A Short Course in Differential Geometry and Topology

A Short Course in DifferentialGeometry and TopologyA.T. Fomenko and A.S. Mishchenkoс s P

A Short CourseinDifferential Geometryand TopologyA.T. Fomenko and A.S. MishchenkoFaculty of Mechanics and Mathematics, Moscow State UniversityС S PCambridge Scientific Publishers

2009 Cambridge Scientific PublishersCover design: Clare TurnerAll rights reserved. No part of this book may be reprinted or reproduced or utilised inany form or by any electronic, mechanical, or other means, now known or hereafterinvented, including photocopying and recording, or in any information storage orretrieval system, without prior permission in writing from the publisher.British Library Cataloguing in Publication DataA catalogue record for this book has been requestedLibrary of Congress Cataloguing in Publication DataA catalogue record has been requestedISBN 978-1-904868-32-3Cambridge Scientific Publishers LtdP.O. Box 806Cottenham, Cambridge CB24 8RTUKwww.cambridgescientificpublishers.comPrinted and bound in Great Britain byCPI Antony Rowe, Chippenham and Eastbourne

ContentsPrefaceixIntroduction to Differential Geometry 11.1 Curvilinear Coordinate Systems. Simplest Examples 11.1.1 Motivation 11.1.2 Cartesian and curvilinear coordinates 31.1.3 Simplest examples of curvilinear coordinate systems . . 71.2 The Length of a Curve in Curvilinear Coordinates 101.2.1 The length of a curve in Euclidean coordinates 101.2.2 The length of a curve in curvilinear coordinates 121.2.3 Concept of a Riemannian metric in a domain ofEuclidean space 151.2.4 Indefinite metrics 181.3 Geometry on the Sphere and Plane 201.4 Pseudo-Sphere and Lobachevskii Geometry 25General Topology 392.1 Definitions and Simplest Properties of Metric andTopological Spaces 392.1.1 Metric spaces 392.1.2 Topological spaces 402.1.3 Continuous mappings 422.1.4 Quotient topology 442.2 Connectedness. Separation Axioms 452.2.1 Connectedness 452.2.2 Separation axioms 472.3 Compact Spaces 492.3.1 Compact spaces 492.3.2 Properties of compact spaces 502.3.3 Compact metric spaces 512.3.4 Operations on compact spaces 51

viCONTENTS2.4 Functional Separability. Partition of Unity 512.4.1 Functional separability 522.4.2 Partition of unity 543 Smooth Manifolds (General Theory) 573.1 Concept of a Manifold 593.1.1 Main definitions 593.1.2 Functions of change of coordinates. Definitionof a smooth manifold 623.1.3 Smooth mappings. Diffeomorphism 653.2 Assignment of Manifolds by Equations 683.3 Tangent Vectors. Tangent Space 723.3.1 Simplest examples 723.3.2 General definition of tangent vector 743.3.3 Tangent space T Po (M) 753.3.4 Directional derivative of a function 763.3.5 Tangent bundle 793.4 Submanifolds 813.4.1 Differential of a smooth mapping 813.4.2 Local properties of mappings and the differential .... 843.4.3 Embedding of manifolds in Euclidean space 853.4.4 Riemannian metric on a manifold 873.4.5 Sard theorem 894 Smooth Manifolds (Examples) 934.1 Theory of Curves on the Plane and in the Three-Dimensional Space 934.1.1 Theory of curves on the plane. Frenet formulae 934.1.2 Theory of spatial curves. Frenet formulae 984.2 Surfaces. First and Second Quadratic Forms 1024.2.1 First quadratic form 1024.2.2 Second quadratic form 1044.2.3 Elementary theory of smooth curves ona hypersurface 1084.2.4 Gaussian and mean curvatures of two-dimensionalsurfaces 1124.3 Transformation Groups 1214.3.1 Simplest examples of transformation groups 1214.3.2 Matrix transformation groups 1314.3.3 General linear group 1324.3.4 Special linear group 132

