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 $K\equiv 1$. 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 $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of $\hat{G}=G\times T$ on the free loop space $\lambda M$ of $M$. In particular, for each closed geodesic, we have a $\hat{G}$-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 ${\mathcal{B}}_i$ of prime closed geodesics, such that for each $i$, the union $B_i$ of geodesics in ${\mathcal{B}}_i$ 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 $K\equiv 1$, provide examples with the sharp estimate for our main theorem.