Periodic Solutions of Hilbert's Fourth Problem

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.