Closed geodesics on positively curved spheres $S^n$ with Finsler metric induced by $(\mathbb{R}P^n,F)$

TL;DR

This paper proves the existence of multiple prime closed geodesics on positively curved Finsler spheres induced by real projective spaces, with specific bounds depending on the dimension and curvature conditions.

Contribution

It establishes new lower bounds on the number of prime closed geodesics on Finsler spheres derived from real projective spaces under certain curvature and reversibility conditions.

Findings

At least n-1 prime closed geodesics exist on the sphere for n≥3.

If finitely many closed geodesics exist, at least 2[ n/2 ] - 1 are non-hyperbolic.

Results depend on curvature bounds involving reversibility and flag curvature.

Abstract

It's well known that the n-sphere $S^n$ is the universal double covering of the $n$-dimensional real projective space $\mathbb{R}P^n$ and then any Finsler metric on $\mathbb{R}P^n$ induces a Finsler metric of $S^n$. In this paper, we prove that for every Finsler $(S^n, F)$ for $n\geq3$ whose metric is induced by irreversible Finsler $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\leq 1$, there exist at least $n-1$ prime closed geodesics on $(S^n, F)$. Furthermore, if there exist finitely many distinct closed geodesics on $(S^n, F)$, then there exist at least $2[\frac{n}{2}]-1$ of them are non-hyperbolic.