Finsler Metrics and Closed Geodesics

一方面，Finsler度量的曲率性质与最短闭测地线的长度之间的关系。另一方面，我们将考虑具有Finsler度量的紧致流形上闭测地线的内射性半径和存在性结果。更详细地，我们将研究以下问题：1.申请人给出的关于正旗曲率的不可逆芬斯勒度量的最短闭测地线的长度估计相等是否意味着旗曲率是常数的？2.有可能改进申请人给出的具有正旗曲率的紧致单连通黎曼流形的内射半径估计的下界吗？是否存在一个不依赖于可逆性的下界？3.紧一阶对称空间上两条闭测地线的存在性。

On one hand the connection between curvature properties of Finsler metrics and the length of the shortest closed geodesic resp. the injectivity radius and on the other hand existence results for closed geodesics on compact manifolds carrying a Finsler metric will be considered. In more detail the following problems will be investigated:1. Does equality in the length estimate given by the applicant for the shortest closed geodesic of a non-reversible Finsler metric of positive flag curvature imply that the flag curvature is constant?2. Is is possible to improve the lower bound in the estimate presented by the applicant for the injectivity radius of a compact and simply-connected Riemannian manifold with positive flag curvature? Is there a lower bound which does not depend on the reversibility?3. The existence of two closed geodesics on a compact rank one symmetric space witha bumpy non-reversible Finsler metric.