  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.

Differential Geometry