Descending maps between slashed tangent bundles

Abstract

Suppose $TM\setminus \{0\}$ and $T\tilde{M}\setminus \{0\}$ are slashed tangent bundles of two smooth manifolds $M$ and $\tilde{M}$, respectively. In this paper we characterize those diffeomorphisms $F:TM\setminus \{0\}\to T\tilde{M}\setminus \{0\}$ that can be written as $F=(D\varphi ){|}_{TM\setminus \{0\}}$ for a diffeomorphism $ϕ\colon M\to \wt M$. When $F=(D\varphi ){|}_{TM\setminus \{0\}}$ 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.