On the existence of infinitely many closed geodesics on non-compact manifolds

TL;DR

This paper proves that non-compact complete Riemannian manifolds have infinitely many closed geodesics under certain topological and geometric conditions, extending classical results to broader settings.

Contribution

It generalizes the Gromoll-Meyer theorem to non-compact manifolds by establishing conditions for infinitely many closed geodesics.

Findings

Infinitely many closed geodesics exist on certain non-compact manifolds.

The existence depends on unbounded Betti numbers of the free loop space.

No close conjugate points at infinity are required.

Abstract

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni, and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds.