM theory developed by Witten[Witten (1988a)] also has a nature of TQFT. This YM theory can beinterpreted as Donaldson theory and the correlation functions are identifiedwith Donaldson polynomials, which classify smooth structures of


<strong>Geom</strong>etrical Path Integrals and Their Applications 1119topological 4–manifolds. A new TQFT on 4–manifolds was discoveredin SW studies of electric–magnetic duality of supersymmetric gauge theory.As discussed before, Seiberg and Witten [Seiberg and Witten (1994a);Seiberg and Witten (1994b)] studied the electric–magnetic duality of N = 2supersymmetric SU(2) YM gauge theory, by using a version of Montonen–Olive duality and obtained exact solutions. According to this result, theexact low energy effective action can be determined by a certain ellipticcurve with a parameter u = 〈Tr(φ) 2 〉, where φ is a complex scalar fieldin the adjoint representation of the gauge group, describing the quantummoduli space. For large u, the theory is weakly–coupled and semi–classical,but at u = ±Λ 2 corresponding to strong coupling regime, where Λ is thedynamically generated mass scale, the elliptic curve becomes singular andthe situation of the theory changes drastically. At these singular points,magnetically charged particles become massless. Witten showed that atu = ±Λ 2 the TQFT was related to the moduli problem of counting the solutionof the (Abelian) ‘Seiberg–Witten monopole equations’ [Witten (1994)]and it gave a dual description for the SU(2) Donaldson theory.It turns out that in 3D a particular TQFT of Bogomol’nyi monopolescan be obtained from a dimensional reduction of Donaldson theory and thepartition function of this theory gives the so–called Casson invariant of3–manifolds [Atiyah and Jeffrey (1990)].Ohta [Ohta (1998)] discussed TQFTs associated with the 3D versionof both Abelian and non–Abelian SW–monopoles, by applying Batalin–Vilkovisky quantization procedure. In particular, Ohta constructed thetopological actions, topological observables and BRST transformation rules.In this subsection, mainly following [Ohta (1998)], we will discussTQFTs associated with both Abelian and non–Abelian SW–monopoles.We will use the following notation.Let X be a compact orientable Spin 4–manifold without boundary andg µν be its Riemannian metric tensor (with g = det g µν ). Here we usex µ as the local coordinates on X. γ µ are Dirac’s gamma matrices andσ µν = [γ µ , γ ν ]/2 with {γ µ , γ ν } = g µν . M is a Weyl fermion and M is acomplex conjugate of M. (We will suppress spinor indices.) The Lie algebrag is defined by [T a , T b ] = if abc T c , where T a is a generator normalizedas Tr(T a T b ) = δ ab . The symbol f abc is a structure constant of g and isantisymmetric in its indices. The Greek indices µ, ν, α etc run from 0 to 3.The Roman indices a, b, c, · · · are used for the Lie algebra indices runningfrom 1 to dim g, whereas i, j, k, · · · are the indices for space coordinates.Space–time indices are raised and lowered with g µν . The repeated indices


1120 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionare assumed to be summed. ɛ µνρσ is an antisymmetric tensor with ɛ 0123 = 1.We often use the abbreviation of roman indices as θ = θ a T a etc., in orderto suppress the summation over Lie algebra indices.Brief Review of TQFTFirstly, we give a brief review of TQFT (compare with Witten’s TQFTpresented in subsection 6.5.1 above).Let φ be any field content. For a local symmetry of φ, we can constructa nilpotent BRST–operator Q B (Q 2 B = 0). The variation of any functionalO of φ is denoted by δO = {Q B , O}, where the bracket {∗, ∗} represnts agraded commutator, that is, if O is bosonic, the bracket means a commutator[∗, ∗] and otherwise it is an anti–bracket. Now, we can give the definitionof topological field theory, as given in [Birmingham et. al. (1991)]:A topological field theory consists of:(1) a collection of Grassmann graded fields φ on an nD Riemannian manifoldX with a metric g,(2) a nilpotent Grassmann odd operator Q,(3) physical states to be Q−cohomology classes,(4) an energy–momentum tensor T αβ which is Q−exact for some functionalV αβ such asT αβ = {Q, V αβ (φ, g)}.In this definition, Q is often identified with Q B and is in general independentof the metric. Now, recall that there are two broad types ofTQFTs satisfying this definition and they are classified into Witten–type [Witten (1994)] or Schwarz–type [Schwarz ((1978))].For Witten–type TQFT, the quantum action S q which comprises theclassical action, ghost and gauge fixing terms, can be represented by S q ={Q B , V }, for some function V of metric and fields and BRST charge Q B .Under the metric variation δ g of the partition function Z, it is easy to seethat∫ (δ g Z = Dφ e −Sq − 1 ∫d n x √ )gδg αβ T αβ2 X∫= Dφ e −Sq {Q, χ} ≡ 〈{Q, χ}〉 = 0, (6.187)where χ = − 1 ∫d n x √ gδg αβ V αβ .2X


<strong>Geom</strong>etrical Path Integrals and Their Applications 1121The last equality in (6.187) follows from the BRST invariance of the vacuumand means that Z is independent of the local structure of X, that is, Z isa topological invariant of X.In general, for Witten type theory, Q B can be constructed by an introductionof a topological shift with other local gauge symmetry [Ohta(1998)]. For example, in order to get the topological YM theory on fourmanifold M 4 , we introduce the shift in the gauge transformation for thegauge field A a µ such as δA a µ = D µ θ a +ɛ a µ, where D µ is a covariant derivative,θ a and ɛ a µ are the (Lie algebra valued) usual gauge transformation parameterand topological shift parameter, respectively. In order to see the roleof this shift, let us consider the first Pontryagin class on M 4 given byS = 1 ∫ɛ µνρσ F8µνF a ρσd a 4 x, (6.188)M 4where Fµν a is a field strength of the gauge field. We can easily check theinvariance of (6.188) under the action of δ. In this sense, (6.188) has a largersymmetry than the usual YM gauge symmetry. Taking this into account,we can construct the topological YM gauge theory. We can also considersimilar ‘topological’ shifts for matter fields.In addition, in general, Witten type topological field theory can be obtainedfrom the quantization of some Langevin equations. This approachhas been used for the construction of several topological field theories, e.g.,supersymmetric quantum mechanics, topological sigma models or Donaldsontheory (see [Birmingham et. al. (1991)]).On the other hand, Schwarz–type TQFT [Schwarz ((1978))] begins withany metric independent classical action S c as a starting point, but S c isassumed not to be a total derivative. Then the quantum action (up togauge fixing term) can be written byS q = S c + {Q, V (φ, g)}, (6.189)for some function V . For this quantum action, we can easily check thetopological nature of the partition function, but note that the energy–momentum tensor contributes only from the second term in (6.189). Oneof the differences between Witten type and Schwarz type theories can beseen in this point. Namely, the energy–momentum tensor of the classicalaction for Schwarz type theory vanishes because it is derived as a result ofmetric variation.Finally, we comment on the local symmetry of Schwarz type theory. Let


1122 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionus consider the Chern–Simons theory as an example. The classical action,∫S CS = d 3 x(A ∧ dA + 2 )M 3 A ∧ A ∧ A , (6.190)3is a topological invariant, which gives the second Chern class of 3–manifoldM 3 . As is easy to find, S CS is not invariant under the topological gaugetransformation, although it is YM gauge invariant. Therefore the quantizationis proceeded by the standard BRST method. This is a general featureof Schwarz–type TQFT.6.5.3.1 Dimensional ReductionFirst, let us recall the SW monopole equations in 4D. We assume that Xhas Spin structure. Then there exist rank two positive and negative spinorbundles S ± . For Abelian gauge theory, we introduce a complex line bundleL and a connection A µ on L. The Weyl spinor M(M) is a section of S + ⊗L(S + ⊗ L −1 ), hence M satisfies the positive chirality condition γ 5 M = M.If X does not have Spin structure, we introduce Spin c structure and Spin cbundles S ± ⊗ L, where L 2 is a line bundle. In this case, M should beinterpreted as a section of S + ⊗ L. Below, we assume Spin structure.Recall that the 4D Abelian SW monopole equations are the followingset of differential equationsF + µν + i 2 Mσ µνM = 0, iγ µ D µ M = 0, (6.191)where F + µν is the self–dual part of the U(1) curvature tensorF µν = ∂ µ A ν − ∂ ν A µ , F + µν = P + µνρσF ρσ , (6.192)while P µνρσ + is the self–dual projector defined byP µνρσ + = 1 ( √ ) gδ µρ δ νσ +22 ɛ µνρσ .Note that the second term in the first equation of (6.191) is also self–dual. On the other hand, the second equation in (6.191) is a twisted Diracequation, whose covariant derivative D µ is given byD µ = ∂ µ + ω µ − iA µ , where ω µ = 1 4 ωαβ µ [γ α , γ β ]is the spin connection 1–form on X.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1123In order to perform a reduction to 3D, let us first assume that X is aproduct manifold of the form X = Y × [0, 1], where Y is a 3D compactmanifold which has Spin structure. We may identify t ∈ [0, 1] as a ‘time’variable, or, we assume t as the zero–th coordinate of X, whereas x i (i =1, 2, 3) are the coordinates on (space manifoild) Y . Then the metric is givenbyds 2 = dt 2 + g ij dx i dx j .The dimensional reduction is proceeded by assumnig that all fields areindependent of t. (Below, we suppress the volume factor √ g of Y forsimplicity.)First, let us consider the Dirac equation. After the dimensional reduction,the Dirac equation will beγ i D i M − iγ 0 A 0 M = 0.As for the first monopole equation, using (6.192) we find thatF i0 + 1 2 ɛ i0jkF jk = −iMσ i0 M, F ij + ɛ ijk0 F k0 = −iMσ ij M. (6.193)Since the above two equations are dual each other, the first one, for instance,can be reduced to the second one by a contraction with the totally anti–symmetric tensor. Thus, it is sufficient to consider one of them. Here, wetake the first equation in (6.193).After the dimensional reduction, (6.193) will be∂ i A 0 − 1 2 ɛ ijkF jk = −iMσ i0 M, (6.194)where we have set ɛ ijk ≡ ɛ 0ijk .Therefore, the 3D version of the SW equations are given by∂ i b − 1 2 ɛ ijkF jk + iMσ i0 M = 0, i(γ i D i − iγ 0 b)M = 0, (b ≡ A 0 ).(6.195)It is now easy to establish the non–Abelian 3D monopole equations as∂ i b a + f abc A b ib c − 1 2 ɛ ijkF ajk + iMσ i0 T a M = 0, i(γ i D i − iγ 0 b)M = 0,where we have abbreaviated Mσ µν T a M ≡ M i σ µν (T a ) ij M j , subscripts of(T a ) ij run from 1 to dim g and b a ≡ A a 0.


1124 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNext, let us find an action which produces (6.195). The simplest one isgiven byS = 1 ∫ [ ( ]∂ i b − 1 22 Y 2 ɛ ijkF jk + iMσ i0 M)+ |i(γ i D i − iγ 0 b)M| 2 d 3 x.(6.196)Note that the minimum of (6.196) is given by (6.195). In this sense, the 3Dmonopole equations are not equations of motion but rather of constraints.Furthermore, there is a constraint for b. To see this, let us rewrite (6.196)as∫ [ ( ]2S = d 3 1 1x2 2 ɛ ijkF jk − iMσ i0 M)+ 1 2 |γi D i M| 2 + 1 4 (∂ ib) 2 + 1 2 b2 |M| 2 .YThe minimum of this action is clearly given by the 3D monopole equationswith b = 0, for non–trivial A i and M. However, for trivial A i and M, wemay relax the condition b = 0 to ∂ i b = 0, i.e., b is in general a non–zeroconstant. This can be also seen from (6.194). Therefore, we get12 ɛ ijkF jk − iMσ i0 M = 0, iγ i D i M = 0,with b = 0 or ∂ i b = 0, (6.197)as an equivalent expression to (6.195), but we will rather use (6.195) forconvenience. The Gaussian action will be used in the next subsection inorder to construct a TQFT by Batalin–Vilkovisky quantization algorithm.The non–Abelian version of (6.196) and (6.197) would be obvious.6.5.3.2 TQFTs of 3D MonopolesIn this subsection, we construct TQFTs associated with both the Abelianand non–Abelian 3D monopoles by Batalin–Vilkovisky quantization algorithm.Abelian CaseA 3D action for the Abelian 3D monopoles was found by the direct dimensionalreduction of the 4D one [Zhang et. al. (1995)], but here we rathershow that the 3D topological action can be directly constructed from the3D monopole equations [Ohta (1998)].


<strong>Geom</strong>etrical Path Integrals and Their Applications 1125Topological Bogomol’nyi ActionA topological Bogomol’nyi action was constructed by using Batalin–Vilkovisky quantization algorithm [Birmingham et. al. (1989)], or quantizationof a magnetic charge [Baulieu and Grossman (1988)]. The formeris based on the quantization of a certain Langevin equation (‘Bogomol’nyimonopole equation’) and the classical action is quadratic, but the latter isbased on the ‘quantization’ of the pure topological invariant by using theBogomol’nyi monopole equation as a gauge fixing condition.In order to compare the action to be constructed with those of Bogomol’nyimonopoles [Birmingham et. al. (1989); Baulieu and Grossman(1988)], we take Batalin–Vilkovisky procedure (see also [Birmingham et. al.(1991)]).In order to get the topological action associated with 3D monopoles,we introduce random Gaussian fields G i and ν(ν) and then start with theactionS c = 1 ∫[(G i −∂ i b+ 1 2 Y2 ɛ ijkF jk −iMσ i0 M) 2 +|(ν −iγ i D i M −γ 0 bM)| 2 ]d 3 x.(6.198)Note that G i and ν(ν) are also regarded as auxiliary fields. This actionreduces to (6.196) in the gaugeG i = 0, ν = 0. (6.199)Firstly, note that (6.198) is invariant under the topological gauge transformationδA i = ∂ i θ + ɛ i , δb = τ, δM = iθM + ϕ,δG i = ∂ i τ − ɛ ijk ∂ j ɛ k + i(ϕσ i0 M + Mσ i0 ϕ),δν = iθν + γ i ɛ i M + iγ i D i ϕ + γ 0 bϕ + γ 0 τM, (6.200)where θ is the parameter of gauge transformation, ɛ i and τ ≡ ɛ 4 areparameters which represent the topological shifts and ϕ the shift on thespinor space. The brackets for indices means anti–symmetrization, i.e.,A [i B j] = A i B j − A j B i .Here, let us classify the gauge algebra (6.200). This is necessary to useBatalin–Vilkovisky algorithm. Let us recall that the local symmetry forfields φ i can be written generally in the form δφ i = Rα(φ)ɛ i α , where theindices mean the label of fields and ɛ α is a some local parameter. Whenδφ i = 0 for non–zero ɛ α , this symmetry is called first–stage reducible. In the


1126 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionreducible theory, we can find zero–eigenvectors Z α a satisfying R i αZ α a = 0.Moreover, when the theory is on–shell reducible, we can find such eigenvectorsby using equations of motion.For the case at hand, under the identificationsθ = Λ, ɛ i = −∂ i Λ, ϕ = −iΛM,and τ = 0, so that (6.200) will be (6.201)δA i = 0, δb = 0, δM = 0, δG i = 0,δν = iΛ(ν − iγ i D i M − γ 0 bM)| on−shell = 0. (6.202)Then for δA i , for example, the R coefficients and the zero–eigenvectors arederived fromδA i = R AiθZθ Λ + Rɛ AijZ ɛjΛ= 0, that isR Aiθ= ∂ i , R Aiɛ j= δ ij , Z θ Λ = 1, Z ɛjΛ = −∂ j.Obviously, similar relations hold for other fields. The reader may think thatthe choice (6.201) is not suitable as a first stage reducible theory, but notethat the zero–eigenvectors appear on every point where the gauge equivalenceand the topological shift happen to coincide. In this three dimensionaltheory, b(A 0 ) is invariant for the usual infinitesimal gauge transformationbecause of its ‘time’ independence, so (6.201) means that the existence ofthe points on spinor space where the topological shift trivializes indicatesthe first stage reducibility.If we carry out BRST quantization via Faddeev–Popov procedure in thissituation, the Faddeev–Popov determinant will have zero modes. Thereforein order to fix the gauge further we need a ghost for ghost. This reflects onthe second generation gauge invariance (6.202) realized on–shell. However,since b is irrelevant to Λ, the ghost for τ will not couple to the secondgeneration ghost. With this in mind, we use Batalin–Vilkovisky algorithmin order to make BRST quantization (for details, see [Ohta (1998)] andreferences therein).Let us assign new ghosts carrying opposite statistics to the local parameters.The assortment is given byθ −→ c, ɛ i −→ ψ i , τ −→ ξ, ϕ −→ N, Λ −→ φ. (6.203)Ghosts in (6.203) are first generations, in particular, c is Faddeev–Popovghost, whereas φ is a second generation ghost. Their Grassmann parity and


<strong>Geom</strong>etrical Path Integrals and Their Applications 1127ghost number (U number) are given byc ψ i ξ N φ1 − 1 − 1 − 1 − 2 + ,(6.204)where the superscript of ghost number denotes the Grassmann parity. Notethat the ghost number counts the degree of differential form on the modulispace M of the solution to the 3D monopole equations. The minimal setΦ min of fields consists ofA i b M G i ν0 + 0 + 0 + 0 + 0 + ,and (6.204).On the other hand, the set of anti–fields Φ ∗ min carrying opposite statisticsto Φ min is given byΦ ∗ minA ∗ i b ∗ M ∗ G ∗ i ν ∗ c ∗ ψ ∗ i N ∗ φ ∗−1 − −1 − −1 − −1 − −1 − −2 + −2 + −2 + −3 − .Next step is to find a solution to the master equation with Φ min and, given by∂r S ∂ l S∂Φ A ∂Φ ∗ − ∂ rS ∂ l SA∂Φ ∗ = 0, (6.205)A∂ΦA where r(l) denotes right (left) derivative.The general solution for the first stage reducible theory at hand can beexpressed byS = S c +Φ ∗ i RαC i 1 α +C1α(Z ∗ β α C β 2 +T βγC α γ 1 Cβ 1 )+C∗ 2γA γ βα Cα 1 C β 2 +Φ∗ i Φ ∗ j Bα ji C2 α +· · · ,(6.206)where C1 α (C2 α ) denotes generally the first (second) generation ghost andonly relevant terms in our case are shown. We often use Φ A min =(φ i , C1 α , C β 2 ), where φi denote generally the fields. In this expression, theindices should be interpreted as the label of fields. Do not confuse withspace–time indices. The coefficients Zβ α, T βγ α , etc can be directly determinedfrom the master equation. In fact, it is known that these coefficientssatisfy the following relationsR i αZ α β C β 2 − 2∂ rS c∂φ j Bji α C α 2 (−1) |i| = 0,∂ r R i αC α 1∂φ j R j β Cβ 1 + Ri αT α βγC γ 1 Cβ 1 = 0,∂ r Z α β Cβ 2∂φ j R j γC γ 1 + 2T α βγC γ 1 Zβ δ Cδ 2 + Z α β A β δγ Cγ 1 Cδ 2 = 0, (6.207)


1128 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere |i| means the Grassmann parity of the ith field.In these expansion coefficients, Rα i and Zβ α are related to the local symmetry(6.200). On the other hand, as Tβγ α is related to the structure constantof a given Lie algebra for a gauge theory, it is generally called asstructure function. Of course if the theory is Abelian, such structure functiondoes not appear. However, for a theory coupled with matters, all ofthe structure functions do not always vanish, even if the gauge group isAbelian. At first sight, this seems to be strange, but the expansion (6.206)obviously detects the coupling of matter fields and ghosts. In fact, the appearanceof this type of structure function is required in order to make theaction to be constructed being full BRST invariant.After some algebraic works, we will find the solution to be∫S(Φ min , Φ ∗ min) = S c + ∆Sd 3 x, whereY∆S = A ∗ i (∂ i c + ψ i ) + b ∗ ξ + M ∗ (icM + N) + M ∗ (−icM + N)[∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) ]+G ∗ i+ν ∗ (icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)+ν ∗ (icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)+c ∗ φ − ψ ∗ i ∂ i φ − iN ∗ (φM + cN) + iN ∗ (φM + cN)+ 2iν ∗ ν ∗ φ.We augment Φ min by new fields χ i , d i , µ(µ), ζ(ζ), λ, ρ, η, e and the correspondinganti–fields. Their ghost number and Grassmann patity are givenbyχ i d i µ ζ λ ρ η e−1 − 0 + −1 − 0 + −2 + −1 − −1 − 0 + ,andχ ∗ i µ ∗ λ ∗ ρ ∗0 + 0 + 1 − 0 + .Then we look for the solution∫S ′ = S(Φ min , Φ ∗ min) + (χ ∗i d i + µ ∗ ζ + µ ∗ ζ + ρ ∗ e + λ ∗ η)d 3 x, (6.208)Ywhere d i , ζ, e, η, are Lagrange multiplier fields.In order to get the quantum action we must fix the gauge. The bestchoice for the gauge fixing condition, which can reproduce the action obtainedfrom the dimensional reduction of the 4D one, is found to be [Ohta


<strong>Geom</strong>etrical Path Integrals and Their Applications 1129(1998)]G i = 0, ν = 0, ∂ i A i = 0,−∂ i ψ i + i (NM − MN) = 0.2Thus we can get the gauge fermion carrying the ghost number -1 andodd Grassmann parity,Ψ = −χ i G i − µν − µν + ρ∂ i A i − λ[−∂ i ψ i + i ]2 (NM − MN) .The quantum action S q can be obtained by eliminating anti–fields and arerestricted to lie on the gauge surface Φ ∗ = ∂rΨ∂Φ. Therefore the anti–fieldswill beG ∗ i = −χ i , χ ∗ i = −G i , ν ∗ = −µ, ν ∗ = −µ, µ ∗ = −ν,µ ∗ = −ν, M ∗ = − i 2 λN, M ∗ = i 2 λN, N ∗ = i 2 λM,N ∗ = − i 2 λM, ρ∗ = ∂ i A i , A ∗ i = −∂ i ρ, ψ ∗ i = −∂ i λ, (6.209)λ ∗ = −[−∂ i ψ i + i ]2 (NM − MN) , c ∗ = φ ∗ = b ∗ = ζ ∗ (ζ ∗ ) = 0.Then the quantum action S q is given by S q = S ′ (Φ, Φ ∗ = ∂ r Ψ/∂Φ) . Substituting(6.209) into S q , we find that∫S q = S c + ˜∆Sd 3 x, whereY˜∆S = (−△φ + φMM − iNN)λ −[−∂ i ψ i + i ]2 (NM − MN) η− µ(icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)+ (icν [ + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)µ + 2iφµµ]− χ i ∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N)+ ρ(△c + ∂ i ψ i ) − d i G i − ζν − νζ + e∂ i A i . (6.210)Using the condition (6.199) with c = 0, we can arrive at∫S q ′ = S c | Gi=ν(ν)=0 + ˜∆S| c=0 d 3 x, where (6.211)Y


1130 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction˜∆S| c=0 = (−△φ + φMM − iNN)λ −[−∂ i ψ i + i ]2 (NM − MN) η− µ(iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)+ (iγ [ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)µ + 2iφµµ]− χ i ∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) + ρ∂ i ψ i + e∂ i A i .It is easy to find that (6.211) is consistent with the action found by thedimensional reduction of the 4D topological action [Zhang et. al. (1995)].BRST TransformationThe Batalin–Vilkovisky algorithm also facilitates to construct BRST transformationrule. The BRST transformation rule for a field Φ is defined byδ B Φ = ɛ ∂ ∣rS ′ ∣∣∣Φ∗∂Φ ∗ , (6.212)= ∂r Ψ∂Φwhere ɛ is a constant Grassmann odd parameter. With this definition for(6.210), we getδ B A i = −ɛ(∂ i c + ψ i ), δ B b = −ɛξ, δ B M = −ɛ(icM + N),[]δ B G i = −ɛ ∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) ,δ B ν = −ɛ(icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM − iµφ),δ B c = ɛφ, δ B ψ i = −ɛ∂ i φ, δ B ρ = ɛe, δ B λ = −ɛη,δ B µ = ɛζ, δ B N = −iɛ(φM + cN), δ B χ i = ɛd i ,δ B φ = δ B ξ = δ B d i = δ B e = δ B ζ = δ B η = 0. (6.213)It is clear at this stage that (6.213) has on-shell nilpotency, i.e., the quantumequation of motion for ν must be used in order to have δ 2 B = 0. This is dueto the fact that the gauge algebra has on–shell reducibility. Accordingly, theBatalin–Vilkovisky algorithm gives a BRST invariant action and on–shellnilpotent BRST transformation. Note that the equations∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) = 0,iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM = 0can be recognized as linearizations of the 3D monopole equations and thenumber of linearly independent solutions gives the dimension of the modulispace M.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1131It is now easy to show that the global supersymmetry can be recoveredfrom (6.213). In Witten type theory, Q B can be interpreted as a supersymmetricBRST charge. We define the supersymmetry transformation as[Ohta (1998)]Off–Shell Actionδ S Φ := δ B Φ| c=0 .As was mentioned before, the quantum action of Witten type TQFT can berepresented by BRST commutator with nilpotent BRST charge Q B . However,since our BRST transformation rule is on-shell nilpotent, we shouldintegrate out ν and G i in order to get off–shell BRST transformation andoff–shell quantum action.For this purpose, let us consider the following terms in (6.210),12 (G i − X i ) 2 + 1 2 |ν − A|2 − iµcν + icνµ − ζν − νζ − d i G i , (6.214)whereHere, let us defineX i = ∂ i b − 1 2 ɛ ijkF jk + iMσ i0 M, A = iγ i D i M + γ 0 bM.ν ′ = ν − A, B = −icµ − ζ.ν ′ (ν ′ ) and G i can be integrated out and then (6.214) will be− 1 2 d id i − d i X i − 2|B| 2 + BA + BA.Consequently, we get the off–shell quantum action{S q = Q, ˜Ψ}, where (6.215)˜Ψ ( = −χ i X i + α )2 d i − µ(iγ i D i M + γ 0 bM − βB)−µ(iγ i D i M + γ 0 bM − βB) + ρ∂ i A i − λ[−∂ i ψ i + i ]2 (NM − MN) .α and β are arbitrary gauge fixing paramaters. Convenience choice forthem is α = β = 1. The BRST transformation rule for X i and B fields canbe easily obtained, although we do not write down here.


1132 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionObservablesWe can now discuss the observables. For this purpose, let us define [Baulieuand Grossman (1988)]A = A + c, F = F + ψ − φ, K = db + ξ,where we have introduced differential form notations, but their meaningswould be obvious. A and c are considered as a (1, 0) and (0, 1) part of1–form on (Y, M). Similarly, F, ψ and φ are (2, 0), (1, 1) and (0, 2) part ofthe 2–form F, and db and ξ are (1, 0) and (0, 1) part of the 1–form K. ThusA defines a connection 1–form on (Y, M) and F is a curvature 2–form.Note that the exterior derivative d maps any (p 1 , p 2 )−form X p of totaldegree p = p 1 + p 2 to (p 1 + 1, p 2 )−form, but δ B maps any (p 1 , p 2 )−form to(p 1 , p 2 + 1)−form. Also note that X p X q = (−1) pq X q X p . Then the actionof δ B is(d + δ B )A = F, (d + δ B )b = K. (6.216)F and K also satisfy the following Bianchi identities in Abelian theory:(d + δ B )F = 0, (d + δ B )K = 0. (6.217)Equations (6.216) and (6.217) mean anti–commuting property between theBRST variation δ B and the exterior differential d, i.e., {δ B , d} = 0.The BRST transformation rule in geometric sector can be easily readfrom (6.213), i.e., δ B A, δ B ψ, δ B c and δ B φ. (6.217) implies(d + δ B )F n = 0, (6.218)and expanding the above expression by ghost number and form degree, weget the following (i, 2n − i)−form W n,i ,W n,0 = φnn! , W n,1 = φn−1(n − 1)! ψ,W n,2 =W n,3 =φn−22(n − 2)! ψ ∧ ψ − φn−1(n − 1)! F,φn−36(n − 3)! ψ ∧ ψ ∧ ψ − φn−2F ∧ ψ, (6.219)(n − 2)!where 0 = δ B W n,0 , dW n,0 = δ B W n,1 , (6.220)dW n,1 = δ B W n,2 , dW n,2 = δ B W n,3 , dW n,3 = 0.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1133Picking a certain k−cycle γ as a representative and defining the integral∫W n,k (γ) = W n,k ,we can easily prove that∫δ B W n,k (γ) = −γγ∫dW n,k−1 = − W n,k−1 = 0,∂γas a consequence of (6.220). Note that the last equality follows from thefact that the cycle γ is a simplex without boundary, i.e., ∂γ = 0. Therefore,W n,k (γ) indeed gives a topological invariant associated with n−th Chernclass on Y × M.On the other hand, since we have a scalar field b and its ghosts, wemay construct topological observables associated with them. Therefore,the observables can be obtained from the ghost expansion of(d + δ B )F n ∧ K m = 0.Explicitly, for m = 1, for example, we get0 = δ B W n,1,0 , dW n,1,0 = δ B W n,1,1 , dW n,1,1 = δ B W n,1,2 ,dW n,1,2 = δ B W n,1,3 , dW n,1,3 = 0, whereW n,1,0 = φnn! ξ, W n,1,1 = φn−1 φnψξ −(n − 1)! n! db,W n,1,2 =W n,1,3 =φn−2φn−1ψ ∧ ψξ −2(n − 2)! (n − 1)! F ξ − φn−1(n − 1)! ψ ∧ db,φn−3φn−1ψ ∧ ψ ∧ ψξ +6(n − 3)! (n − 1)! F ∧ db+ φn−2(2ψ ∧ F ξ + ψ ∧ ψ ∧ db). (6.221)2(n − 2)!These relations correspond to the cocycles [Baulieu and Grossman (1988)]in U(1) case.Next, let us look for the observables for matter sector. The BRSTtransformation rules in this sector is given by δ B , δ B N, δ B c and δ B φ. Atfirst sight, the matter sector does not have any observable, but we can findthe combined form˜W = iφMM + NN (6.222)


1134 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis an observable. However, unfortunately, as ˜W is cohomologically trivialbecause δ B ˜W = 0 but d˜W ≠ δB ˜W ′ for some ˜W ′ . Accordingly, ˜W does notgive any new topological invariant [Ohta (1998)].In topological Bogomol’nyi theory, there is a sequence of observablesassociated with a magnetic charge. For the Abelian case, it is given by∫W = F ∧ db. (6.223)YAs is pointed out for the case of Bogomol’nyi monopoles [Birmingham et. al.(1989)], we can not get the observables related with this magnetic chargeby the action of δ B as well, but we can construct those observables byanti–BRST variation δ B which maps (m, n)−form to (m, n − 1)−form. δ Bcan be obtained by a discrete symmetry which is realized as ‘time reversalsymmetry’ in 4D. In our 3D theory, the discrete symmetry is given byφ −→ −λ, λ −→ −φ, N −→ i √ 2µ, µ −→ i √2N,ψ i −→ χ i√ , χ i −→ √ 2ψ i , η −→ √ 2ξ, ξ −→ −√ η (6.224)2 2with b −→ −b, (6.225)where (6.225) represents an additional symmetry [Birmingham et. al.(1989)]. Note that we must also change N and µ (and their conjugates).The positive chirality condition for M should be used in order to check theinvariance of the action. In this way, we can get anti–BRST transformationrule by substituting (6.224) and (6.225) into (6.213) and then we can getthe observables associated with the magnetic charge by using the action ofthis anti–BRST variation [Birmingham et. al. (1989)].The topological observables available in this theory are the same withthose of topological Bogomol’nyi monopoles.Finally, let us briefly comment on our three dimensional theory. Firstnote that Lagrangian L and Hamiltonian H in dimensional reduction canbe considered as equivalent. This is because the relation between them isdefined byH = p ˙q − L,where q is any field, the overdot means time derivative and p is a canonicalconjugate momentum of q, and the dimensional reduction requires the timeindependence of all fields, thus H = −L in this sense. Though we haveconstructed the three dimensional action directly from the 3D monopole


<strong>Geom</strong>etrical Path Integrals and Their Applications 1135equations, our action may be interpreted essentially as the Hamiltonian ofthe four dimensional SW theory.6.5.3.3 Non–Abelian CaseIt is easy to extend the results obtained in the previous subsection tonon–Abelian case. In this subsection, we summarize the results for thenon–Abelian 3D monopoles (for details, see [Ohta (1998)] and referencestherein).Non–Abelian Topological ActionWith the auxiliary fields G a µν and ν, we considerS c = 1 2∫Yd 3 x [ (G a i − K a i ) 2 + |ν − iγ i D i M − γ 0 bM| 2] , (6.226)where K a i = ∂ i b a + f abc A b ib c − 1 2 ɛ ijkF a jk + iMσ i0 T a M.Note that the minimum of (6.226) with the gauge G a i = ν = 0 are givenby the non–Abelian 3D monopoles. We take the generator of Lie algebrain the fundamental representation, e.g., for SU(n),(T a ) ij (T a ) kl = δ il δ jk − 1 n δ ijδ kl .Extension to other Lie algebra and representation is straightforward.The gauge transformation rule for (6.226) is given byδA a i = ∂ i θ a + f abc A b iθ c + ɛ a i , δb a = f abc b b θ c + τ a , δM = iθM + ϕ,δG a i = f abc G b iθ c + [ −ɛ ijk (∂ j ɛ ak + f abc ɛ jb A ck )+ ∂ i τ a + f abc (ɛ b ib c − τ b A c i) + i(ϕσ i0 T a M + Mσ i0 T a ϕ) ] ,δν = iγ i D i ϕ + γ i ɛ i M + γ 0 bϕ + γ 0 τM + iθν. (6.227)Note that we have a G a i term in the transformation of Ga i , while it did notappear in Abelian theory.The gauge algebra (6.227) possesses on–shell zero modes as in theAbelian case. Settingθ a = Λ a , ɛ a i = −∂ i Λ a − f abc A b iΛ c , τ a = −f abc b b Λ c , ϕ = −iΛM,


1136 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwe can easily find that (6.227) closes, i.e.,δA a i = 0, δb a = 0, δM = 0,δG a i = f abc Λ c [G b i − K b i ]| on−shell = 0,δν = iΛ[ν − i(γ i D i − iγ 0 b)M]| on−shell = 0, (6.228)when the equations of motion of G a i and ν are used. Note that we must useboth equations of motion of G a i and ν in the non–Abelian case, while only‘ν’ was needed for the Abelian theory. Furthermore, as ϕ is a parameter inthe spinor space, ϕ is not g−valued, in other words, ϕ ≠ ϕ a T a . (6.227) isfirst stage reducible.The assortment of ghost fields, the minimal set Φ min of the fields andthe ghost number and the Grassmann parity, furthermore those for Φ ∗ minwould be obvious.Then the solution to the master equation will be∫S(Φ min , Φ ∗ min) = S c + Tr (∆S n ) d 3 x, whereY∆S n = A ∗ i (D i c + ψ i ) + b ∗ (i[b, c] + ξ) + M ∗ (icM + N) + M ∗ (−icM + N) + G ∗ i ˜G i−iN ∗ (φM + cN) + iN ∗ (φM + cN) + ν ∗ (icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)Here+ν ∗ (icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM) + 2iν ∗ ν ∗ φ + ψ ∗ i (−D i φ − i{ψ i , c})+c ∗ (φ − i 2 {c, c} )− iφ ∗ [φ, c] − i 2 {G∗ i , G ∗i }φ + iξ ∗ ([b, φ] − {ξ, c}).˜G i = i[c, G i ] − ɛ ijk D j ψ k + D i ξ + [ψ i , ξ] + i(Nσ i0 T a T a M + Mσ i0 T a T a N).The equations−ɛ ijk D j ψ k + D i ξ + [ψ i , ξ] + i(Nσ i0 T a T a M + Mσ i0 T a T a N) = 0,iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM = 0,can be seen as linearizations of non–Abelian 3D monopoles.We augment Φ min by new fields χ a i , da i , µ(µ), ζ(ζ), λ, ρ, η, e and the correspondinganti–fields, but Lagrange multipliers fields d a i , ζ(ζ), e, η, are assumednot to have anti–fields for simplicity and therefore their BRST transformationrules are set to zero. This simplification means that we do nottake into account of BRST exact terms. In this sense, the result to be


<strong>Geom</strong>etrical Path Integrals and Their Applications 1137obtained will correspond to those of the dimensionally reduced version ofthe 4D theory up to these terms, i.e., topological numbers.From the gauge fixing conditionG a i = 0, ν = 0, ∂ i A i = 0, −D i ψ i + i (NM − MN) = 0,2the gauge fermion will beΨ = −χ i G i − µν − µν + ρ∂ i A i − λThe anti–fields are then given by[−D i ψ i + i 2 (NM − MN) ].G ∗ i = −χ i , χ ∗ i = −G i , ν ∗ = −µ, ν ∗ = −µ, µ ∗ = −ν, µ = −ν,M ∗ = − i 2 λN, M ∗ = i 2 λN, N ∗ = i 2 λM, N ∗ = − i 2 λM,ρ ∗ = ∂ i A i , A ∗ i = −∂ i ρ + i[λ, ψ i ], b ∗ = c ∗ = ξ ∗ = φ ∗ = ζ ∗ (ζ ∗ ) = 0,λ ∗ = −[−D i ψ i + [b, ξ] + i ]2 (NM − MN) , ψ ∗ i = −D i λ.Therefore we find the quantum action∫ ( )S q = S c + Tr ˜∆Sn d 3 x, where (6.229)Y˜∆S n = −[−D i ψ i + [b, ξ] + i ]2 (NM − MN) η − λ(D i D i φ + iD i {ψ i , c}),+iλ{ψ i , D i c + ψ i } + (φMM − iNN)λ,−χ[i[c, i G i ] + ɛ ijk D j ψ k + D k ξ + [ψ k , ξ] + i ]2 (Nσij T a T a M + Mσ ij T a T a N) ,−µ(iγ µ D µ N + γ µ ψ µ M + icν) + (iγ i D i N + γ µ ψ µ M + icν)µ, (6.230)+2iφµµ − i 2 {χ i, χ i }φ + ρ(∂ i D i c + ∂ i ψ i ) − d i G i − ζν − νζ + e∂ i A i .,In this quantum action, settingM(M) = N(N) = µ(µ) = ν(ν) = 0,we can find that the resulting action coincides with that of Bogomol’nyimonopoles [Birmingham et. al. (1989)].Finally, in order to get the off–shell quantum action, both the auxiliaryfields should be integrated out by the similar technique presented in Abeliancase.


