Two closed geodesics on compact bumpy Finsler manifolds

In this paper, we prove there are at least two closed geodesics on any compact bumpy Finsler $n$-manifold with finite fundamental group and $n \geq 2$. 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 $(S^2 / \mathbf{Z}_2, F)$. cf. [Kat].

Keywords

closed geodesic, Finsler manifold, bumpy

2010 Mathematics Subject Classification

53C22, 58E05, 58E10

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The author was partially supported by NSFC No. 12025101, 11431001.

Received 12 November 2018

Accepted 11 February 2020

Published 3 September 2021