Isometry-invariant geodesics and the fundamental group

TL;DR

This paper proves that on certain closed Riemannian manifolds with specific fundamental groups, any identity-homotopic isometry has infinitely many invariant geodesics, and extends this idea to contact manifolds and Reeb orbits.

Contribution

It establishes the existence of infinitely many invariant geodesics under specific topological conditions and proposes generalizations related to contact geometry and the Weinstein conjecture.

Findings

Infinitely many invariant geodesics exist on manifolds with infinite abelian, non-cyclic fundamental groups.

Conjecture: the result holds for manifolds with infinite cyclic fundamental group.

Generalization of the Weinstein conjecture for contactomorphisms and Reeb orbits.

Abstract

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the celebrated Weinstein conjecture: on a closed contact manifold with a selected contact form, any strict contactomorphism that is contact-isotopic to the identity possesses an invariant Reeb orbit.