The number of geometrically distinct reversible closed geodesics on a Finsler sphere with $K\equiv 1$

Abstract

In this paper we study the Finsler sphere $(S^n,F)$ with $n>1$, which has constant flag curvature $K\equiv 1$ and only finite prime closed geodesics. In this case, the connected isometry group $I_0(S^n,F)$ must be a torus which dimension satisfies $0<\dim I(S^n,F) \leq[\frac{n+1}{2}]$. We will prove that the number of geometrically distinct reversible closed geodesics on $(S^n,F)$ is at least $\dim I(S^n,F)$. When $\dim I_0(S^n,F)=[\frac{n+1}{2}]$, the equality happens, and there are exactly $2[\frac{n+1}{2}]$ prime closed geodesics, which verifies Anosov conjecture in this special case.