The Fadell–Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler [formula omitted

Abstract

In this paper, we prove that for every irreversible Finsler n-dimensional real projective space (RPn,F) with reversibility λ and flag curvature K satisfying 169(λ1+λ)2<K≤1 with λ<3, there exist at least n−1 non-contractible closed geodesics. In addition, if the metric F is bumpy with 6425(λ1+λ)2<K≤1 and λ<53, then there exist at least 2[n+12] non-contractible closed geodesics, which is the optimal lower bound due to Katok's example. The main ingredients of the proofs are the Fadell–Rabinowitz index theory of non-contractible closed geodesics on (RPn,F) and the S1-equivariant Poincaré series of the non-contractible component of the free loop space on RPn.