Ivancevic_Applied-Diff-Geom

486 Applied Differential Geometry: 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