Ivancevic_Applied-Diff-Geom

496 Applied Differential Geometry: A Modern Introduction(1) κ M ◦ T 2 f = T 2 f ◦ κ M for each f ∈ C k (M, N);(2) T (π M ) ◦ κ M = π T M ;(3) π T M ◦ κ M = T (π M );(4) κ −1M = κ M ;(5) κ M is a linear isomorphism from the bundle (T T M, T (π M ), T M) to(T T M, π T M , T M), so it interchanges the two vector bundle structureson T T M;(6) κ M is the unique smooth map T T M → T T M which, for each γ : R →M, satisfies∂ t ∂ s γ(t, s) = κ M ∂ t ∂ s γ(t, s).In a similar way the second cotangent bundle of a manifold M can bedefined. Even more, for every manifold there is a geometrical isomorphismbetween the bundles T T ∗ M = T (T ∗ M) and T ∗ T M = T ∗ (T M) [Modugnoand Stefani (1978)].4.3.2 The Natural Vector BundleIn this section we mainly follow [Michor (2001); Kolar et al. (1993)].A vector bundle functor or natural vector bundle is a functor F whichassociates a vector bundle (F(M), π M , M) to each n−manifold M and avector bundle homomorphismF(M)F(ϕ)✲ F(N)π M❄Mϕπ N❄✲ Nto each ϕ : M → N in M, which covers ϕ and is fiberwise a linear isomorphism.Two common examples of the vector bundle functor F are tangentbundle functor T and cotangent bundle functor T ∗ (see section 3.5).The space of all smooth sections of the vector bundle (E, π M , M) is denotedby Γ (E, π M , M). Clearly, it is a vector space with fiberwise additionand scalar multiplication.Let F be a vector bundle functor on M. Let M be a smooth manifoldand let X ∈ X (M) be a vector–field on M. Then the flow F t of X for fixed