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 reversible$C^2$ Finsler metric on $M$ whose unparametrized geodesics coincide with those of $g$, then $(M,F)$ is a Finsler symmetric space.

中文翻译：

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希尔伯特第四个问题的周期解

结果表明，一个可能不可逆的 $C^2$ 环面或任何其他紧凑欧几里得空间形式上的 Finsler 度量，其测地线是直线是平面度量和封闭度量的总和 $1$-形式。这用于证明如果$(米,G)$ 是一个秩大于 1 的紧黎曼对称空间，并且 $F$是可逆的$C^2$ 芬斯勒度量 $米$ 其非参数化测地线与 $G$， 然后 $(米,F)$ 是 Finsler 对称空间。