1138 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionBRST transformationThe BRST transformation rule is given byδ B A i = −ɛ(D i c + ψ i ), δ B b = −ɛ(i[c, b] + ξ), δ B ξ = iɛ([b, φ] − {ξ, c}),δ B M = −ɛ(icM + N),δ B G i = −ɛ( ˜G i − i[χ i , φ]),δ B ν = −ɛ(icν + γ µ D µ N + γ µ ψ µ M − iµφ),δ B c = ɛδ B ψ i = −ɛ (D i φ + i{ψ i , c}) , δ B ρ = ɛe, δ B λ = −ɛη,δ B µ = ɛζ, δ B N = −iɛ(φM + cN), δ B χ i = ɛd i ,(φ − i 2 {c, c} ),δ B φ = iɛ[φ, c], δ B d i = δ B e = δ B ζ = δ B η = 0. (6.231)It is easy to get supersymmetry also in this case. However, as we haveomitted the BRST exact terms, the supersymmetry in our constructiondoes not detect them.ObservablesWe have already constructed the topological observables for Abelian case.Also in non-Abelian case, the construction of observables is basically thesame. But the relation (6.216) and (6.217) are required to modify(d + δ B )A − i 2 [A, A] = F, (d + δ B)b − ii[A, b] = K, and (6.232)(d + δ B )F − i[A, F] = 0, (d + δ B )K − i[A, K] = i[F, b], (6.233)respectively, where [∗, ∗] is a graded commutator. The observables in geometricand matter sector are the same as before, but we should replacedb by d A b in (6.221) as well as (6.232) and (6.233), where d A is a exteriorcovariant derivative and trace is required. In addition, the magnetic chargeobservables are again obtained by anti-BRST variation as outlined berfore.The observables in geometric sector are those in (6.219) and follow thecohomological relation (6.220). In this way, the topological observablesavailable in this three dimensional theory are precisely the Bogomol’nyimonopole cocycles [Baulieu and Grossman (1988)].6.5.4 Stringy Actions and AmplitudesNow we give a brief review of modern path–integral methods in superstringtheory (mainly following [Deligne et. al. (1999)]). Recall that the funda-


<strong>Geom</strong>etrical Path Integrals and Their Applications 1139mental quantities in quantum field theory (QFT) are the transition amplitudesAmp : IN =⇒ OUT, describing processes in which a number IN ofincoming particles scatter to produce a number OUT of outgoing particles.The square modulus of the transition amplitude yields the probability forthis process to take place.6.5.4.1 StringsRecall that in string theory, elementary particles are not described as 0–dimensional points, but instead as 1D strings. If M s and M(∼ R × M s )denote the 3D space and 4D space–time manifolds respectively, then wepicture strings as in Figure 6.14.Fig. 6.14 Basic geometrical objects of string theory: (a) a space with fixed time; (b) aspace–time picture; (c) a point–particle; (d) a world–line of a point–particle; (e) a closedstring; (f) a world–sheet of a closed string; (g) an open string; (h) a world–sheet of anopen string.While the point–particle sweeps out a 1D world–line, the string sweepsout a world–sheet, i.e., a 2D real surface. For a free string, the topology ofthe world–sheet is a cylinder (in the case of a closed string) or a sheet (foran open string).Roughly, different elementary particles correspond to different vibrationmodes of the string just as different minimal notes correspond to differentvibrational modes of musical string instruments.It turns out that the physical size of strings is set by gravity, moreprecisely the Planck length l P ∼ 10 −33 cm. This scale is so small that weeffectively only see point–particles at our distance scales. Thus, for lengthscales much larger than l P , we expect to recover a QFT–description of


1140 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionpoint–particles, plus typical string corrections that represent physics at thePlanck scale.6.5.4.2 InteractionsWhile the string itself is an extended 1D object, the fundamental stringinteractions are local, just as for point–particles. The interaction takesplace when strings overlap in space at the same time. In case of closed stringtheories the interactions have a form depicted in Figure 6.15, while in caseof open string theories the interactions have a form depicted in Figure 6.16.Other interactions result from combining the interactions defined above.Fig. 6.15Interactions in closed string theories (left 2D–picture and right 3D–picture).Fig. 6.16Interactions in open string theories (left 2D–picture and right 3D–picture).In point–particle theories, the fundamental interactions are read off fromthe QFT–Lagrangian. An interaction occurs at a geometrical point, wherethe world–lines join and cease to be a manifold. In Lorentz–invariant theories(where manifold M is a flat Minkowski space–time), the interactionpoint is Lorentz–invariant. To specify how the point–particles interact, additionaldata must be supplied at the interaction point, giving rise to manypossible distinct quantum field theories.In string theory, the interaction point depends upon the Lorentz frame


<strong>Geom</strong>etrical Path Integrals and Their Applications 1141chosen to observe the process. In the Figure above, equal time slices areindicated from the point of view of two different Lorentz frames, schematicallyindicated by t and t ′ . The closed string interaction, as seen fromframes t and t ′ , occurs at times t 2 and t ′ 2 and at (distinct) points P and P ′respectively.Lorentz invariance of interaction forbids that any point on the world–sheet be singled out as interaction point. Instead, the interaction resultspurely from the joining and splitting of strings. While free closed strings arecharacterized by their topology being that of a cylinder, interacting stringsare characterized by the fact that their associated world–sheet is connectedto at least 3 strings, incoming and/or outgoing.As a result, the free string determines the nature of the interactionscompletely, leaving only the string coupling constant undetermined.The orientation is an additional structure of closed strings, dividingthem into two categories: (i) oriented strings, in which all world–sheets areassumed to be orientable; and (ii) non–oriented strings, in which world–sheets are non–orientable, such as the Möbius strip, Klein bottle, etc.Fig. 6.17 Boundary components and handles of closed oriented system of M incomingstrings, interacting through internal loops, to produce N outgoing strings. Notethe striking similarity with MIMO–systems of nonlinear control theory, with M inputprocesses and N output processes(see section 4.9.1 below).6.5.4.3 Loop Expansion – Topology of Closed SurfacesFor simplicity, here we consider closed oriented strings only, so that theassociated world–sheet is also oriented. A general string configuration de-


1142 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionscribing the process in which M incoming strings interact and produce Noutgoing strings looks at the topological level like a closed surface withM + N = E boundary components and any number of handles (see Figure6.17). This picture is a kind of topological generalization of nonlinear controlMIMO–systems with M inputs, N outputs X states (see section 4.9.1below).The internal loops may arise when virtual particle pairs are produced,just as in quantum field theory. For example, a Feynman diagram in quantumfield theory that involves a loop is shown in Figure 6.18 together withthe corresponding string diagram.Fig. 6.18 A QFT Feynman diagram that involves an internal loop (left), with thecorresponding string diagram (right).Surfaces associated with closed oriented strings have two topologicalinvariants: (i) the number of boundary components E = M +N (which maybe shrunk to punctures, under certain conditions), and (ii) the number hof handles on the surface, which equals the surface genus.Fig. 6.19 Number h of handles on the surface of closed oriented strings, which equalsthe string–surface genus: (a) h = 0 for sphere S 2 ; (b) h = 1 for torus T 2 ; (c) h = 2 forstring–surfaces with higher genus, etc.When E = 0, we just have the topological classification of compact orientedsurfaces without boundary. Rendering E > 0 is achieved by removingE discs from the surface.Recall that in QFT, an expansion in powers of Planck’s constant yieldsan expansion in the number of loops of the associated Feynman diagram,


<strong>Geom</strong>etrical Path Integrals and Their Applications 1143for a given number of external states:⎧⎨ E+h−1 = −1⎩−1for every propagatorfor every vertexfor overall momentum conservationThus, in string theory we expect that, for a given number of external stringsE, the topological expansion genus by genus should correspond to a loopexpansion as well.Recall that in QFT, there are in general many Feynman diagrams thatcorrespond to an amplitude with a given number of external particles anda given number of loops. For example, for E = 4 external particles andh = 1 loop in φ 3 theory are given in Figure 6.20, together with the sameprocess in string theory (for closed oriented strings), where it is describedby just a single diagram (right).Fig. 6.20 Feynman QFT–diagrams for φ 3 theory with E = 4 external particles andh = 1 loop (left), and a single corresponding string diagram (right). In this way the usualFeynman diagrams of quantum field theory are generalized by arbitrary Riemanniansurfaces.Much of recent interest has been focused on the so–called D−branes.A D−brane is a submanifold of space–time with the property that stringscan end or begin on it.6.5.5 Transition Amplitudes for StringsThe only way we have today to define string theory is by giving a rulefor the evaluation of transition amplitudes, order by order in the loop expansion,i.e., genus by genus. The rule is to assign a relative weight to agiven configuration and then to sum over all configurations [Deligne et. al.(1999)]. To make this more precise, we first describe the system’s configurationmanifold M (see Figure 6.21).


1144 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFig. 6.21 The embedding map x from the reference surface Σ into the pseudo–Riemannian configuration manifold M (see text for explanation).We assume that Σ and M are smooth manifolds, of dimensions 2 andn respectively, and that x is a continuous map from Σ to M. If ξ m , (form = 1, 2), are local coordinates on Σ and x µ , (µ = 1, . . . , n), are localcoordinates on M then the map x may be described by functions x µ (ξ m )which are continuous.To each system configuration we can associate a weight e −S[x,Σ,M] , (forS ∈ C) and the transition amplitude Amp for specified external strings(incoming and outgoing) is get by summing over all surfaces Σ and allpossible maps x,∑ ∑Amp =e −S[x,Σ,M] .surfaces ΣWe now need to specify each of these ingredients:(1) We assume M to be an nD Riemannian manifold, with metric g. Aspecial case is flat Euclidean space–time R n . The space–time metric isassumed fixed.ds 2 = (dx, dx) g = g µν (x)dx µ ⊗ dx ν .(2) The metric g on M induces a metric on Σ: γ = x ∗ (g),γ = γ mn dξ m ⊗ dξ n , γ mn = g µν∂x µ∂ξ m ∂x ν∂ξ n .This metric is non–negative, but depends upon x. It is advantageous tointroduce an intrinsic Riemannian metric g on Σ, independently of x; inlocal coordinates, we haveg = g mn (ξ)dξ m ⊗ dξ n .x


<strong>Geom</strong>etrical Path Integrals and Their Applications 1145A natural intrinsic candidate for S is the area of x(Σ), which gives theso–called Nambu–Goto action 20∫ ∫Area (x (Σ)) = dµ γ = n 2 ξ √ det γ mn , (6.234)Σwhich depends only upon g and x, but not on g [Goto (1971)]. However,the transition amplitudes derived from the Nambu–Goto action are notwell–defined quantum–mechanically.Otherwise, we can take as starting point the so–called Polyakov action 21∫∫S[x, g] = κ (dx, ∗dx) g = κ dµ g g mn ∂ m x µ ∂ n x ν g µν (x), (6.235)ΣΣwhere κ is the string tension (a positive constant with dimension of inverselength square). The stationary points of S with respect to g are at g 0 = e φ γfor some function φ on Σ, and thus S[x, g 0 ] ∼ Area (x (Σ)).The Polyakov action leads to well–defined transition amplitudes, get byintegration over the space Met(Σ) of all positive metrics on Σ for a giventopology, as well as over the space of all maps Map(Σ, M). We can definethe path integralAmp =∑topologiesΣ∫Met(Σ)Σ∫1D[x] e −S[x,g,g] ,N(g) Map(Σ,M)where N is a normalization factor, while the measures D[g] and D[x] areconstructed from <strong>Diff</strong> + (Σ) and <strong>Diff</strong>(M) invariant L 2 norms on Σ andM. For fixed metric g, the action S is well–known: its stationary pointsare the harmonic maps x: Σ → M (see, e.g., [Eells and Lemaire (1978)]).However, g here varies and in fact is to be integrated over. For a generalmetric g, the action S defines a nonlinear sigma model, which is renormalizablebecause the dimension of Σ is 2. It would not in general berenormalizable in dimension higher than 2, which is usually regarded asan argument against the existence of fundamental membrane theories (see[Deligne et. al. (1999)]).20 Nambu–Goto action is the starting point of the analysis of string behavior, usingthe principles of ordinary Lagrangian mechanics. Just as the Lagrangian for a free pointparticle is proportional to its proper timei.e., the ‘length’ of its world–line, a relativisticstring’s Lagrangian is proportional to the area of the sheet which the string traces as ittravels through space–time.21 The Polyakov action is the 2D action from conformal field theory, used in stringtheory to describe the world–sheet of a moving string.


1146 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe Nambu–Goto action (6.234) and Polyakov action (6.235) representthe core of the so–called bosonic string theory, the original version ofstring theory, developed in the late 1960s. Although it has many attractivefeatures, it also predicts a particle called the tachyon possessing some unsettlingproperties, and it has no fermions. All of its particles are bosons,the matter particles. The physicists have also calculated that bosonic stringtheory requires 26 space–time dimensions: 25 spatial dimensions and onedimension of time. In the early 1970s, supersymmetry was discovered in thecontext of string theory, and a new version of string theory called superstringtheory (i.e., supersymmetric string theory) became the real focus, asit includes also fermions, the force particles. Nevertheless, bosonic stringtheory remains a very useful ‘toy model’ to understand many general featuresof perturbative string theory (see section 6.7 below).6.5.6 Weyl Invariance and Vertex Operator FormulationThe action S is also invariant under Weyl rescalings of the metric g bya positive function on σ : Σ → R, given by g → e 2σ g. In general, Weylinvariance of the full amplitude may be spoiled by anomalies. AssumingWeyl invariance of the full amplitude, the integral defining Amp may besimplified in two ways.1) The integration over Met(Σ) effectively collapses to an integration overthe moduli space of surfaces, which is finite dimensional, for each genus h.2) The boundary components of Σ — characterizing external string states— may be mapped to regular points on an underlying compact surfacewithout boundary by conformal transformations. The data, such as momentaand other quantum numbers of the external states, are mapped intovertex operators. The amplitudes are now given by the path integralAmp =∞∑∫h=0Met(Σ)∫1D[g]N(g)for suitable vertex operators V 1 , . . . V N .Map(Σ,M)D[x] V 1 . . . V N e −S ,6.5.7 More General ActionsGeneralizations of the action S given above are possible when M carriesextra structure.


<strong>Geom</strong>etrical Path Integrals and Their Applications 11471) M carries a 2−form B ∈ Ω (2) (M). The resulting contribution to theaction is also that of a ‘nonlinear sigma model’∫ ∫S B [x, B] = x ∗ (B) = dx µ ∧ dx ν B µν (x)Σ2) M may carry a dilaton field Φ ∈ Ω (0) (M) so that∫S Φ [x, Φ] = dµ g R g Φ(x).where R g is the Gaussian curvature of Σ for the metric g.3) There may be a tachyon field T ∈ Ω (0) (M) contributing∫S T [x, T ] = dµ g T (x).6.5.8 Transition Amplitude for a Single Point ParticleΣThe transition amplitude for a single point–particle could in fact be get ina way analogous to how we prescribed string amplitudes. Let space–timebe again a Riemannian manifold M, with metric g. The prescription forthe transition amplitude of a particle travelling from a point y ∈ M to apoint y ′ to M is expressible in terms of a sum over all (continuous) pathsconnecting y to y ′ :Amp(y, y ′ ) =ΣΣ∑e −S[path] .pathsjoining y and y ′Paths may be parametrized by maps from C = [0, 1] into M with x(0) = y,x(1) = y ′ . A simple world–line action for a massless particle is get byintroducing a metric g on [0, 1]S[x, g] = 1 ∫dτ g(τ) −1 ẋ µ ẋ ν g µν (x),2Cwhich is invariant under <strong>Diff</strong> + (C) and <strong>Diff</strong>(M).Recall that the analogous prescription for the point–particle transitionamplitude is the path integral∫Amp(y, y ′ ) = D[g] 1 ∫D[x] e −S[x,g] .NMet(C)Map(C,M)


1148 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionNote that for string theory, we had a prescription for transition amplitudesvalid for all topologies of the world–sheet. For point–particles,there is only the topology of the interval C, and we can only describe asingle point–particle, but not interactions with other point–particles. Toput those in, we would have to supply additional information.Finally, it is very instructive to work out the amplitude Amp by carryingout∫the integrations. The only <strong>Diff</strong> + (C) invariant of g is the length L =10 dτ g(τ); all else is generated by <strong>Diff</strong>+ (C). Defining the normalizationfactor to be the volume of <strong>Diff</strong>(C): N = Vol(<strong>Diff</strong>(C)) we have D[g] =D[v] dL and the transition amplitude becomes∫ ∞ ∫∫Amp(y, y ′ ) = dL D[x] e − 1 R ∞12L 0 dτ(ẋ,ẋ)g = dL 〈 y ′ |e −L∆ |y 〉 =〈y ′ | 1 〉∆ |y .0Thus, the amplitude is just the Green function at (y, y ′ ) for the Laplacian∆ and corresponds to the propagation of a massless particle (see [Deligneet. al. (1999)]).06.5.9 Witten’s Open String Field TheoryNoncommutative nature of space–time has often appeared in non–perturbative aspects of string theory. It has been used in a formulationof interacting open string field theory by Ed Witten [Witten (1986b);Witten (1986a)]. Witten has written a classical action of open string fieldtheory in terms of noncommutative geometry, where the noncommutativityappears in a product of string fields. Later, the Dirichlet branes (or,D–branes) have been recognized as solitonic objects in superstring theory[Polchinski (1995)]. Further, it has been found that the low energy behaviorof the D–branes are well described by supersymmetric Yang–Millstheory (SYM) [Witten (1996)]. In the situation of some D–branes coinciding,the space–time coordinates are promoted to matrices which appear asthe fields in SYM. Then the size of the matrices corresponds to the numberof the D–branes, so noncommutativity of the matrices is related to thenoncommutative nature of space–time.In this subsection, mainly following [Sugino (2000)], we review some basicproperties of Witten’s bosonic open string field theory [Witten (1986b)]and its explicit construction based on a Fock space representation of stringfield functional and δ−function overlap vertices [Gross and Jevicki(1987a);Gross and Jevicki(1987b); Cremmer et. al. (1986)].Witten introduced a beautiful formulation of open string field theory in


<strong>Geom</strong>etrical Path Integrals and Their Applications 1149terms of a noncommutative extension of differential geometry, where stringfields, the BRST operator Q and the integration over the string configurations∫ in string field theory are analogs of differential forms, the exteriorderivative d and the integration over the manifold ∫ in the differentialMgeometry, respectively. The ghost number assigned to the string field correspondsto the degree of the differential form. Also the (noncommutative)product between the string fields ∗ is interpreted as an analog of the wedgeproduct ∧ between the differential forms.The axioms obeyed by the system of ∫ , ∗ and Q are∫QA = 0, Q(A ∗ B) = (QA) ∗ B + (−1) n AA ∗ (QB),∫∫(A ∗ B) ∗ C = A ∗ (B ∗ C), A ∗ B = (−1) n An BB ∗ A,where A, B and C are arbitrary string fields, whose ghost number is half–integer valued: The ghost number of A is defined by the integer n A asn A + 1 2 .Then Witten discussed the following string–field–theory actionS = 1 G s∫ ( 12 ψ ∗ Qψ + 1 3 ψ ∗ ψ ∗ ψ ), (6.236)where G s is the open string coupling constant and ψ is the string field withthe ghost number - 1 2. The action is invariant under the gauge transformationδψ = QΛ + ψ ∗ Λ − Λ ∗ ψ,with the gauge parameter Λ of the ghost number - 3 2 .6.5.9.1 Operator Formulation of String Field TheoryThe objects defined above can be explicitly constructed by using the operatorformulation, where the string field is represented as a Fock space, andthe integration ∫ as an inner product on the Fock space. It was consideredby [Gross and Jevicki(1987a); Gross and Jevicki(1987b)] in the case of theNeumann boundary condition. We will heavily use the notation of [Grossand Jevicki(1987a); Gross and Jevicki(1987b)]. In the operator formulation,the action (6.236) is described asS = 1 ( 1〈V 2 ||ψ〉 1 Q|ψ〉 2 + 1 )〈V 3 ||ψ〉 1 |ψ〉 2 |ψ〉 3 , (6.237)G s 2 123 123


1150 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere the structure of the product ∗ in the kinetic and potential termsis encoded to that of the overlap vertices 〈V 2 | and 〈V 3 | respectively (here,subscripts put to vectors in the Fock space label the strings concerning thevertices).As a preparation for giving the explicit form of the overlaps, let usconsider open strings in 26-dimensional space–time with the constant metricG ij in the Neumann boundary condition. The world sheet action is givenbyS W S = 1 ∫ ∫ π4πα ′ dτ dσG ij (∂ τ X i ∂ τ X j − ∂ σ X i ∂ σ X j ) + S gh , (6.238)0where S gh is the action of the bc−ghosts:S gh =i ∫ ∫ πdτ dσ[c + (∂ τ − ∂ σ )b + + c − (∂ τ + ∂ σ )b − ]. (6.239)2π0Under the Neumann boundary condition, the string coordinates have thestandard mode expansions:X j (τ, σ) = x j + 2α ′ τp j + i √ 2α ′ ∑ n≠0also the mode expansions of the ghosts are given by1n αj ne −inτ cos(nσ), (6.240)c ± (τ, σ) = ∑ n∈Zc n e −in(τ±σ) ≡ c(τ, σ) ± iπ b (τ, σ),b ± (τ, σ) = ∑ n∈Zb n e −in(τ±σ) ≡ π c (τ, σ) ∓ ib(τ, σ).As a result of the quantization, the modes obey the commutation relatons:[x i , p j ] = iG ij , [α i n, α j m] = nG ij δ n+m,0 , {b n , c m } = δ n+m,0 ,while the other therms vanish.The overlap |V N 〉 = |V N 〉 X |V N 〉 gh , (N = 1, 2, · · · ) is the state satisfyingthe continuity conditions for the string coordinates and the ghostsat the N−string vertex of the string field theory. The superscripts X andgh show the contribution of the sectors of the coordinates and the ghostsrespectively. The continuity conditions for the coordinates are(X (r)j (σ)−X (r−1)j (π−σ))|V N 〉 X = 0, (P (r)i (σ)+P (r−1)i (π−σ))|V N 〉 X = 0,(6.241)for 0 ≤ σ ≤ π 2and r = 1, · · · , N. Here P i(σ) is the momentum conjugateto the coordinate X j (σ) at τ = 0, and the superscript (r) labels the string


<strong>Geom</strong>etrical Path Integrals and Their Applications 1151(r) meeting at the vertex. In the above formulas, we regard r = 0 asr = N because of the cyclic property of the vertex. For the ghost sector,we impose the following conditions on the variables c(σ), b(σ) and theirconjugate momenta π c (σ), π b (σ):(π (r)c (σ) − π (r−1)c (π − σ))|V N 〉 gh = 0, (b (r) (σ) − b (r−1) (π − σ))|V N 〉 gh = 0,(c (r) (σ) + c (r−1) (π − σ))|V N 〉 gh = 0, (π (r)b(σ) + π (r−1)b(π − σ))|V N 〉 gh = 0,for 0 ≤ σ ≤ π 2and r = 1, · · · , N.6.5.9.2 Open Strings in Constant B−Field BackgroundWe consider a constant background of the second–rank antisymmetric tensorfield B ij in addition to the constant metric g ij where open strings propagate.Then the boundary condition at the end points of the open stringschanges from the Neumann type, and thus the open string has a differentmode expansion from the Neumann case (6.240). As a result, the endpoint is to be noncommutative, in the picture of the D–branes which impliesnoncommutativity of the world volume coordinates on the D–branes.Here we derive the mode–expanded form of the open string coordinates asa preparation for a calculation of the overlap vertices in the next section.We start with the world sheet actionSW B S = 1 ∫ ∫ π4πα ′ dτ dσ[g ij (∂ τ X i ∂ τ X j − ∂ σ X i ∂ σ X j )0− 2πα ′ B ij (∂ τ X i ∂ σ X j − ∂ σ X i ∂ τ X j )] + S gh . (6.242)Because the term proportional to B ij can be written as a total derivativeterm, it does not affect the equation of motion but does the boundarycondition, which requiresg ij ∂ σ X j − 2πα ′ B ij ∂ τ X j = 0 (6.243)on σ = 0, π. This can be rewritten to the convenient formE ij ∂ − X j = (E T ) ij ∂ + X j , (6.244)where E ij ≡ g ij + 2πα ′ B ij ,and ∂ ± are derivatives with respect to the light cone variables σ ± = τ ± σ.We can easily see that X j (τ, σ) satisfying the boundary condition (6.244)


1152 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionhas the following mode expansion:X j (τ, σ = ˜x j + α ′ [ (E −1 ) jk g kl p l σ − + (E −1T ) jk g kl p l σ +] (6.245)√α′ ∑ 1[+ i(E −1 ) jk g kl α l2 nne −inσ− + (E −1T ) jk g kl α l ne −inσ+] .n≠0We will get the commutators between the modes from the propagator ofthe open strings, which gives another derivation different from the methodby [Chu and Ho (1999)] based on the quantization via the Dirac bracket.When performing the Wick rotation: τ → −iτ and mapping the worldsheet to the upper half planez = e τ+iσ ,¯z = e τ−iσ (0 ≤ σ ≤ π),the boundary condition (6.244) becomesE ij ∂¯z X j = (E T ) ij ∂ z X j , (6.246)which is imposed on the real axis z = ¯z. The propagator 〈X i (z, ¯z)X j (z ′ , ¯z ′ )〉satisfying the boundary condition (6.246)is determined as〈X i (z, ¯z)X j (z ′ , ¯z ′ )〉 = −α ′ [ g ij ln |z − z ′ | − g ij ln |z − ¯z ′ |where G ij and θ ij are given by+ G ij ln |z − ¯z ′ | 2 + 12πα ′ θij ln z − ¯z ′¯z − z ′ + Dij ,G ij = 1 2 (ET −1 + E −1 ) ij = (E T −1 gE −1 ) ij = (E −1 gE T −1 ) ij , (6.247)θ ij = 2πα ′ · 12 (ET −1 − E −1 ) ij = (2πα ′ ) 2 (E T −1 BE −1 ) ij (6.248)= −(2πα ′ ) 2 (E −1 BE T −1 ) ij .Also the constant D ij remains unknown from the boundary condition alone.However it is an irrelevant parameter, so we can fix an appropriate value.The mode-expanded form (6.245) is mapped toX j (z, ¯z) = ˜x j − iα ′ [(E −1 ) jk p k ln ¯z + (E −1T ) jk p k ln z]√α′ ∑ 1 [+ i(E −1 ) jk α n,k ¯z −n + (E −1T ) jk α n,k z −n] .2 nn≠0


<strong>Geom</strong>etrical Path Integrals and Their Applications 1153Note that the indices of p l and α l n were lowered by the metric g ij not G ij .Recall the definition of the propagator〈X i (z, ¯z)X j (z ′ , ¯z ′ )〉 ≡ R(X i (z, ¯z)X j (z ′ , ¯z ′ )) − N(X i (z, ¯z)X j (z ′ , ¯z ′ )),(6.249)where R and N stand for the radial ordering and the normal orderingrespectively. We take a prescription for the normal ordering which pushesp i to the right and ˜x j to the left with respect to the zero–modes p i and ˜x j .It corresponds to considering the vacuum satisfyingp j |0〉 = α n,j |0〉 = 0 (n > 0), 〈0|α n,j = 0 (n < 0), (6.250)which is the standard prescription for calculating the propagator of themassless scalar field in 2D conformal field theory from the operator formalism.Making use of (6.249), (6.250) and techniques of the contour integration,it is easy to get the commutators[α n,i , α m,j ] = nδ n+m,0 G ij , [˜x i , p j ] = iδ i j,where the first equation holds for all integers with α 0,i ≡ √ 2α ′ p i . Theconstant D ij is written as α ′ D ij = −〈0|˜x i˜x j |0〉. Let us fix D ij as α ′ D ij =− i 2 θij , which is the convention taken in [Seiberg and Witten (1999)]. Thenthe coordinates ˜x i become noncommutative:[˜x i , ˜x j ] = iθ ij ,but the center of mass coordinates x i ≡ ˜x i + 1 2 θij p j can be seen to commuteeach other.Now we have the mode–expanded form of the string coordinates andthe commutation relations between the modes, which areX j (τ, σ) = x j + 2α ′ (G jk τ + 12πα ′ θjk (σ − π 2 ) )p k+ i √ 2α ∑ []′ 1n e−inτ G jk 1cos(nσ) − i2πα ′ θjk sin(nσ) α n,k ,n≠0[α n,i , α m,j ] = nδ n+m,0 G ij , [x i , p j ] = iδ i j,with all the other commutators vanishing.Also, due to the formula∞∑n=12n sin(n(σ + σ′ )) ={ π − σ − σ ′ , (σ + σ ′ ≠ 0, 2π)0, (σ + σ ′ = 0, 2π),


1154 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwe can see by a direct calculation that the end points of the string becomenoncommutative⎧⎨ iθ ij , (σ = σ ′ = 0)[X i (τ, σ), X j (τ, σ ′ )] = −iθ ij , (σ = σ ′ = π)⎩0, (otherwise).On the other hand, it is noted that the conjugate momenta have the modeexpansion identical with that in the Neumann case:P i (τ, σ) = 12πα ′ (g ij∂ τ − 2πα ′ B ij ∂ σ )X j (τ, σ)= 1 π p i + 1π √ ∑e −inτ cos(nσ)α n,i .2α ′Note that the relations (6.247) and (6.248) are in the same form as aT–duality transformation, although the correspondence is a formal sense,because we are not considering any compactification of space–time. Thegeneralized T–duality transformation, namely O(D, D)−transformation, isdefined byn≠0E ′ = (aE + b)(cE + d) −1 , (6.251)with a, b, c and d being D ׄD«real matrices. (D is the dimension ofa bspace–time.) The matrix h = is O(D, D) matrix, which satisfiesc dh T Jh = J, where J =( ) 0 1D.1 D 0The relations (6.247) and (6.248) correspond to the case of the inversiona = d = 0, b = c = 1 D .6.5.9.3 Construction of Overlap VerticesHere we construct Witten’s open string theory in the constant B−fieldbackground by obtaining the explicit formulas of the overlap vertices. As isunderstood from the fact that the action of the ghosts (6.239) contains nobackground fields, the ghost sector is not affected by turning on the B−fieldbackground. Thus we may consider the coordinate sector only. First, let us


<strong>Geom</strong>etrical Path Integrals and Their Applications 1155see the mode-expanded forms of the coordinates and the momenta at τ = 0X j (σ) = G jk y k + 1 π θjk (σ − π 2 )p k+ 2 √ ∑∞ [α ′ G jk cos(nσ)x n,k + 12πα ′ θjk sin(nσ) 1 ]n p n,k ,n=1P i (σ) = 1 π p i + 1π √ α ′∞ ∑n=1cos(nσ)p n,i ,where x j = G jk y k , the coordinates and the momenta for the oscillatormodes arex n,k = i √22 n (a n,k − a † n,k ) = √ i (α n,k − α −n,k ),2n√ np n,k =2 (a n,k + a † n,k ) = √ 1 (α n,k + α −n,k ).2The nonvanishing commutators are given by[x n,k , p m,l ] = iG kl δ n,m , [y k , p l ] = iG kl . (6.252)We should note that the metric appearing in eqs. (6.252) is G ij , insteadof g ij . So it can be seen that if we employ the variables with the loweredspace–time indices y k , p k , x n,k and p n,k , the metric used in the expressionof the overlaps is G ij not g ij .The continuity condition (6.241) is universal for any background, andthe mode expansion of the momenta P i (σ)’s is of the same form as in theNeumann case, thus the continuity conditions for the momenta in terms ofthe modes p n,i are identical with those in the Neumann case. Also, sincep n,i ’s mutually commute, it is natural to find a solution of the continuitycondition, assuming the following form for the overlap vertices:| ˆV N 〉 X 1···N = exp[i4πα θijN ∑r,s=1p (r)n,i Zrs nmp (s)m,j]|V N 〉 X 1···N , (6.253)where | ˆV N 〉 X 1···N and |V N〉 X 1···N are the overlaps in the background correspondingto the world sheet actions (6.242) and (6.238) respectively, theexplicit form of the latter is given in appendix A. Clearly the expression(6.253) satisfies the continuity conditions for the modes of the momenta,and the coefficients Znm rs are determined so that the continuity conditionsfor the coordinates are satisfied [Sugino (2000)].


1156 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction• |Î〉X ≡ | ˆV 1 〉 XFor the N = 1 case, we consider the identity overlap |Î〉X ≡ | ˆV 1 〉 X . Thecontinuity conditions for the momenta require thatP i (σ) + P i (π − σ) = 2 π p i + 2π √ ∑cos(nσ)p n,iα ′should vanish for 0 ≤ σ ≤ π 2 , namely,n=2,4,6,···p i = 0, p n,i = 0 (n = 2, 4, 6, · · · ), (6.254)which is satisfied by the overlap in the Neumann case |I〉. In addition, theconditions for the coordinates are thatX j (σ) − X j (π − σ) = 2 π θjk (σ − π 2 )p k + (6.255)4 √ α ′ ∑n=1,3,5,···G jk cos(nσ)x n,k + 4 √ α ′∑n=2,4,6,···12πα ′ θjk sin(nσ) 1 n p n,k,should vanish for 0 ≤ σ ≤ π 2. The first and third lines in the r. h. s.can be put to zero by using (6.254). So what we have to consider is theremaining condition x n,k = 0 for n = 1, 3, 5, · · · , which however is nothingbut the continuity condition for the coordinates in the Neumann case. Itcan be understood from the point that the second line in (6.255) does notdepend on θ ij . Thus it turns out that the continuity conditions in the caseof the B−field turned on are satisfied by the identity overlap made in theNeumann case. The solution is [Sugino (2000)][]|Î〉X = |I〉 X = exp− 1 2 Gij∞∑n=0(−1) n a † n,i a† n,j|0〉, (6.256)where also the zero modes y i and p i are written by using the creation andannihilation operators a † 0,i and a 0,i asy i = i 2√2α′ (a 0,i − a † 0,i ), p i = 1 √2α′ (a 0,i + a † 0,i ).


