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Periodic solutions of Hilbert’s fourth problem
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-10-22, DOI:
J. C. Álvarez Paiva, J. Barbosa Gomes

It is shown that a possibly irreversible $C2$ 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 reversible$C2$ Finsler metric on $M$ whose unparametrized geodesics coincide with those of $g$, then $(M,F)$ is a Finsler symmetric space.