Isometry-invariant geodesics and the fundamental group, II

TL;DR

This paper proves that on certain closed Riemannian manifolds with fundamental group isomorphic to rac12;Z, any isometry homotopic to the identity (except on the circle) has infinitely many invariant geodesics, extending previous work.

Contribution

It establishes the existence of infinitely many invariant geodesics for a broad class of manifolds with fundamental group rac12;Z, completing prior results by the second author.

Findings

Infinite invariant geodesics exist on these manifolds.

The result applies to all such manifolds except the circle.

It confirms the conjecture for manifolds with fundamental group rac12;Z.

Abstract

We show that on a closed Riemannian manifold with fundamental group isomorphic to $\mathbb{Z}$, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result of the second author.