Periodic Solutions of Hilbert's Fourth Problem

TL;DR

The paper characterizes Finsler metrics on compact spaces with straight-line geodesics, showing they decompose into flat metrics plus closed forms, and extends results to reversible Finsler metrics on symmetric spaces.

Contribution

It proves that Finsler metrics with straight-line geodesics on compact spaces are sums of flat metrics and closed 1-forms, and characterizes reversible Finsler metrics on symmetric spaces.

Findings

Finsler metrics with straight-line geodesics decompose into flat metrics plus closed 1-forms.

Reversible Finsler metrics on higher-rank symmetric spaces are Finsler symmetric spaces.

Geodesic structures determine the Finsler metric's form on compact spaces.

Abstract

It is shown that a possibly irreversible $C^2$ Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed $1$-form. This is used to prove that if $(M,g)$ is a compact Riemannian symmetric space of rank greater than one and $F$ is a {\sl reversible} $C^2$ Finsler metric on $M$ whose unparametrized geodesics coincide with those of $g$, then $(M,F)$ is a Finsler symmetric space.