Multiplicity of closed geodesics on bumpy Finsler manifolds with elliptic closed geodesics

TL;DR

This paper investigates the number of closed geodesics on certain bumpy Finsler manifolds, establishing conditions under which there are finitely many or infinitely many such geodesics, based on the manifold's topology and geodesic properties.

Contribution

It provides new results on the multiplicity of closed geodesics on bumpy Finsler manifolds with specific topological and geometric conditions, especially when all prime geodesics are elliptic.

Findings

Exactly rac{dn(n+1)}{2} or (d+1) closed geodesics exist under certain conditions

Either finitely many or infinitely many closed geodesics occur

Results depend on the parity of d and the elliptic nature of geodesics

Abstract

Let $M$ be a compact simply connected manifold satisfying $H^*(M;\mathbf{Q})\cong T_{d,n+1}(x)$ for integers $d\ge 2$ and $n\ge 1$. If all prime closed geodesics on $(M,F)$ with an irreversible bumpy Finsler metric $F$ are elliptic, either there exist exactly $\frac{dn(n+1)}{2}$ (when $d\ge 2$ is even) or $(d+1)$ (when $d\ge 3$ is odd) distinct closed geodesics, or there exist infinitely many distinct closed geodesics.