Metrizability of holonomy invariant projective deformation of sprays

TL;DR

This paper investigates when projective deformations of Finsler geodesic sprays, altered by holonomy invariant functions, remain metrizable, revealing that most such deformations are not Finsler metrizable except in special cases involving principal curvatures.

Contribution

It characterizes the conditions under which holonomy invariant projective deformations of Finsler sprays are metrizable, identifying principal curvatures as key factors.

Findings

Most deformations are not Finsler metrizable for generic parameters.

Metrizability occurs only when the holonomy invariant function is a principal curvature.

The paper provides criteria to determine when a deformation preserves metrizability.

Abstract

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\lambda P C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $\lambda\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal curvatures of the geodesic structure.