Sprays metrizable by Finsler functions of constant flag curvature

TL;DR

This paper characterizes when sprays can be derived from Finsler functions with constant flag curvature, providing tensorial conditions to determine Finsler metrizability for certain differential equations.

Contribution

It offers necessary and sufficient tensorial conditions for Finsler metrizability with constant flag curvature, solving a specific case of the Finsler metrizability problem.

Findings

Tensorial equations characterize Finsler metrizability

Conditions determine when a spray corresponds to a Finsler function of constant flag curvature

Identifies classes where Finsler metrizability aligns with specific geometric properties

Abstract

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary differential equations represents the Euler-Lagrange equations of a Finsler function of constant flag curvature. The conditions we provide are tensorial equations on the Jacobi endomorphism. We identify the class of homogeneous SODE where the Finsler metrizability is equivalent with the metrizability by a Finsler function of constant flag curvature.