<strong>Geom</strong>etrical Path Integrals and Their Applications 1157• | ˆV 2 〉 X 12For the N = 2 case, we are to do the same argument as in the N = 1 case.The continuity conditions mean thatP (1)i (σ) + P (2)i (π − σ) = 1 π (p(1) i + p (2)i ) + 1π √ α ′∞ ∑n=1cos(nσ)(p (1)n,i + (−1)n p (2)n,i ),X (1)j (σ) − X (2)j (π − σ) = G jk (y (1)k− y (2)k) + 1 π θjk (σ − π 2 )(p(1) k+ p (2)k )+ 2 √ ∑∞ [α ′ G jk cos(nσ)(x (1)n,k − (−1)n x (2)n,k )n=1+ 12πα ′ θjk sin(nσ) 1 ]n (p(1) n,k + (−1)n p (2)n,k )should be zero for 0 ≤ σ ≤ π. It turns out again that the conditions forthe modes are identical with those in the Neumann case:p (1)i + p (2)i = 0, p (1)n,i + (−1)n p (2)n,i = 0,y (1)i − y (2)i = 0, x (1)n,i − (−1)n x (2)n,i = 0,for n ≥ 1. Thus we have the solution [Sugino (2000)][]| ˆV 2 〉 X 12 = |V 2 〉 X 12 = exp−G ij∞∑n=0(−1) n a (1)†n,i a(2)† n,j|0〉 12 . (6.257)


1158 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction• | ˆV 4 〉 X 1234We find a solution of the continuity conditions (6.241) in the N = 4 caseassuming the form[]| ˆV 4 〉 X i4∑1234 = exp4πα θij p (r)n,i Zrs nmp (s)m,j |V 4 〉 X 1234. (6.258)r,s=1When considering the continuity conditions, it is convenient to employ theZ 4 −Fourier transformed variables:Q j 1 (σ) = 1 2 [iX(1)j (σ) − X (2)j (σ) − iX (3)j (σ) + X (4)j (σ)] ≡ Q j (σ),Q j 2 (σ) = 1 2 [−X(1)j (σ) + X (2)j (σ) − X (3)j (σ) + X (4)j (σ)],Q j 3 (σ) = 1 2 [−iX(1)j (σ) − X (2)j (σ) + iX (3)j (σ) + X (4)j (σ)] ≡ ¯Q j (σ),Q j 4 (σ) = 1 2 [X(1)j (σ) + X (2)j (σ) + X (3)j (σ) + X (4)j (σ)].For the momentum variables we also define the Z 4 −Fourier transformedvariables P 1,i (σ)(≡ P i (σ)), P 2,i (σ), P 3,i (σ)(≡ ¯P i (σ)) and P 4,i (σ) by thesame combinations of P (r)i (σ)’s as the above. These variables have thefollowing mode expansionsP t,i (σ) =1π √ 2α ′ P t,0,i + 1π √ α ′∞∑n=1cos(nσ)P t,n,i ,Q j t(σ) = G jk√ 2α ′ Q t,0,k + 1 π θjk (σ − π 2 ) 1√2α′ P t,0,k (6.259)+ √ 2α ′ ∞∑n=1[G jk cos(nσ)Q t,n,k + 12πα ′ θjk sin(nσ) 1 ]n P t,n,k ,where t = 1, 2, 3, 4. From now on, we frequently omit the subscript t forthe t = 1 case, and at the same time we employ the expression with a barinstead of putting the subscript t for the t = 3 case.Using those variables, the continuity conditions are written asQ j 4 (σ) − Qj 4 (π − σ) = 0, P 4,i(σ) + P 4,i (π − σ) = 0,Q j 2 (σ) + Qj 2 (π − σ) = 0, P 2,i(σ) − P 2,i (π − σ) = 0,Q j (σ) − iQ j (π − σ) = 0, P i (σ) + iP i (π − σ) = 0,¯Q j (σ) + i ¯Q j (π − σ) = 0, ¯Pi (σ) − i ¯P i (π − σ) = 0 (6.260)


<strong>Geom</strong>etrical Path Integrals and Their Applications 1159for 0 ≤ σ ≤ π 2. In terms of the modes, the conditions for the sectors oft = 2 and 4 are identical with the Neumann case(1 − C)|Q 4,k )| ˆV 4 〉 X = (1 + C)|P 4,k )| ˆV 4 〉 X = 0,(1 + C)|Q 2,k )| ˆV 4 〉 X = (1 − C)|P 2,k )| ˆV 4 〉 X = 0,which can be seen from the point that the conditions (6.260) for the sectorsof t = 2 and 4 lead the same relations between the modes as those withoutthe terms containing θ jk . Here we adopted the vector notation for themodes⎡ ⎤⎡ ⎤⎢|Q t,k ) = ⎣Q t,0,kQ t,1,k.⎥⎢⎦ , |P t,k ) = ⎣P t,0,kP t,1,kand C is a matrix such that (C) nm = (−1) n δ nm (n, m ≥ 0). Thus there isneeded no correction containing θ ij for the sectors of t = 2 and 4, so it isnatural to assume the form of the phase factor in (6.258) as14∑2 θij(p (r)ir,s=1.⎥⎦ ,|Z rs |p (s)j ) = θ ij (P i |Z| ¯P j ) (6.261)with Z being anti–Hermitian.Next let us consider the conditions for the sectors of t = 1 and 3. Werewrite the mode expansions of Q j (σ) and ¯Q j (σ) as [Sugino (2000)]Q j (σ) = G jk ( √ 2α ′ Q 0,k + 2 √ α ′+ θ jk [ ∫ σ≡ θ jk ∫ σπ/2π/2∞∑n=1dσ ′ P i (σ ′ ) + 1π √ α ′¯Q j (σ) = G jk ( √ 2α ′ ¯Q 0,k + 2 √ α ′+ θ jk [ ∫ σπ/2cos(nσ)Q n,k )∑n=1,3,5,···]1n (−1)(n−1)/2 P n,kdσ ′ P i (σ ′ ) + ∆Q j (σ), (6.262)∞ ∑n=1dσ ′ ¯Pi (σ ′ ) + 1π √ α ′cos(nσ) ¯Q n,k )∑n=1,3,5,···]1n (−1)(n−1)/2 ¯Pn,k∫ σ≡ θ jk dσ ′ ¯Pi (σ ′ ) + ∆ ¯Q j (σ). (6.263)π/2


1160 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionUsing the conditions for P i (σ) and ¯P i (σ) in (6.260), we can reduce theconditions for Q j (σ) and ¯Q j (σ) to those for ∆Q j (σ) and ∆ ¯Q j (σ):{ i∆Q∆Q j j (π − σ) (0 ≤ σ ≤ π(σ) =2 )−i∆Q j (π − σ) ( π 2{ ≤ σ ≤ π),∆ ¯Q −i∆ j (σ) =¯Qj (π − σ) (0 ≤ σ ≤ π 2 )i∆ ¯Q j (π − σ) ( π 2 ≤ σ ≤ π).These formulas are translated to the relations between the modes via theFourier transformation. The result is expressed in the vector notation as(1 − X)|Q i )| ˆV 4 〉 X = (1 + X)|Q i )| ˆV 4 〉 X = 0, (6.264)where the vectors |Q i ) and |Q i ) stand for⎡Q 0,i +|Q i ) = ⎢⎣i4αG ′ ik θ kj ∑ ∞Q 1,iQ 2,in=0 X 0nP n,j⎤⎥⎦ ,.⎡¯Q 0,i +i4αG ′ ik θ kj ∑ ∞n=0 X ¯P ⎤0n n,j¯Q 1,i|Q i ) = ⎢⎣¯Q 2,i⎥⎦ .In (6.264), passing the vectors through the phase factor of the | ˆV 4 〉 andusing the continuity conditions in the Neumann case.(1 + X)|P i )|V 4 〉 X = (1 − X)| ¯P i )|V 4 〉 X = 0, (6.265)(1 − X)|Q i )|V 4 〉 X = (1 + X)| ¯Q i )|V 4 〉 X = 0,we get the equations, which the coefficients Z nm ’s should satisfy,∑∞[(1 − X) m0 ( ¯Z 0n + i π 2 ¯X∞∑∑∞0n )P n,j + (1 − X) mn¯Z nn ′P n ′ ,j]|V 4 〉 X = 0n=0n=0n=1n ′ =0∑∞[(1 + X) m0 (Z 0n − i π 2 X 0n) ¯P∞∑∑∞n,j + (1 + X) mn Z nn ′¯Pn′ ,j]|V 4 〉 X = 0n=1n ′ =0for m ≥ 0. Now all our remaining task is to solve these equations. It is


<strong>Geom</strong>etrical Path Integrals and Their Applications 1161easy to see that a solution of them is given by [Sugino (2000)]Z mn = −i π 2 (1 − X) mn + iβ π 2 C mn, (m, n ≥ 0, except for m = n = 0),Z 00 = iβ π 2 ,if we pay attention to (6.265). Here β is an unknown real constant, which isnot fixed by the continuity conditions alone. This ambiguity of the solutioncomes from the property of the matrix X: XC = −CX. However it willbecome clear that the term containing the constant β does not contributeto the vertex | ˆV 4 〉 X .Therefore, we have the expression of the phase (6.261)θ ij (P i |Z| ¯P j ) = θ ij [i π 2 P 0,i ¯P 0,j + iβ π 2− i π 2∞∑m,n=0∞∑(−1) n P n,i ¯Pn,jn=0P m,i (1 − X) mn ¯Pn,j ].Then recalling (6.265) again, the last term in the r. h. s. can be discarded.Also we can rewrite the term containing β− θij4α ′ β(P i|C| ¯P j ) = + θij4α ′ β(P i|X T CX| ¯P j ) = + θij4α ′ β(P i|C| ¯P j ),on |V 4 〉 X . The above formula means that the term containing β can be setto zero on |V 4 〉 X . After all, the form of the 4-string vertex becomes[]| ˆV 4 〉 X 1234 = exp − θij4α ′ P ¯P 0,i 0,j |V 4 〉 X 1234.Note that the phase factor has the cyclic symmetric form− θij4α ′ P 0,i ¯P 0,j = i θij8α ′ (p(1) 0.i p(2) 0,j + p(2) 0.i p(3) 0,j + p(3) 0.i p(4) 0,j + p(4) 0.i p(1) 0,j ),which is a property the vertices should have 22 .22 Here the momentum p (r)0,i is given by p(r)0,i = √ 2α ′ p (r)i .


1162 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction• | ˆV 3 〉 X 123We can get the 3-string overlap in the similar manner as in the 4-stringcase. First, we introduce the Z 3 −Fourier transformed variablesQ j 1 (σ) = 1 √3[eX (1)j (σ) + ēX (2)j (σ) + X (3)j (σ)] ≡ Q j (σ),Q j 2 (σ) = 1 √3[ēX (1)j (σ) + eX (2)j (σ) + X (3)j (σ)] ≡ ¯Q j (σ),Q j 3 (σ) = 1 √3[X (1)j (σ) + X (2)j (σ) + X (3)j (σ)],where e ≡ e i2π/3 , ē ≡ e −i2π/3 . The momenta P 1,i (σ)(≡ P i (σ)), P 2,i (σ)(≡¯P i (σ)) and P 3,i (σ) are defined in the same way. The mode expansionstake the same form as those in (6.259). In these variables, the continuityconditions requireQ j (σ) − eQ j (π − σ) = 0, P i (σ) + eP i (π − σ) = 0,¯Q j (σ) − ē ¯Q j (π − σ) = 0, ¯Pi (σ) + ē ¯P i (π − σ) = 0,Q j 3 (σ) − Qj 3 (π − σ) = 0, P 3,i(σ) + P 3,i (π − σ) = 0for 0 ≤ σ ≤ π 2. The conditions imposed to the modes with respect to thet = 3 component are identical with those in the Neumann case(1 + C)|P 3,i )| ˆV 3 〉 X = (1 − C)|Q 3,i )| ˆV 3 〉 X = 0.Thus the t = 3 component does not couple with θ ij , so we can find thesolution by determining the single anti–Hermitian matrix Z in the phasefactor whose form is assumed as [Sugino (2000)]13∑2 θij(p (r)ir,s=1|Z rs |p (s)j ) = θ ij (P i |Z| ¯P j ). (6.266)For the sectors of t = 1 and 2, the same argument goes on as in the4-string case. Q j (σ) and ¯Q j (σ) have the mode expansions same as in eqs.(6.262) and (6.263). The conditions we have to consider are{ e∆Q∆Q j j (π − σ), (0 ≤ σ ≤ π(σ) =2 )ē∆Q j (π − σ), ( π 2{ ≤ σ ≤ π),∆ ¯Q ē∆ j (σ) =¯Qj (π − σ), (0 ≤ σ ≤ π 2 )e∆ ¯Q j (π − σ), ( π 2 ≤ σ ≤ π),


<strong>Geom</strong>etrical Path Integrals and Their Applications 1163which are rewritten as the relations between the modes(1 − Y )|Q i )| ˆV 3 〉 X = (1 − Y T )|Q i )| ˆV 3 〉 X = 0. (6.267)Recalling the conditions in the Neumann case(1 + Y )|P i )|V 3 〉 X = (1 + Y T )| ¯P i )|V 3 〉 X = 0, (6.268)(1 − Y )|Q i )|V 3 〉 X = (1 − Y T )| ¯Q i )|V 3 〉 X = 0,we end up with the following equations∑∞[(1 − Y ) m0 ( ¯Z 0n + π 2 ¯X∞∑ ∑∞0n )P n,j + (1 − Y ) mn¯Z nn ′P n ′ ,j]|V 3 〉 X = 0,n=0n=0n=1n ′ =0∑ ∞[(1 − Y T ) m0 (Z 0n − i π 2 X 0n) ¯P∞∑∑ ∞n,j + (1 − Y T ) mn Z nn ′¯Pn′ ,j]|V 3 〉 X = 0n=1n ′ =0for m ≥ 0. It can be easily found out that the expressionZ mn = −i π √3(1 + Y T ) mn (m, n ≥ 0, except for m = n = 0),Z 00 = 0,satisfies the above equations. It should be noted that in this case, becauseof CY C = Ȳ ≠ −Y , it does not contain any unknown constant differentlyfrom the 4–string case.Owing to the condition (6.268) we can write the phase factor only interms of the zero-modes. Finally we have [Sugino (2000)][]| ˆV 3 〉 X 123 = exp −θij4 √ 3α P ¯P ′ 0,i 0,j |V 3 〉 X 123[]= exp i θij12α ′ (p(1) 0,i p(2) 0,j + p(2) 0,i p(3) 0,j + p(3) 0,i p(1) 0,j ) |V 3 〉 X 123. (6.269)It is not clear whether the solutions we have obtained here are uniqueor not. However we can show that the phase factors are consistent with therelations between the overlaps which they should satisfy,3〈Î| ˆV 3 〉 123 = | ˆV 2 〉 12 , 4〈Î| ˆV 4 〉 1234 = | ˆV 3 〉 123 , 34〈 ˆV 2 || ˆV 3 〉 123 | ˆV 3 〉 456 = | ˆV 4 〉 1256 ,by using the momentum conservation on the vertices (p (1)i + · · · +p (N)i )| ˆV N 〉 X 1···N = 0. Furthermore we can see that the phase factors successfullyreproduce the Moyal product structures of the correlators among


1164 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionvertex operators obtained in the perturbative approach to open string theoryin the constant B−field background [Seiberg and Witten (1999)]. Thesefacts convince us that the solutions obtained here are physically meaningful.6.5.9.4 Transformation of String FieldsIn the previous section, we have explicitly constructed the overlap verticesin the operator formulation under the constant B−field background. Thenwe have obtained the vertices with a new noncommutative structure ofthe Moyal type originating from the constant B−field, in addition to theordinary product ∗ of string fields. Denoting the product with the newstructure by ⋆, the action of the string field theory is written asS B = 1 ∫ ( 1G s 2 ψ ⋆ Qψ + 1 )3 ψ ⋆ ψ ⋆ ψ= 1 ( 1〈G s 2 ˆV 2 ||ψ〉 1 Q|ψ〉 2 + 1 )〈123 ˆV 3 ||ψ〉 1 |ψ〉 2 |ψ〉 3 , (6.270)123where the BRST charge Q is constructed from the world sheet action(6.242). The theory (6.270) gives the noncommutative U(1) Yang–Millstheory in the low energy region in the same sense as Witten’s open stringfield theory in the case of the Neumann boundary condition leads to theordinary U(1) Yang–Mills theory in the low energy limit. 23In [Seiberg and Witten (1999)] the authors argued that open string theoryin the constant B−field background leads to either commutative ornoncommutative Yang–Mills theories, corresponding to the different regularizationscheme (the so–called Pauli–Villars regularization or the point–splitting regularization) in the world sheet formulation. They discusseda map between the gauge fields in the commutative and noncommutativeYang–Mills theories. In string field theory perspective, there also shouldbe a certain transformation (hopefully simpler than the Yang–Mills case)from the string field ψ in (6.270) to a string field in a new string field theorywhich leads to the commutative Yang–Mills theory in the low energy limit.Here we get the new string field theory by finding a unitary transformationwhich absorbs the noncommutative structure of the Moyal type inthe product ⋆ into a redefinition of the string fields. There are used thetwo vertices | ˆV 2 〉 and | ˆV 3 〉 in the action (6.270). Recall that the 2–stringvertex is in the same form as in the Neumann case and has no Moyal type23 It can be explicitly seen by repeating a similar calculation as that carried out in[Dearnaley (1990)].


<strong>Geom</strong>etrical Path Integrals and Their Applications 1165noncommutative structure. First, we consider the phase factor of the 3–string vertex which multiplies in front of |V 3 〉 (see (6.269)). Making use ofthe continuity conditions∞∑∞∑P 0,i = −2 Y 0n P n,i , ¯P0,i = −2 Ȳ 0n ¯Pn,i , (6.271)n=1it can be rewritten as [Sugino (2000)]n=1−θij4 √ 3α P ¯P ′ 0,i 0,j = θij4 √ ∑∞ (P 0,i Ȳ 0n ¯Pn,j + P n,i Y 0n ¯P0,j )3α ′= − θij24α ′n=1∑ ∞n=1X 0n [(−p (2)0,i − p(3) 0,i + 2p(1) 0,i )p(1) n,j+ (−p (3)0,i − p(1) 0,i + 2p(2) 0,i )p(2) n,j + (−p(1) 0,i − p(2) 0,i + 2p(3) 0,i )p(3)3∑ ∞∑= − θij8α ′r=1 n=1X 0n p (r)0,i p(r) n,j ,where we used the property of the matrix Y : Y 0n = −Ȳ0n = √ 32 X 0n forn ≥ 1 and the momentum conservation on |V 3 〉: p (1)0,i + p(2) 0,i + p(3) 0,i = 0. Wemanage to represent the phase factor of the Moyal type as a form factorizedinto the product of the unitary operators(θ ij ∑U r = exp8α ′n=1,3,5,···X 0n p (r)0,i p(r) n,j)n,j ]. (6.272)Note that the unitary operator acts to a single string field. So the Moyaltype noncommutativity can be absorbed by the unitary rotation of thestring field123〈 ˆV 3 ||ψ〉 1 |ψ〉 2 |ψ〉 3 = 123 〈V 3 |U 1 U 2 U 3 |ψ〉 1 |ψ〉 2 |ψ〉 3 = 123 〈V 3 ||˜ψ〉 1 |˜ψ〉 2 |˜ψ〉 3 ,(6.273)with |˜ψ〉 r = U r |ψ〉 r . It should be remarked that this manipulation hasbeen suceeded owing to the factorized expression of the phase factor, whichoriginates from the continuity conditions relating the zero-modes to thenonzero-modes (6.271). It is a characteristic feature of string field theorythat can not be found in any local field theories.Next let us see the kinetic term. In doing so, it is judicious to write thekinetic term as follows:12〈 ˆV 2 ||ψ〉 1 (Q|ψ〉 2 ) = 123 〈 ˆV 3 ||ψ〉 1 (Q L |I〉 2 |ψ〉 3 + |ψ〉 2 Q L |I〉 3 ), (6.274)


1166 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere Q L is defined by integrating the BRST current j BRST (σ) with respectto σ over the left half regionQ L =∫ π/20dσj BRST (σ).Equation (6.274) is also represented by the product ⋆ asψ ⋆ (Qψ) = ψ ⋆ [(Q L I) ⋆ ψ + ψ ⋆ (Q L I)]. (6.275)Here, I stands for the identity element with respect to the ⋆−product,carrying the ghost number - 3 2, which corresponds to |I〉 in the operatorformulation. As is discussed by [Horowitz et. al. (1986)], in order to showthe relation (6.275) we need the formulasQ R I = −Q L I, (Q R ψ) ⋆ ξ = −(−1) n ψψ ⋆ (Q L ξ) (6.276)for arbitrary string fields ψ and ξ, where Q R is the integrated BRST currentover the right half region of σ. n ψ stands for the ghost number of thestring field ψ minus 1 2, and takes an integer value. The first formula meansthat the identity element is a physical quantity, also the second does theconservation of the BRST charge. By using these formulas, the first termin the bracket in r. h. s. of (6.275) becomes(Q L I) ⋆ ψ = −(Q R I) ⋆ ψ = I ⋆ (Q L ψ) = Q L ψ.Also, it turns out that the second term is equal to Q R ψ. Combining these,we can see that (6.275) holds.Further, we should remark that because the BRST current does notcontain the center of mass coordinate x j , it commute with the momentump i . From the continuity condition p i |I〉 = 0, it can be seen that p i Q L |I〉 = 0.Expanding the exponential in the expression of the unitary operator (6.272)and passing the momentum p 0,i to the right, we getUQ L |I〉 = Q L |I〉. (6.277)Now we can write down the result of the kinetic term. As a result of the


<strong>Geom</strong>etrical Path Integrals and Their Applications 1167same manipulation as in eq. (6.273) and the use of eq. (6.277), we have 2412〈 ˆV 2 ||ψ〉 1 (Q|ψ〉 2 ) = 123 〈 ˆV 3 ||ψ〉 1 (Q L |I〉 2 |ψ〉 3 + |ψ〉 2 Q L |I〉 3 )= 123 〈V 3 ||˜ψ〉 1 (Q L |I〉 2 |˜ψ〉 3 + |˜ψ〉 2 Q L |I〉 3 ) = 12 〈V 2 ||˜ψ〉 1 (Q|˜ψ〉 2 ). (6.278)Here we have a comment [Sugino (2000)]. If we considered the kineticterm itself without using (6.274), what would be going on? Let us see this.From the continuity conditions for | ˆV 2 〉 X 12 = |V 2 〉 X 12:p (1)0,i + p(2) 0,i = 0, p(1) n,i + (−1)n p (2)n,i = 0 (n = 1, 2, · · · ),it could be shown that the 2–string overlap is invariant under the unitaryrotationU 1 U 2 |V 2 〉 12 = |V 2 〉 12 .So we would find the expression for the kinetic term after the rotation12〈V 2 ||ψ〉 1 Q|ψ〉 2 = 12 〈V 2 ||˜ψ〉 1 ˜Q|˜ψ〉2 ,where ˜Q is the BRST charge similarity transformed by U˜Q = UQU † . (6.279)However, after some computations of the r. h. s. of (6.279), we could seethat ˜Q has divergent term proportional to∑1n=1,3,5,···and thus it is not well–defined. It seems that this procedure is not correctand needs some suitable regularization, which preserves the conformalsymmetry 25 . It is considered that the use of eq. (6.274) gives that kind of24 Strictly speaking, in general this formula holds in the case that both of the stringfields |ψ〉 and |˜ψ〉 belong to the Fock space which consists of states excited by finitenumber of creation operators. This point is subtle for giving a proof. However, for theinfinitesimal θ case, by keeping arbitrary finite order terms in the expanded form of theexponential of U, we can make the situation of both |ψ〉 and |˜ψ〉 being inside the Fockspace, and thus clearly eq. (6.278) holds. From this fact, it is plausible to expect thateq. (6.278) is correct in the finite θ case.25 That divergence comes from the mid–point singularity of the string coordinatestransformed by U. In fact, after some calculations, we haveUX j (σ)U † = X j θ jk(σ) − i4 √ XX n0 p n,k − θjk “2α ′ 4 p k sgn σ − π ”. (6.280)2n=1,3,5,···The last term leads to the mid-point sigularity in the energy–momentum tensor and


1168 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionregularization, which will be justified at the end of the next section.Therefore, the string field theory action (6.270) with the Moyal typenoncommutativity added to the ordinary noncommutativity is equivalentlyrewritten as the one with the ordinary noncommutativity alone [Sugino(2000)]:S B =1 ∫ ( 1G s 2 ˜ψ ∗ Q˜ψ + 1 )3 ˜ψ ∗ ˜ψ ∗ ˜ψ= 1 ( 1〈V 2 ||˜ψ〉 1 Q|˜ψ〉 2 + 1 )〈V 3 ||˜ψ〉 1 |˜ψ〉 2 |˜ψ〉 3 . (6.281)G s 2 123 123It is noted that the BRST charge Q, which is constructed from the worldsheet action (6.242), has the same form as the one obtained from the action(6.238) with the relation (6.247). So all the B−dependence has been stuffedinto the string fields except that existing in the metric G ij . Furthermore,recalling that the relation between the metrics G ij and g ij is the sameform as the T–duality inversion transformation, which was pointed out atthe end of section 3, we can make the metric g ij appear in the overlapvertices, instead of the metric G ij . To do so, we consider the followingtransformation for the modes:ˆα i n = (E T −1 ) ik α n,k , ˆp i = (E T −1 ) ik p k , ˆx i = E ik x k . (6.282)By this transformation, the commutators become[ˆα i n, ˆα j m] = ng ij δ n+m,0 , [ˆp i , ˆx j ] = −iδ i j,and the bilinear form of the modesG ij α n,i α m,j = g ij ˆα i nˆα j m, G ij p i α m,j = g ij ˆp iˆα j m, G ij p i p j = g ij ˆp i ˆp j .(6.283)6.6 Application: Dynamics of Strings and BranesDynamics of Nambu–Goto strings and branes was analyzed recently in[Golovnev and Prokhorov (2005)]. It was shown that they could be contheBRST charge Q. It seems that the use of (6.274) corresponds to taking the pointsplitting regularization with respect to the mid–point. Because of the discontinuity ofthe last term in (6.280), it is considered that the transformed string coordinates haveno longer a good picture as a string. It could be understood from the point that thetransformation U drives states around a perturbative vacuum to those around highlynon–perturbative one like coherent states.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1169sidered as continuous limits of ordered discrete sets of relativistic particlesfor which the tangential velocities were excluded from the action. The authorshave proved that p−branes might be considered as continuous limitsof discrete sets of relativistic particles and that the p−brane action∫S = −γ d p+1 σ √ (−1) p g,g = det ∂xµ∂σ a∂x µ∂σ b, (6.284)is a continuous limit of sum of properly modified relativistic particle actions.Here a, b = 0, 1, . . . , p and µ = 0, 1, . . . , n where p + 1 and n + 1 are thebrane world–sheet and the bulk space dimensions respectively; g is theinduced metric determinant on the world–sheet. Strings correspond to thep = 1 case in (6.284). The modification is such that particle motions alongthe brane hypersurface become un–physical. It yields p constraints; theremaining constraint (H = 0) is a consequence of arbitrariness of ‘time’ σ 0 .The authors have also found the linear constraints in un–physical momenta,which allowed them to derive the evolution operators for the objects underconsideration from the ‘first principles’.6.6.1 A Relativistic ParticleFollowing [Golovnev and Prokhorov (2005)], we start with the simplest caseof one relativistic particle which can be formally considered as a 0−brane.Recall that the motion of free relativistic particle is defined by the well–known action∫ √S = −m 1 − −→ v 2 dt,where −→ v = d−→ x (t)dt. The canonical momentum is−→ ∂L p ≡∂ −→ v = m−→ v√1 − −→ ≡ E −→ v 2 p v ,with L being the Lagrangian, and the Hamiltonian is√H = E p = m 2 + −→ p 2 .We can write down the action in the explicitly relativistic invariant form byparametrization of the world line: x µ = x µ (σ 0 ) (usually one uses τ insteadof σ 0 ) withx µ (σ 0 ) = (t(σ 0 ), −→ x (σ 0 )),(µ = 0, . . . , n).


1170 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionWe denoteẋ µ ≡ dxµdσ 0,so that−→ v = ˙−→ xdσ 0dt ,andS = −m∫ √ẋµẋµ dσ 0 .It is the p = 0 case of (6.284) in which we put m (particle’s mass) insteadof γ. The vector of canonical momentum isp µ ≡ ∂L ẋµ= −m√ẋ2 ∂ẋµ . (6.285)The obtained theory is invariant under reparametrization group σ 0 →˜σ 0 = f(σ 0 )), hence its Hamiltonian is zero and by squaring the equation(6.285) one gets a constraint:p 2 − m 2 = 0. (6.286)Note that the information about the sign of p 0 is lost. On the other handone can prove that any constraint has to be linear in un–physical momenta[Golovnev and Prokhorov (2005)]. One can get the solution in the followingway.From the constraint (6.286) one finds√p 0 = ± m 2 + −→ p 2 ,and it is obvious from (6.285) that sgnp 0 = −sgnẋ 0 . Combining these factswe havep 0 + E p sgn(ẋ 0 ) = 0, (6.287)with E p ( −→ √p ) = m 2 + −→ p 2 .The Hamiltonian is zero and the total Dirac Hamiltonian is [Dirac (1982)]H T = v ( p 0 + E p sgn(ẋ 0 ) ) .Here v is the Lagrange multiplier. Strictly speaking (6.287) is not a constraintbecause it contains velocity (and H T is not a Hamiltonian due tothe same reason). But it depends only upon the sign of ẋ, and this factallows us to formulate the quantum theory.We fix the σ 0 ‘time arrow’ by condition∂x 0∂σ 0> 0, (6.288)


<strong>Geom</strong>etrical Path Integrals and Their Applications 1171which forces us to admit that v > 0, as∆x 0 = v∆σ 0 .We use the evolution operator〈〉U ω (x, ˜x) = x| exp(−iωĤT )|˜x ,∫ψ(x, σ 0 + ω) = d 4˜x U ω (x, ˜x)ψ(˜x, σ 0 )and the following relation for infinitesimal ω:〈〉 ∫x| exp(−iωĤT )|˜x = d 4 p 〈x| exp(−iωH T (x, p))|p〉 〈p|˜x〉 .Integrating over p 0 and (with use of δ−function δ(x 0 − ˜x 0 − ωv)) over ˜x 0we get for the wave function [Golovnev and Prokhorov (2005)]:ψ(x 0 , −→ ∫ d 3 p d 3˜xx ) =(2π) 3 exp ( i[p i ∆x i − ∆x 0 E p ] ) ψ(˜x 0 , −→˜x ), (6.289)where ∆x 0 = vω, and we omit the argument σ 0 because all the informationon σ 0 is accumulated in x 0 . Using (6.289) one can get the right Feynmanpropagator for a relativistic particle.6.6.2 A StringNow we show that any string and brane can be described as a system of particles.More precisely, the action (6.284) may be regarded as a continuouslimit of sum of free relativistic particle actions provided that we take −→ v ⊥instead of −→ v , where −→ v ⊥ is the part of velocity orthogonal to the constanttime hypersurface of the brane world–sheet. We deal here with a kind ofindirectly introduced particle interaction.In this section we consider a string (1−brane) and reproduce the proofof our statement by an explicit calculation [Barbashov and Nesterenko(1990)]. We consider N + 1 particles with the position vectors −→ x k (x 0 ),(k = 0, 1, . . . , N) and the actionN∑∫S = −m dx√1 0 − −→ v 2 k⊥ (x0 ), in whichk=0−→ v k = d−→ x kdx 0 ,(6.290)and−→ v k⊥ = −→ v k − (−→ v k · ∆ −→ x k )(∆ −→ x k ) 2 ∆ −→ x k , with ∆ −→ x k ≡ −→ x k+1 − −→ x k .


1172 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIn the continuous limit we definekAN → σ 1,∆ −→ x kA/N → −→ k (σ 1 ),m|∆ −→ x k | → γ.Here σ 1 ∈ [0, A]; usually one takes A = π, but for our purposes it may benatural to consider A as the string length and|∆ −→ x k | = A N , |−→ k | = 1.In any case we haveS = −γN∑∫k=0with−→ k =∂ −→ x (x 0 , σ 1 )∂σ 1,√dx 0 ∆l 1 − −→ ∫v 2 k⊥ → −γ−→ v =∂ −→ x (x 0 , σ 1 )∂x 0 ,The string length is equal to∫L = | −→ k |dσ 1 .dx 0 | −→ k |dσ 1√1 − −→ v 2 ⊥,−→ v ⊥ = −→ v − (−→ v −→ k )k 2 −→ k .After that we parameterize the world–sheetx 0 = x 0 (σ 0 , σ 1 ),by introducing a new parameter σ 0x 0 t and σ 0 τ). We get−→ x =−→ x (σ0 , σ 1 )(again the standard notations areẋ ≡ ∂x(σ 0, σ 1 )= (1, −→ v )ẋ 0 , x ′ ≡ ∂x(σ 0, σ 1 )= (x ′ 0 , −→ k + −→ v x ′ 0 ),∂σ 0 ∂σ∫1and S = −γ d 2 σẋ 0 | −→ √k | 1 − −→ v 2 ⊥.The last expression is equal to the Nambu–Goto action:∫ √S = −γ d 2 σ (ẋx ′ ) 2 − ẋ 2 x ′2 .Note that one could start with it and get∫ √S = −γ dx 0 dl 1 − −→ 2v ⊥ ,which is a continuous limit of (6.290).


