Isomorphism classes for higher order tangent bundles

The tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.