The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$

TL;DR

This paper establishes lower bounds on the number of non-contractible closed geodesics on certain Finsler real projective spaces using Fadell-Rabinowitz index theory and equivariant topology, extending previous results.

Contribution

It provides new lower bounds for non-contractible closed geodesics on Finsler projective spaces under specific curvature and reversibility conditions, utilizing advanced topological methods.

Findings

At least n-1 non-contractible closed geodesics exist under certain curvature conditions.

Under stronger conditions, at least 2[ (n+1)/2 ] non-contractible closed geodesics are guaranteed.

The results match the optimal bounds given by Katok's example.

Abstract

In this paper, we prove that for every irreversible Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\frac{16}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ with $\lambda<3$, there exist at least $n-1$ non-contractible closed geodesics. In addition, if the metric $F$ is bumpy with $\frac{64}{25}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ and $\lambda<\frac{5}{3}$, then there exist at least $2[\frac{n+1}{2}]$ 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 $(\mathbb{R}P^n,F)$ and the $S^1$-equivariant Poincar$\acute{e}$ series of the non-contractible component of the free loop space on $\mathbb{R}P^n$.