Two short closed geodesics on a sphere of odd dimension

TL;DR

This paper proves that on odd-dimensional spheres with generic non-reversible Finsler metrics, there always exists a second closed geodesic with bounded Morse index, expanding understanding of geodesic multiplicity in Finsler geometry.

Contribution

It demonstrates the existence of a second closed geodesic with bounded Morse index for a generic class of non-reversible Finsler metrics on odd-dimensional spheres.

Findings

Existence of a second closed geodesic on odd-dimensional spheres.

Bounded Morse index for the second geodesic.

Applicability to an open and dense set of non-reversible Finsler metrics.

Abstract

We show that for an open and dense set non-reversible Finsler metrics on a sphere of odd dimension $n=2m-1 \ge 3$ there is a second closed geodesic with Morse index $\le 4(m+2)(m-1)+2.$