Isomorphism classes for higher order tangent bundles

The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq \infty$, admits a vector bundle structure on $M$ if and only if $M$ is endowed with a linear connection or equivalently a connection map on $T^kM$ 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 notion of the $k$'th order differential $T^kg:T^kM\longrightarrow T^kN$ for a given differentiable map $g$ between manifolds $M$ and $N$. As we shall see, $T^kg$ 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 $g:M\longrightarrow M$ is a diffeomorphism, follows invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combination of connection maps and manifold of $C^r$ maps we offer three examples to support our theory and reveal its interaction with the known problems such as Sasaki lift of metrics.