Homogeneous Finsler spaces with only one orbit of prime closed geodesics

Abstract

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To extend the results of Duan, Long, Rademacher, Wang and others on the existence of two prime closed geodesics to the equivariant situation, we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space. In this paper, we study this problem in homogeneous Finsler geometry, and get a positive answer when the dimension is even or the metric is reversible. We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.