Descending maps between slashed tangent bundles

TL;DR

This paper characterizes when diffeomorphisms between slashed tangent bundles descend from base manifold diffeomorphisms, providing conditions for descent to totally geodesic maps and isometries in Riemannian geometry.

Contribution

It offers a characterization of descending maps between slashed tangent bundles and establishes conditions for such maps to be totally geodesic or isometric.

Findings

Characterization of diffeomorphisms that descend between slashed tangent bundles.

Sufficient conditions for descent to totally geodesic maps.

Conditions for descent to isometries in Riemannian geometry.

Abstract

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.