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On geodesically reversible Finsler manifolds
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-10-25, DOI: 10.1142/s1793525321500576
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 g be a Riemannian–Finsler metric on a closed surface Σ, and p be a small antisymmetric potential on Σ that is a natural generalization of 1-form (see Sec. 1). If the Randers-type Finsler metric gp is geodesically reversible, and the geodesic flow of 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.