<strong>Geom</strong>etrical Path Integrals and Their Applications 1173We can also propose another discrete analogue of the Nambu–Gotostring. Let’s consider the following action:N∑∫S = −m dσ 0√ẋ µ k⊥ẋµk⊥,k=0where ẋ µ k⊥ is the part of ẋµ k perpendicular to xµ k+1 − xµ k. The continuouslimit isN → ∞,with the invariant intervalWe haveẋ µ ⊥ = ẋµ − ẋν x ′ νx ′ρ x ′ x ′µ ,ρ∫ √S = −γ dτ|ds|kAN → σ 1,m|∆s k | → γ,(∆s k ) 2 = (x µ k+1 − xµ k )(x µ k+1− x µk ).( dsdσ 1) 2= x ′2 ,ẋ 2 − (ẋx′ ) 2x ′ 2and∫= −γ dσ 0 dσ 1√(ẋx ′ ) 2 − ẋ 2 x ′2 .In contrast to the previous paragraph the presented discrete theory hasthe relativistic invariant form from the very beginning but even the senseof ẋ ⊥ depends upon the parametrization of the world–sheet. In the gaugeσ 0 = x 0 these two approaches coincide.6.6.3 A BraneNow we shall prove our statement for p > 1. We consider (N +1) p particlesarranged into some pD lattice with the position vectors −→ x i1i 2...i pand theaction [Golovnev and Prokhorov (2005)]∫S = −mdx 0N ∑i 1=0· · ·N∑i p=0√1 − −→ v 2 i 1,...i p⊥ ,where−→ v i1,...i p⊥ is the component of−→ v i1...i pperpendicular to−→ x i1...i k +1...i p− −→ x i1...i k ...i pfor all k. In continuous limit we demandAi kN → σ k and m ∆V→ γ with ∆V being volume of a cell of the lattice.The action takes the form∫ √S = −γ dx 0 dV 1 − −→ v 2 ⊥,


1174 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand we need to prove that S is equal to (6.284).For the sake of simplicity first we consider some special coordinatesystem on the brane. Let the brane be parametrized by σ 0 , σ 1 , . . . , σ p ,σ i ∈ [0, A] for i = 1, . . . , p. We choose a coordinate system in whichσ 0 = x 0 , so that ∂x0∂σ i= 0, and ∂xµ ∂x µ∂σ i ∂σ j= 0, i ≠ j. It is always possible,locally at least. Let’s denote−→ k i ≡ ∂−→ x (x 0 , σ i )∂σ iand−→ v ≡∂ −→ x (x 0 , σ i )∂x 0on the world–sheet. In our coordinate system −→ k i is orthogonal to −→ k j ,i ≠ j and−→ v ⊥ = −→ v −p∑i=1( −→ v −→ k i ) −→2 k i .k iNote that one can find −→ v ⊥ without the orthogonality condition with theuse of standard orthogonalization procedure.We get∂x µ∂σ 0= (1, −→ v )and∂x µ∂σ i= (0, −→ k i ),so the determinant in (6.284) is1 − −→ v 2 − −→ v −→ k 1 − −→ v −→ k 2 . . . − −→ v −→ k pdet ∂xµ ∂x − −→ v −→ k 1 −k 2 1 0 . . . 0µ=− −→ v −→ k 2 0 −k2 2 . . . 0∂σ a ∂σ b . . . . .. .∣ − −→ v −→ k p 0 0 . . . −kp2 ∣( p∏) ⎛ ( −→v −→) 2⎞= (−k 2 ⎜p∑i ) ⎝1 −−→ 2 k i⎟v +2 ⎠ki=1iwhere a, b = 0, 1, . . . , p. It is not also difficult to see that the volume elementat the constant time hypersurface on the brane world–sheet isdV =p∏|k i |dσ i .i=1i=1


<strong>Geom</strong>etrical Path Integrals and Their Applications 1175Thus, we can conclude that the action is∫S = −γ√ ∣∣∣∣ ∣ ∫dσ 0 d n σ i det ∂xµ ∂x µ ∣∣∣= −γ∂σ a ∂σ b√dx 0 dV 1 − −→ v 2 ⊥. (6.291)6.6.4 String DynamicsAs it was mentioned above, the evolution operator U ω can be constructedfor strings and branes in the same way as above [Goto (1971)]. For p =1 action (6.284) is the action of free bosonic string with the Lagrangiandensity (see, e.g., [Kaku (1988)])L = −γ √ √−g = −γ (ẋx ′ ) 2 − ẋ 2 x ′2 . (6.292)We assume that ẋ is time–like and x ′ is space–like, so that σ 0 can beregarded as a time parameter. In this case the momentump µ ≡ ∂L∂ẋ µ = −γ (ẋx′ )x ′ µ − x ′2ẋ µ√(ẋx′ ) 2 − ẋ 2 x ′2 (6.293)will also be time–like. One easily gets two constraintsp µ x ′ µ = 0, (6.294)p 2 + γ 2 x ′ 2 = 0. (6.295)The second one is obtained by squaring the (6.293) and hence some informationis lost. As in the case of a pointlike particle (6.295) yields p 0 = ±E pwithE p =√−→p 2 − γ2 x ′2 .The sign of p 0 follows from the definition of the momentum. We havep 0 + E p (x, −→ p )sgn(y 0 ) = 0,wherey µ = (ẋx ′ )x ′ µ − x ′ 2ẋµ .The vector y is obviously time–like (indeed, y 2 = x ′2 g > 0 and y µ ẋ µ =−g > 0, so that sgnẋ 0 = sgny 0 ). We get the ‘constraint’ analogous to(6.287):p 0 + E p (x, −→ p )sgn(ẋ 0 ) = 0. (6.296)


1176 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe ‘Hamiltonian’ of the theory,H = p 0 + E p sgnẋ 0 ,is zero, and demanding again sgnẋ 0 > 0, we get the total HamiltonianH T = u(p 0 + E p ) + vp µ x ′ µ .We have two un–physical momenta and need to exclude them from E p .Let’s exclude p 0 and p 1 (we denote the remaining components by the lowerindex ‘ > ’: p µ = (p 0 , p 1 , p > )). One can find p 1 from (6.294) if x ′ 1 ≠ 0.Above we restricted ourselves to the case x ′ 0 = 0 and hadE p (x, p > ) =√ (p>x ′ >x ′ 1) 2+ p 2 > − γ 2 x ′2 .In general case one can substitute p 1 from (6.294) to (6.295) and get aquadratic equation for p 0 :⎛ ( ) ⎞2 xp 0⎝1 ′ 0 2 (− ⎠(p > x ′x ′ 1+ 2p>)x ′ 0( ) )x0(x ′1 ) 2 − p 2 ′ 2> 1 + >x ′ 1+ γ 2 x ′ 2 = 0.If |x ′1 | > |x ′0 |, it has two real roots of opposite signs, and we can choose aproper one following (6.296). Otherwise we have to try to exclude anothercomponent of p µ from E p . It’s always possible because x ′ is space–like and| −→ x ′ | > |x ′0 |. Above we have seen that the un–physical degree of freedom isrelated to the motion of particles along the string.Now we write down the evolution equation [Golovnev and Prokhorov(2005)]∫〈〉ψ(x) = D n+1˜x(σ 1 ) x(σ 1 )| exp(−iωĤT )|˜x(σ 1 ) ψ(˜x(σ 1 )) =∫= D n+1 pD n+1˜x exp(i[p µ ∆x µ − ωu( p 0 + E p (˜x, p ⊥ )) − ωvp µ x ′ µ ])ψ(˜x) =∫= D n−1 p > D n+1˜x exp(i[−p > ∆x > − ωuE p (˜x, p > ) + ωvp > x ′ >])×× δ(∆x 0 − ωu − ωvx ′ 0 )δ(−∆x1 + ωvx ′ 1)ψ(˜x).Here D˜x and Dp denote differentials in the functional spaces and all theintegrals are path integrals.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1177δ−Functions determine the Lagrange multipliersωv = ∆x 1x ′ 1and ωu = ∆x 0 − x′0 ∆x 1x ′ ,1giving the final result∫ψ(x) = D n−1 p > D n−1˜x > exp(i[−p > ∆x > −()−If x ′0 = 0, (6.288) implies u > 0.∆x 0 − x′0 ∆x 1x ′ 1E p (˜x, p > ) + ∆x 1x ′ p > x ′ >])ψ(˜x).16.6.5 Brane DynamicsWe turn to the general case of action (6.284). We denote the σ 0 , σ 1 , . . . , σ pderivatives of x by x ,0 , x ,1 , . . . , x ,p . Again we assume that the vector x µ ,0 istime–like, and vectors x µ ,i are space–like (here and hereafter in this chapteri, k, l = 1, . . . , p while a, b = 0, . . . , p). Now in the action (6.284)g(σ) =1(p + 1)! ɛ a 0...a pɛ b0...bp x α0,b 0x α0,a0 · · · x αp,b px αp,ap ,with ɛ being the unit antisymmetric Levi–Civita symbol, and the canonicalmomentum is [Golovnev and Prokhorov (2005)]p µ =−γ(−1)pp! √ (−1) p g ɛ 0a 1...a pɛ b0...bp x µ,b0 x ,b1 · x ,a1 . . . x ,bp · x ,ap .Evidently p µ x µ ,i = 0 due to antisymmetry of ɛ, and using the equalityɛ a0...ap g(σ) = ɛ b0...bp x ,b0 · x ,a0 . . . x ,bp · x ,apone getsp 2 = (−1) p γ 2 ζ(x), with ζ(x) = detx µ ,i x µ,k.So, with the loss of information about the sign of p 0 , the constraints arep µ x µ ,i = 0, (i = 1, 2, . . . , p), (6.297)p 2 − (−1) p γ 2 ζ(x) = 0. (6.298)From (6.298) we have p 0 = ±E p withE p =√−→p 2 + (−1)p γ 2 ζ(x).


1178 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionAgain, the p 0 sign can be easily found: p µ and ẋ µ are time–like andp µ ẋ µ = −γ √ (−1) p g < 0, so that sgn(p 0 ) = −sgn(ẋ 0 ).The result is similar to (6.287):p 0 + E p sgn(ẋ 0 ) = 0,the ‘Hamiltonian’ is equal to zero, and the total Hamiltonian isH T = u(p 0 + E p (x, −→ p )) + v i p µ x µ ,i .Here, p+1 momenta are un–physical ones. We assume that det(x i,k ) ≠ 0(x i,k is an p × p matrix) and exclude momenta p 1 , . . . , p p from E p . Due to(6.297) we havep i = ([x .,. ] −1 ) il (p 0 x 0,l − p > x >,l ).Here we denoted all the components of p µ with µ > p by the lower index‘ > ’, and ([x .,. ] −1 ) il ≡ d il is a matrix inverse of x l,i . Then (6.298) turns intoquadratic equationp 02 ( 1 − d il x 0,l d ik x 0,k)+ 2p0 d il ( p > x >,l )d ik x 0,k−d il (p > x >,l )d ik (p > x >,k ) − (−1) p γ 2 ζ(x) = 0.It has two real roots of opposite signs iff d il x 0,l d ik x 0,k < 1. The sufficientcondition is that the norm of x i,k as a linear operator is greater than thelength of pD vector x 0,l . If x 0,l = 0 the simple answer exists:∑E p = E p (x, p > ) = √ p (p i (x, p > )) 2 + p 2 > + (−1) p γ 2 ζ(x).i=1For the wave function, taking (6.288) into account, we get path integrals∫ψ(x) = D n+1 pD n+1˜x exp(i[p µ ∆x µ −∫=− ωu(p 0 + E p (˜x, p ⊥ )) − ωv i p µ x µ ,i ])ψ(˜x) =D n−p p > D n+1˜x exp(i[−p > ∆x > − ωuE p (˜x, p > ) + ωv i p > x >,i ])r∏× δ(∆x 0 − ωu − ωv i x ′ 0,i) δ(−∆x l + ωv i x l,i )ψ(˜x).l=1


<strong>Geom</strong>etrical Path Integrals and Their Applications 1179Again δ−functions determine the Lagrange multipliersωv i = d il ∆x l , and ωu = ∆x 0 − ωv i x 0,i ,and reduce the number of integrals over x:∫ψ(x) = D n−p p > D n−p˜x > exp(i[−p > ∆x >− (∆x 0 − d il (∆x l )x 0,i )E p (˜x, p > ) + d il (∆x l )p > x >,i ])ψ(˜x).Now we are ready to prove this statement without fixing any specialcoordinate system. We just need to find out the general formula for −→ v ⊥ .Notice that by definition p µ lies in a hyperplane of x µ ,0 and xµ ,i. Then, dueto the constraints (6.297), p µ is proportional to x µ ⊥,0. We havev µ = ∂xµ (x 0 , σ i )∂x 0 = xµ ,0,and hence p µ is also proportional to v µ ⊥ : vµ ⊥ = αpµ . To find the coefficientα we takex 0 ,0v µ ⊥ p µ = xµ ⊥,0 p µx 0 ,0= −γ√ (−1) p gx 0 ,,0(the last equality is just the Euler’s homogeneous function theorem) andp µ p µ = (−1) p γ 2 ζ(x).However [Golovnev and Prokhorov (2005)],∫√as v µ ⊥ p µ = αp µ p µ , we have v µ ⊥ = − (−1)p g(−1) p γζx 0 p µ ,,0and 1 − −→ v 2 ⊥ = v µ ⊥ v µ⊥ = (−1)p g p 2dx 0 dV√1 − −→ ∫v 2 ⊥ =√ ∫ g× d p σ iζx 0 2 =,0γ 2 ζ 2 x 0 ,0x 0 ,0 dσ 0√|detx µ ,i x µ,k| ×d p+1 σ √ |g|.2 = gζx 0 2, so we get,0


1180 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction6.7 Application: Topological String Theory6.7.1 Quantum <strong>Geom</strong>etry FrameworkTo start our review on topological string theory, here we depict a generalquantum geometry framework (see e.g., [Witten (1998a)]).SPECIAL RELATIVITYQUANTUM FIELD THEORYv/c ↑→CLASSICAL DYNAMICSQUANTUM MECHANICSFig. 6.22 The deformation from classical dynamics to quantum field theory (see textfor explanation).The relationship between non–relativistic classical mechanics and quantumfield theory (see [Coleman (1988)]) can be summarized as in Figure6.22. We see that the horizontal axis corresponds to the Planck constant (divided by the typical action of the system being studied), while thevertical axis corresponds to v/c, the ration of motion velocity and lightvelocity.Similarly, in the superstring theory there are also two relevant expansionparameters, as shown in Figure 6.23. Here we see that the horizontal axiscorresponds to the value of the string coupling constant, g s , while the verticalaxis corresponds to the value of the dimensionless sigma model couplingα ′ /R 2 with R being a typical radius of a compactified portion of space). Inthe extreme α ′ = g s = 0 limit, for instance, we recover relativistic particledynamics. For nonzero g s we recover point particle quantum field theory.For g s = 0 and nonzero α ′ we are studying classical string theory. In generalthough, we need to understand the theory for arbitrary values of theseparameters (see [Greene (1996)]).Quantum stringy geometry postulates the existence of 6D Calabi–Yaumanifolds at every point of the space–time (see, e.g., [Candelas et. al.(1985)]). These curled–up local manifolds transform according to the generalorbifolding procedure, as will be described below.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1181CLASSICAL STRINGYGEOMETRYQUANTUM STRINGYGEOMETRYα ′ /R 2CLASSICAL RIEMANNIANGEOMETRY↑→g sQUANTUM RIEMANNIANGEOMETRYFig. 6.23 The deformation from classical Riemannian geometry to quantum stringygeometry (see text for explanation).6.7.2 Green–Schwarz Bosonic Strings and BranesHere, we briefly describe the world–sheet dynamics of the Green–Schwarzbosonic string theory, and (more generally), bosonic p−brane theory, thepredecessor of the current superstring theory (see [Schwarz (1993); Greenet. al. (1987)] for details).World–Line Description of a Point ParticleRecall that a point particle sweeps out a trajectory called world–line inspace–time. This can be described by functions x µ (τ), that describe howthe world–line, parameterized by τ, is embedded in the space–time, whosecoordinates are denoted x µ (µ = 0, 1, 2, 3). For simplicity, let us assumethat the space–time is flat Minkowski space with a Lorentz metric tensor⎛ ⎞η µν =⎜⎝−1 0 0 00 1 0 0⎟0 0 1 0 ⎠ .0 0 0 1Then, the Lorentz–invariant line element (metric form) is given byds 2 = −η µν dx µ dx ν .In normal units ( = c = 1), the action for a particle of mass m is given by∫S = −m ds.


1182 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis could be generalized to a curved space–time by replacing η µν by aRiemannian metric tensor g µν (x), but (for simplicity) we will not do so here.In terms of the embedding functions, x µ (t), the action can be rewritten as∫ √S[x] = −m dτ −η µν ẋ µ ẋ ν ,where overdot represents the derivative with respect to τ. An importantproperty of this action is invariance under local reparametrizations. Thisis a kind of gauge invariance, whose meaning is that the form of S is unchangedunder an arbitrary reparametrization of the world–line τ → τ(˜τ).Actually, one should require that the function τ(˜τ) is smooth and monotonic( dτd˜τ > 0) . The reparametrization invariance is a 1D analog of the 4Dgeneral coordinate invariance of general relativity. Mathematicians refer tothis kind of symmetry as diffeomorphism invariance.The reparametrization invariance of S allows us to choose a gauge. Anice choice is the static gauge, x 0 = τ. In this gauge (renaming theparameter to t) the action becomes∫ √S = −m 1 − vi 2dt, where v i = dx idt .Requiring this action to be stationary under an arbitrary variation of x i (t)gives the Euler–Lagrangian equationsdp idt = 0, where p i = δS =mv i√ ,δv i 1 − v2iwhich is the usual result. So we see that usual relativistic kinematics followsfrom the action S = −m ∫ ds.p−Branes and World–Volume ActionsWe can now generalize the analysis of the massive point particle to ageneric p−brane, which is characterized by its tension T p . The action inthis case involves the invariant (p + 1)D volume and is given by∫S p = −T p dµ p+1 ,where the invariant volume element is√dµ p+1 = − det(−η µν ∂ α x µ ∂ β x ν )d p+1 σ.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1183Here the embedding of the p−brane into dD space–time is given by functionsx µ (σ α ). The index α = 0, . . . , p labels the p + 1 coordinates σ α ofthe p−brane world–volume and the index µ = 0, . . . , d − 1 labels the dcoordinates x µ of the dD space–time. We have defined ∂ α x µ = ∂xµ∂σ.Theαdeterminant operation acts on the (p + 1) × (p + 1) matrix whose rows andcolumns are labelled by α and β. The tension T p is interpreted as the massper unit volume of the p−brane. For a 0−brane, it is just the mass.Let us now specialize to the string, p = 1. Evaluating the determinantgives the Nambu–Goto action (see subsection 6.5.4 above)∫S[x] = −T dσdτ √ ẋ 2 x ′2 − (ẋ · x ′ ) 2 ,where we have defined σ 0 = τ, σ 1 = σ , and ẋ µ = ∂xµ∂τ , x′µ = ∂xµ∂σ . Theabove action is equivalent to the actionS[x, h] = − T ∫d 2 σ √ −hh αβ η2µν ∂ α x µ ∂ β x ν , (6.299)where h αβ (σ, τ) is the world–sheet metric, h = det h αβ , and h αβ = (h αβ ) −1is the inverse of h αβ . The Euler–Lagrangian equations obtained by varyingh αβ areT αβ = ∂ α x · ∂ β x − 1 2 h αβh γδ ∂ γ x · ∂ δ x = 0.In addition to reparametrization invariance, the action S[x, h] has anotherlocal symmetry, called conformal invariance, or, Weyl invariance.Specifically, it is invariant under the replacementh αβ → Λ(σ, τ)h αβ , x µ → x µ .This local symmetry is special to the p = 1 case (strings).The two reparametrization invariance symmetries of S[x, h] allow us tochoose a gauge in which the three functions h αβ (this is a symmetric 2 × 2matrix) are expressed in terms of just one function. A convenient choice isthe conformally flat gaugeh αβ = η αβ e φ(σ,τ) .Here, η αβ denoted the 2D Minkowski metric of a flat world–sheet. However,h αβ is only ‘conformally flat’, because of the factor e φ . Classically,


1184 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsubstitution of this gauge choice into S[x, h] leaves the gauge–fixed actionS = T ∫d 2 ση αβ ∂ α x · ∂ β x. (6.300)2Quantum mechanically, the story is more subtle. Instead of eliminatingh via its classical field equations, one should perform a Feynman pathintegral, using standard machinery to deal with the local symmetries andgauge fixing. When this is done correctly, one finds that in general φ doesnot decouple from the answer. Only for the special case d = 26 does thequantum analysis reproduce the formula we have given based on classicalreasoning. Otherwise, there are correction terms whose presence can betraced to a conformal anomaly (i.e., a quantum–mechanical breakdown ofthe conformal invariance).The gauge–fixed action is quadratic in the x’s. Mathematically, it is thesame as a theory of d free scalar fields in two dimensions. The equations ofmotion obtained by varying x µ are free 2D wave equations:ẍ µ − x ′′µ = 0.However, this is not the whole story, because we must also take accountof the constraints T αβ = 0, which evaluated in the conformally flat gauge,readT 01 = T 10 = ẋ · x ′ = 0, T 00 = T 11 = 1 2 (ẋ2 + x ′2 ) = 0.Adding and subtracting givesBoundary Conditions(ẋ ± x ′ ) 2 = 0. (6.301)To go further, one needs to choose boundary conditions. There are threeimportant types. For a closed string one should impose periodicity in thespatial parameter σ. Choosing its range to be π (as is conventional)x µ (σ, τ) = x µ (σ + π, τ).For an open string (which has two ends), each end can be required to satisfyeither Neumann or Dirichlet boundary conditions for each value of µ,Neumann :Dirichlet :∂x µ= 0∂σat σ = 0 or π,∂x µ= 0∂τat σ = 0 or π.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1185The Dirichlet condition can be integrated, and then it specifies a space–time location on which the string ends. The only way this makes sense isif the open string ends on a physical object – it ends on a D–brane. 26 If allthe open–string boundary conditions are Neumann, then the ends of thestring can be anywhere in the space–time. The modern interpretation isthat this means that there are space–time–filling D–branes present.Let us now consider the closed–string case in more detail. The generalsolution of the 2D wave equation is given by a sum of ‘right–movers’ and‘left–movers’: x µ (σ, τ) = x µ R (τ − σ) + xµ L(τ + σ). These should be subjectto the following additional conditions:• x µ (σ, τ) is real,• x µ (σ + π, τ) = x µ (σ, τ), and• (x ′ L )2 = (x ′ R )2 = 0 (these are the T αβ = 0 constraints in (6.301)).The first two of these conditions can be solved explicitly in terms ofFourier series:x µ R = 1 2 xµ + l 2 sp µ (τ − σ) +x µ L = 1 2 xµ + l 2 sp µ (τ + σ) +√ i ∑l s2n≠01n αµ ne −2in(τ−σ)√ i ∑ 1l s2 n ˜αµ ne −2in(τ−σ) ,where the expansion parameters α µ n, ˜α µ n satisfy α µ −n = (α µ n) † , ˜α µ −n =(˜α µ n) † .The center–of–mass coordinate x µ and momentum p µ are also real.The fundamental string length scale l s is related to the tension T byn≠0T = 12πα ′ , α′ = l 2 s.The parameter α ′ is called the universal Regge slope, since the string modeslie on linear parallel Regge trajectories with this slope.Canonical QuantizationThe analysis of closed–string left–moving modes, closed–string right–moving modes, and open–string modes are all very similar. Therefore, toavoid repetition, we focus on the closed–string right–movers. Starting withthe gauge–fixed action in (6.300), the canonical momentum of the string is26 D here stands for Dirichlet.p µ (σ, τ) = δSδẋ µ = T ẋµ .


1186 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionCanonical quantization (this is just free 2D field theory for scalar fields)gives[p µ (σ, τ), x ν (σ ′ , τ)] = −iη µν δ(σ − σ ′ ).In terms of the Fourier modes (setting = 1) these become[p µ , x ν ] = −iη µν , [α µ m, α ν n] = mδ m+n,0 η µν , [˜α µ m, ˜α ν n] = mδ m+n,0 η µν ,and all other commutators vanish.Recall that a quantum–mechanical harmonic oscillator can be describedin terms of raising and lowering operators, usually called a † and a, whichsatisfy [a, a † ] = 1. We see that, aside from a normalization factor, the expansioncoefficients α µ −m and α µ m are raising and lowering operators. Thereis just one problem. As η 00 = −1, the time components are proportional tooscillators with the wrong sign ([a, a † ] = −1). This is potentially very bad,because such oscillators create states of negative norm, which could lead toan inconsistent quantum theory (with negative probabilities, etc.). Fortunately,as we will explain, the T αβ = 0 constraints eliminate the negative–norm states from the physical spectrum.The classical constraint for the right–moving closed–string modes,(x ′ R )2 = 0, has Fourier componentsL m = T 2∫ π0e −2imσ (x ′ R) 2 dσ = 1 2∞∑n=−∞α m−n · α n ,which are called Virasoro operators.α µ −m, L 0 needs to be normal–ordered:Since α µ m does not commute withL 0 = 1 ∞∑2 α2 0 + α −n · α n .n=1Here α µ 0 = l sp µ / √ 2, where p µ is the momentum.6.7.3 Calabi–Yau Manifolds, Orbifolds and Mirror SymmetryCalabi–Yau ManifoldsFundamental geometrical objects in string theory are the Calabi–Yaumanifolds [Calabi (1957); Yau (1978)]. Recall (from subsection 3.14.2


<strong>Geom</strong>etrical Path Integrals and Their Applications 1187above) that a Calabi–Yau manifold is a compact Ricci–flat Kähler manifoldwith a vanishing first Chern class. A Calabi–Yau manifold of complexdimension n is also called a Calabi–Yau n−fold, which is a manifold withan SU(n) holonomyi.e., it admits a global nowhere vanishing holomorphic(n, 0)−form.For example, in one complex dimension, the only examples are family oftori. Note that the Ricci–flat metric on the torus is actually a flat metric, sothat the holonomy is the trivial group SU(1). In particular, 1D Calabi–Yaumanifolds are also called elliptic curves.In two complex dimensions, the torus T 4 and the K3 surfaces 27 are theonly examples. T 4 is sometimes excluded from the classification of being aCalabi–Yau, as its holonomy (again the trivial group) is a proper subgroupof SU(2), instead of being isomorphic to SU(2). On the other hand, theholonomy group of a K3 surface is the full SU(2) group, so it may properlybe called a Calabi–Yau in 2D.In three complex dimensions, classification of the possible Calabi–Yaumanifolds is an open problem. One example of a 3D Calabi–Yau is thequintic threefold in CP 4 .In string theory, the term compactification refers to ‘curling up’ the extradimensions (6 in the superstring theory), usually on Calabi–Yau spacesor on orbifolds. The mechanism behind this type of compactification isdescribed by the Kaluza–Klein theory.In the most conventional superstring models, 10 conjectural dimensionsin string theory are supposed to come as 4 of which we are aware, carryingsome kind of fibration with fiber dimension 6. Compactification onCalabi–Yau n−folds are important because they leave some of the originalsupersymmetry unbroken. More precisely, compactification on a Calabi–Yau 3−fold (with real dimension 6) leaves one quarter of the original supersymmetryunbroken.OrbifoldsRecall that in topology, an orbifold is a generalization of a manifold, atopological space (called the underlying space) with an orbifold structure.The underlying space locally looks like a quotient of a Euclidean spaceunder the action of a finite group of isometries.27 Recall that K3 surfaces are compact, complex, simply–connected surfaces, withtrivial canonical line bundle, named after three algebraic geometers, Kummer, Kählerand Kodaira. Otherwise, they are hyperkähler manifolds of real dimension 4 with SU(2)holonomy.


1188 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe formal orbifold definition goes along the same lines as a definitionof manifold, but instead of taking domains in R n as the target spaces ofcharts one should take domains of finite quotients of R n . A (topological)orbifold O, is a Hausdorff topological space X with a countable base, calledthe underlying space, with an orbifold structure, which is defined by orbifoldatlas, given as follows.An orbifold chart is an open subset U ⊂ X together with open setV ⊂ R n and a continuous map ϕ : U → V which satisfy the followingproperty: there is a finite group Γ acting by linear transformations on Vand a homeomorphism θ : U → V/Γ such that ϕ = θ ◦ π, where π denotesthe projection V → V/Γ. A collection of orbifold charts, {ϕ i = U i → V i },is called the orbifold atlas if it satisfies the following properties:(i) ∪ i U i = X;(ii) if ϕ i (x) = ϕ j (y) then there is a neighborhood x ∈ V x ⊂ V i andy ∈ V y ⊂ V j as well as a homeomorphism ψ : V x → V y such that ϕ i = ϕ j ◦ψ.The orbifold atlas defines the orbifold structure completely and we regardtwo orbifold atlases of X to give the same orbifold structure if they canbe combined to give a larger orbifold atlas. One can add differentiabilityconditions on the gluing map in the above definition and get a definition ofsmooth (C ∞ ) orbifolds in the same way as it was done for manifolds.The main example of underlying space is a quotient space of a manifoldunder the action of a finite group of diffeomorphisms, in particular manifoldwith boundary carries natural orbifold structure, since it is Z 2 −factor ofits double. A factor space of a manifold along a smooth S 1 −action withoutfixed points cares an orbifold structure. The orbifold structure gives anatural stratification by open manifolds on its underlying space, where onestrata corresponds to a set of singular points of the same type.Note that one topological space can carry many different orbifold structures.For example, consider the orbifold O associated with a factor spaceof a 2−sphere S 2 along a rotation by π. It is homeomorphic to S 2 , butthe natural orbifold structure is different. It is possible to adopt most ofthe characteristics of manifolds to orbifolds and these characteristics areusually different from the correspondent characteristics of the underlyingspace. In the above example, its orbifold fundamental group of O is Z 2 andits orbifold Euler characteristic is 1.Manifold orbifolding denotes an operation of wrapping, or folding inthe case of mirrors, to superimpose all equivalent points on the originalmanifold – to get a new one.In string theory, the word orbifold has a new flavor. In physics, the


<strong>Geom</strong>etrical Path Integrals and Their Applications 1189notion of an orbifold usually describes an object that can be globally writtenas a coset M/G where M is a manifold (or a theory) and G is a group of itsisometries (or symmetries). In string theory, these symmetries do not needto have a geometric interpretation. The so–called orbifolding is a generalprocedure of string theory to derive a new string theory from an old stringtheory in which the elements of the group G have been identified with theidentity. Such a procedure reduces the number of string states because thestates must be invariant under G, but it also increases the number of statesbecause of the extra twisted sectors. The result is usually a new, perfectlysmooth string theory.Mirror SymmetryThe so–called mirror symmetry is a surprising relation that can exist betweentwo Calabi–Yau manifolds. It happens, usually for two such 6Dmanifolds, that the shapes may look very different geometrically, but neverthelessthey are equivalent if they are employed as hidden dimensions of a(super)string theory. More specifically, mirror symmetry relates two manifoldsM and W whose Hodge numbers h 1,1 = dim H 1,1 and h 1,2 = dim H 1,2are swapped; string theory compactified on these two manifolds leads toidentical physical phenomena (see [Greene (2000)]).Strominger showed in [Strominger (1990)] that mirror symmetry is aspecial example of the so–called T −duality: the Calabi–Yau manifold maybe written as a fiber bundle whose fiber is a 3D torus T 3 = S 1 × S 1 × S 1 .The simultaneous action of T −duality on all three dimensions of this torusis equivalent to mirror symmetry.Mirror symmetry allowed the physicists to calculate many quantitiesthat seemed virtually incalculable before, by invoking the ‘mirror’ descriptionof a given physical situation, which can be often much easier. Mirrorsymmetry has also become a very powerful tool in mathematics, and althoughmathematicians have proved many rigorous theorems based on thephysicists’ intuition, a full mathematical understanding of the phenomenonof mirror symmetry is still lacking.6.7.4 More on Topological Field TheoriesUnfortunately, there is no such thing as a crash course in string theory, butthe necessary background can be found in the classic two–volume monographs[Green et. al. (1987)] and [Polchinski (1998)]. A good introduction


1190 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionto conformal field theory is given in the lecture notes [Schellekens (1996)].The basics of topological string theory were laid out in a series of beautifulpapers by E. Witten in 1990s [Witten (1988b); Witten (1988a); Witten(1988d); Witten (1989); Witten (1990); Witten (1991); Witten (1992);Witten (1995a); Witten (1991)], and more or less completed in a seminalpaper [Bershadsky et. al. (1994)]. Reviews about topological string theory,usually also contain a discussion of topological field theory. The firstone of these is the review [Dijkgraaf et. al. (1991)] from the same period.However, if we want to dig even deeper, there is the 900–page book [Horiet. al. (2003)], which discusses topological string theory from the point ofview of mirror symmetry.In particular, there exists a mathematically rigorous, axiomatic definitionof topological field theories due to [Atiyah (1988a)]. Instead of givingthis definition, we will define topological field theory in a more physically–intuitive way, but as a result somewhat less rigorous way.Recall (from subsection 6.5.1 above) that the output of a QFT is givenby its observables: correlation functions of products of operators,〈O 1 (x 1 ) · · · O n (x n )〉 b . (6.302)Here, the O i (x) are physical operators of the theory. What one calls ‘physical’is part of the definition of the theory, but it is important to realize thatin general not all combinations of fields are viewed as physical operators.For example, in a gauge theory, we usually require the observables to arisefrom gauge–invariant operators. That is, Tr F would be one of the O i , butTr A or A itself would not.The subscript b in the above formula serves as a reminder that thecorrelation function is usually calculated in a certain background. That is,the definition of the theory may involve a choice of a Riemannian manifoldM on which the theory lives, it may involve choosing a metric on M, itmay involve choosing certain coupling constants, and so on.The definition of a topological field theory is now as follows. Supposethat we have a quantum field theory where the background choices involvea choice of manifold M and a choice of metric h on M. Then the theoryis called a topological field theory if the observables (6.302) do not dependon the choice of metric h. Let us stress that it is part of the definitionthat h is a background field – in particular, we do not integrate over hin the path integral. One may wonder what happens if, once we have atopological field theory, we do make the metric h dynamical and integrate


<strong>Geom</strong>etrical Path Integrals and Their Applications 1191over it. This is exactly what we will do once we start considering topologicalstring theories.Note that the word ‘topological’ in the definition may be somewhat ofa misnomer [Vonk (2005)]. The reason is that the above definition doesnot strictly imply that the observables depend only on the topology of M –there may be other background choices hidden in b on which they dependas well. For example, in the case of a complex manifold M, correlationfunctions will in general not only depend on the topology of M and itsmetric, but also on our specific way of combining the 2D real coordinateson M into d complex ones. This choice, a complex structure, is part ofthe background of the quantum field theory, and correlation functions in atopological field theory will in general still depend on it.If our quantum field theory has general coordinate invariance, as we willusually assume to be the case, then the above definition has an interestingconsequence. The reason is that in such a case we can do an arbitrarygeneral coordinate transformation, changing both the coordinates on Mand its metric, under which the correlation functions should be invariant.Then, using the topological invariance, we can transform back the metricto its old value. The combined effect is that we have only changed the x i in(6.302). That is, in a generally coordinate invariant topological field theory,the observables do not depend on the insertion points of the operators.Chern ClassesInspired by the identification of a connection with a gauge field, let usconsider the analogue of the non–Abelian field strength,F = dA − A ∧ A,where A ∧ A is a shorthand for A I Ji AJ Kj dxi ∧ dx j (note that A ∧ A ≠ 0).A short calculation shows that on the overlap of two patches of M (orequivalently, under a gauge transformation), this quantity transforms asF (b) = Λ (ba) F (a) Λ −1(ba) ,from which we see that F can be viewed as a section of a true vector bundleof Lie algebra valued 2–forms. In particular, we can take its trace and geta genuine 2–form:c 1 =i Tr(F ),2π


1192 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere the prefactor is convention. It can be seen that this 2–form is closed:d(Tr(F )) = Tr(dF ) = − Tr(d(A ∧ A)) = − Tr(dA ∧ A − A ∧ dA) = 0.Therefore, we can take its cohomology class, for which we would like toargue that it is a topological invariant. Note that this construction is independentof the choice of coordinates on M. Moreover, it is independent ofgauge transformations. However, on a general vector bundle there may beconnections which cannot be reached in this way from a given connection.Changing to such a connection is called a ‘large gauge transformation’, andfrom what we have said it is not clear a priori that the Chern classes do notdepend on this choice of equivalence class of connections. However, withsome work we can also prove this fact. The invariant [c 1 ] is called the firstChern class. In fact, it might be better to call it a ‘relative topologicalinvariant’: given a base manifold M of fixed topology, we can topologicallydistinguish vector bundles over it by calculating the above cohomologyclass.By taking the trace of F , we loose a lot of information. There turns outto be a lot more topological information in F , and it can be extracted byconsidering the expression(c(F ) = det 1 + iF ),2πwhere 1 is the identity matrix of the same size as the elements of the Liealgebra of G. Again, it can be checked that this expression is invariantunder a change of coordinates for M and under a change of connection.Since the matrix components inside the determinant consist of the 0–form1 and the 2–form F , expanding the determinant will lead to an expressionconsisting of forms of all even degrees. One writes this asc(F ) = c 0 (F ) + c 1 (F ) + c 2 (F ) + . . .The sum terminates either at the highest degree encountered in expandingthe determinant, or at the highest allowed even form on M. Note thatc 0 (F ) = 1, and [c 1 (F )] is exactly the first Chern class we defined above.The cohomology class of c n is called the nth Chern class.As an almost trivial example, let us consider the case of a productbundle M × W . In this case there is a global section g(x) of the principalbundle P , and we can use this to construct a connection A = −gdg −1 , so


<strong>Geom</strong>etrical Path Integrals and Their Applications 1193that [Vonk (2005)]F = −dgdg −1 − gdg −1 gdg −1 = −dgdg −1 + dgdg −1 = 0.Thus, for a trivial bundle with this connection, c 0 = 1 and c n = 0 for alln > 0.Chern–Simons TheoryThe easiest way to construct a topological field theory is to construct atheory where both the action S (or, quantum measure e iS ) and the fieldsdo not include the metric at all. Such topological field theories are called‘Schwarz–type’ topological field theories. This may sound like a trivialsolution to the problem, but nevertheless it can lead to quite interestingresults. To see this, let us consider familiar example: Chern–Simons gaugetheory on a 3D manifold M – now from a physical point of view.Recall from subsection 5.11.8 that Chern–Simons theory is a gauge theory– that is, it is constructed from a vector bundle E over the base spaceM, with a structure group (gauge group) G and a connection (gauge field)A. The Lagrangian of Chern–Simons theory is then given byL = Tr(A ∧ dA − 2 3 A ∧ A ∧ A).It is a straightforward exercise to check how this Lagrangian changes underthe gauge transformationand one finds˜L ≡ Tr(à ∧ dà − 2 3à ∧ à ∧ Ã)à = gAg −1 − gdg −1 ,= Tr(A ∧ dA − 2 3 A ∧ A ∧ A) − d Tr(gA ∧ dg−1 ) + 1 3 Tr(gdg−1 ∧ dg ∧ dg −1 ).The second term is a total derivative, so if M does not have a boundary,the action, being the integral of L over M, does not get a contribution fromthis term. The last term is not a total derivative, but its integral turns outto be a topological invariant of the map g(x), which is quantized as124π 2 ∫MTr(gdg −1 ∧ dg ∧ dg −1 ) = m ∈ Z.


