Homotheties of Finsler manifolds

TL;DR

This paper provides a new proof that complete, connected finite-dimensional Finsler manifolds with a proper homothety are essentially Minkowski spaces, using the exponential map of the canonical spray.

Contribution

It offers a complete and alternative proof of Laugwitz's theorem, establishing the Minkowski space structure under the given conditions.

Findings

Finsler manifolds with proper homothety are Minkowski spaces

The exponential map of the canonical spray provides the isometry

The result characterizes the structure of such Finsler manifolds

Abstract

We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent Minkowski vector space of the fixed point of the homothety via the exponential map of the canonical spray of the Finsler manifold.