Closed geodesics on compact orbifolds and on noncompact manifolds

TL;DR

This paper proves the existence of nontrivial closed geodesics on certain compact orbifolds and noncompact manifolds with cocompact isometric group actions, extending geodesic existence results to these settings.

Contribution

It establishes new existence results for closed geodesics on compact orbifolds and noncompact manifolds with specific symmetry properties.

Findings

Noncontractible manifolds with cocompact isometric actions have closed geodesics.

Odd-dimensional compact orbifolds possess nontrivial closed geodesics.

Results extend classical geodesic existence theorems to orbifolds and symmetric noncompact manifolds.

Abstract

We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an action, and every odd-dimensional, compact Riemannian orbifold has a nontrivial closed geodesic.