A remark on the Morse Theorem about infinitely many geodesics between two points

TL;DR

This paper proves that on certain Finsler manifolds, two non-conjugate points can be connected by infinitely many geodesics if the Betti numbers of the loop space grow unbounded, extending Morse theory results.

Contribution

It establishes a new link between the topology of the loop space and the existence of infinitely many geodesics on Finsler manifolds, generalizing Morse Theorem.

Findings

Infinitely many geodesics exist between two points under given conditions.

Growth of Betti numbers implies abundance of geodesics.

Extends classical Morse Theorem to Finsler geometry.

Abstract

We show that any two non-conjugate points on a forward or backward complete connected Finsler manifold can be joined by infinitely many geodesics which are not covered by finitely many closed ones, provided that the Betti numbers of the based loop space grow unbounded.