CONTENTSvii4.3.5 Orthogonal group 1334.3.6 Unitary group and special unitary group 1344.3.7 Symplectic compact and noncompact groups 1374.4 Dynamical Systems 1404.5 Classification of Two-Dimensional Surfaces 1494.5.1 Manifolds with boundary 1504.5.2 Orientable manifolds 1514.5.3 Classification of two-dimensional manifolds 1534.6 Two-Dimensional Manifolds as Riemann Surfacesof Algebraic Functions 1635 Tensor Analysis 1735.1 General Concept of Tensor Field on a Manifold 1735.2 Simplest Examples of Tensor Fields 1775.2.1 Examples 1775.2.2 Algebraic operations on tensors 1805.2.3 Skew-symmetric tensors 1835.3 Connection and Covariant Differentiation 1895.3.1 Definition and properties of affine connection 1895.3.2 Riemannian connections 1955.4 Parallel Translation. Geodesies 1985.4.1 Preparatory remarks 1985.4.2 Equation of parallel translation 1995.4.3 Geodesies 2015.5 Curvature Tensor 2105.5.1 Preparatory remarks 2105.5.2 Coordinate definition of the curvature tensor 2105.5.3 Invariant definition of the curvature tensor 2115.5.4 Algebraic properties of the Riemannian curvaturetensor 2125.5.5 Some applications of the Riemannian curvaturetensor 2156 Homology Theory 2196.1 Calculus of Differential Forms. Cohomologies 2206.1.1 Differential properties of exterior forms 2206.1.2 Cohomologies of a smooth manifold (de Rhamcohomologies) 2256.1.3 Homotopic properties of cohomology groups 2276.2 Integration of Exterior Forms 2316.2.1 Integral of a differential form over a manifold 231

6.2.2 Stokes formula 2326.3 Degree of a Mapping and Its Applications 2366.3.1 Degree of a mapping 2366.3.2 Main theorem of algebra 2386.3.3 Integration of forms 2396.3.4 Gaussian mapping of a hypersurface 239Simplest Variational Problems of Riemannian Geometry 2417.1 Concept of Functional. Extremal Functions. Euler Equation . . 2417.2 Extremality of Geodesies 2477.3 Minimal Surfaces 2507.4 Calculus of Variations and Symplectic Geometry 253Bibliography 267Index 269

PrefaceA Short Course on Differential Geometry and Topology by ProfessorA.T. Fomenko and Professor A.S. Mishchenko is based on the course taughtat the Faculty of Mechanics and Mathematics of Moscow State University.It is intended for students of mathematics, mechanics and physics and alsoprovides a useful reference text for postgraduates and researchers specialisingin modern geometry and its applications.The course is structured in seven chapters and covers the basic material ongeneral topology, nonlinear coordinate systems, theory of smooth manifolds,theory of curves and surfaces, transformation groups, tensor analysis andRiemannian geometry, theory of integration and homologies, fundamentalgroups and variational problems of Riemannian geometry. All of thechapters are highly illustrated and the text is supplemented by a large numberof definitions, examples, problems, exercises to test the concepts introduced.The material has been carefully selected and is presented in a concise mannerwhich is easily accessible to students.Chapter 1: Introduction to Differential GeometryThis chapter consists of four sections and includes definitions, examples, problemsand illustrations to aid the reader.1.1 Curvilinear Coordinate Systems. Simplest Examples1.2 The Length of a Curve in Curvilinear Coordinates1.3 Geometry on the Sphere and Plane1.4 Pseudo-Sphere and Lobachevskii GeometryChapter 2: General TopologyThis chapter is an introduction to topology and the development of topology.Topology is a field of mathematics which studies the properties of geometricthat are not changed under a "deformation" or other transformations similarto deformations. General topology arose as a result of studying the mostgeneral properties of geometric spaces and their transformations related tothe convergence and continuity properties.ix

