Ivancevic_Applied-Diff-Geom

498 Applied Differential Geometry: A Modern Introduction4.3.3 Vertical Tangent and Cotangent Bundles4.3.3.1 Tangent and Cotangent Bundles RevisitedRecall (from section 3.5 above) that the most important vector bundles arefamiliar tangent and cotangent bundles. The fibres of the tangent bundleπ M : T M → M of a manifold M are tangent spaces to M. The peculiarityof the tangent bundle T M in comparison with other vector bundles overM lies in the fact that, given an atlas Ψ M = {(U ξ , φ ξ )} of a manifoldM, the tangent bundle of M admits the holonomic atlas Ψ = {(U ξ , ψ ξ =T φ ξ )}, where by T φ ξ is denoted the tangent map to φ ξ . Namely, givencoordinates x α on a manifold M, the associated bundle coordinates on T Mare holonomic coordinates (ẋ α ) with respect to the holonomic frames {∂ α }for tangent spaces T x M, x ∈ M. Their transition functions readẋ ′α = ∂x′α∂x µ ẋµ .Every manifold map f : M → M ′ induces the linear bundle map over f ofthe tangent bundles (4.2).The cotangent bundle of a manifold M is the dual π ∗M : T ∗ M →M of the tangent bundle T M → M. It is equipped with the holonomiccoordinates (x α , ẋ α ) with respect to the coframes {dx α } for T ∗ M whichare the duals of {∂ α }. Their transition functions readẋ ′ α = ∂xµ∂x ′ α ẋµ.Recall that a tensor product of tangent and cotangent bundles over M,T = (⊗ m T M) ⊗ (⊗ k T ∗ M), (m, k ∈ N), (4.11)is called a tensor bundle. Given two vector bundles Y and Y ′ over the samebase X, their tensor product Y ⊗ Y ′ is a vector bundle over X whose fibresare the tensor products of those of the vector bundles Y and Y ′ .Tangent, cotangent and tensor bundles belong to the category BUN ofnatural fibre bundles which admit the canonical lift of any diffeomorphismf of a base to a bundle automorphism, called the natural automorphism[Kolar et al. (1993)]. For example, the natural automorphism of the tangentbundle T M over a diffeomorphism f of its base M is the tangent map T f(4.2) over f. In view of the expression (4.2), natural automorphisms arealso called holonomic transformations or general covariant transformations(in gravitation theory).