Periodic solutions of Hilbert’s fourth problem

It is shown that a possibly irreversible [Formula: see text] 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 [Formula: see text]-form. This is used to prove that if [Formula: see text] is a compact Riemannian symmetric space of rank greater than one and [Formula: see text] is a reversible [Formula: see text] Finsler metric on [Formula: see text] whose unparametrized geodesics coincide with those of [Formula: see text], then [Formula: see text] is a Finsler symmetric space.