Journal of Topology and Analysis ( IF 0.5 ) Pub Date : 2021-10-25 , DOI: 10.1142/s1793525321500576 Yong Fang 1
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 -forms. Another family of well-known examples are projectively flat Finsler metrics on the -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 be a Riemannian–Finsler metric on a closed surface , and be a small antisymmetric potential on that is a natural generalization of -form (see Sec. 1). If the Randers-type Finsler metric is geodesically reversible, and the geodesic flow of is topologically transitive, then we prove that must be a closed -form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the -sphere of constant positive curvature.
中文翻译:
关于测地线可逆芬斯勒流形
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-形式。另一个著名的例子是投影平坦芬斯勒度量-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:让是闭合曲面上的黎曼-芬斯勒度量, 和是一个小的反对称势这是一个自然的概括-form(参见第 1 节)。如果 Randers 型 Finsler 度量是测地线可逆的,并且测地线流是拓扑传递的,那么我们证明必须是一个封闭的-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。- 恒定正曲率的球体。