Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 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