Metrizable isotropic second-order differential equations and Hilbert's fourth problem

TL;DR

This paper characterizes when second-order differential equations (sprays) are metrizable by Finsler functions of scalar flag curvature, providing algorithms for construction and exploring solutions related to Hilbert's fourth problem.

Contribution

It offers new criteria and algorithms to determine and construct Finsler functions of scalar flag curvature for metrizing sprays, extending to conic and degenerate cases.

Findings

Characterization of isotropic sprays for Finsler metrizability

Algorithm for constructing Finsler functions of scalar flag curvature

Application to solutions of Hilbert's fourth problem

Abstract

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. The proof of Theorem 3.1 provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. One condition of Theorem 3.1, regarding the regularity of the sought after Finsler function, can be relaxed. By relaxing this condition, we provide examples of sprays that are metrizable by conic pseudo-Finsler functions as well as degenerate Finsler functions. Hilbert's fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert's fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the conditions of [11, Theorem 4.1] and Theorem 3.1 to test when the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the proofs of [11, Theorem 4.1] and Theorem 3.1 to construct solutions to Hilbert's fourth problem.