Two closed geodesics on compact bumpy Finsler manifolds

TL;DR

This paper proves that any compact bumpy Finsler manifold with finite fundamental group and dimension at least 2 has at least two closed geodesics, extending known results and confirming the minimal number in specific cases.

Contribution

It establishes the existence of at least two closed geodesics on broad classes of compact Finsler manifolds, including 2-manifolds, with specific examples achieving this bound.

Findings

At least two closed geodesics exist on compact bumpy Finsler manifolds with finite fundamental group.

The minimal number of closed geodesics is two for certain 2-manifolds, such as the Katok 2-sphere.

The result generalizes previous theorems and confirms the minimal bound in specific cases.

Abstract

In this paper, we prove there are at least two closed geodesics on any compact bumpy Finsler $n$-manifold with finite fundamental group and $n\ge 2$. Thus generically there are at least two closed geodesics on compact Finsler manifolds with finite fundamental group. Furthermore, there are at least two closed geodesics on any compact Finsler $2$-manifold, and this lower bound is achieved by the Katok 2-sphere $(S^2, F)$ and 2-real projective space $(S^2/\Z_2, F)$, cf. \cite{Kat}.