Yong Fang

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.