Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics

Abstract

In this paper, we consider a Finsler sphere (M, F ) = (S n , F ) with the dimension n > 1 and the flag curvature K ≡ 1. The action of the connected isometry group G = I o (M, F ) on M , together with the action of T = S 1 shifting the parameter t ∈ R/Z of the closed curve c(t), define an action ofĜ = G × T on the free loop space λM of M . In particular, for each closed geodesic, we have aĜ-orbit of closed geodesics. We assume the Finsler sphere (M, F ) described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the subgroup H of all isometries preserving each close geodesic has a dimension m, then there exists m geometrically distinct orbits B i of prime closed geodesics, such that for each i, the union B i of geodesics in B i is a totally geodesic sub-manifold in (M, F ) with a non-trivial H o -action. This theorem generalizes and slightly refines the one in a previous work, which only discussed the case of finite prime closed geodesics. At the end, we show that, assuming certain generic conditions, the Katok metrics, i.e. the Randers metrics on spheres with K ≡ 1, provide examples with the sharp estimate for our main theorem.Mathematics Subject Classification (2000): 22E46, 53C22, 53C60.

Reversible homogeneous Finsler metrics with positive flag curvature

“…The Riemannian case and the (non-Riemannian) Randers case are well understood. They provide models and motivations for our previous works estimating the number of orbits of prime closed geodesics on Finsler spheres with K ≡ 1 [23,25] and homogeneous Finsler spaces [24]. We show that many properties of the geodesics on a homogeneous Randers sphere with K ≡ 1 can be generalized to the non-Randers case.…”

“…This observation inspire us to ask if they are the only ones. A partial answer for this rigidity problem from the positive side has been given in homogenous Finsler geometry (see Theorem 1.4 in [24]).…”

“…When the F 0 -length of V is an irrational multiple of π, there are no other prime closed geodesics except those twoĜ-orbits of prime closed geodesics generated by ±V , which lengths l ± are irrational multiples of π. This is an important basic model for studying Finsler spheres with K ≡ 1 and only finite orbits of prime closed geodesics [24].…”

Homogeneous Finsler spaces with only one orbit of prime closed geodesics

“…In this paper, we will assume the Finsler manifold (M, F ) admits nontrivial continuous isometries and discuss an equivalent analog of above theorems. Some thought and technique were purposed in [5] and further developed in [28] and [29], while studying the geodesics in a Finsler sphere of constant curvature.…”

“…It was suggested in [29] that, when the connected isometry group G = I 0 (M, F ) has a positive dimension, estimating the number of prime closed geodesic seems more reasonable to be switched to estimating the number of orbits of prime closed geodesics, with respect to the action ofĜ = G × S 1 (the precise description for this action will be explained at the end of Section 2). Though there are examples of compact Finsler manifolds with only one orbit of prime closed geodesics, they are very rare.…”