1194 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFrom this, we see that if we define the action asS = k ∫L,4πwith k an integer, the action changes by 2πkm under gauge transformations,and the quantum measure e iS is invariant.So from this discussion we seem to arrive at the conclusion that thepartition function∫Z = D[A] e iS[A]for a line bundle of a fixed topology E is a topological invariant of M,as are the correlation functions of gauge–invariant operators such as Tr F .However, there is one more detail we have to worry about: there may be ananomaly in the quantum theory. That is, it may not be possible to definethe path integral measure D[A] in a gauge–invariant way.One way to see what problems can arise is to note that to actuallycalculate the path integral, one has to pick a gauge condition on A. Thatis, we have to pick one representative of A in each equivalence class undergauge transformations. To make such a choice will in general require achoice of metric. For example, from electromagnetism (where E is a 1Dcomplex line bundle and G = U(1)) we know that a useful gauge is theFeynman gauge, in which the equation of motion for A becomesM∆A = 0.As we have seen before, the Laplacian ∆ is an operator which, throughthe Hodge star, depends on the metric, and hence the results we find willa priori be metric dependent. To show that the results are truly metricindependent, one needs to show that the quantum results do not dependon our arbitrary choice of gauge.We will not go into the details of this, but state that one can show thatChern–Simons theory on a compact 3–manifold is anomaly–free, so ournaive argument above was correct, and one can indeed calculate topologicalinvariants of M in this way.Let us briefly discuss the kind of topological invariants that Chern–Simons theory can lead to. Recall that one can construct a Lie groupelement g from a Lie algebra element A as g = e A . Now suppose we have apath γ(t) inside M. Suppose that we chop up γ into very small line elementsgiven by tangent vectors ˙γδt. Then we can insert this tangent vector into


<strong>Geom</strong>etrical Path Integrals and Their Applications 1195the connection 1–form A, and get a Lie algebra element. As we have seen,it is precisely this Lie algebra element which transports vectors in E alongthis small distance: we have to multiply these vectors by 1 + A. This is alinear approximation to the finite transformation e A . So if we transport avector along the entire closed curve γ, it will return multiplied by a groupelementg = limδt→0[exp (A ( ˙γ(0)) δt) exp (A ( ˙γ(δt)) δt) exp (A ( ˙γ(2δt)) δt) · · · ] .Now it is tempting to add all the exponents and write their sum in thelimit as an integral, but this is not quite allowed since the different groupelements may not commute, so e X e Y ≠ e X+Y . Therefore, one uses thefollowing notation,g = P exp∫A,γwhere P stands for path ordering, while the element g is called the holonomyof A around the closed curve γ. An interesting gauge and metricindependent object turns out to be the trace of this group element. Thistrace is called the Wilson loop W γ (A), given by∫W γ (A) = Tr(P ) exp A.The topological invariants we are interested in are now the correlation functionsof such Wilson loops in Chern–Simons theory. Since these correlationfunctions are independent of the parametrization of M, we can equivalentlysay that they will be independent of the precise location of the loop γ; wehave in fact constructed a topological invariant of the embedding of γ insideM. This embedding takes the shape of a knot, so the invariants wehave constructed are knot invariants. One can show that the invariants areactually polynomials in the variable y = exp 2πi/(k + 2), where k is theinteger ‘coupling constant’ of the Chern–Simons theory.The above construction is due to E. Witten, and was carried out in[Witten (1988a)]. Before Witten’s work, several polynomial invariants ofknots were known, one of the simplest ones being the so–called Jones polynomial.It can be shown that many of these polynomials arise as specialcases of the above construction, where one takes a certain structure groupG, SU(2) for the Jones polynomial, and a certain vector bundle (representation)E, the fundamental representation for the Jones polynomial. Thatis, using this ‘trivial’ topological field theory, Witten was able to reproduceγ


1196 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiona large number of the known knot invariants in a unified framework, andconstruct a great number of new invariants as well [Vonk (2005)].Cohomological Field TheoriesEven though the above example leads to quite interesting topological invariants,the construction itself is somewhat trivial: given the absence ofanomalies that we mentioned, the independence of the metric is completelymanifest throughout the procedure. There exists a different way of constructingtopological field theories in which the definition of the theorydoes use a metric, but one can still show that the partition function and thephysical correlation functions of the theory are metric–independent. Thetheories constructed in this way are called topological theories of ‘Witten–type’, or cohomological field theories.Cohomological field theories are field theories that possess a very specialtype of symmetry. Recall that from Noether’s Theorem a global symmetryof a theory leads to a conserved charge S, and that after quantizing thetheory, the symmetry is generated by the corresponding operator:δ ɛ O i = iɛ[S, O i ], or δ ɛ O i = iɛ{S, O i },depending on whether S and O i are fermionic or bosonic. Furthermore, thesymmetry–invariant states |j〉 satisfyS |j〉 = 0. (6.303)In particular, if the symmetry is not spontaneously broken, the vacuum ofsuch a theory will be symmetric, i.e., S |0〉 = 0, and expectation values ofoperators will be unchanged after a symmetry transformation:〈0|O i + δO i |0〉 = 〈0|O i |0〉 + iɛ〈0|SO i ± O i S|0〉 = 〈0|O i |0〉,since S annihilates the vacuum. Note that, to linear order in a ‘smallparameter’ ɛ, the second term in the first line (with |0〉 replaced by anarbitrary state |ψ〉) can also be obtained if instead of on the operators, welet the symmetry operator S act on the state as|ψ〉 → |ψ〉 + iɛS|ψ〉. (6.304)In case S is the Hamiltonian, this is the infinitesimal version of the well–known transition between the Schrödinger and Heisenberg pictures. Note


<strong>Geom</strong>etrical Path Integrals and Their Applications 1197that the equations (6.303) and (6.304) already contain some flavor of cohomology.Cohomological field theories are theories where this analogy canbe made exact.With this in mind, the first property of a cohomological field theory willnot come as a surprise: it should contain a fermionic symmetry operator Qwhich squares to zero, Q 2 = 0. This may seem like a strange requirementfor a field theory, but symmetries of this type occur for example when wehave a gauge symmetry and fix it by using the Faddeev–Popov procedure;the resulting theory will then have a global BRST symmetry, which satisfiesprecisely this constraint. Another example is found in supersymmetry,where one also encounters symmetry operators that square to zero, as wewill see in detail later on.The second property a cohomological field theory should have is reallya definition: we define the physical operators in this theory to be theoperators that are closed under the action of this Q−operator 28 :{Q, O i } = 0. (6.305)Again, this may seem to be a strange requirement for a physical theory,but again it naturally appears in BRST quantization, and for examplein conformal field theories, where we have a 1–1 correspondence betweenoperators and states. In such theories, the symmetry requirement (6.303)on the states translates into the requirement (6.305) on the operators.Thirdly, we want to have a theory in which the Q−symmetry is notspontaneously broken, so the vacuum is symmetric. Note that this impliesthe equivalenceO i ∼ O i + {Q, Λ}. (6.306)The reason for this is that the expectation value of an operator productinvolving a Q−exact operator {Q, Λ} takes the form〈0|O i1 · · · O ij {Q, Λ}O ij+1 · · · O in |0〉 = 〈0|O i1 · · · O ij (QΛ−ΛQ)O ij+1 · · · O in |0〉,and each term vanishes separately, e.g.,〈0|O i1 · · · O ij QΛO ij+1 · · · O in |0〉 = ±〈0|O i1 · · · QO ij ΛO ij+1 · · · O in |0〉 (6.307)= ±〈0|QO i1 · · · O ij ΛO ij+1 · · · O in |0〉 = 0,28 From now on, we denote both the commutator and the anti-commutator by curlybrackets, unless it is clear that one of the operators inside the brackets is bosonic.


1198 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere we made repeated use of (6.305). Together, (6.305) and (6.306) meanthat our physical operators are Q−cohomology classes.The fourth and final requirement for a cohomological field theory is thatthe metric SEM–tensor is Q−exact,T αβ ≡δSδh αβ = {Q, G αβ} (6.308)for some operator G αβ . The physical interpretation of this is the following.The integrals of the components T 0α over a space–like hypersurface are conservedquantities. For example, the integral of T 00 gives the Hamiltonian:∫H = T 00 .spaceSimilarly, T 0a for a ≠ 0 give the momentum charges. Certainly, the HamiltonianH should commute with all symmetry operators of the theory, andone usually takes the other space–time symmetries to commute with theinternal symmetries as well. The choice of the first lower index 0 here is relatedto a choice of Lorentz frame, so in general the integrals of all T αβ willcommute with the internal symmetries. In a local theory, it is then naturalto assume that also the densities commute with Q. However, (6.308) is aneven stronger requirement: the SEM densities should not only commutewith Q (that is, be Q−closed), but they have to do so in a trivial way(they should be Q−commutators, that is, Q−exact) for the theory to becohomological.This fourth requirement is the crucial one in showing the topologicalinvariance of the theory. Let us consider the functional h αβ −derivative ofan observable:)Dφ O i1 · · · O in e iS[φ]∫= iδδh αβ 〈O i 1· · · O in 〉 =δ (∫δh αβδSDφ O i1 · · · O inδh αβ eiS[φ] = i〈O i1 · · · O in {Q, G αβ }〉 = 0,where in the last line we used the same argument as in (6.307). One mightbe worried about the operator ordering in going from the operator formalismto the path integral formalism and back. We should really have inserteda time ordering operator on the r.h.s. in the first step above. However, theresult then shows us that we can arbitrarily change the metric – and hencethe time ordering of the operators, so with hindsight we may actually thinkof the operators as being arbitrarily ordered. Finally, we have assumed thatour operators do not depend explicitly on the metric.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1199A very practical way to ensure (6.308) is to use a Lagrangian whichitself is Q−exact, L = {Q, V }, for some operator V . This choice has anextra virtue, which we can see if we explicitly include Planck’s constant inour description: the quantum measure then readsexp i {Q,Then, we can use exactly the same argument as before to show that∫MV}.dd 〈O i 1 · · · O in 〉 = 0.That is, the correlators we are interested in are independent of , and wecan therefore calculate them exactly in the classical limit.Descent equationsAn important property of cohomological field theories is that, given a scalarphysical operator on M – where by ‘scalar’ we mean an operator that doesnot transform under coordinate transformations of M, so in particular ithas no α−indices – we can construct further operators which behave likep−forms on M. The basic observation is that we can integrate (6.308) overa spatial hypersurface to get a similar relation for the momentum operators:P α = {Q, G α },where G α is a fermionic operator. Now consider the operatorO (1)α = i{G α , O (0) },where O (0) (x) is a scalar physical operator: {Q, O (0) (x)} = 0.calculateddx α O(0) = i[P α , O (0) ] = [{Q, G α }, O (0) ]= ±i{{G α , O (0) }, Q} − i{{O (0) , Q}, G α } = {Q, O (1)α }.Let usIn going from the second to the third line, we have used the Jacobi identity.The first sign in the third line depends on whether O (0) is bosonic orfermionic, but there is no sign ambiguity in the last line. By defining the1–form operatorO (1) = O α(1) dx α


1200 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwe can write this result asdO (0) = {Q, O (1) }.Then, we can integrate this equation over a closed curve γ ⊂ M to find∫{Q, O (1) } = 0.γThat is, by constructing a ∫ γ O(1) for each O (0) , we have found a wholeclass of new, non–local, physical operators.The above procedure can now be repeated in exactly the same way startingfrom O (1) , and doing this we find a whole tower of p−form operators:{Q, O (0) } = 0, {Q, O (1) } = dO (0) , {Q, O (2) } = dO (1) ,· · · , {Q, O (n) } = dO (n−1) , 0 = dO (n) .The last equation is trivial, since there are no (n + 1)−forms on an nDsmooth manifold.Following the same reasoning, the integral of O (p) over a pD submanifoldof M is now a physical operator. This gives us a large class of new physicaloperators, starting from the scalar ones. Note that these operators, beingintegrated over a submanifold, are inherently nonlocal. Nevertheless, theycan have a very physical interpretation. Particularly important examplesof this are the ‘top–form’ operators O (n) that can be integrated over thewhole manifold, leading to{ ∫ }Q, O (n) = 0.MThis implies that we can add terms t a O (n)a , with t a arbitrary couplingconstants, to our Lagrangian without spoiling the fact that the theory iscohomological. These deformations of the theory will be important to uslater.2D Cohomological Field TheoriesSince string theories are 2D field theories, we will in particular be interestedin cohomological field theories in two dimensions (see, e.g., [Witten(1995b)]). These theories have some extra properties which will be importantin our discussion. Let us begin by reminding the reader of the relationbetween states in the operator formalism of quantum field theories, and


<strong>Geom</strong>etrical Path Integrals and Their Applications 1201boundary conditions in the path–integral formalism. In its simplest form,this relation looks like∫ BC2BC1Dφ · · · e iS[φ] = 〈BC1| T (· · · ) |BC2〉. (6.309)Here, we included a time–ordering operator for completeness, but as wehave stated before, for the theories we are interested in this ordering isirrelevant. The notation |BCi〉 indicates the state corresponding to theincoming or outgoing boundary condition. For example, if on the pathintegral side we prescribe all fields at a certain initial time, φ(t = t 1 ) = f(t 1 ), on the operator side this corresponds to the incoming state satisfyingφ(t 1 )|BC1〉 = f(t 1 )|BC1〉. (6.310)where on the l.h.s. we have an operator acting, but on the r.h.s. there isa simple scalar multiplication. More generally, in the operator formalismwe can have linear combinations of states of the type (6.310). Therefore,we should allow for linear combinations on the l.h.s. of (6.309) as well. Inother words, in the path–integral formalism, a state is an operator whichadds a number (a weight) to each possible boundary condition on the fields.From this point of view, the states in (6.310) are like ‘delta–functionals’:they assign weight 1 to the boundary condition φ(t 1 ) = f(t 1 ), and weight0 to all other boundary conditions.Now, let us specialize to 2D field theories. Here, the boundary of acompact manifold is a set of circles. Let us for simplicity assume that the‘incoming’ boundary consists of a single circle. We can now define a statein the above sense by doing a path integral over a second surface withthe topology of a hemisphere. This path integral gives a number for everyboundary condition of the fields on the circle, and this is exactly whata state in the path–integral formalism should do. In particular, one canuse this procedure to define a state corresponding to every operator O aby inserting O a on the hemisphere and then stretching this hemisphere toinfinite size. An expectation value in the operator formalism, such as〈O a |O b (x 2 )O c (x 3 )|O d 〉 cyl , (6.311)on a cylinder of finite length, can then schematically be drawn as in Figure6.24. Here, instead of first doing the path integrals over the semi–infinitehemispheres and inserting the result in the path integral over the cylinder,one can just as well integrate over the whole surface at once. However, notethat in topological field theories, there is no need to do the stretching, since


1202 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionthe path integral only depends on the topology of the surface, and hencenot on the size of the hemisphere [Vonk (2005)].Fig. 6.24 A graphical representation of the correlation function (6.311).We assume that all states that we are interested in are of the above form,which in particular means that we will integrate only over Riemann surfaceswithout boundary. Moreover, we assume these surfaces to be orientable.An important property of topological field theories in two dimensions isnow that its correlation functions factorize in the following way:〈O 1 · · · O n 〉 Σ = ∑ a,b〈O 1 · · · O i O a 〉 Σ1 η ab 〈O b O i+1 · · · O n 〉 Σ2 , (6.312)where the genus of Σ is the sum of the genera of Σ 1 and Σ 2 . This statementis explained in Figure 6.25. By using the topological invariance, we candeform a Riemann surface Σ with a set of operator insertions in such a waythat it develops a long tube. From general quantum field theory, we knowthat if we stretch this tube long enough, only the asymptotic states – thatis, the states in the physical part of the Hilbert space – will propagate. Butas we have just argued, instead of inserting these asymptotic states, we mayjust as well insert the corresponding operators at a finite distance. However,to conclude that this leads to (6.312), we have to show that this definitionof ‘physical states’ – being the ones that need to be inserted as asymptoticstates – agrees with our previous definition in terms of Q−cohomology. Letus argue that it does by first writing the factorization as〈O 1 · · · O n 〉 Σ = ∑ A,B〈O 1 · · · O i |O A 〉 Σ1 η AB 〈O B |O i+1 · · · O n 〉 Σ2 . (6.313)where the O A with capital index now correspond to a complete basis ofasymptotic states in the Hilbert space. The reader may be more used tothis type of expressions in the case where η AB = δ AB , but since we havenot shown that with our definitions 〈O A |O B 〉 = δ AB , we have to work withthis more general form of the identity operator, where η is a metric thatwe will determine in a moment.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1203Now, we can write the Hilbert space as a direct sum, H = H 0 ⊕ H 1 ,where H 0 consists of states |ψ〉 for which Q|ψ〉 = 0 and H 1 is its orthogonalcomplement. SinceQ (O 1 . . . O i |0〉) = 0, (6.314)the states in H 1 are in particular orthogonal to states of the formO 1 . . . O i |0〉, and hence the states in H 1 do not contribute to the sum in(6.313). Moreover, changing O A to O A + {Q, Λ} does not change the resultin (6.313), so we only need to sum over a basis of H 0 /I({Q, ·}), which isexactly the space H phys of ‘topologically physical’ states.Fig. 6.25 A correlation function on a Riemann surface factorizes into correlation functionson two Riemann surfaces of lower genus (see text for explanation).Finally, let us determine the metric η ab . We can deduce its form byfactorizing the 2–point functionin the above way, resulting inC ab = 〈O a O b 〉 (6.315)C ab = C ac η cd C db . (6.316)In other words, we find that the metric η ab is the matrix inverse of the2–point function C ab , which for this reason we will write as η ab from nowon.


1204 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionOne can apply a similar procedure to ‘cut open’ internal loops in aRiemann surface Σ, so that we get〈O 1 · · · O n 〉 Σ = (−1) Fa η ab ∑ a,b〈O a O b O 1 · · · O n 〉 Σ ′, (6.317)where the genus of Σ ′ is one less than the genus of Σ. The factor (−1) Famultiplies the expression on the r.h.s. by -1 if the inserted operator O a(and hence also O b ) is fermionic. Proving that it needs to be included isnot straightforward, but one can think of it as the ‘stringy version’ of thewell–known statement from quantum field theory that fermion loops addan extra minus sign to a Feynman diagram.The reader should convince himself that together, the equations (6.312)and (6.317) imply that we can reduce all n−point correlation functions toproducts of 3–point functions on the sphere. We denote these importantquantities byc abc ≡ 〈O a O b O c 〉 0 , (6.318)where the label 0 denotes the genus of the sphere. By using (6.312) toseparate two insertion points on a sphere, we see that〈· · · O a O b · · · 〉 Σ = ∑ c,d〈· · · O c · · · 〉 Σ η cd c abd = ∑ c〈· · · O c · · · 〉 Σ c ab c ,where we raised an index of c abc with the metric η. We can view the aboveresult as the definition of an operator algebra with structure constants c ab c :O a O b = ∑ cc ab c O c . (6.319)From the metric η ab and the structure constants c ab c , we can now calculateany correlation function in the cohomological field theory.6.7.5 Topological StringsThe 2D field theories we have constructed are already very similar to stringtheories. However, one ingredient from string theory is missing: in stringtheory, the world–sheet theory does not only involve a path integral overthe maps φ i to the target space and their fermionic partners, but also apath integral over the world–sheet metric h αβ . So far, we have set thismetric to a fixed background value.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1205We have also encountered a drawback of our construction. Even thoughthe theories we have found can give us some interesting ‘semi–topological’information about the target spaces, one would like to be able to definegeneral nonzero n−point functions at genus g instead of just the partitionfunction at genus one and the particular correlation functions we calculatedat genus zero.It turns out that these two remarks are intimately related. In this sectionwe will go from topological field theory to topological string theory byintroducing integrals over all metrics, and in doing so we will find interestingnonzero correlation functions at any genus (see [Vonk (2005)]).Coupling to Topological GravityIn coupling an ordinary field theory to gravity, one has to perform thefollowing three steps.• First of all, one rewrites the Lagrangian of the theory in a covariantway by replacing all the flat metrics by the dynamical ones, introducingcovariant derivatives and multiplying the measure by a factor of √ det h.• Secondly, one introduces an Einstein–Hilbert term as the ‘kinetic’ termfor the metric field, plus possibly extra terms and fields to preserve thesymmetries of the original Lagrangian.• Finally, one has to integrate the resulting theory over the space of allmetrics.Here we will not discuss the first two steps in this procedure. As wehave seen in our discussion of topological field theories, the precise form ofthe Lagrangian only plays a comparatively minor role in determining theproperties of the theory, and we can derive many results without actuallyconsidering a Lagrangian. Therefore, let us just state that it is possible tocarry out the analog of the first two steps mentioned above, and construct aLagrangian with a ‘dynamical’ metric which still possesses the topologicalQ−symmetry we have constructed. The reader who is interested in thedetails of this construction is referred to the paper [Witten (1990)] and tothe lecture notes [Dijkgraaf et. al. (1991)].The third step, integrating over the space of all metrics, is the one wewill be most interested in here. Naively, by the metric independence of ourtheories, integrating their partition functions over the space of all metrics,and then dividing the results by the volume of the topological ‘gauge group’,


1206 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwould be equivalent to multiplication by a factor of 1,?Z[h 0 ] = 1 ∫D[h] Z[h], (6.320)G topfor any arbitrary background metric h 0 . There are several reasons why thisnaive reasoning might go wrong:• There may be metric configurations which cannot be reached from agiven metric by continuous changes.• There may be anomalies in the topological symmetry at the quantumlevel preventing the conclusion that all gauge fixed configurations areequivalent.• The volume of G top is infinite, so even if we could rigorously definea path integral the above multiplication and division would not bemathematically well–defined.For these reasons, we should really be more careful and precisely definewhat we mean by the ‘integral over the space of all metrics’. Let us notethe important fact that just like in ordinary string theory (and even beforetwisting), the 2D sigma models become conformal field theories when weinclude the metric in the Lagrangian. This means that we can borrow thetechnology from string theory to integrate over all conformally equivalentmetrics. As is well known, and as we will discuss in more detail later, theconformal symmetry group is a huge group, and integrating over conformallyequivalent metrics leaves only a nD integral over a set of world–sheetmoduli. Therefore, our strategy will be to use the analogy to ordinary stringtheory to first do this integral over all conformally equivalent metrics, andthen perform the integral over the remaining nD moduli space.In integrating over conformally equivalent metrics, one usually has toworry about conformal anomalies. However, here a very important factbecomes our help. To understand this fact, it is useful to rewrite ourtwisting procedure in a somewhat different language (see [Vonk (2005)]).Let us consider the SEM–tensor T αβ , which is the conserved Noethercurrent with respect to global translations on C. From conformal fieldtheory, it is known that T z¯z = T¯zz = 0, and the fact that T is a conservedcurrent, ∂ α T α β = 0, means that T zz ≡ T (z) and T¯z¯z ≡ ¯T (¯z) are (anti–)holomorphic in z. One can now expand T (z) in Laurent modes,T (z) = ∑ L m z −m−2 . (6.321)


<strong>Geom</strong>etrical Path Integrals and Their Applications 1207The L m are called the Virasoro generators, and it is a well–known resultfrom conformal field theory that in the quantum theory their commutationrelations are[L m , L n ] = (m − n)L m+n + c12 m(m2 − 1)δ m+n .The number c depends on the details of the theory under consideration,and it is called the central charge. When this central charge is nonzero, oneruns into a technical problem. The reason for this is that the equation ofmotion for the metric field readsδSδh αβ = T αβ = 0.In conformal field theory, one imposes this equation as a constraint in thequantum theory. That is, one requires that for physical states |ψ〉,L m |ψ〉 = 0(for all m ∈ Z).However, this is clearly incompatible with the above commutation relationunless c = 0. In string theory, this value for c can be achieved by takingthe target space of the theory to be 10D. If c ≠ 0 the quantum theory isproblematic to define, and we speak of a ‘conformal anomaly’ [Vonk (2005)].The whole above story repeats itself for ¯T (¯z) and its modes ¯L m . Atthis point there is a crucial difference between open and closed strings.On an open string, left–moving and right–moving vibrations are relatedin such a way that they combine into standing waves. In our complexnotation, ‘left–moving’ translates into ‘z−dependent’ (i.e. holomorphic),and ‘right–moving’ into ‘¯z−dependent’ (i.e. anti–holomorphic). Thus, on anopen string all holomorphic quantities are related to their anti–holomorphiccounterparts. In particular, T (z) and ¯T (¯z), and their modes L m and ¯L m ,turn out to be complex conjugates. There is therefore only one independentalgebra of Virasoro generators L m .On a closed string on the other hand, which is the situation we have beenstudying so far, left– and right–moving waves are completely independent.This means that all holomorphic and anti–holomorphic quantities, and inparticular T (z) and ¯T (¯z), are independent. One therefore has two sets ofVirasoro generators, L m and ¯L m .Let us now analyze the problem of central charge in the twisted theories.To twist the theory, we have used the U(1)−symmetries. Any globalU(1)−symmetry of our theory has a conserved current J α . The fact thatit is conserved again means that J z ≡ J(z) is holomorphic and J¯z ≡ ¯J(¯z)


1208 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionis anti–holomorphic. Once again, on an open string J and ¯J will be related,but in the closed string theory we are studying they will be independentfunctions. In particular, this means that we can view a globalU(1)−symmetry as really consisting of two independent, left– and right–moving, U(1)−symmetries, with generators F L and F R .Note that the sum of U(1)−symmetries F V + F A only acts on objectswith a + index. That is, it acts purely on left–moving quantities. Similarly,F V − F A acts purely on right–moving quantities. From our discussionabove, it is therefore natural to identify these two symmetries with the twocomponents of a single global U(1) symmetry:F V = 1 2 (F L + F R ) F A = 1 2 (F L − F R ).A more detailed construction shows that this can indeed be done.Let us expand the left–moving conserved U(1)−current into Laurentmodes,J(z) = ∑ J m z −m−1 . (6.322)The commutation relations of these modes with one another and with theVirasoro modes can be calculated, either by writing down all of the modesin terms of the fields of the theory, or by using more abstract knowledgefrom the theory of superconformal symmetry algebras. In either case, onefinds[L m , L n ] = (m − n)L m+n + c12 m(m2 − 1)δ m+n [L m , J n ]= −nJ m+n [J m , J n ] = c 3 mδ m+n.Note that the same central charge c appears in the J− and in theL−commu-tators. This turns out to be crucial.Following the standard Noether procedure, we can now construct a conservedcharge by integrating the conserved current J(z) over a space–likeslice of the z−plane. In string theory, the physical time direction is theradial direction in the z−plane, so a space–like slice is just a curve aroundthe origin. The integral is therefore calculated using the Cauchy Theorem,∮F L = J(z)dz = 2πiJ 0 .z=0In the quantum theory, it will be this operator that generates theU(1) L −symmetry. Now recall that to twist the theory we want to introduce


<strong>Geom</strong>etrical Path Integrals and Their Applications 1209new Lorentz rotation generators,M A = M − F V = M − 1 2 (F L + F R )M B = M − F A = M − 1 2 (F L − F R ).A well–known result from string theory (see [Vonk (2005)]) is that thegenerator of Lorentz rotations is M = 2πi(L 0 − ¯L 0 ). Therefore, we findthat the twisting procedure in this new language amounts toA : L 0,A = L 0 − 1 2 J 0, ¯L0,A = ¯L 0 + 1 2 ¯J 0 ,B : L 0,B = L 0 − 1 2 J 0, ¯L0,B = ¯L 0 − 1 2 ¯J 0 .Let us now focus on the left–moving sector; we see that for both twistingsthe new Lorentz rotation generator is the difference of L 0 and 1 2 J 0. Thenew Lorentz generator should also correspond to a conserved 2–tensor, andfrom (6.321) and (6.322) there is a very natural way to get such a current:which clearly satisfies ¯∂ ˜T = 0 and˜T (z) = T (z) + 1 ∂J(z), (6.323)2˜L m = L m − 1 2 (m + 1)J m, (6.324)so in particular we find that ˜L 0 can serve as L 0,A or L 0,B . We should applythe same procedure (with a minus sign in the A−model case) in the right–moving sector. Equations (6.323) and (6.324) tell us how to implementthe twisting procedure not only on the conserved charges, but on the wholeN = 2 superconformal algebra – or at least on the part consisting of the J−and L−modes, but a further investigation shows that this is the only partthat changes. We have motivated, but not rigorously derived (6.323); for acomplete justification the reader is referred to the original papers [Lercheet. al. (1989)] and [Cecotti and Vafa (1991)].Now, we come to the crucial point. The algebra that the new modes˜L m satisfy can be directly calculated from (6.323), and we find[˜L m , ˜L n ] = (m − n)˜L m+n .That is, there is no central charge left. This means that we do not haveany restriction on the dimension of the theory, and topological strings willactually be well–defined in target spaces of any dimension.


1210 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionFrom this result, we see that we can integrate our partition functionover conformally equivalent metrics without having to worry about theconformal anomaly represented by the nonzero central charge. After havingintegrated over this large part of the space of all metrics, it turns out thatthere is a nD integral left to do. In particular, it is known that one canalways find a conformal transformation which in the neighborhood of achosen point puts the metric in the form h αβ = η αβ , with η the usual flatmetric with diagonal entries ±1. (Or, +1 in the Euclidean setting.) Onthe other hand, when one considers the global situation, it turns out thatone cannot always enforce this gauge condition everywhere. For example,if the world–sheet is a torus, there is a left–over complex parameter τ thatcannot be gauged away. The easiest way to visualize this parameter (see[Vonk (2005)]) is by drawing the resulting torus in the complex plane andrescaling it in such a way that one of its edges runs from 0 to 1; the otheredge then runs from 0 to τ, see Figure 6.26. It seems intuitively clear that aconformal transformation – which should leave all angles fixed – will neverdeform τ, and even though intuition often fails when considering conformalmappings, in this case this can indeed be proven. Thus, τ is really a modularparameter which we need to integrate over. Another fairly intuitive resultis that any locally flat torus can, after a rescaling, be drawn in this form,so τ indeed is the only modulus of the torus.Fig. 6.26 The only modulus τ of a torus T 2 .More generally, one can show that a Riemann surface of genus g hasm g = 3(g − 1) complex modular parameters. As usual, this is the virtualdimension of the moduli space. If g > 1, one can show that this virtualdimension equals the actual dimension. For g = 0, the sphere, we have anegative virtual dimension m g = −3, but the actual dimension is 0: there


<strong>Geom</strong>etrical Path Integrals and Their Applications 1211is always a flat metric on a surface which is topologically a sphere (justconsider the sphere as a plane with a point added at infinity), and afterhaving chosen this metric there are no remaining parameters such as τ inthe torus case. For g = 1, the virtual dimension is m g = 0, but as we haveseen the actual dimension is 1.We can explain these discrepancies using the fact that, after we haveused the conformal invariance to fix the metric to be flat, the sphere and thetorus have leftover symmetries. In the case of the sphere, it is well known instring theory that one can use these extra symmetries to fix the positionsof three labelled points. In the case of the torus, after fixing the metricto be flat we still have rigid translations of the torus left, which we canuse to fix the position of a single labelled point. To see how this leads to adifference between the virtual and the actual dimensions, let us for exampleconsider tori with n labelled points on them. Since the virtual dimension ofthe moduli space of tori without labelled points is 0, the virtual dimensionof the moduli space of tori with n labelled points is n. One may expectthat at some point (and in fact, this happens already when n = 1), onereaches a sufficiently generic situation where the virtual dimension reallyis the actual dimension. However, even in this case we can fix one of thepositions using the remaining conformal (translational) symmetry, so thepositions of the points only represent n − 1 moduli. Hence, there must bean nth modulus of a different kind, which is exactly the shape parameterτ that we have encountered above. In the limiting case where n = 0, thisparameter survives, thus causing the difference between the virtual and thereal dimension of the moduli space.For the sphere, the reasoning is somewhat more formal: we analogouslyexpect to have three ‘extra’ moduli when n = 0. In fact, three extraparameters are present, but they do not show up as moduli. They must beviewed as the three parameters which need to be added to the problem tofind a 0D moduli space.Since the cases g = 0, 1 are thus somewhat special, let us begin bystudying the theory on a Riemann surface with g > 1. To arrive at thetopological string correlation functions, after gauge fixing we have to integrateover the remaining moduli space of complex dimension 3(g − 1). Todo this, we need to fix a measure on this moduli space. That is, given aset of 6(g − 1) tangent vectors to the moduli space, we want to produce anumber which represents the size of the volume element spanned by thesevectors, see Figure 6.27. We should do this in a way which is invariant undercoordinate redefinitions of both the moduli space and the world–sheet.


1212 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIs there a ‘natural’ way to do this?Fig. 6.27 A measure on the moduli space M assigns a number to every set of threetangent vectors. This number is interpreted as the volume of the element spanned bythese vectors.To answer this question, let us first ask how we can describe the tangentvectors to the moduli space (see [Vonk (2005)]). In two dimensions,conformal transformations are equivalent to holomorphic transformations:z ↦→ f(z). It thus seems natural to assume that the moduli space we haveleft labels different complex structures on Σ, and indeed this can be shownto be the case. Therefore, a tangent vector to the moduli space is aninfinitesimal change of complex structure, and these changes can be parameterizedby holomorphic 1–forms with anti–holomorphic vector indices,dz ↦→ dz + ɛµ z¯z(z)d¯z.The dimension counting above tells us that there are 3(g − 1) independent(µ i ) z¯z, plus their 3(g − 1) complex conjugates which change d¯z. So thetangent space is spanned by these µ i (z, ¯z), ¯µ i (z, ¯z). How do we get a numberout of a set of these objects? Since µ i has a z and a ¯z index, it seems naturalto integrate it over Σ. However, the z−index is an upper index, so we needto lower it first with some tensor with two z−indices. It turns out thata good choice is to use the Q−partner G zz of the SEM–tensor componentT zz , and thus to define the integration over moduli space as3g=3∏∫M gi=1(dm i d ¯m i ∫G zz (µ i ) z¯zΣ∫G¯z¯z (¯µ )¯z i zΣ). (6.325)Note that by construction, this integral is also invariant under a change ofbasis of the moduli space. There are several reasons why using G zz is anatural choice. First of all, this choice is analogous to what one does inbosonic string theory. There, one integrates over the moduli space using


<strong>Geom</strong>etrical Path Integrals and Their Applications 1213exactly the same formula, but with G replaced by the conformal ghost b.This ghost is the BRST–partner of the SEM–tensor in exactly the sameway as G is the Q−partner of T . Secondly, one can make the not unrelatedobservation that since {Q, G} = T, we can still use the standard argumentsto show independence of the theory of the parameters in a Lagrangian ofthe form L = {Q, V }. The only difference is that now we also have tocommute Q through G to make it act on the vacuum, but since T αβ itself isthe derivative of the action with respect to the metric h αβ , the terms we getin this way amount to integrating a total derivative over the moduli space.Therefore, apart from possible boundary terms these contributions vanish.Note that this reasoning also gives us an argument for using G zz insteadof T zz (which is more or less the only other reasonable option) in (6.325):if we had chosen T zz then all path integrals would have been over totalderivatives on the moduli space, and apart from boundary contributionsthe whole theory would have become trivial.If we consider the vector and axial charges of the full path integralmeasure, including the new path integral over the world–sheet metric h,we find a surprising result. Since the world–sheet metric does not transformunder R−symmetry, naively one might expect that its measure doesnot either. However, this is clearly not correct since one should also takeinto account the explicit G−insertions in (6.325) that do transform underR−symmetry. From the N = 2 superconformal algebra (or, more down–to–earth, from expressing the operators in terms of the fields), it followsthat the product of G and Ḡ has vector charge zero and axial charge 2.Therefore, the total vector charge of the measure remains zero, and theaxial charge gets an extra contribution of 6(g − 1), so we find a total axialR−charge of 6(g − 1) − 2m(g − 1). From this, we see that the case ofcomplex target space dimension 3 is very special: here, the axial charge ofthe measure vanishes for any g, and hence the partition function is nonzeroat every genus. If m > 3 and g > 1, the total axial charge of the measureis negative, and we have seen that we cannot cancel such a charge withlocal operators. Therefore, for these theories only the partition function atg = 1 and a specific set of correlation functions at genus zero give nonzeroresults. Moreover, for m = 2 and m = 1, the results can be shown to betrivial by other arguments. Therefore, a Calabi–Yau threefold is by far themost interesting target space for a topological string theory. It is a ‘happycoincidence’ (see [Vonk (2005)]) that this is exactly the dimension we aremost interested in from the string theory perspective.Finally, let us come back to the special cases of genus 0 and 1. At genus


1214 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionzero, the Riemann surface has a single point as its moduli space, so there areno extra integrals or G−insertions to worry about. Therefore, we can copythe topological field theory result saying that we have to introduce localoperators with total degree (m, m) in the theory. The only remnant of thefact that we are integrating over metrics is that we should also somehow fixthe remaining three symmetries of the sphere. The most straightforwardway to do this is to consider 3–point functions with insertions on threelabelled points. As a gauge choice, we can then for example require thesepoints to be at the points 0, 1 and ∞ in the compactified complex plane. Forexample, in the A−model on a Calabi–Yau threefold, the 3–point functionof three operators corresponding to (1, 1)−forms would thus give a nonzeroresult.In the case of the torus, we have seen that there is one ‘unexpected’modular parameter over which we have to integrate. This means we haveto insert one G− and one Ḡ−operator in the measure, which spoils theabsence of the axial anomaly we had for g = 1 in the topological field theorycase. However, we also must fix the one remaining translational symmetry,which we can do by inserting a local operator at a labelled point. Thus, wecan restore the axial R−charge to its zero value by choosing this to be anoperator of degree (1, 1).Summarizing, we have found that in topological string theory on a targetCalabi–Yau 3–fold, we have a non–vanishing 3–point function of totaldegree (3,3) at genus zero; a non–vanishing 1–point function of degree (1,1)at genus one, and a non–vanishing partition (‘zero–point’) function at allgenera g > 1.Nonlocal OperatorsIn one respect, what we have achieved is great progress: we can now forany genus define a nonzero partition function (or for low genus a correlationfunction) of the topological string theory. On the other hand, we would alsolike to define correlation functions of an arbitrary number of operators atthese genera. As we have seen, the insertion of extra local operators inthe correlation functions is not possible, since any such insertion will spoilour carefully constructed absence of R−symmetry anomalies. Therefore,we have to introduce nonlocal operators.There is one class of nonlocal operators which immediately becomesmind. Before we saw, using the descent equations, that for every localoperator we can define a corresponding 1–form and a 2–form operator. If


