On a conjecture of Anosov

TL;DR

This paper proves that certain Finsler spheres with specific curvature and reversibility conditions have a guaranteed number of prime closed geodesics, confirming a conjecture of Anosov for generic cases.

Contribution

It establishes the existence of multiple prime closed geodesics on Finsler spheres under new curvature and reversibility conditions, confirming a longstanding conjecture.

Findings

Existence of at least 2[(n+1)/2] prime closed geodesics

Valid for Finsler n-spheres with specific curvature bounds

Confirms Anosov's conjecture in a generic setting

Abstract

In this paper, we prove that for every bumpy Finsler $n$-sphere $(S^n,\,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, there exist $2[\frac{n+1}{2}]$ prime closed geodesics. This gives a confirmed answer to a conjecture of D. V. Anosov \cite{Ano} in 1974 for a generic case.