xPREFACEThe chapter consists of four sections and includes examples, definitions, problemsand illustrations.2.1 Definitions and Simplest Properties of Metric and Topological Spaces2.2 Connectedness. Separation Axioms2.3 Compact Spaces2.4 Functional Separability. Partition of UnityChapter 3: Smooth Manifolds: General TheoryThis chapter covers the general theory of smooth manifolds, introduces themanifold as a special concept in geometry and includes definitions, examplesand problems to test understanding.3.1 Concept of a Manifold3.2 Assignment of Manifolds in Equations3.3 Tangent Vectors. Tangent Space3.4 SubmanifoldsChapter 4: Smooth Manifolds: ExamplesThis chapter continues the study of smooth manifolds and focuses onexamples.4.1 Theory of Curves on the Plane and in the Three-Dimensional Space4.2 Surfaces4.3 Transformation Groups4.4 Dynamical Systems4.5 Classification of Two-Dimensional Surfaces4.6 Two-Dimensional Manifolds on Riemann SurfacesChapter 5: Tensor Analysis and Riemannian GeometryThis chapter studies local properties of smooth manifolds and includes definitions,illustrations, exercises and examples throughout.5.1 General Concept of Tensor Field on a Manifold5.2 Simplest Examples of Tensor Fields5.3 Connection and Covariant Differentiation5.4 Parallel Translation. Geodesies5.5 Curvature TensorChapter 6: Homology TheoryThis chapter focuses on the properties of manifolds on which other functionsand mapping depend. Examples and illustrations are included throughoutthe chapter.6.1 Calculus of Differential Forms6.2 Integration of Exterior Forms6.3 Degree of a Mapping and its Applications

PREFACExiChapter 7: Simplest Variational Problems of Riemannian GeometryThis chapter begins with the general concept of functional and its variationand subsequent sections cover extremality of geodesies, minimal surfaces, calculusof variations and symplectic geometry.7.1 Concept of Functional. Extremal Functions. Euler Equation7.2 Extremality of Geodesies7.3 Minimal Surfaces7.4 Calculus of Variations and Symplectic GeometryA Short Course in Differential Geometry and Topology is intendedfor students of mathematics, mechanics and physics and also provides a usefulreference text for postgraduates and researchers specialising in moderngeometry and its applications.Selected Problems in Differential Geometry and TopologyA.T. Fomenko, A.S. Mischenko and Y.P. SolovyevISBN: 978-1-904868-33-0 Cambridge Scientific Publishers 2008 is designed asan associated companion volume to A Short Course in Differential Geometryand Topology and is based on seminars conducted at the Faculty of Mechanicsand Mathematics of Moscow State University for second and third yearmathematics and mechanics students. The two volumes complement eachother and can be used as a two volume set.

A Short Course in DifferentialGeometry and TopologyA.T. Fomenko and A.S. MishchenkoMoscow State UniversityThis volume is intended for graduates and research students in mathematicsand physics. It covers general topology, nonlinear co-ordinate systems, theoryof smooth manifolds, theory of curves and surfaces, transformation groups,tensor analysis and Riemannian geometry, theory of integration and homologies,fundamental groups and variational principles in Riemannian geometry. The textis presented in a form that is easily accessible to students and is supplemented bya large number of examples, problems, drawings and appendices.ABOUT THE AUTHORSAnatoly T. Fomenko is a full member of the Russian Academy of Sciences andProfessor, Head of Department of Differential Geometry and Applications,Faculty of Mathematics and Mechanics at Moscow State University. He is a wellknownspecialist and the author of fundamental results in the fields of geometry,topology, multidimensional calculus of variations, Hamiltonian mechanics andcomputer geometry.Alexander S.Mishchenko is Professor in the Department of Higher Geometryand Topology at the Faculty of Mathematics and Mechanics, Moscow StateUniversity. He is a well-known author and specialist in the field of algebraictopology, symplectic topology, functional analysis, differential equations andapplications.С S PCAMBRIDGE SCIENTIFIC PUBLISHERS