Two closed geodesics on compact bumpy Finsler manifolds

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 $(S^2/{\mathbb{Z}}_2,F)$, cf. \cite{Kat}.

10 pages. arXiv admin note: substantial text overlap with arXiv:1803.08350; text overlap with arXiv:1504.07007 by other authors