<strong>Geom</strong>etrical Path Integrals and Their Applications 1215we check the axial and vector charges of these operators, we find that if westart with an operator of degree (1, 1), the 2–form operator we end up withactually has vanishing axial and vector charges. This has two importantconsequences. First of all, we can add the integral of this operator to ouraction [Vonk (2005)],∫S[t] = S 0 + t a O a (2) ,without spoiling the axial and vector symmetry of the theory. Secondly, wecan insert the integrated operator into correlation functions,∫ ∫〈 O (2)1 · · · O n (2) 〉and still get a nonzero result by the vanishing of the axial and vectorcharges. These two statements are related: one obtains such correlatorsby differentiating S[t] with respect to the appropriate t’s, and then settingall t a = 0.A few remarks are in place here. First of all, recall that the integrationover the insertion points of the operators can be viewed as part of theintegration over the moduli space of Riemann surfaces, where now we labela certain number of points on the Riemann surface. From this point ofview, the g = 0, 1 cases fit naturally into the same framework. We couldunite the descendant fields into a world–sheet super–field,Φ a = O (0)a + O (1)aα θ α + O (2)aαβ θα θ βwhere we formally replaced each dz and d¯z by corresponding fermioniccoordinates θ z and θ¯z . Now, one can write the above correlators as integralsover n copies of this super–space,∫∏ nd 2 z s d 2 θ s 〈Φ a1 (z 1 , θ 1 ) · · · Φ an (z n , θ n )〉s=1The integration prescription at genus 0 and 1 tells us to fix 3 and 1 pointsrespectively, so we need to remove this number of super–space integrals.Then, integrating over the other super–space coordinates, the genus 0 correlatorsindeed become〈O (0)a 1O (0)a 2O (0)a 3∫∫O a (2)4 · · ·O (2)a n〉From this prescription we note that these expressions are symmetric in theexchange of all a i and a j . In particular, this means that the genus zero


1216 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introduction3–point functions at arbitrary t,have symmetric derivatives:c abc [t] = 〈O (0)a O (0)∂c abc∂t dbO c(0)= ∂c abd∂t c ,and similarly with permuted indices. These equations can be viewed asintegrability conditions, and using the Poincaré lemma we see that theyimply thatc ijk [t] =∂Z 0[t]∂t i ∂t j ∂t k .for some function Z 0 [t]. Following the general philosophy that n−pointfunctions are nth derivatives of the t−dependent partition function, we seethat Z 0 [t] can be naturally thought of as the partition function at genuszero. Similarly, the partition function at genus 1 can be defined by integratingup the one-point functions once.The quantities we have calculated above should be semi–topological invariants,meaning that they only depend on ‘half’ of the moduli (eitherthe Kähler ones or the complex structure ones) of the target space. Forexample, in the A−model we find the Gromov–Witten invariants. In theB−model, it turns out that F 0 [t] = ln Z 0 [t] is actually a quantity we alreadyknew: it is the prepotential of the Calabi–Yau manifold. A discussion ofwhy this is the case can be found in the paper [Bershadsky et. al. (1994)].The higher genus partition functions can be thought of as ‘quantum corrections’to the prepotential.Finally, there is a type of operator we have not discussed at all sofar. Recall that in the topological string theory, the metric itself is now adynamical field. We could not include the metric in our physical operators,since this would spoil the topological invariance. However, the metric ispart of a Q−multiplet, and the highest field in this multiplet is a scalarfield which is usually labelled ϕ. (It should not be confused with the fieldsφ i .) We can get more correlation functions by inserting operators ϕ k andthe operators related to them by the descent equations into the correlationfunctions. These operators are called ‘gravitational descendants’. Even thecase where the power is k = 0 is nontrivial; it does not insert any operator,but it does label a certain point, and hence changes the moduli space oneintegrates over. This operator is called the ‘puncture operator’.〉[t]


<strong>Geom</strong>etrical Path Integrals and Their Applications 1217All of this seems to lead to an enormous amount of semi-topologicaltarget space invariants that can be calculated, but there are many recursionrelations between the several correlators. This is similar to how we showedbefore that all correlators for the cohomological field theories follow fromthe 2–and 3–point functions on the sphere. Here, it turns out that the setof all correlators has a structure which is related to the theory of integrablehierarchies. Unfortunately, a discussion of this is outside the scope of boththese lectures and the author’s current knowledge.The Holomorphic AnomalyWe have now defined the partition function and correlation functions oftopological string theory, but even though the expressions we obtained aremuch simpler than the path integrals for ordinary quantum field or stringtheories, it would still be very hard to explicitly calculate them. Fortunately,it turns out that the t−dependent partition and correlation functionsare actually ‘nearly holomorphic’ in t, and this is a great aid in exactlycalculating these quantities.Let us make this ‘near holomorphy’ more precise. As we have seen,calculating correlation functions of primary operators in topological stringtheories amounts to taking t−derivatives of the corresponding perturbedpartition function Z[t] and consequently setting t = 0. Recall that Z[t] isdefined through adding terms to the action of the formt a ∫ΣO (2)a , (6.326)Let us for definiteness consider the B−twisted model. We want to showthat the above term is Q B −exact. For simplicity, we assume that O a(2) is abosonic operator, but what we are about to say can by inserting a few signsstraightforwardly be generalized to the fermionic case. From the descentequations we studied in the subsection 6.7.4 above, we know that(O (2)a ) +− = −{G + , [G − , O (0)a ]}, (6.327)where G + is the charge corresponding to the current G zz , and G − the onecorresponding to G¯z¯z . We can in fact express G ± in terms of the N = (2, 2)supercharges Q. So, according to [Vonk (2005)], we haveH = 2πi(L 0 + ¯L 0 ) = 1 2 {Q +, ¯Q + } − 1 2 {Q −, ¯Q − }P= 2πi(L 0 − ¯L 0 ) = 1 2 {Q +, ¯Q + } + 1 2 {Q −, ¯Q − }.


1218 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThus, we find that the left– and right–moving SEM charges satisfyT + = 2πiL 0 = 1 2 {Q +, ¯Q + }T − = 2πi¯L 0 = − 1 2 {Q −, ¯Q − }.To find G in the B−model, we should write these charges as commutatorswith respect to Q B = ¯Q + + ¯Q − , which givesT + = 1 2 {Q B, Q + }T − = − 1 2 {Q B, Q − },so we arrive at the conclusion that for the B−model,Now, we can rewrite (6.327) asG + = 1 2 Q +G − = − 1 2 Q −.(O (2)a ) +− = −{G + , [G − , O (0)a ]} = 1 4 {Q +, [Q − , O (0)a ]} (6.328)= 1 8 { ¯Q B , [(Q − − Q + ), O (0)a ]},which proves our claim that O a(2) is Q B −exact.An N = (2, 2) sigma model with a real action does, apart from the term(6.326), also contain a term∫tā Ō a (2) , (6.329)Σwhere tā is the complex conjugate of t a . It is not immediately clear that Ō(2) ais a physical operator: we have seen that physical operators in the B−modelcorrespond to forms that are ¯∂−closed, but the complex conjugate of sucha form is ∂−closed. However, by taking the complex conjugate of (6.328),we see that(Ō(2) a ) +− = 1 8 {Q B, [( ¯Q − − ¯Q + ), Ō(0) a ]},so not only is the operator Q B −closed, it is even Q B −exact. Thismeans that we can add terms of the form (6.329) to the action, and takingtā−derivatives inserts Q B −exact terms in the correlation functions.Naively, we would expect this to give a zero result, so all the physical quantitiesseem to be t−independent, and thus holomorphic in t. We will see ina moment that this naive expectation turns out to be almost right, but notquite.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1219However, before doing so, let us comment briefly on the generalizationof the above argument in the case of the A−model. It seems that astraightforward generalization of the argument fails, since Q A is its owncomplex conjugate, and the complex conjugate of the de Rham operator isalso the same operator. However, note that the N = (2, 2)−theory has adifferent kind of ‘conjugation symmetry’: we can exchange the two supersymmetries,or in other words, exchange θ + with ¯θ + and θ − with ¯θ − . Thisexchanges Q A with an operator which we might denote as Q Ā ≡ Q + + ¯Q − .Using the above argument, we then find that the physical operators O a(2)are Q Ā −exact, and that their conjugates in the new sense are Q A −exact.We can now add these conjugates to the action with parameters tā, andwe again naively find independence of these parameters. In this case itis less natural to choose t a and tā to be complex conjugates, but we arefree to choose this particular ‘background point’ and study how the theorybehaves if we then vary t a and tā independently.Now, let us see how the naive argument showing independence of thetheory of tā fails. In fact, the argument above would certainly hold fortopological field theories. However, in topological string theories (see [Vonk(2005)]), we have to worry about the insertions in the path integral of∫G · µ i ≡ d 2 z G zz (µ i ) z¯z,and their complex conjugates, when commuting the Q B towards the vacuumand making sure it gives a zero answer. Indeed, the Q B −commutator ofthe above factor is not zero, but it gives{Q B , G · µ i } = T · µ i .Now recall that T αβ = ∂ h αβ S. We did not give a very precise definitionof µ i above, but we know that it parameterizes the change in the metricunder an infinitesimal change of the coordinates m i on the moduli space.One can make this intuition precise, and then finds the following ‘chainrule’: T · µ i = ∂ m iS. Inserting this into the partition function, we findthat∂F g∂tā= ∫3g−3∏M gi=1dm i d ¯m i ∑ j,k〈∂ 2∂m j ∂ ¯m k ( ∏ ∫l≠jµ l · G)( ∏ l≠k∫¯µ l · Ḡ) ∫Ō (2)a〉,


1220 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionwhere F g = ln Z g is the free energy at genus g, and the reason F g appears inthe above equation instead of Z g is, as usual in quantum field theory, thatthe expectation values in the r.h.s. are normalized such that 〈1〉 = 1, andso the l.h.s. should be normalized accordingly and equal Zg−1 ∂āZ g = ∂āF g[Vonk (2005)].Thus, as we have claimed before, we are integrating a total derivativeover the moduli space of genus g surfaces. If the moduli space did nothave a boundary, this would indeed give zero, but in fact the moduli spacedoes have a boundary. It consists of the moduli which make the genus gsurface degenerate. This can happen in two ways: an internal cycle of thegenus g surface can be pinched, leaving a single surface of genus g − 1, asin Figure 6.28 (a), or the surface can split up into two surfaces of genus g 1and g 2 = g −g 1 , as depicted in Figure 6.28 (b). By carefully considering theboundary contributions to the integral for these two types of boundaries,it was shown in [Bershadsky et. al. (1994)] that()∂F g= 1∂tā 2 c ∑g−1ā¯b¯ce 2K G¯bd G¯ce D d D e F g−1 + D d F r D e F g−r ,where G is the so–called Zamolodchikov metric on the space parameterizedby the coupling constants t a , tā; K is its Kähler potential, and the D a arecovariant derivatives on this space. The coefficients cā¯b¯c are the 3–pointfunctions on the sphere of the operators Ō(0) a . We will not derive the aboveformula in detail, but the reader should notice that the contributions fromthe two types of boundary are quite clear.r=1Fig. 6.28 At the boundary of the moduli space of genus g surfaces, the surfaces degeneratebecause certain cycles are pinched. This either lowers the genus of the surface (a)or breaks the surface into two lower genus ones (b) (see text for explanation).Using this formula, one can inductively determine the tā dependenceon the partition functions if the holomorphic t a −dependence is known.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1221Holomorphic functions on complex spaces (or more generally holomorphicsections of complex vector bundles) are quite rare: usually, there is onlya nD space of such functions. The same turns out to hold for our topologicalstring partition functions: even though they are not quite holomorphic,their anti–holomorphic behavior is determined by the holomorphicdependence on the coordinates, and as a result there is a finite number ofcoefficients which determines them.Thus, just from the above structure and without doing any path integrals,one can already determine the topological string partition functionsup to a finite number of constants. This leads to a feasible program forcompletely determining the topological string partition function for a giventarget space and at given genus. From the holomorphic anomaly equation,one first has to find the general form of the partition function. Then, allone has left to do is to fix the unknown constants. Here, the fact that inthe A−model the partition function counts a number of points becomesour help: by requiring that the A−model partition functions are integral,one can often fix the unknown constants and completely determine thet−dependent partition function. In practice, the procedure is still quiteelaborate, so we will not describe any examples here, but several have beenworked out in detail in the literature. Once again, the pioneering work forthis can be found in the paper [Bershadsky et. al. (1994)].6.7.6 <strong>Geom</strong>etrical TransitionsConifoldsRecall that a conifold is a generalization of the notion of a manifold. Unlikemanifolds, a conifold can (or, should) contain conical singularities i.e.,points whose neighborhood looks like a cone with a certain base. The baseis usually a 5D manifold.In string theory, a conifold transition represents such an evolution ofthe Calabi–Yau manifold in which its fabric rips and repairs itself, yet withmild and acceptable physical consequences in the context of string theory.However, the tears involved are more severe than those in an ‘weaker’ floptransition (see [Greene (2000)]). The geometrically singular conifolds wereshown to lead to completely smooth physics of strings. The divergencesare ‘smeared out’ by D3–branes wrapped on the shrinking 3–sphere S 3 , asoriginally pointed out by A. Strominger, who, together with D. Morrison


1222 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand B. Greene have also found that the topology near the conifold singularitycan undergo a topological phase–transition (see subsection 6.4.5). Itis believed that nearly all Calabi–Yau manifolds can be connected via these‘critical transitions’.More precisely, the conifold is the simplest example of a non–compactCalabi–Yau 3–fold: it is the set of solutions to the equationx 1 x 2 − x 3 x 4 = 0in C 4 . The resulting manifold is a cone, meaning in this case that anyreal multiple of a solution to this equation is again a solution. The point(0, 0, 0, 0) is the ‘tip’ of this cone, and it is a singular point of the solutionspace. Note that by writingx 1 = z 1 + iz 2 , x 2 = z 1 − iz 2 , x 3 = z 3 + iz 4 , x 4 = −z 3 + iz 4 ,where the z i are still complex numbers, one can also write the equation asz 2 1 + z 2 2 + z 2 3 + z 2 4 = 0.Writing each z i as a i + ib i , with a i and b i real, we get the two equations|a| 2 − |b| 2 = 0, a · b = 0. (6.330)Here a · b = ∑ i a ib i and |a| 2 = a · a. Since the geometry is a cone, let usfocus on a ‘slice’ of this cone given by|a| 2 + |b| 2 = 2r 2 ,for some r ∈ R. On this slice, the first equation in (6.330) becomes|a| 2 = r 2 , (6.331)which is the equation defining a 3–sphere S 3 of radius r. The same holdsfor b, so both a and b lie on 3–spheres. However, we also have to take thesecond equation in (6.330) into account. Let us suppose that we fix an asatisfying (6.331). Then b has to lie on a 3–sphere, but also on the planethrough the origin defined by a · b = 0. That is, b lies on a 2–sphere. Thisholds for every a, so the slice we are considering is a fibration of 2–spheresover the 3–sphere. With a little more work, one can show that this fibrationis trivial, so the conifold is a cone over S 2 × S 3 .Since the conifold is a singular geometry, we would like to find geometrieswhich approximate it, but which are non–singular. There are two


<strong>Geom</strong>etrical Path Integrals and Their Applications 1223interesting ways in which this can be done. The simplest way is to replacethe defining equation byx 1 x 2 − x 3 x 4 = µ 2 . (6.332)From the two equations constraining a and b, we now see that |a| 2 ≥ µ 2 .In other words, the parameter r should be at least µ. At r = µ, thea−sphere still has finite radius µ, but the b−sphere shrinks to zero size.This geometry is called the deformed conifold. Even though this is notclear from the picture, from the equation (6.332) one can straightforwardlyshow that it is nonsingular. One can also show that it is topologicallyequivalent to the cotangent bundle on the 3–sphere, T ∗ S 3 . Here, the S 3 onwhich the cotangent bundle is defined is exactly the S 3 at the ‘tip’ of thedeformed conifold.The second way to change the conifold geometry arises from studyingthe two equationsx 1 A + x 3 B = 0, x 4 A + x 2 B = 0. (6.333)Here, we require A and B to be homogeneous complex coordinates on aCP 1 , i.e.,(A, B) ≠ (0, 0),(A, B) ∼ (λA, λB)where λ is any nonzero complex number. If one of the x i is nonzero, sayx 1 , one can solve for A or B, e.g., A = − x3Bx 1, and insert this in the otherequation to getx 1 x 2 − x 3 x 4 = 0which is the conifold equation. However, if all x i are zero, any A and Bsolve the system of equations (6.333). In other words, we have constructeda geometry which away from the former singularity is completely the sameas the conifold, but the singularity itself is replaced by a CP 1 , which topologicallyis the same as an S 2 . From the defining equations one can againshow that the resulting geometry is nonsingular, so we have now replacedour conifold geometry by the so–called resolved conifold.Topological D–branesSince topological string theories are in many ways similar to an ordinary(bosonic) string theories, one natural question which arises is: are therealso open topological strings which can end on D–branes? To answer the


1224 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionabove question rigorously, we would have to study boundary conditions onworld–sheets with boundaries which preserve the Q−symmetry.In the A−model, one can only construct 3D–branes wrapping so–called‘Lagrangian’ submanifolds of M. Here, ‘Lagrangian’ means that the Kählerform ω vanishes on this submanifold. In the B−model, one can constructD–branes of any even dimension, as long as these branes wrap holomorphicsubmanifolds of M.Just like in ordinary string theory, when we consider open topologicalstrings ending on a D–brane, there should be a field theory on the braneworld–volume describing the low–energy physics of the open strings. Moreover,since we are studying topological theories, one may expect such atheory to inherit the property that it only depends on a restricted amountof data of the manifolds involved. A key example is the case of the A−modelon the deformed conifold, M = T ∗ S 3 , where we wrap ND–branes on the S 3in the base. (One can show that this is indeed a Lagrangian submanifold.)In ordinary string theory, the world–volume theory on ND–branes has aU(N) gauge symmetry, so putting the ingredients together we can makethe guess that the world–volume theory is a 3D topological field theorywith U(N) gauge symmetry. There is really only one candidate for such atheory: the Chern–Simons gauge theory. Recall that it consists of a singleU(N) gauge field, and has the actionS = k4π∫S 3 Tr(A ∧ dA + 2 3 A ∧ A ∧ A ). (6.334)Before the invention of D–branes, E. Witten showed that this is indeedthe theory one gets. In fact, he showed even more: this theory actuallydescribes the full topological string–field theory on the D–branes, evenwithout going to a low–energy limit [Witten (1995a)].Let us briefly outline the argument that gives this result. In his paper,Witten derived the open string–field theory action for the open A−modeltopological string; it reads∫S = Tr(A ∗ Q A A + 2 )3 A ∗ A ∗ A .The form of this action is very similar to Chern–Simons theory, but itsinterpretation is completely different: A is a string–field (a wave functionon the space of all maps from an open string to the space–time manifold),Q A is the topological symmetry generator, which has a natural action on thestring–field, and ∗ is a certain noncommutative product. Witten shows that


<strong>Geom</strong>etrical Path Integrals and Their Applications 1225the topological properties of the theory imply that only the constant mapscontribute, so A becomes a field on M – and since open strings can onlyend on D–branes, it actually becomes a field on S 3 . Moreover, recall thatQ A can be interpreted as a de Rham differential. Using these observationsand the precise definition of the star product one can indeed show that thestring–field theory action reduces to Chern–Simons theory on S 3 .6.7.7 Topological Strings and Black Hole AttractorsTopological string theory is naturally related to black hole dynamics (seesubsection 5.12.3 above). Namely, critical string theory compactified onCalabi–Yau manifolds has played a central role in both the mathematicaland physical development of modern string theory. The physical relevanceof the data provided by the topological string ĉ = 6 (of A and B types) hasbeen that it computes F −type terms in the corresponding four dimensionaltheory [Bershadsky et. al. (1994); Antoniadis et. al. (1994)]. These higher–derivative F −type terms for Type II superstring on a Calabi–Yau manifoldare of the general form∫d 4 xd 4 θ(W ab W ab ) g F g (X Λ ), (6.335)where W ab is the graviphoton super–field of the N = 2 super–gravity andX Λ are the vector multiplet fields. The lowest component of W is F thegraviphoton field strength and the highest one is the Riemann tensor. Thelowest components of X Λ are the complex scalars parameterizing Calabi–Yau moduli and their highest components are the associated U(1) vector–fields. These terms contribute to multiple graviphoton–graviton scattering.(6.335) includes (after θ integrations) an R 2 F 2g−2 term. The topologicalstring partition function Z top represents the canonical ensemble for multi–particle spinning five dimensional black holes [Breckenridge et. al. (1997);Katz et. al. (1999)].Recently, [Ooguri et. al. (2004)] proposed a simple and direct relationshipbetween the second–quantized topological string partition functionZ top and black hole partition function Z BH in four dimensions of the formZ BH (p Λ , φ Λ ) = |Z top (X Λ )| 2 , where X Λ = p Λ + i π φΛin a certain Kähler gauge. The l.h.s. here is evaluated as a function ofinteger magnetic charges p Λ and continuous electric potentials φ Λ , whichare conjugate to integer electric charges q Λ . The r.h.s. is the holomorphic


1226 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsquare of the partition function for a gas of topological strings on a Calabi–Yau whose moduli are those associated to the charges/potentials (p Λ , φ Λ )via the attractor equations [Ooguri et. al. (2004)]. Both sides of (6.336)are defined in a perturbation expansion in 1/Q, where Q is the graviphotoncharge carried by the black hole. 29 The non–perturbative completion ofeither side of (6.336) might in principle be defined as the partition functionof the holographic CFT dual to the black hole, as in [Strominger and Vafa(1996)]. Then we have the triple equality,Z CF T = Z BH = |Z top | 2 .The existence of fundamental connection between 4D black holes andthe topological string might have been anticipated from the following observation.Calabi–Yau spaces have two types of moduli: Kähler and complexstructure. The world–sheet twisting which produces the A (B) model topologicalstring from the critical superstring eliminates all dependence on thecomplex structure (Kähler) moduli at the perturbative level. Hence theperturbative topological string depends on only half the moduli. Blackhole entropy on the other hand, insofar as it is an intrinsic property of theblack hole, cannot depend on any externally specified moduli. What happensat leading order is that the moduli in vector multiplets are driven toattractor values at the horizon which depend only on the black hole chargesand not on their asymptotically specified values. Hypermultiplet vevs onthe other hand are not fixed by an attractor mechanism but simply dropout of the entropy formula. It is natural to assume this is valid to all ordersin a 1/Q expansion. Hence the perturbative topological string and the largeblack hole partition functions depend on only half the Calabi–Yau moduli.It would be surprising if string theory produced two functions on the samespace that were not simply related. Indeed [Ooguri et. al. (2004)] arguedthat they were simply related as in (6.336).Supergravity Area–Entropy FormulaRecall that a well–known hypothesis by J. Bekenstein and S. Hawking statesthat the entropy of a black hole is proportional to the area of its horizon(see [Hawking and Israel (1979)]). This area is a function of the black holemass, or in the extremal case, of its charges. Here we review the leadingsemiclassical area–entropy formula for a general N = 2, d = 4 extremalblack hole characterized by magnetic and electric charges (p Λ , q Λ ), recently29 The string coupling g s is in a hypermultiplet and decouples from the computation.


<strong>Geom</strong>etrical Path Integrals and Their Applications 1227reviewed in [Ooguri et. al. (2004)]. The asymptotic values of the moduliin vector multiplets, parameterized by complex projective coordinatesX Λ , (Λ = 0, 1, . . . , n V ) in the black hole solution, are arbitrary. Thesemoduli couple to the electromagnetic fields and accordingly vary as a functionof the radius. At the horizon they approach an attractor point whoselocation in the moduli space depends only on the charges. The locations ofthese attractor points can be found by looking for supersymmetric solutionswith constant moduli. They are determined by the attractor equations,p Λ = Re[CX Λ ], q Λ = Re[CF 0Λ ], (6.336)where F 0Λ = ∂F 0 /∂X Λ are the holomorphic periods, and the subscript 0distinguishes these from the string loop corrected periods to appear in thenext subsection. Both (p Λ , q Λ ) and (X Λ , F 0Λ ) transform as vectors underthe Sp(2n + 2; Z) duality group.The (2n v + 2) real equations (6.336) determine the (n v + 2) complexquantities (C, X Λ ) up to Kähler transformations, which act asK → K − f(X) − ¯f( ¯X), X Λ → e f X Λ , F 0 → e 2f F 0 , C → e −f C,where the Kähler potential K is given bye −K = i( ¯X Λ F 0Λ − X Λ ¯F0Λ ).We could at this point set C = 1 and fix the Kähler gauge but later weshall find other gauges useful. It is easy to see that (as required) thecharges (p Λ , q Λ ) determined by the attractor equations (6.336) are invariantunder Kähler transformations. Given the horizon attractor values of themoduli determined by (6.336) the Bekenstein–Hawking entropy S BH maybe written asS BH = 1 4 Area = π|Q|2 ,where Q = Q m +iQ e is a complex combination of the magnetic and electricgraviphoton charges and|Q| 2 = i 2(qΛ ¯C ¯XΛ − p Λ ¯C ¯F0Λ)=C ¯C4 e−K .The normalization of Q here is chosen so that |Q| equals the radius of thetwo sphere at the horizon.


1228 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionIt is useful to rephrase the above results in the context of type IIB superstringsin terms of geometry of Calabi–Yau. In this case the attractor equationsfix the complex geometry of the Calabi–Yau. The electric/magenticcharges correlate with three cycles of Calabi–Yau. Choosing a symplecticbasis for the three cycles gives a choice of the splitting to electric and magneticcharges. Let A Λ denote a basis for the electric three cycles, B Σ thedual basis for the magnetic charges and Ω the holomorphic 3–form at theattractor point. Ω is fixed up to an overall multiplication by a complexnumber Ω → λΩ. There is a unique choice of λ such that the resulting Ωhas the property that∫∫p Λ = Re Ω = Re[CX Λ ], q Λ = Re Ω = Re[CF 0Λ ],A Λ B ΛwhereRe Ω = 1 (Ω + Ω).2In terms of this choice, the black hole entropy can be written asS BH = π ∫Ω ∧ Ω.4Higher–Order CorrectionsF −term corrections to the action are encoded in a string loop correctedholomorphic prepotential∞∑F (X Λ , W 2 ) = F h (X Λ )W 2h , (6.337)CYh=0where F h can be computed by topological string amplitudes (as we reviewin the next section) and W 2 involves the square of the anti–self dualgraviphoton field strength. This obeys the homogeneity equationX Λ ∂ Λ F (X Λ , W 2 ) + W ∂ W F (X Λ , W 2 ) = 2F (X Λ , W 2 ). (6.338)Near the black hole horizon, the attractor value of W 2 obeys C 2 W 2 = 256,and therefore the exact attractor equations read(p Λ = Re[CX Λ ], q Λ = Re[CF Λ X Λ , 256 )]C 2 . (6.339)This is essentially the only possibility consistent with symplectic invariance.It has been then argued that the entropy as a function of the charges isS BH = πi2 (q Λ ¯C ¯X Λ − p Λ ¯C ¯FΛ ) + π 2 Im[C3 ∂ C F ], (6.340)


<strong>Geom</strong>etrical Path Integrals and Their Applications 1229where F Λ , X Λ and C are expressed in terms of the charges using (6.339).Topological StringsPartition Functions for Black Hole and Topological Strings. Thenotion of topological string was introduced in [Witten (1990)]. Subsequentlya connection between them and superstring was discovered: It was shown in[Bershadsky et. al. (1994); Antoniadis et. al. (1994)], that the superstringloop corrected F −terms (6.337) can be computed as topological string amplitudes.The purpose of this subsection is to translate the super–gravitynotation of the previous section to the topological string notation.The second quantized partition function for the topological string maybe writtenZ top (t A , g top ) = exp [ F top (t A , g top ) ] ,whereF top (t A , g top ) = ∑ hg 2h−2top F top,h (t A ),and F top,h is the h−loop topological string amplitude. The Kähler moduliare expressed in the flat coordinatest A = XAX 0 = θA + ir A ,where r A are the Kähler classes of the Calabi–Yau M and θ A are periodicθ A ∼ θ A + 1.We would like to determine relations between super–gravity quantitiesand topological string quantities. Using the homogeneity property (6.338)and the expansion (6.337), the holomorphic prepotential in super–gravitycan be expressed as( )XF (CX Λ , 256) = (CX 0 ) 2 ΛFX 0 , 256(CX 0 ) 2∞∑= (CX 0 ) 2−2h f h (t A ), (6.341)h=0where f h (t A ) is related to F h (X Λ ) in (6.337) asf h (t A ) = 16 2h F h( XΛX 0 ).


1230 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThis suggests an identification of the form f h (t A ) ∼ F top,h (t A ) and g top ∼(CX 0 ) −1 . For later purposes, we need precise relations between super–gravity and topological string quantities, including numerical coefficients.These can be determined by studying the limit of a large Calabi–Yau space.In the super–gravity notation, the genus 0 and 1 terms in the largevolume are given byF ( CX Λ , 256 ) = C 2 D ABCX A X B X CX 0− 1 6 c X A2AX 0 + · · ·= (CX 0 ) 2 D ABC t A t B t C − 1 6 c 2At A + · · · ,∫where c 2A = c 2 ∧ α A ,Mwith c 2 being the second Chern class of M, and C ABC = −6D ABC arethe 4–cycle intersection numbers. These terms are normalized so that themixed entropy S BH is given by (6.340). On the other hand, the topologicalstring amplitude in this limit is given byF top = − (2π)3 igtop2 D ABC t A t B t C − πi12 c 2At A + · · · (6.342)The normalization here is fixed by the holomorphic anomaly equations in[Bershadsky et. al. (1994)], which are nonlinear equations for F top,h .Comparing the one–loop terms in (6.341) and (6.342), which are independentof g top , we findF (CX Λ , 256) = − 2iπ F top(t A , g top ).Given this, we can compare the genus 0 terms to findThis impliesg top = ± 4πiCX 0 .ln Z BH = −π Im [ F (CX Λ , 256) ] = F top + ¯F topZ BH (φ Λ , p Λ ) = |Z top (t A , g top )| 2 , witht A = pA + iφ A /πp 0 + iφ 0 /π , g 4πitop = ±p 0 + iφ 0 /π .and


<strong>Geom</strong>etrical Path Integrals and Their Applications 1231Supergravity Approach to Z BH . The above relationZ BH = |Z top | 2 (6.343)can have a simpler super–gravity derivation [Ooguri et. al. (2004)].A main ingredient in this derivation is the observation that the N = 2super–gravity coupled to vector multiplets can be written as the action∫∫S = d 4 xd 4 θ (super − −volume form)+h.c. = d 4 x √ −gR+..., (6.344)where the super–volume form in the above depends non–trivially on curvatureof the fields. This reproduces the ordinary action after integratingover d 4 θ and picking up the θ 4 term in the super–volume. In the contextof black holes the boundary terms accompanying (6.344) give the classicalblack hole entropy.We now become the derivation of (6.343). As was observed in [Bershadskyet. al. (1994); Antoniadis et. al. (1994)], topological string computesthe terms∞∑∫F = d 4 xd 4 θF h (X)(W 2 ) g + c.c. (6.345)h=0There are various terms one can get from the above action after integratingover d 4 θ. Let us concentrate on one of the terms which turns out to bethe relevant one for us: Take the top components of X Λ and W 2 , andabsorb the d 4 θ integral from the super–volume measure as in (6.344). Wewill work in the gauge X 0 ∼ 1 and thus C ∼ 1/g top . As noted beforein the near–horizon black hole geometry in this gauge the top componentW 2 ∼ 1/C 2 ∼ gtop 2 and the X Λ are fixed by the attractor mechanism.Thus, we have the black hole free energy∞∑∫ln Z BH = gtopF 2htop,h (X Λ /X 0 ) d 4 xd 4 θ + c.c.=h=0∞∑(g top ) 2h−2 F top,h (X Λ /X 0 ) + c.c.g=0= 2 Re F top , (using∫d 4 xd 4 θ ∼ 1/g 2 top).Upon exponentiation this leads to (6.343).Here we have shown that if we consider one absorption of θ 4 term in(6.345) upon d 4 θ integral we get the desired result. That there be no other


1232 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionterms is not obvious. For example another way to absorb the θ’s would havegiven the familiar term R 2 F 2g−2 where F is the graviphoton field. However,such terms do not contribute in the black hole background. It would benice to find a simple way to argue why these terms do not contribute andthat we are left with this simple absorption of the θ integrals.6.8 Application: Advanced <strong>Geom</strong>etry and Topology ofString Theory6.8.1 String Theory and Noncommutative <strong>Geom</strong>etryThe idea that the space–time coordinates do not commute is quite old(see [Snyder (1947a); Snyder (1947b)]). It has been studied by many authorsboth from a mathematical and a physical perspective. The theory ofoperator algebras has been suggested as a framework for physics in noncommutativespace–time (see [Connes (1994)] for an exposition of the philosophy).Yang-Mills (YM) theory on a noncommutative torus has beenproposed as an example; though this example at first sight appears to beneither covariant nor causal, it has proved to arise in string theory in adefinite limit [Connes et. al. (1997)] with the non–covariance arising fromthe expectation value of a background field. This analysis involved toroidalcompactification, in the limit of small volume, with fixed and generic valuesof the world–sheet theta angles. This limit is fairly natural in the context ofthe matrix model of M−theory [Matacz (2002)], and the original discussionwas made in this context. Part of the beauty of this analysis in [Conneset. al. (1997)] was that T −duality acts within the noncommutative YMframework, rather than mixing the modes of noncommutative YM theorywith string winding states and other stringy excitations. This makes theframework of noncommutative YM theory seem very powerful.Seiberg and Witten in [Seiberg and Witten (1999)], reexamined thequantization of open strings ending on D−branes in the presence of aB−field. They have showed that noncommutative YM theory is valid forsome purposes in the presence of any nonzero constant B−field, and thatthere is a systematic and efficient description of the physics in terms ofnoncommutative YM theory when B is large. The limit of a torus of smallvolume with fixed theta angle (that is, fixed periods of B) is an examplewith large B, but it is also possible to have large B on R n and therebymake contact with the application of noncommutative YM to instantonson R 4 . An important element in their analysis is a distinction between two


<strong>Geom</strong>etrical Path Integrals and Their Applications 1233different metrics in the problem. Distances measured with respect to onemetric have been scaled to zero. However, the noncommutative theory ison a space with a different metric with respect to which all distances arenonzero. This guarantees that both on R n and on T n we end up with atheory with finite metric.6.8.1.1 Noncommutative Gauge TheoryFor R n with coordinates x i whose commutators are c−numbers, we write[x i , x j ] = iθ ij ,with real θ. Given such a Lie algebra, one seeks to deform the algebra offunctions on R n to a noncommutative, associative algebra A such thatf ∗ g = fg + 1 2 iθij ∂ i f∂ j g + O(θ 2 ),with the coefficient of each power of θ being a local differential expressionbilinear in f and g. The essentially unique solution of this problem (moduloredefinitions of f and g that are local order by order in θ) is given by theexplicit formulaf(x) ∗ g(x) = e i 2 θij∂∂ξ i∂∂ζ j f(x + ξ)g(x + ζ)| ξ=ζ=0 = fg + i 2 θij ∂ i f∂ j g + O(θ 2 ).This formula defines what is often called the Moyal bracket of functions;it has appeared in the physics literature in many contexts, includingapplications to old and new matrix theories [Witten (1986b);Seiberg and Witten (1999)]. We also consider the case of N × N matrix–valued functions f, g. In this case, we define the ∗ product to be the tensorproduct of matrix multiplication with the ∗ product of functions as justdefined. The extended ∗ product is still associative.The ∗ product is compatible with integration in the sense that for functionsf, g that vanish rapidly enough at infinity, so that one can integrateby parts in evaluating the following integrals, one has∫∫Tr f ∗ g = Tr g ∗ f,where Tr is the ordinary trace of the N ×N matrices, and ∫ is the ordinaryintegration of functions.


