Descending maps between slashed tangent bundles

Suppose $TM\setminus \{0\}$ and $T\widetilde M\setminus\{0\}$ are slashed tangent bundles of two smooth manifolds $M$ and $\widetilde M$, respectively. In this paper we characterize those diffeomorphisms $F\colon TM\setminus\{0\} \to T\widetilde M\setminus\{0\}$ that can be written as $F = (D\phi)|_{TM\setminus\{0\}}$ for a diffeomorphism $\phi\colon M\to \wt M$. When $F = (D\phi)|_{TM\setminus\{0\}}$ one say that $F$ \emph{descends}. If $M$ is equipped with two sprays, we use the characterization to derive sufficient conditions that imply that $F$ descends to a totally geodesic map. Specializing to Riemann geometry we also obtain sufficient conditions for $F$ to descent to an isometry.