Ivancevic_Applied-Diff-Geom

Chapter 4Applied Bundle Geometry4.1 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