1234 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionRecall that for ordinary YM–theory, we can write the gauge transformationsand field strength asδ λ A i = ∂ i λ+i[λ, A i ], F ij = ∂ i A j −∂ j A i −i[A i , A j ], δ λ F ij = i[λ, F ij ],(6.346)where A and λ are N × N hermitian matrices. The Wilson line isW (a, b) = P e i R ab A ,where in the path ordering A(b) is to the right. Under the gauge transformation(6.346), we haveδW (a, b) = iλ(a)W (a, b) − iW (a, b)λ(b).For noncommutative gauge theory, one uses the same formulas for thegauge transformation law and the field strength, except that matrix multiplicationis replaced by the ∗ product. Thus, the gauge parameter ˆλ takesvalues in A tensored with N × N hermitian matrices, for some N, andthe same is true for the components Âi of the gauge field Â. The gaugetransformations and field strength of noncommutative Yang–Mills theoryare thusˆδˆλ i = ∂ iˆλ + iˆλ ∗  i − iÂi ∗ ˆλ,ˆF ij = ∂ i  j − ∂ j  i − iÂi ∗ Âj + iÂj ∗ Âi,ˆδˆλF ij = iˆλ ∗ ˆF ij − i ˆF ij ∗ ˆλ.The theory obtained this way reduces to conventional U(N) YM theory forθ → 0. Because of the way that the theory is constructed from associativealgebras, there seems to be no convenient way to get other gauge groups.The commutator of two infinitesimal gauge transformations with generatorsˆλ 1 and ˆλ 2 is, rather as in ordinary YM theory, a gauge transformationgenerated by i(ˆλ 1 ∗ ˆλ 2 − ˆλ 2 ∗ ˆλ 1 ). Such commutators are nontrivial evenfor the rank 1 case, that is N = 1, though for θ = 0 the rank 1 case isthe Abelian U(1) gauge theory. For rank 1, to first order in θ, the aboveformulas for the gauge transformations and field strength readˆδˆλ i = ∂ iˆλ − θ kl ∂ kˆλ∂l  i + O(θ 2 ),ˆF ij = ∂ i  j − ∂ j  i + θ kl ∂ k  i ∂ l  j + O(θ 2 ),ˆδˆλˆFij = −θ kl ∂ kˆλ∂l ˆFij + O(θ 2 ).


<strong>Geom</strong>etrical Path Integrals and Their Applications 12356.8.1.2 Open Strings in the Presence of Constant B−FieldBosonic StringsFollowing [Seiberg and Witten (1999)], we now study strings in flat space,with metric g ij , in the presence of a constant Neveu–Schwarz B–field andwith Dp−branes. The B−field is equivalent to a constant magnetic fieldon the brane.We denote the rank of the matrix B ij as r; r is of course even. Since thecomponents of B not along the brane can be gauged away, we can assumethat r ≤ p + 1. When our target space has Lorentzian signature, we willassume that B 0i = 0, with “0” the time direction. With a Euclidean targetspace we will not impose such a restriction. Our discussion applies equallywell if space is R 10 or if some directions are toroidally compactified withx i ∼ x i + 2πr i . (One could pick a coordinate system with g ij = δ ij , inwhich case the identification of the compactified coordinates may not besimply x i ∼ x i + 2πr i , but we will not do that.) If our space is R 10 , we canpick coordinates so that B ij is nonzero only for i, j = 1, . . . , r and that g ijvanishes for i = 1, . . . , r, j ≠ 1, . . . , r. If some of the coordinates are on atorus, we cannot pick such coordinates without affecting the identificationx i ∼ x i + 2πr i . For simplicity, we will still consider the case B ij ≠ 0 onlyfor i, j = 1, . . . , r and g ij = 0 for i = 1, . . . , r, j ≠ 1, . . . , r.The world–sheet action isS = 1 (gij4πα∫Σ′ ∂ a x i ∂ a x j − 2πiα ′ B ij ɛ ab ∂ a x i ∂ b x j)= 14πα∫Σ ′ g ij ∂ a x i ∂ a x j − i ∫B ij x i ∂ t x j , (6.347)2where Σ is the string world–sheet, which we take to be with Euclideansignature. ∂ t is a tangential derivative along the world–sheet boundary ∂Σ.The equations of motion determine the boundary conditions. For i alongthe Dp−branes they are∂Σg ij ∂ n x j + 2πiα ′ B ij ∂ t x j | ∂Σ = 0, (6.348)where ∂ n is a normal derivative to ∂Σ. These boundary conditions are notcompatible with real x, though with a Lorentzian world–sheet the analogousboundary conditions would be real. Nonetheless, the open string theory canbe analyzed by determining the propagator and computing the correlationfunctions with these boundary conditions. In fact, another approach to the


1236 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionopen string problem is to omit or not specify the boundary term with B inthe action (6.347) and simply impose the boundary conditions (6.348).For B = 0, the boundary conditions in (6.348) are Neumann boundaryconditions. When B has rank r = p and B −→ ∞, or equivalently g ij → 0along the spatial directions of the brane, the boundary conditions becomeDirichlet; indeed, in this limit, the second term in (6.348) dominates, and,with B being invertible, (6.348) reduces to ∂ t x j = 0. This interpolationfrom Neumann to Dirichlet boundary conditions will be important, since wewill eventually take B −→ ∞ or g ij −→ 0. For B very large or g very small,each boundary of the string world–sheet is attached to a single point in theDp−brane, as if the string is attached to a zero–brane in the Dp−brane.Intuitively, these zero–branes are roughly the constituent zero-branes ofthe Dp−brane as in the matrix model of M−theory [Seiberg and Witten(1999)], an interpretation that is supported by the fact that in the matrixmodel the construction of Dp−branes requires a nonzero B−field.Our main focus is the case that Σ is a disc, corresponding to the classicalapproximation to open string theory. The disc can be conformally mappedto the upper half plane; in this description, the boundary conditions (6.348)areg ij (∂ − ¯∂)x j + 2πα ′ B ij (∂ + ¯∂)x j | z=¯z = 0,where ∂ = ∂/∂z, ¯∂ = ∂/∂¯z, and Im z ≥ 0.boundary conditions isThe propagator with these〈x i (z)x j (z ′ )〉 = −α ′ [g ij log |z − z ′ | − g ij log |z − ¯z ′ | (6.349)+ G ij log |z − ¯z ′ | 2 + 12πα ′ θij log z − ¯z′¯z − z ′ + Dij ].Here(G ij 1=g + 2πα ′ B) ijS() ij1=g + 2πα ′ B g 1g − 2πα ′ ,BG ij = g ij − (2πα ′ ) 2 (Bg −1 B) ij ,( ) ij () ijθ ij = 2πα ′ 1g + 2πα ′ = −(2πα ′ ) 2 1BAg + 2πα ′ B B 1g − 2πα ′ ,Bwhere ( ) S and ( ) A denote the symmetric and antisymmetric part of thematrix. The constants D ij in (6.349) can depend on B but are independentof z and z ′ ; they play no essential role and can be set to a convenient value.The first three terms in (6.349) are manifestly single–valued. The fourth


<strong>Geom</strong>etrical Path Integrals and Their Applications 1237term is single–valued, if the branch cut of the logarithm is in the lower halfplane.Our focus is on the open string vertex operators and interactions [Witten(1986b); Seiberg and Witten (1999)]. Open string vertex operators are ofcourse inserted on the boundary of Σ. So to get the relevant propagator,we restrict (6.349) to real z and z ′ , which we denote τ and τ ′ . Evaluatedat boundary points, the propagator is〈x i (τ)x j (τ ′ )〉 = −α ′ G ij log(τ − τ ′ ) 2 + i 2 θij ɛ(τ − τ ′ ), (6.350)where we have set D ij to a convenient value. ɛ(τ) is the function that is 1or –1 for positive or negative τ.The object G ij has a very simple intuitive interpretation: it is the effectivemetric seen by the open strings. The short distance behavior of thepropagator between interior points on Σ is〈x i (z)x j (z ′ )〉 = −α ′ g ij log |z − z ′ |.The coefficient of the logarithm determines the anomalous dimensions ofclosed string vertex operators, so that it appears in the mass shell conditionfor closed string states. Thus, we will refer to g ij as the closed string metric.G ij plays exactly the analogous role for open strings, since anomalousdimensions of open string vertex operators are determined by the coefficientof log(τ − τ ′ ) 2 in (6.350), and in this coefficient G ij enters in exactly theway that g ij would enter at θ = 0. We will refer to G ij as the open stringmetric.The coefficient θ ij in the propagator also has a simple intuitive interpretation.In conformal field theory, one can compute commutators of operatorsfrom the short distance behavior of operator products by interpretingtime ordering as operator ordering. Interpreting τ as time, we see that[x i (τ), x j (τ)] = T ( x i (τ)x j (τ − ) − x i (τ)x j (τ + ) ) = iθ ij . (6.351)That is, x i are coordinates on a noncommutative space with noncommutativityparameter θ.Consider the product of tachyon vertex operators e ip·x (τ) and e iq·x (τ ′ ).With τ > τ ′ , we get for the leading short distance singularitye ip·x (τ) · e iq·x (τ ′ ) ∼ (τ − τ ′ ) 2α′ G ij p iq je − 1 2 iθij p iq je i(p+q)·x (τ ′ ) + . . . .If we could ignore the term (τ − τ ′ ) 2α′ p·q , then the formula for the operator


1238 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionproduct would reduce to a ∗ product; we would gete ip·x (τ)e iq·x (τ ′ ) ∼ e ip·x ∗ e iq·x (τ ′ ).This is no coincidence. If the dimensions of all operators were zero, theleading terms of operator products O(τ)O ′ (τ ′ ) would be independent ofτ − τ ′ for τ → τ ′ , and would give an ordinary associative product of multiplicationof operators. This would have to be the ∗ product, since thatproduct is determined by associativity, translation invariance, and (6.351)(in the form x i ∗ x j − x j ∗ x i = iθ ij ).Now, consider an operator on the boundary of the disc that is of thegeneral form P (∂x, ∂ 2 x, . . . )e ip·x , where P is a polynomial in derivativesof x, and x are coordinates along the Dp−brane (the transverse coordinatessatisfy Dirichlet boundary conditions). Since the second term in thepropagator (6.350) is proportional to ɛ(τ −τ ′ ), it does not contribute to contractionsof derivatives of x. Therefore, the expectation value of a productof k such operators, of momenta p 1 , . . . , p k , satisfies〈∏ k〉P n (∂x(τ n ), ∂ 2 x(τ n ), . . . )e ipn·x(τ n)= (6.352)e − i 2n=1〈 k〉∏Pn>m pn i θij p m j ɛ(τ n−τ m) P n (∂x(τ n ), ∂ 2 x(τ n ), . . . )e ipn·x(τ n)n=1G,θG,θ=0where 〈. . . 〉 G,θ is the expectation value with the propagator (6.350)parametrized by G and θ. We see that when the theory is described interms of the open string parameters G and θ, rather than in terms of gand B, the θ dependence of correlation functions is very simple. Note thatbecause of momentum conservation ( ∑ m pm = 0), the crucial factor()exp − i ∑p n i θ ij p m j ɛ(τ n − τ m )(6.353)2n>mdepends only on the cyclic ordering of the points τ 1 , . . . , τ k around thecircle [Witten (1986b); Seiberg and Witten (1999)].The string theory S−matrix can be obtained from the conformal fieldtheory correlators by putting external fields on shell and integrating overthe τ’s. Therefore, it has a structure inherited from (6.352). To be veryprecise, in a theory with N × N Chan–Paton factors, consider a k pointfunction of particles with Chan–Paton wave functions W i , i = 1, . . . , k, momentap i , and additional labels such as polarizations or spins that we will,


<strong>Geom</strong>etrical Path Integrals and Their Applications 1239generically call ɛ i . The contribution to the scattering amplitude in whichthe particles are cyclically ordered around the disc in the order from 1 to kdepends on the Chan–Paton wave functions by a factor Tr(W 1 W 2 . . . W k ).We suppose, for simplicity, that N is large enough so that there are noidentities between this factor and similar factors with other orderings. Bystudying the behavior of the S−matrix of massless particles of small momenta,one can extract order by order in α ′ a low energy effective actionfor the theory. If Φ i is an N × N matrix-valued function in space–timerepresenting a wavefunction for the i th field, then at B = 0 a general termin the effective action is a sum of expressions of the form∫d p+1 x √ det G Tr(∂ n1 Φ 1 ∂ n2 Φ 2 . . . ∂ n kΦ k ). (6.354)Here ∂ ni is, for each i, the product of n i partial derivatives with respectto some of the space–time coordinates; which coordinates it is has notbeen specified in the notation. The indices on fields and derivatives arecontracted with the metric G, though this is not shown explicitly in theformula.Now to incorporate the B−field, at fixed G, is very simple: if the effectiveaction is written in momentum space, we need only incorporate thefactor (6.353). Including this factor is equivalent to replacing the ordinaryproduct of fields in (6.354) by a ∗ product (in this formulation, one canwork in coordinate space rather than momentum space). So the term correspondingto (6.354) in the effective action is given by the same expressionbut with the wave functions multiplied using the ∗ product:∫d p+1 x √ det G Tr(∂ n1 Φ 1 ∗ ∂ n2 Φ 2 ∗ · · · ∗ ∂ n kΦ k ).It follows, then, that the B dependence of the effective action for fixedG and constant B can be obtained in the following very simple fashion:replace ordinary multiplication by the ∗ product. We will make presentlyan explicit calculation of an S−matrix element to illustrate this statement,and we will make a detailed check of a different kind in section 4 usingalmost constant fields and the Dirac–Born–Infeld theory.Now, recall that background gauge fields couple to the string world–sheet by adding∫− i dτA i (x)∂ τ x i (6.355)


1240 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionto the action (6.347). We assume for simplicity that there is only a rankone gauge field; the extension to higher rank is straightforward. Comparing(6.347) and (6.355), we see that a constant B−field can be replaced by thegauge field A i = − 1 2 B ijx j , whose field strength is F = B. When we areworking on R n , we are usually interested in situations where B and F areconstant at infinity, and we fix the ambiguity be requiring that F is zeroat infinity.Naively, (6.355) is invariant under ordinary gauge transformations asδA i = ∂ i λ, (6.356)because (6.355) transforms by a total derivative∫∫∫δ dτA i (x)∂ τ x i = dτ∂ i λ∂ τ x i =dτ∂ τ λ.However, because of the infinities in quantum field theory, the theory hasto be regularized and we need to be more careful. We will examine a pointsplitting regularization, where different operators are never at the samepoint.Then expanding the exponential of the action in powers of A and usingthe transformation law (6.356), we find that the functional integral transformsby∫−∫dτA i (x)∂ τ x i ·dτ ′ ∂ τ ′λ, (6.357)plus terms of higher order in A. The product of operators in (6.357) canbe regularized in a variety of ways. We will make a point-splitting regularizationin which we cut out the region |τ − τ ′ | < δ and take the limitδ → 0. Though the integrand is a total derivative, the τ ′ integral contributessurface terms at τ − τ ′ = ±δ. In the limit δ → 0, the surface termscontribute∫−∫= −dτA i (x(τ))∂ τ x i (τ) ( λ(x(τ − )) − λ(x(τ + )) )dτ (A i (x) ∗ λ − λ ∗ A i (x)) ∂ τ x i .Here we have used the relation of the operator product to the ∗ product,and the fact that with the limit boundary propagator [Seiberg and Witten(1999)]〈x i (τ)x j (0)〉 = i 2 θij ɛ(τ),


<strong>Geom</strong>etrical Path Integrals and Their Applications 1241there is no contraction between ∂ τ x and x. To cancel this term, we mustadd another term to the variation of the gauge field; the theory is invariantnot under (6.356), but underˆδ i = ∂ i λ + iλ ∗ Âi − iÂi ∗ λ. (6.358)This is the gauge invariance of noncommutative YM theory, and in recognitionof that fact we henceforth denote the gauge field in the theory definedwith point splitting regularization as Â. A sigma model expansionwith Pauli–Villars regularization would have preserved the standard gaugeinvariance of open string gauge field, so whether we get ordinary or noncommutativegauge fields depends on the choice of regulator.We have made this derivation to lowest order in Â, but it is straightforwardto go to higher orders. At the nth order in Â, the variation isi n+1 ∫Â(x(t 1 )) . . .n!Â(x(t n))∂ t λ(x(t)) (6.359)∫()+ in+1Â(x(t 1 )) . . .(n − 1)!Â(x(t n−1)) λ ∗ Â(x(t n)) −  ∗ λ(x(t n)) ,where the integration region excludes points where some t’s coincide. Thefirst term in (6.359) arises by using the naive gauge transformation (6.356),and expanding the action to nth order in  and to first order in λ. Thesecond term arises from using the correction to the gauge transformationin (6.358) and expanding the action to the same order in  and λ. Thefirst term can be written asi n+1 ∑∫Â(x(t 1 )) . . .n!Â(x(tj−1))Â(x(t j+1)) . . .j). . . Â(x(t n))( ∗ λ(x(tj )) − λ ∗ Â(x(t j))∫)= in+1Â(x(t 1 )) . . .(n − 1)!Â(x(t n−1))( ∗ λ(x(tn )) − λ ∗ Â(x(t n)) ,making it clear that (6.359) vanishes. Therefore, there is no need to modifythe gauge transformation law (6.358) at higher orders in Â. For furthertechnical details, see [Witten (1986b); Seiberg and Witten (1999)].6.8.2 K−Theory Classification of StringsIn this subsection, following [Witten (1998c); Witten (2000)], we will revisitthe relation between strings, D−brane and K−theory, which we started in


1242 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionsection 4.5.As Ed. Witten has pointed out, K−theory provides a framework forclassifying Ramond–Ramond (RR) charges and fields. K−theory of manifoldshas a natural extension to K−theory of noncommutative algebras,such as the algebras considered in noncommutative Yang–Mills (YM) theory,or in open string field theory. In a number of concrete problems, theK−theory analysis proceeds most naturally if one starts out with an infiniteset of D−branes, reduced by tachyon condensation to a finite set.Recall that a D−brane wrapped on a submanifold S of space–time maycarry a nonzero RR–charge. As RR–fields are p−forms, superficially itseems that the conserved charge should be measured by the cohomologyclass of the RR–form (or of the cycle S itself). However, D−branes carrygauge fields; and gauge fields are not natural in ordinary (co)homologytheories. Instead, they are natural in K−theory.More precisely, if X is space–time and A(X) is the commutative, associativealgebra of continuous complex–valued functions on X, then theK−theory of X can be defined in terms of representations of A(X). Arepresentation of a ring is usually called a module. Here are some examplesof A(X)−modules.The most obvious example of an A(X)−module is A(X) itself. Forf ∈ A(X) (regarded as a ring) and g ∈ A(X) (regarded as a module), wedefine f(g) = fg, where on the right hand side the multiplication occursin A(X). This obviously obeys the defining condition of a module, whichis that(f 1 f 2 )(g) = f 1 (f 2 (g)).More generally, consider a Dp−brane (or a collection of N Dp−branesfor some N > 0) wrapped on a submanifold S of X, with any Chan–Patongauge bundle W on the D−brane. Let M(S) be the space of sections of W ,that is, the space of one–particle states for a charged scalar coupled to thebundle W . Then M(S) is an A(X)−module; for f ∈ A(X), g ∈ M(S), wesimply set again f(g) = fg. On the right hand side, the multiplication isdefined by restricting f, which is a function on X, to S and then multiplyingf and g.So in, say, Type IIB superstring theory, a collection of D9−branes definesa representation or module E of A(X). A collection of D9−branesdefines another module F . So any configuration of D9 and D9−branesdetermines a pair (E, F ).To classify D−brane charge, we want to classify pairs (E, F ) modulophysical processes. An important process [Sen (1998)] is brane–antibrane


<strong>Geom</strong>etrical Path Integrals and Their Applications 1243creation and annihilation – the creation or annihilation of a set of D9’s andD9’s each bearing the same gauge bundle G. This amounts to(E, F ) ↔ (E ⊕ G, F ⊕ G). (6.360)The equivalence classes make up a group called K(X) (or K(A(X))if we want to make the interpretation in terms of A(X)−modules moreexplicit). The addition law in this group is just(E, F ) + (E ′ F ′ ) = (E ⊕ E ′ , F ⊕ F ′ ).The inverse of (E, F ) is (F, E); note that (E, F ) ⊕ (F, E) = (E ⊕ F, E ⊕ F ),and using the equivalence relation (6.360), this is equivalent to zero.D−branes of Type IIB carry conserved charges that take values in K(X)[Witten (1998c)]. In the above definition of K(X), we used only ninebranes,even though, as we explained earlier, an A(X) module can be constructedusing Dp−branes (or antibranes) for any p. In fact, we can classifyD−brane charge just using the ninebranes, and then build the Dp−branesof p < 9 via pairs (E, F ) with a suitable tachyon condensate.Wherever one looks closely at topological properties of RR charges (orfields), one sees effects that reflect the K−theory structure. For example,there are stable D−brane states (like the nonsupersymmetric D0−branesof Type I) that would not exist if D−brane charge were classified by cohomologyinstead of K−theory. Conversely, it is possible to have a D−branestate that would be stable if D−brane charge were measured by cohomology,but which is in fact unstable (via a process that involves nucleation ofD9−D9 pairs in an intermediate state). This occurs in Type II superstringtheory, in which a D−brane wrapped on a homologically nontrivial cyclein space–time is in fact in certain cases unstable.According to Witten, there is a deeper reason that it is good to knowabout the K−theory interpretation of D−branes: it may be naturallyadapted for stringy generalizations. In fact, we can define K(X) in termsof representations of the algebra A(X) of functions on space–time. Wecan similarly define K(A) for any noncommutative algebra A, in terms ofpairs (E, F ) of A−modules. By contrast, we would not have an equallyuseful and convenient notion of ‘cohomology’ if the algebra of functions onspace–time is replaced by a noncommutative ring.For example, turning on a Neveu–Schwarz B–field, we can make A(X)noncommutative; the associated K(A) was used by Connes, Douglas, andSchwarz in the original paper on noncommutative YM–theory applied to


1244 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionstring theory [Connes et. al. (1997)]. This is an interesting example, eventhough it involves only the zero modes of the strings. One would muchlike to have a fully stringy version involving a noncommutative algebraconstructed using all of the modes of the string, not just the zero modes.At this stage, we do not know what is the right noncommutative algebrathat uses all of the modes. One concrete candidate is the ∗−algebra of openstring field theory, defined in terms of gluing strings together. If we call thisalgebra A st , it seems plausible that D−brane charge is naturally labelledby K(A st ). For a manifold of very large volume compared to the stringscale, we would conjecture that K(A st ) is the same as the ordinary K(X)of topological K−theory.K−Theory and RR–FieldsNaively speaking, an RR p−form field G p obeys a Dirac quantization lawaccording to which, for any p−cycle U in space–time,∫G p= integer. (6.361)2πUIf that were the right condition, then RR fields would be classified by cohomology.However, that is not the right answer, because the actual quantizationcondition on RR periods is much more subtle than (6.361). Thereare a variety of corrections to (6.361) that involve space–time curvature andthe gauge fields on the brane, as well as self-duality and global anomalies.The answer, for Type IIB superstrings, turns out to be that RR fieldsare classified by K 1 (X). For our purposes, K 1 (X) can be defined as thegroup of components of the group of continuous maps from X to U(N), forany sufficiently large N. This means that topological classes of RR fields onX are classified by a map U : X → U(N) for some large N. The relation ofG p to U is roughly G p ∼ Tr (U −1 dU) p , where we have ignored correctionsdue to space–time curvature and subtleties associated with self-duality ofRR fields [Fabinger and Horava (2000)].The physical meaning of U is not clear. For Type IIA, the analog isthat RR fields are classified by a U(N) gauge bundle (for some large N)with connection A and curvature F A , the relation being G p ∼ Tr F p/2A. Theanalog for M−theory involves E 8 gauge bundles with connection. Again,the physical meaning of the U(N) or E 8 gauge fields is not clear.The value of using K 1 to classify RR fields of Type IIB is that this givesa concise way to summarize the otherwise rather complicated quantization


<strong>Geom</strong>etrical Path Integrals and Their Applications 1245conditions obeyed by the RR fields. In addition, this framework is usefulin describing subtle phase factors that enter in the RR partition function.In hindsight, once it is known that RR charges are classified by K−theory,one should have suspected a similar classification for RR fields. After all,RR charges produce RR fields! So the geometry used to classify RR chargesmust be similar to the geometry used to classify RR fields.Just like K(X), K 1 (X) has an analog for any noncommutative algebraA. Given A, we let A N denote the group of invertible N × N matriceswhose matrix elements are elements of A. Then K 1 (A) is the group ofcomponents of A N , for large N.For example, for A = A(X) the ring of complex-valued continuousfunctions on X, A N is the group of maps of X to GL(N, C). This iscontractible to the group of maps of X to U(N), so for large N the groupof components of A N is the same as K 1 (X), as we defined it initially.The existence of a generalization of K 1 (X) for noncommutative ringsmeans that the description of Type IIB RR fields by K 1 (X) in the longdistance limit may be a useful starting point for stringy generalizations.N −→ ∞In the previous subsection, N was a sufficiently large but finite integer. Ournext task will be to describe some things that depend on setting N equalto infinity.Before doing so, let us recall the role of the N −→ ∞ limit in physics.It is important in the conjectured link of gauge theory with strings; in theold matrix models that are used to give soluble examples of string theory;in the matrix model of M−theory; and in the correspondence betweengravity in an asymptotically AdS space–time and conformal field theory onthe boundary.For Type IIB superstrings, [Witten (1998c); Witten (2000)] used K(X)to classify RR charges, and K 1 (X) to classify RR fields.The T −dual statement is that for Type IIA, K 1 (X) should classifyRR–charges, and K(X) should classify RR fields. Recall that, by Bottperiodicity, K i+2 (X) = K i (X), so the only K−groups of X are K 0 (X),which we have called simply K(X), and K 1 (X).The most concrete and natural attempt to explain in general why K 1 (X)classifies RR charges for Type IIA is that of [Horava (1999)]. The startingpoint here is to consider a system of N unstable D9−branes of Type IIA.The branes support a U(N) gauge field and a tachyon field T in the adjoint


1246 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionrepresentation of U(N). There is a symmetry T −→ −T .The effective potential for the tachyon field is believed to have the generalformV (T ) = 1g stTr F (T ),where the function F (T ) is non-negative and, after scaling T correctly,vanishes precisely if T = ±1. Hence V (T ) is minimized if and only if everyeigenvalue of T is ±1.It was argued in [Horava (1999)] that, in flat R 10 , one can make supersymmetricDp−branes (for even p) as solitons of T . For example, to makea D6−brane, we set N = 2. Let ⃗x be the three coordinates normal to theD6−brane, and setT =⃗σ · ⃗x|x| f(|x|),where f(r) ∼ r for small r, and f(r) → ∞ for r → ∞. So for |x| → ∞,the eigenvalues of T are everywhere ±1. Near x = 0, there is a topologicalknot that we interpret as the D6−brane.In flat R 10 , one can similarly make Dp−branes for other even p. Buton a general space–time, this does not work for arbitrary Dp-branes unlesswe set N = ∞. The problem is most obvious if X, or at least its spatialpart, is compact. The tachyon field T , being adjoint-valued, maps X tothe Lie algebra of U(N); since the Lie algebra is contractible, T carriesno topology. So a map from X to the Lie algebra does not represent anelement of K 1 (X); indeed, it does not carry topological information at all.To define an element of K 1 (X), we need the group, not the Lie algebra; amap U : X −→ U(N) does the job.As Atiyah and Singer showed long ago, we get back the right topologyfrom the Lie algebra if we set N = ∞! We have to interpret U(∞) to bethe unitary group U of a separable Hilbert space H of countably infinitedimension. We interpret the N = ∞ analog of the space of hermitian N ×Nmatrices to be the space of bounded self-adjoint operators T on H whosespectrum is as follows: there are infinitely many positive eigenvalues andinfinitely many negative ones, and zero is not an accumulation point of thespectrum. The last condition makes T a Fredholm operator. Physically, Tshould be required to obey these conditions, since they are needed to makethe energy and the D8−brane charge finite. In fact, to make the energyfinite, almost all the eigenvalues of T are very close to ±1. Anyway, with


<strong>Geom</strong>etrical Path Integrals and Their Applications 1247these conditions imposed on T , it turns out that the space of T ’s has thesame topology as that of U(N) for large N.So we can use tachyon condensation on a system of D9−branes to describeRR charges for Type IIA. But we have to start with infinitely manyD9−branes, which then undergo tachyon condensation down to a configurationof finite energy.Let us now explain in more concrete terms the obstruction to makingD−branes in this way for finite N, and how it vanishes for N = ∞. Letus go back to the example of a D6−brane constructed with 2 D9−branes.We took the transverse directions to be a copy of R 3 , and the tachyon fieldto beT =⃗σ · ⃗x|x| f(|x|).If we try to compactify the transverse directions to § 3 , we run into troublebecause T is not constant at infinity. The conjugacy class of T is constantat infinity – the eigenvalues of T are everywhere 1 and -1 – but T itself isnot constant.Moreover, T is not homotopic to a constant at infinity. If T were homotopicto a constant near infinity, we would deform it to be constant andthen extend it over § 3 . But it is not homotopic to a constant.The basic obstruction to making T constant at infinity is the ‘magneticcharge’. Let § 2 be a sphere at infinity in R 3 . Over § 2 , we can define aline bundle ̷L + whose fiber is the +1 eigenspace of T , and a line bundle̷L − whose fiber is the -1 eigenspace of T . The line bundles ̷L + and ̷L − aretopologically nontrivial – their first Chern classes are respectively 1 and -1.As long as we try to deform T preserving the fact that its eigenvalues are1 and -1, the line bundles ̷L ± are well-defined, and their first Chern classesare invariant. So the nontriviality of ̷L + (or ̷L − ) prevents us from makinga homotopy to constant T .Let us add some additional ‘spectator’ D9−branes, and see if anythingchanges. Suppose there are M = 2k additional branes, so that the totalnumber of branes is N = 2 + M = 2 + 2k. Let the tachyon field beT ′ = T ⊕ U, where T is as above and U is the sum of k copies of the matrix( ) 1 0(6.362)0 −1acting on the 2k additional branes.Thus T ′ has near infinity k + 1 eigenvalues +1 and k + 1 eigenvalues -1.


1248 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe +1 eigenspace of T ′ is a vector bundle V + of first Chern class 1 (since itis constructed by adding a trivial bundle to ̷L + ), and the -1 eigenspace of T ′is similarly a vector bundle V − of first Chern class -1. In particular, V + andV − are nontrivial, so we have not gained anything by adding the spectatorbranes: T ′ is not homotopic to a constant, and cannot be extended overinfinity. The nontriviality of V + is controlled by π 1 (U(k + 1)) = Z, whichis associated with the existence of a first Chern class.Instead, what happens if we set k = ∞? To be more precise, we takethe number of spectator D9−branes to be countably infinite, and assumeT ′ = T ⊕ U, where U is the direct sum of countably many copies of thematrix in (6.362). We can still define the bundles V + and V − ; their fibersare separable Hilbert spaces (that is, Hilbert spaces of countably infinitedimension). U(k + 1) is replaced by U, the unitary group of a separableHilbert space. Now we run into the fundamental fact (Kuiper’s Theorem)that U is contractible; its homotopy groups are all zero. Thus, any bundleof separable Hilbert spaces is trivial. In particular, V + and V − are trivial,so T ′ is homotopic to a constant and can be extended over infinity.So if the total number of unstable D9−branes is N = ∞, we can makea D6−brane localized at a point in § 3 . More generally, in view of the resultof Atiyah and Singer, we can starting at N = ∞ build an arbitrary class inK 1 (X) via tachyon condensation.In terms of applying this result to physics, there are a few issues that weshould worry about. One question is simply whether it is physically sensibleto start with infinitely many branes and rely on tachyon condensation toget us down to something of finite energy.Quite a different question is whether the answer that we have obtainedby setting N = ∞ is the right one for physics. In the field of a D6−branethat is localized at a point on § 3 , the equation for the RR two-form field G 2(of which the D6−brane is a magnetic source) has no solution, since “theflux has nowhere to go.”It seems that the situation is that N = ∞ corresponds to the correctanswer in classical open string theory, where the effective action comesfrom world–sheets with the topology of a disc. The RR fields enter as acorrection of relative order g st (the closed string coupling constant) comingfrom world–sheets with cylinder topology, and should be ignored in theclassical approximation.The classification of D−branes by brane creation and annihilation holdsat g st = 0, and leads for Type IIB to a classification of D−brane chargeby K(X). To get the analogous answer, namely K 1 (X), for Type IIA via


<strong>Geom</strong>etrical Path Integrals and Their Applications 1249unstable D9−branes and tachyon condensation, we need to start at N = ∞.Intuitively, in the absence of tachyon condensation, N = ∞ shouldcorrespond to g st = 0, since the effective expansion parameter for openstrings is g st N. If N is infinite, then prior to tachyon condensation, g stmust be zero, or the quantum corrections diverge. If we want g st to benonzero, we need tachyon condensation to reduce to an effective finite valueof N.A somewhat analogous problem is to consider D−branes when theNeveu–Schwarz 3–form field H is topologically nontrivial. We will carryout this discussion in Type IIB (for Type IIA, we would have to combinewhat follows with what we said above in the absence of the H−field).Just as at H = 0, we would like to classify D−brane states by pairs(E, F ) (where E is a D9 state and F is a D9 state) subject to the usualsort of equivalence relation. But there is a problem in having a D9 state inthe presence of an H−field.In fact, when H is topologically non-trivial, one cannot have a singleD9−brane. On the D9−brane, there is a U(1) gauge field with field strengthF . The relation dF = H shows, at the level of de Rham cohomology,that H must be topologically trivial if a single D9−brane is present. Thisconclusion actually holds precisely, not just in de Rham cohomology.There is a special case in which there is a comparatively elementarycure for this difficulty. If H is torsion, that is if there is an integer M > 0such that MH is topologically trivial, then it is possible to have a set of MD9−branes whose ‘gauge bundle’ actually has structure group U(M)/Z M ,rather than U(M). (The obstruction to lifting the U(M)/Z M bundle toa U(M) bundle is determined by H.) We will call such a gauge bundle atwisted bundle. More generally, for any positive integer m, we can have N =mM D9−branes with the structure group of the bundle being U(mM)/Z M .In such a situation, D−brane charge is classified, as one would guess, bypairs (E, F ) of twisted bundles (or D9 and D9 states) subject to the usualequivalence relation. The equivalence classes make a group K H (X).If one wishes to interpret K H (X) as the K−theory of representationsof an algebra, one must pick a particular twisted bundle W and considera D−brane state with boundary conditions determined by W . The W −Wopen strings transform in the adjoint representation, so the gauge parametersof the zero mode sector of the open strings are sections of W ⊗ ¯W . Noticethat although W is a twisted bundle (with structure group U(M)/Z Mrather than U(M)), W ⊗ ¯W is an ordinary bundle, since the center actstrivially in the adjoint representation.


1250 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionThe sections of W ⊗ ¯W form an algebra, defined as follows: if s i j and t k lare sections of W ⊗ ¯W – where the upper and lower indices are respectivelyW − and ¯W −valued – then their product is (st) i l = ∑ k si kt k l. This isthe algebra A W (X) of all endomorphisms or linear transformations of thebundle W . The algebra of open string field theory, for W −W open strings,reduces to A W if one looks only at the zero modes of the strings. This is asensible approximation at low energies in a limit in which X is very largecompared to the string scale.If H is zero and W is a trivial rank one complex bundle, then A W (X)is our friend A(X). If H is zero and W is a trivial rank N complex bundle,then including W means simply that there are N ×N Chan–Paton matriceseverywhere. So in this case, A W (X) = A(X) ⊗ M N , where M N is thealgebra of N × N complex-valued matrices. In general, whatever H is, Wis always trivial locally, so locally A W (X) is isomorphic to A(X) ⊗ M N .A twisted bundle is equivalent to an A W −module, and the groupK H (X) of pairs (E, F ) of twisted bundles (modulo the usual equivalence)coincides with K(A W ), the K−group of A W −modules. This assertionleads to an immediate puzzle; K H (X) as defined in terms of pairs (E, F )of twisted bundles is manifestly independent of W while K(A W ) appearsto depend on W . Indeed, given any two distinct twisted bundles W andW ′ , the corresponding algebras A W and A W ′ are distinct, but at the sametime Morita–equivalent.So far, we have only considered the case that H is torsion. A typicalexample, important in the AdS/CFT correspondence, is the space–timeX = AdS 5 × RP 5 , where a torsion H−field on RP 5 is used to describeSp(n) rather than SO(2n) gauge theory in the boundary CFT.However, in most physical applications, H is not torsion. In that case,we must somehow take a large M limit of what has been said above. Theright way to do this has been shown by [Bouwknegt and Mathai (2000)].The suitable large M limit of U(M)/Z M is P U(H) = U(H)/U(1). In otherwords, for M = ∞, one replaces U(M) by the unitary group U(H) of aseparable Hilbert space H; and one replaces Z M by U(1). This means, inparticular, that when H is not torsion, one cannot have a finite set of D9−or D9−branes, but one can have an infinite set, with a suitable infiniterank twisted gauge bundle E or F . Then D−brane charge is classifiedby the group K H of pairs (E, F ) modulo the usual equivalence relation.A detailed explanation can be found in [Bouwknegt and Mathai (2000)].Here, the Kuiper’s Theorem – the contractibility of U = U(H) – plays an


<strong>Geom</strong>etrical Path Integrals and Their Applications 1251important role.This construction, in the M = ∞ limit, has the beautiful property,explained in [Bouwknegt and Mathai (2000)], that the noncommutativealgebra whose K−group is K H is unique, independent of any arbitrarychoice of twisted bundle W or W ′ . This really depends on the number ofD9 and D9−branes being infinite.


