Projective and Finsler metrizability: parameterization-rigidity of the geodesics

TL;DR

This paper demonstrates that most projective deformations of Finsler geodesic sprays result in non-metrizable sprays, highlighting the rigidity of Finsler metrizability under certain reparameterizations.

Contribution

It proves that the natural projective deformation of a Finsler geodesic spray generally destroys Finsler metrizability, revealing the rigidity of this property.

Findings

Most projective deformations lead to non-Finsler metrizable sprays.

The projective class of any spray contains infinitely many non-metrizable sprays.

Finsler metrizability is highly sensitive to specific reparameterizations.

Abstract

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 \lambda F\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $\lambda \in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.