A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed $1$-forms. Another family of well-known examples are projectively flat Finsler metrics on the $2$-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let $\sqrt{g}$ be a Riemannian–Finsler metric on a closed surface $\mathrm{\Sigma }$, and $p$ be a small antisymmetric potential on $\mathrm{\Sigma }$ that is a natural generalization of $1$-form (see Sec. 1). If the Randers-type Finsler metric $\sqrt{g}-p$ is geodesically reversible, and the geodesic flow of $\sqrt{g}$ is topologically transitive, then we prove that $p$ must be a closed $1$-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the $2$-sphere of constant positive curvature.

如果任何测地线的反转曲线仍然是几何测地线，则称芬斯勒流形是测地线可逆的。测地线可逆 Finsler 度量的著名示例是带有闭环的 Randers 度量$1$-形式。另一个著名的例子是投影平坦芬斯勒度量$2$-具有恒定正曲率的球体。在本文中，我们证明了测地线可逆芬斯勒度量的一些几何和动力学特征，并证明了一系列所谓的兰德斯型芬斯勒度量的几个刚性结果。我们的结果之一如下：让$\sqrt{G}$是闭合曲面上的黎曼-芬斯勒度量$\mathrm{\Sigma }$， 和$p$是一个小的反对称势$\mathrm{\Sigma }$这是一个自然的概括$1$-form（参见第 1 节）。如果 Randers 型 Finsler 度量$\sqrt{G}-p$是测地线可逆的，并且测地线流$\sqrt{G}$是拓扑传递的，那么我们证明$p$必须是一个封闭的$1$-形式。我们还证明，对于投影平坦芬斯勒度量系列，这种刚性结果并不成立。$2$- 恒定正曲率的球体。