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Index1−jet bundle, 8271−jet lift, 8051−jet space, 800, 8022−jet lift, 8202−jet space, 800, 819c−split, 650k−jet, 797k−jet space, 822absolute covariant derivative, 273, 280absolute time derivative, 156action, 23, 24, 83, 1098, 1181, 1194action of a Lie group, 207action paradigm, 87action principle, 24, 84action–amplitude picture, 983, 1079action–angle variables, 344, 365, 370adaptive path integral, 997, 1080,1082, 1083adjoint action, 671adjoint bundle, 917adjoint functors, 111adjoint Jacobi equation, 572admissible Dynkin diagrams, 242affine 1−jet bundle, 827affine bundle, 500affine connection, 19, 292, 808affine control system, 556affine information structure, 334affine jet bundle, 803AG–structures, 693agent’s momentum phase–space, 603agent’s velocity phase–space, 603algebra homomorphism, 202algebra of noncommutative functions,460algebraic cycle theory, 511algebraic K–theory, 510algebraic properties, 28all, 638almost complex structure, 656Ambrose–Singer Theorem, 269American option, 1035Amid I peptide groups, 381amplitude, 970, 991analytic map, 28anti–self–dual basis, 681antipode, 463any, 649, 650approximate feedback linearization,545arc–element, 275area functional, 877Arnold conjecture, 28Ashtekar connection, 709Ashtekar phase–space formalism, 708associated graded algebra, 671associativity of morphisms, 103associator, 133Atiyah axioms, 302Atiyah–Bott theorem, 653Atiyah–Singer Index Theorem, 518atlas, 2, 6, 144attracting set, 3421295


1296 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionautomorphism of the Hamiltonianstructure, 629autonomous agents, 603Axioms of Lie super–algebra, 671background fields, 887Baker–Campbell–Hausdorff formula,29, 461Banach manifold, 147Banach space, 146basin of attraction, 341Batalin–Vilkovisky quantization,1119, 1125Bekenstein–Hawking entropy, 1227Bekenstein–Hawking formula, 294,709Bendixon criterion, 343Betti numbers, 5, 28, 105, 188, 1076Bianchi covariant derivative, 290, 379Bianchi identity, 21, 35, 528, 743, 913Bianchi relation, 124dual, 124extended, 125Bianchi symmetry condition, 278biholomorphism, 429bijection, 2black hole dynamics, 1225Black–Scholes–Merton formula, 1025blow–down map, 653body–fixed frame, 354Bogomol’nyi monopoles, 1119Bolza problem, 570Borel–Hirzebruch theorem, 653Bose–Einstein condensate, 992Bose–Einstein statistics, 34bosonic string theory, 40, 1146Bott Periodicity, 494Bott Periodicity Theorem, 509, 516Bott tower, 653boundary operator, 183BPS formula, 736braided monoidal category, 129brain–like control functor, 573branched polymer, 301brane, 1143, 1182brane world–sheet, 1171brane–world scenario, 37brane–world theory, 45Brouwer degree, 284Brownian dynamics, 986Brownian motion, 1026BRST quantization, 1126BRST symmetry, 1054, 1197BRST–operator, 1105, 1120bulk, 37, 41bundle diffeomorphism, 490Burgers dynamical system, 268Calabi–Yau manifold, 45, 436Calabi–Yau manifolds, 1180, 1186,1187calculus of variations, 23, 84calibrated submanifolds, 435calibration form, 435Campbell–Baker–Hausdorff formula,195canonical 3–form, 843canonical lift, 608canonical momenta indicators, 1019canonical Poisson structure, 843canonical polysymplectic form, 905canonical soldering form, 612canonical tangent–valued 1–form, 611canonical transformation, 336Cartan equations, 842Cartan magic formula, 198Cartan relation, 610Cartan subalgebra, 242, 734Cartan Theorem, 263Cartesian product, 4Casimir effect, 45Casimir form, 368Casimir function, 775Casson invariant, 1119categorification, 127category, 28, 116category of Hopf algebras, 462Cauchy Theorem, 351, 1208Cauchy–Riemann equation, 655Cauchy–Riemann equations, 428cellular chain complex, 640chain rule, 182


Index 1297Chan–Paton factor, 1238Chan–Paton gauge bundle, 1242change of coordinates, 7chaotic behavior, 506Chapman–Kolmogorov equation, 988,1030Chapman–Kolmogorovintegro–differential equation, 989Chapman–Kolmogorov law, 165, 173characteristic classes, 511chart, 2, 6Chech cohomology, 509Chern character, 511, 517, 522Chern class, 761, 1122, 1133, 1192,1230Chern classes, 511, 516Chern form, 512Chern–Pontryagin Lagrangian, 927Chern–Simons action, 35Chern–Simons gauge theory, 35, 924,1098, 1193, 1224Chern–Simons Lagrangian, 35, 925Chern–Simons Lagrangian density,893, 926Chern–Simons theory, 1103chirality operator, 675Christoffel symbols, 20, 48, 60, 274,290, 389, 434, 559, 834, 872, 936circle, 2circle bundle, 530circle group, 357classical Lie algebra, 770classical theory of Lie groups, 235Clebsch–Gordan condition, 715Clifford, 5Clifford action, 668, 726Clifford algebra, 666, 727Clifford algebras, 958Clifford module, 668closed form, 182closed manifold, 10closed string theories, 1140co–area formula, 1070cobasis, 179coboundary operator, 517cocycle condition, 489, 492codifferential, 191coframing, 179coherence laws, 133cohomology, 182cohomology class, 792cohomology theory, 508collision detection, 262commutative monoid, 508commutative ring, 14, 509commutator subalgebra, 653compact Abelian Lie group, 29compact manifold, 4compactness, 29complementary map, 803complex Clifford algebra, 958complex coordinates, 434complex Laplacian, 439complex manifold, 428complex phase–space manifold, 348complex structure, 429, 432complexified tangent space, 430composite bundle, 960composite connection, 814composite fibration, 814conatural projection, 154configuration bundle, 825, 830configuration manifold, 137, 143, 150,801configuration space of sections, 801conformal field theory, 1145conformal invariance, 1183conformal Killing form, 442conformal Killing tensor–field, 443conformally flat gauge, 1183conjugation, 672connected components, 9connected Lie group, 246connected space, 9connectedness, 28connection, 16, 34, 467, 825connection form, 625connection homotopy, 278connection matrix, 455conservation law, 879constant of motion, 339


1298 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionconstrained Hamiltonian equations,901contact form, 880contact forms, 804contact manifold, 27, 880, 882contact map, 803contact transformation, 879contraction, 178, 670contravariant acceleration functor,385control affine system, 538control vector–fields, 537controlled trajectory, 538controlled Van der Pol oscillator, 543coordinate 1−forms, 175coordinate ball, 145coordinate chart, 145coordinate domain, 145coordinate map, 145coproduct, 463correlation functions, 1118correspondence principle, 991cosmology, 974cotangent bundle, 25, 153cotangent space, 15, 153counit, 463coupling constant, 43coupling form, 626, 638covariant derivative, 467, 810covariant differential, 810, 838covariant differentiation, 273covariant force functor, 49, 384, 573,1097covariant force law, 76, 77, 384, 573,1097covariant forces, 290covariant formalism on smoothmanifolds, 46covariant Hamiltonian, 900covariant Hamiltonian equations, 900cover, 2covering space, 487covertical connection, 809Coxeter graph, 243Coxeter number, 525Coxeter–Dynkin diagram, 243critical point, 312cross–section, 150cumulative distribution function, 985current conservation, 35curvature, 5curvature form, 512curvature of a connection, 811curvature operator, 278curvature–free connection, 812cuspy patterns, 1072CW–complex, 14D’Alambert–Lagrangian principle,535D–branes, 36, 41Darboux’s Theorem, 459de Rham cohomology, 435, 508, 513,636de Rham cohomology class, 336de Rham cohomology group, 182,187, 844de Rham complex, 185de Rham Theorem, 186, 187deformed Einstein–Hilbert action, 472defuzzification, 586degree of non–holonomicity, 534derivation, 163deterministic chaos, 391diffeomorphism, 7, 29, 39, 148, 429diffeomorphism invariance, 707differentiability, 13differentiable manifold, 6, 12differentiable manifolds, 8differential system, 534dilaton, 44dilaton field, 1147dimension, 9Dimension Axiom, 508Dirac condition, 623Dirac constraint systems, 837Dirac equation, 442, 761Dirac Hamiltonian, 1170Dirac matrices, 200Dirac operator, 305, 726, 758Dirac quantization, 526, 614Dirac quantization rule, 87


Index 1299Dirac–Born–Infeld theory, 1239directional derivative, 29Dirichlet boundary conditions, 36Dirichlet branes, 1148disjoint union, 657dissipative structures, 983distribution, 473, 533distribution function, 984divergence of the stress tensor, 81divergence term, 879Dolbeault cohomology, 438Donaldson polynomials, 1118Donaldson theory, 757, 1103, 1118drift vector–field, 537dual Coxeter number, 776Duffing oscillator, 342, 347dynamical chaos, 1072dynamical connections, 827dynamical equation, 825dynamical intuition, 143Dynkin diagram, 240Dynkin index, 776edges, 5effective group action, 208Ehresmann connection, 572, 807Eilenberg–MacLane space, 521, 642Eilenberg–Steenrod Axioms, 508Einstein, 31Einstein equation, 87, 138, 142, 951,965, 1056Einstein tensor, 22Einstein–Hilbert action, 32, 87, 300,469, 972, 1056electro–weak interaction, 34elliptic Calogero–Moser system, 769elliptic geometry, 5emotion field, 603energy conservation law, 928energy function, 26energy functional, 317equal probabilities, 86equations of mathematical physics,473equivariance condition, 531equivariant theory , 518Erlangen programme, 245Euclidean 3D space, 5Euclidean chart, 144Euclidean geometry, 1Euclidean image, 144Euclidean metric, 147, 979Euclidean nD space, 5Euclidean spaces, 2, 7, 8Euclidean triangulations, 1067Euclidean–Schwarzschild metric, 970Euler, 5Euler angles, 28Euler beta function, 40Euler character, 1118Euler characteristic, 5, 28, 271, 284,1072, 1075, 1096, 1188Euler characteristics, 515Euler class, 514, 1117Euler number, 973Euler–Lagrange equations, 22Euler–Lagrangian equations, 84, 278,879, 900, 1183Euler–Lagrangian functionalderivative, 257Euler–Lagrangian operator, 842Euler–Poincaré characteristics, 188Euler–Poincaré equations, 291European option, 1026evolution operator, 165exact form, 182exact sequence, 508exponential, 672exponential law, 30exponential map, 29, 205, 248exponentiation of a vector–field, 247extended supersymmetry, 35extensive quantities, 473exterior algebra, 669exterior bundle, 493exterior calculus, 16exterior derivative, 180exterior differential, 609exterior differential forms, 174exterior differential system, 179, 879exterior horizontal form, 612exterior powers, 435


1300 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionexterior product, 669external configuration manifold, 47extraordinary cohomology theory, 508extremal, 25, 84extremal path, 84extrinsic view, 4faces, 5Faddeev–Popov procedure, 1107, 1126family of probability distributions,333Faraday line, 706Faraday tensor, 124feature–space, 159feedback linearization, 539Fermi–Dirac statistics, 34Feynman, 31Feynman diagram, 1142Feynman kernel, 996Feynman path integral, 86, 983, 989,994, 1079Feynman quantization, 85Feynman–Vernon formalism, 1041fibre, 150, 487fibre bundle, 15fibre–derivative, 258field strength, 1234fields, 84filtration, 670final state, 85finite–time probability distribution,1033Finsler curvature tensor, 319Finsler energy function, 316Finsler information structure, 334Finsler manifold, 316Finsler metric, 334Finsler–Lagrangian field theory, 321first Cauchy law of motion, 81first integral, 339first superstring revolution, 41first variation, 878first variational formula, 841, 883,884, 919first–order Lagrangian formalism,801, 929, 957first–stage reducible, 1125fixed point, 340flag, 534flow, 172, 607flow line, 168flow property, 172flux class, 627flux group, 625flux homomorphism, 644Fock space, 527, 1148Fock state, 991Fokker–Planck, 87Fokker–Planck equation, 591, 988,989, 1023foliation, 533, 553force equation, 342force–field psychodynamics, 1079forced Lagrangian equation, 535formal exponential, 173four fundamental forces, 33Fourier transform, 461Frölicher–Nijenhuis bracket, 613, 913frame bundle, 530Fredholm operator, 520, 1114, 1246free group action, 208free motion equation, 832free string, 1139Freed–Hopkins–Teleman Theorem,526Frobenius Theorem, 553Frobenius–Cartan criterion, 263functional, 83functional manifold, 268Fundamental Theorem of calculus,482Galilei group, 210gauge condition, 1014gauge field, 34, 527gauge form, 513gauge group, 30, 513, 731, 869gauge Lie group, 527gauge potentials, 921gauge theories, 34gauge transformation, 35, 490gauge transformations, 1234


Index 1301Gauss, 5Gauss map, 284Gauss–Bonnet formula, 271, 284,1077Gauss–Bonnet Theorem, 188, 512Gauss–Kronecker curvature, 1077Gaussian curvature, 271, 1061general linear group, 211general linear Lie algebra, 196general principal vector–fields, 896general relativity, 292, 448general theory of systems, 393generalized function, 473generalized vector–field, 256generic energy condition, 966genus, 725, 1076geodesic, 17, 25, 170, 278geodesic deviation equation, 139, 140geodesic equation, 20, 278, 825geodesic flow, 290geodesic spray, 170, 290geometrical intuition, 144geometrical invariance group, 869geometrodynamical functor, 1082gerbe, 524ghost number, 1149global space–time hyperbolicity, 964graph, 485Gravity, 33Green–Gauss–OstrogradskyDivergence Theorem, 480Green–Schwarz bosonic string theory,1181Green–Schwarz mechanism, 41Gromov–Witten invariants, 632, 654Grothendieck Additivity Axiom, 515Grothendieck group, 508Grothendieck’s Riemann–RochTheorem, 516group action, 98group homomorphism, 28group identity element, 203group inversion, 203group multiplication, 203group of rotations, 30group of symplectomorphisms, 625group orbit space, 208Haar measure, 207Hairy Ball Theorem, 513Hamel equations, 292Hamilton’s equations, 27Hamilton’s principle, 23Hamilton–de Donder equations, 843,852Hamilton–Jacobi equation, 27Hamiltonian, 24, 26, 84Hamiltonian action, 372Hamiltonian automorphism, 628Hamiltonian bundle, 624Hamiltonian connection, 840, 844,845, 931Hamiltonian conservation laws, 846Hamiltonian density, 84Hamiltonian dynamics, 154Hamiltonian energy function, 338Hamiltonian equations, 840Hamiltonian extension class, 628Hamiltonian flow, 27Hamiltonian form, 840, 841, 844, 845,854, 900, 938, 946Hamiltonian jet–field, 938Hamiltonian map, 854Hamiltonian mechanical system, 338Hamiltonian mechanics, 13, 25, 26Hamiltonian structure, 626Hamiltonian vector–field, 26, 338harmonic forms, 436Hausdorff space, 145Hawking’s Euclidean quantumgravity, 302heat equation, 250, 261Heisenberg picture, 1007Heisenberg uncertainty relations, 87Heisenberg’s uncertainty principle, 41helicity, 502Hermitian inner product, 432, 686Hermitian metric, 432Hermitian operator, 85, 87Hermitian Riesz representation, 687,688Hermitian structure, 686


1302 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionHermiticity condition, 432Hessian, 276heuristic action paradigm, 83heuristic quantization rule, 711higher–order contact, 797higher–order tangency, 797Hilbert, 31Hilbert 5th problem, 236Hilbert action principle, 142Hilbert manifold, 147Hilbert space, 85, 87, 147, 992Hirzebruch’s Riemann–RochTheorem, 516Hodge decomposition, 436Hodge diamond, 441Hodge identities, 435Hodge numbers, 440, 1189Hodge star operator, 125, 178, 191,681Hodge Theorem, 439, 520Hodge theory, 417Hodge’s Theorem, 436holomorphic 1–form, 791holomorphic cotangent space, 431holomorphic function, 725holomorphic tangent space, 431holomorphicity constraints, 719holonomic atlas, 163holonomic coframes, 164holonomic connections, 827holonomic fibre bases, 486holonomic frames, 163holonomous frame field, 163holonomy, 125, 1195holonomy group, 738homeomorphism, 6–9, 429homoclinic orbits, 343homological algebra, 104, 510homology group, 105, 183homotopy fibration, 645homotopy lifting property, 486, 738homotopy operators, 186, 546Hopf algebra, 463Hopf–Rinow Theorem, 17horizontal density, 801, 883, 900, 957horizontal distribution, 812horizontal foliation, 812horizontal forms, 611horizontal splitting, 960human crowd, 603human–robot team, 154hyperbolic geometry, 5hyperkähler manifolds, 1187idealdifferential, 881Immirzi parameter, 299imprecision of measurement, 391independence condition, 881index, 655inertial metric tensor, 833infinite–dimensional neural network,1080infinite–order jet space, 822infinitesimal generators, 29, 253information gain, 1018initial state, 85initial–date coordinates, 862inner product, 16inner product space, 992input vector–fields, 537insertion operator, 178integrable Hamiltonian system, 262integrable in the sense of Liouville,771integrable system, 790integral curve, 168integral manifold, 533, 879, 881intensive quantities, 473interior product, 178, 670internal configuration manifold, 46intertwining tensor, 714intrinsic definition for differentiablemanifolds, 6intrinsic view, 4invariant interval, 1173invariant of Poincaré–Cartan, 841invariant symplectic form, 690invariant tori, 343involutive closure, 552involutive distribution, 533irreducible representation, 527


Index 1303isometric, 17isomorphic, 28isotropy group, 208Itô lemma, 1026iterated fibration of Kähler manifolds,653Ito prescription, 1019Ito stochastic integral, 987Jacobi equation of geodesic deviation,280Jacobi fields, 280, 313Jacobi identity, 126, 202, 362Jacobi operator, 879jet, 798, 880jet bundle, 16, 801, 803jet field, 807jet functor, 804jet space, 251, 275, 797Jones polynomial, 1195K–theory, 494, 508, 515Kähler form, 740Kähler manifold, 733Kähler metric, 742Kähler potential, 733, 736Kähler condition, 417Kähler form, 432Kähler gauge, 1225Kähler identities, 435Kähler manifold, 27, 416, 433Kähler metric, 433Kähler potential, 434, 724, 1227Kähler structure, 433Kählerity condition, 433Kaluza–Klein theory, 44Kepler problem, 27Killing equation, 441Killing form, 246, 354, 785Killing spinor–field, 442Killing tensor–field, 443, 447Killing vector–field, 441, 446Killing–Riemannian geometry, 441Killing–Yano equation, 442, 450Killing–Yano tensor, 450Klein bottle, 487Klein–Gordon Lagrangian, 1013Kodaira Embedding Theorem, 435Korteveg–de Vries equation, 262, 598Kostant–Souriau prequantization, 622Kuiper’s Theorem, 521Lagrange multipliers, 1179Lagrange’s equations, 22Lagrange–Poincaré equations, 292Lagrangian, 15, 84, 289, 877Lagrangian constraint space, 841Lagrangian density, 84, 801, 888,1013, 1085, 1175Lagrangian dynamics, 153Lagrangian mechanics, 13, 22, 26Lagrangian submanifold, 28Landau pole, 736Langevin, 87Laplace equation, 252Laplace–Beltrami operator, 191, 277Laplacian operator, 482Laplacian symmetry, 445large, 646laws of motion, 166Lax equation, 771, 780Lax operator, 790Lax pair equation, 450Lax tensor, 450Lax type representation, 262leaf space, 553Lebesgue measure, 207Lefschetz condition, 637Lefschetz Theorem, 436left ideal, 202left–invariant Lagrangian, 291left–invariant Riemannian metric, 354Legendre bundle, 900Legendre map, 258, 370, 841, 843, 885Legendre submanifold, 881transverse, 881, 883Legendre transformation, 24Leibniz rule, 181, 464Lepagean equivalent, 842Leray cohomology spectral sequence,665Leray–Hirsch basis., 663


1304 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionLeray–Serre cohomology spectralsequence, 638level vector, 775Levi–Civita connection, 19, 48, 273,379, 416, 434, 450, 451, 502, 535,559, 572, 728, 957Levi–Civita symbol, 1177Lewinian force–field theory, 1081Lewinian psychodynamics, 602Lie, 6, 28Lie algebra, 27, 29, 196, 362Lie algebra homomorphism, 202Lie algebra simple root, 242Lie bracket, 19, 29, 194, 463, 606Lie bracket property, 256Lie derivative, 29, 163, 192, 338, 446,610, 837, 838, 840Lie Functor, 30Lie functor, 205Lie group, 4, 203Lie product, 553Lie structural constants, 362Lie subalgebra, 202Lie super–algebra, 671, 1105Lie–derivative neuro–classifier, 200Lie–group–homomorphism, 28Lie–invariant geometric objects, 263Lie–Lagrangian biodynamicalfunctor, 385Lie–Poisson bracket, 361Lie–Poisson neuro–classifier, 383lifted action, 373limit set, 341line bundle, 514, 524, 691line bundles, 494linear connection, 807linear controllability, 544linear isotopy, 648linearized Hamiltonian dynamics,1073Liouville equation, 27, 988Liouville integrability, 451Liouville measure, 25Liouville operator, 192Liouville Theorem, 363Liouville’s Theorem, 27Liouville–Arnold Theorem, 367Lipschitz condition, 171local connection form, 815local coordinate system, 6local geodesics, 137locally accessible system, 556locally topologically equivalent, 137loop algebra, 706loop quantum gravity, 527, 702loop representation, 717loop variables, 297Lorentz metric tensor, 1181Lorentz–invariant theories, 1140Lorentzian dynamical triangulations,1059Lorentzian–de Sitter metric, 979Lorenz dynamics, 506Lorenz flow, 506low energy effective action, 1119Lyapunov exponent, 361, 1072Lyapunov stable, 865, 866Möbius strip, 487manifold, 1, 8, 137, 143manifold structure, 145manifold with boundary, 186manifoldness, 5marked symplectic manifold, 630Markov assumption, 988Markov chain, 985Markov stochastic process, 985, 1083Markov–Gaussian stochastic systems,87mass conservation principle, 78Master equation, 988Mathai–Quillen formula, 1115mathematical induction, 6matrix representation, 239Maupertius action principle, 277Maurer–Cartan connection, 1117Maurer–Cartan equations, 263maximal atlas, 7maximal geodesic, 170maximal integral curve, 168Maxwell equations, 33, 124Maxwell gauge field theory, 34


Index 1305Maxwell Lagrangian, 34Maxwell–Dirac duality, 724Mayer–Vietoris sequence, 660Mayer–Vietoris Theorem, 508mean curvature, 877mean–field theory, 1075Melnikov function, 348mental force law, 1097meromorphic 1–form, 725, 736mesoscopic order parameters, 1016metric, 468metric manifold, 705metric SEM–tensor, 924, 930, 936,1198metric space, 17metric tensor, 20midpoint discretization, 1019mirror symmetry, 1189model space, 147module, 1242moduli space, 35, 524, 720, 724momentum constraint equation, 977momentum map, 354, 372, 446momentum phase–space manifold,258, 340, 387, 837, 839, 844, 848monodromy, 738monodromy group, 738monoid, 129monoidal category, 129Montonen–Olive duality, 719, 1119Montonen–Olive mass formula, 722Morita–equivalent, 1250morphism of vector–fields, 173Morse function, 312, 1073, 1077Morse lemma, 1073Morse numbers, 1078Morse theory, 311, 1073, 1077, 1098Moser’s homotopy argument, 632motion planning, 262motivation–behavior conservationlaw, 606motivational factor manifold, 603motivational factor–structure, 603Moyal bracket, 1233Moyal product, 458, 470multi–index, 250, 879multiindex, 177multimomentum Hamiltonian, 938multivector–field, 608Nambu–Goto action, 1145, 1172, 1183natural geometrical structures, 128natural projection, 152Navier–Stokes equations, 83Neumann boundary condition, 1149neural path integral, 1045Neveu–Schwarz B–field, 1235, 1243Newman–Penrose equation, 965Newton’s Second Law, 23Newtonian equation of motion, 166Newtonian fluid, 82Newtonian mechanics, 22Nijenhuis differential, 614, 913Noether conservation lawweak, 923Noether conservation laws, 446Noether current, 923, 925Noether identities, 923Noether Theorem, 24, 879Noether–Lagrangian symmetry, 257Noetherian ring, 100non–Abelian field strength, 1191non–equilibrium nonlinearmultivariate, 1016Non–Euclidean geometry, 5noncommutative coordinates, 459noncommutative gravity theory, 463noncommutative phase–space, 459noncommutative product, 215noncommutative Yang–Mills theory,1234nondegenerate quadratic forms, 958nonholonomic connection, 536nonholonomic coordinates, 275nonlinear control system, 537nonlinear control theory, 1141nonlinear controllability criterion, 551nonlinear factor analysis, 603nonlinear MIMO–systems, 537nonlinear Schrödinger equation, 597nonlinear sigma model, 1145normal bundle, 490


1306 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionnormal vector–field, 167obstruction theory, 642on–shell reducible, 1126one–parameter group ofdiffeomorphisms, 172, 341open manifold, 10open string theories, 1140operator algebras, 511orbifold, 13, 1187order parameter equations, 1016oriented strings, 1141orthogonal group, 30overlap, 7parabolic Einstein equation, 285parallel transport, 20, 139, 290, 379,423, 527parameter–space of probabilitydistributions, 333particles, 84partition function, 1070, 1097, 1194path, 9path integral, 86, 970, 971, 1020path measure, 86, 1017path–connected space, 9path–integral expression, 1011path–integral formalism, 1008path–integral formulation, 24, 1007path–integral quantization, 1007path–ordered exponential, 527Pauli exclusion principle, 34perturbation, 656perturbation theory, 345perturbative path integral, 1055perturbative string theory, 1146Peyrard–Bishop system, 1074Pfaff Theorem, 880Pfaffian forms, 608Pfaffian system, 387, 533, 879phase, 991phase space, 5phase trajectory, 340phase transition, 1069, 1074phase–flow, 341phase–space, 25phase–space path integral, 1003Philip Hall basis, 553Picard groupoid, 524Planck constant, 459Planck length, 36, 1139Poincaré, 6Poincaré conjecture, 6, 150Poincaré duality, 440Poincaré lemma, 73, 74, 182Poincaré maps, 346Poincaré–Cartan form, 842, 882Poincaré–Hopf Theorem, 188Poincaré algebra, 462Poincaré dual, 528Poincaré duality, 516Poincaré–Birkhoff–Witt property, 460point evaluation map, 635Poisson bivector–field, 609Poisson bracket, 26, 27, 84, 260, 339,375, 451, 837, 907Poisson detection statistics, 992Poisson evolution equation, 361Poisson manifold, 27, 361Poisson structure, 459Poisson tensor–field, 369polarization, 676Polyakov action, 40, 1145Polyakov equation, 45polysymplectic phase–space, 840Pontryagin class, 512, 1121Pontryagin Maximum Principle, 570,1020Ponzano–Regge ansatz, 301Ponzano–Regge quantization, 302positive chirality condition, 1122prepoint discretization, 1019prepotential, 722, 725presymplectic Hamiltonian systems,837principal bundle, 488, 529principal connections, 815Principle of Democracy, 975Principle of stationary action, 24probability amplitude, 85, 1007, 1097probability distance, 333probability divergence, 333


Index 1307probability manifold, 333product manifold, 1123projectable vector–field, 612, 823projected connection, 536projective bundles, 517prolongation, 250, 502prolonged group action, 253propagator, 86, 996pull–back, 164pull–back vector bundle, 492pull–back–valued forms, 613push–forward, 164qualitative ODE theory, 360quantization, 85quantum algebra, 619quantum brain modelling, 1079quantum bundle, 622quantum chromodynamics, 34, 40quantum coherent state, 991quantum evolution pictures, 85quantum field theory, 34, 35, 292,527, 1097quantum fields, 86quantum gauge theory, 722quantum geometry, 1180quantum gravity, 38, 1055quantum Hilbert space, 520quantum measure, 1193quantum operator, 716Quantum superposition, 85quantum–mechanical particles, 86quaternions, 215Quillen’s plus construction, 511Quillen–Suslin Theorem, 510quotient representation, 240Ramond–Ramond field, 526random variable, 984random walk, 985rank condition, 548ray bundle, 692Ray–Singer torsion, 1118reachable sets, 538real Lie group, 28reduced curvature 1–form, 270reduced phase–space, 373reducible constraints, 854Regge calculus, 1061Regge geometries, 1057Regge simplicial action, 1063regular divergence, 334regularization, 655Reidemeister moves, 132related vector–fields, 164relative degree, 542relativistic mechanics, 618repeated jet space, 819representation of a Lie group, 239representative point, 143Rho–tensor, 694Ricci flow, 285Ricci scalar curvature, 22Ricci tensor, 22, 140, 272, 278, 435Riemann, 5Riemann curvature tensor, 21, 56, 71,139, 141, 271, 278, 835, 950, 1095Riemann sphere, 18, 510, 513Riemann surface, 17, 433, 725Riemann tensor, 435Riemann–Roch Theorem, 515Riemannian geometry, 19Riemannian manifold, 16, 166, 178Riemannian manifolds, 5Riemannian metric, 379Riemannian metric tensor, 271right translation, 204rigid body with a fixed point, 354root system, 240rotation group, 28S–duality, 44saddle point approximation, 978Sasakian metric, 322, 504scalar curvature, 272, 279scalar Gaussian curvature, 281scalar–field, 79scattering, 976Schouten–Nijenhuis bracket, 608Schrödinger equation, 991, 1002Schrödinger operators, 619Schrödinger picture, 1007


1308 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern IntroductionSchrödinger quantization, 616Schwarz–type TQFT, 1121Schwarzschild metric form, 970second superstring revolution, 44second tangent bundle, 495second tangent maps, 156second variation, 879second variation formula, 281second–countable space, 145sectional curvature, 271Seiberg–Witten gauge theory, 35self–dual basis, 681SEM conservation laws, 935SEM–tensorsHamiltonian, 941Lagrangian, 933semisimple representation, 246separatrix, 343Serre fibration, 645Serre’s Conjecture, 510set, 116shape operator, 1077shear viscosity, 82sheaves, 14signature, 958simple Lie group, 245simplicial set, 135Sine–Gordon equation, 598sliding filament theory of muscularcontraction, 381small–time local controllability, 556small–time locally controllable, 538smooth homomorphism, 203smooth manifolds, 4, 8smoothness, 7Sobolev norm, 521Sobolev–Schwartz functional analysis,473soldering curvature, 811soldering form, 612space of paths, 86space of quantum states, 523space–time, 5space–time manifold, 137special Lagrangian, 437sphere, 3sphere bundle, 487spin, 34spin connection, 708spin connection 1–form, 1122spin foam models, 308Spin group, 672spin network, 713spin network theory, 714spin networks, 1056Spin–connection, 728spinor, 200Spinor bundle, 725spinor bundle, 727spinor contraction, 691spinor Lie group, 956spinor module, 676spinor–twistor object, 699stable equivalence, 509stack, 524Standard Model, 30, 33, 292, 702state feedback, 555state manifold, 537static gauge, 1182Steenrod operation, 522Stiefel–Whitney class, 512stochastic forces, 391stochastic integral, 986Stoke’s Theorem, 529Stokes fluid, 82Stokes formula, 187Stokes Theorem, 480, 514strange homology group, 631Stratonovich prescription, 1019string corrections, 1140string cosmology, 296string coupling constant, 44string length, 1172string tension, 1145string–field, 1224string–field–theory action, 1149strong energy condition, 965structural group, 625structure equations, 287submanifold, 28super–algebra, 667super–field, 1215


Index 1309super–space, 1215supercovariant derivative, 743superstring theory, 35, 1146supersymmetry, 33, 38, 1146support of a vector–field, 172surface forces, 80surface of Earth, 1surgery theory, 511surjection lemma, 651SW–spectral curve, 775symbol map, 670symmetric affine connection, 273symmetric Weyl ordering, 460symmetry, 879symmetry group, 250symplectic bundle, 626symplectic condition, 666symplectic connection form, 649symplectic fibre bundle, 625symplectic foliation, 843symplectic form, 335, 610, 689symplectic group, 30, 335symplectic manifold, 15, 25, 336, 338symplectic manifolds, 5symplectic map, 336symplectic potential, 615symplectic Riesz representation, 689,690symplectic structure, 689symplectic volume form, 25symplectomorphism, 27, 335synthetic differential geometry, 473system variables, 83T–duality, 43tachyon field, 1147tangent bundle, 14, 17, 152tangent dynamics equation, 1072tangent map, 151, 152tangent space, 15, 16, 150tangent vector–field, 167tangent–valued r−forms, 611tangent–valued horizontal form, 612tangle, 129tangle diagram, 131tautological bundle, 510tautological section, 692Taylor’s Theorem, 12tensor, 15, 16tensor algebra, 667tensor bundle, 15, 163, 498tensor product of vector bundles, 509tensor–field, 163tensor–product connection, 812tetrad gravitational field, 955the twisted elliptic CM–systems, 771thermodynamic partition function,971three–body problem, 357time–dependent flow, 166time–dependent mechanics, 830time–dependent state of a quantumsystem, 85time–dependent vector–field, 168, 173Toda chain, 790Toda molecule, 353topological Bogomol’nyi action, 1125topological group, 203topological hypothesis, 1070topological invariant, 5, 1076, 1121,1122topological K–theory, 508, 510topological manifold, 4, 6, 8topological operads, 134topological property, 5topological quantum field theory,1097topological space, 8topological structure, 6topological Theorem, 1072torsion, 19torsion of a connection, 811torsion tensor, 536torus, 3total Hamiltonian, 1176transformationcontact, 878point, 878transformation classical, 878transformation gauge, 878transformation of coordinates, 7transition, 85


1310 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductiontransition amplitude, 993, 1001transition function, 7transition functions, 144, 489transition map, 2, 7transition probability, 993transition probability distribution,1031transitive group action, 208transversal bundle, 533tricategories, 134trivial bundle, 15trivial fibration, 487trivial flux, 628twisted, 773, 782, 783twisted Dirac equation, 1122twisted K–groups, 517twisted K–theory, 515twistor bundle, 695twistor equation, 442twistors, 691unified field theory, 31universal bundle, 653vacuum state, 992vector bundle, 487, 491velocity equation, 342velocity phase–space manifold, 150,258, 386, 824, 830velocity vector–field, 150vertical bundle, 495vertical connection, 809vertical cotangent bundle, 499vertical covariant differential, 815vertical lift, 495vertical tangent bundle, 499vertical vector–field, 612vertical–valued horizontal form, 612vertices, 5vierbein, 468Virasoro operators, 1186visual physical intuition, 32volatility, 1026volume element, 1174volume forces, 80volume form, 191volume viscosity, 82von Neumann dimension theory, 518Wang differential, 665Wang sequence, 664wave psi–function, 85wave–particle duality, 991weak conservation law, 893weak energy condition, 457, 965wedge product, 177weighted function, 691Weitzenböck formula, 728Weyl, 6Weyl deformation quantization, 459Weyl group, 732Weyl homomorphism, 927Weyl invariance, 1183Weyl spinor, 721, 1122Weyl tensor, 141Weyl vector, 776Wheeler–DeWitt equation, 977Whitham deformations, 725Whitney, 6Wick rotation, 970, 1065, 1152Wiener process, 988Wigner function, 1040Wilson line, 527, 1234Wilson loop, 527, 1118, 1195winding number, 350Wirtinger’s inequality, 435Witten, 31, 1120Witten’s TQFT, 1098world–sheet, 1139world–sheet dynamics, 1181world–volume, 36Yang–Baxter equation, 1103Yang–Lee Theorem, 1069Yang–Mills action, 39Yang–Mills covariant derivative, 126Yang–Mills gauge field theory, 34Yang–Mills gauge theory, 13, 899,1053, 1099Yang–Mills Lagrangian, 893, 915,920, 921Yang–Mills relation, 126


Index 1311Yang–Mills theory, 39, 297higher, 127Zamolodchikov metric, 1220Zorn’s lemma, 18

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