Ivancevic_Applied-Diff-Geom

530 Applied Differential Geometry: 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.