The multiplier approach to the projective Finsler metrizability problem

TL;DR

This paper investigates the projective Finsler metrizability problem by using a multiplier approach, analyzing local and global conditions for a spray class to derive from a Finsler function, including convexity and positivity aspects.

Contribution

It introduces a multiplier method to address both local and global Finsler metrizability, connecting various conditions like Rapcsák and geodesic convexity.

Findings

Multiplier approach effectively characterizes Finsler metrizability.

Conditions for positivity and convexity are clarified.

Special considerations for two-dimensional cases.

Abstract

This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results entirely in terms of a multiplier, that is, a type (0,2) tensor field along the tangent bundle projection. In the course of the analysis we consider several related issues of interest including the positivity and strong convexity of positively-homogeneous functions, the relation to the so-called Rapcs\'ak conditions, some peculiarities of the two-dimensional case, and geodesic convexity for sprays.