Homogeneous Finsler spaces with only one orbit of prime closed geodesics
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https://doi.org/10.1007/s11425-018-9454-y
Homogeneous Finsler spaces with only one orbit of prime closed geodesics Ming Xu School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Email: [email protected] Received August 31, 2018; accepted December 13, 2018
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. Keywords
homogeneous Finsler space, closed geodesic, compact rank-one symmetric space, connected isom-
etry group, Killing vector field MSC(2010)
53C60, 53C30, 53C22
Citation: Xu M. Homogeneous Finsler spaces with only one orbit of prime closed geodesics. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-9454-y
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Introduction
In Riemannian geometry, it has been conjectured for many decades that on any closed manifold M with dim M > 1, there exist infinitely many prime closed geodesics. In Finsler geometry, this is not true because of the Katok spheres [17, 35] found in 1973. Katok spheres are Randers spheres of constant flag curvature which were much recently classified in [5]. Based on the Katok spheres, Anosov proposed ann other conjecture, claiming the existence of 2[ n+1 2 ] prime closed geodesics on the Finsler sphere (S , F ) [2]. See [3, 26] for some recent progress on this conjecture. Generally speaking, finding the first prime closed geodesic on a compact Finsler manifold is relatively easy (see [14, 18]). Finding the second is already a hard problem if no topological obstacles from [15, 21] are accessible. It was relatively recent that Duan and Long [12] and Rademacher [23,24] provided different proofs of the following theorem. Theorem 1.1. A bumpy and irreversible Finsler metric on a sphere Sn of dimension n greater than or equal to 3 carries two prime closed geodesics. More generally, when Sn is changed to other compact manifolds, Duan et al. proved the following theorem in [13]. c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019 ⃝
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Xu M
Sci China Math
Theorem 1.2. There exist always at least two prime closed geodesics on every compact simply connected bumpy irreversible Finsler manifold (M, F ). In this paper, we will assume that the Finsler manifold (M, F ) admits nontrivial con
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