Remark on non-contractible closed geodesics and homotopy groups

TL;DR

The paper establishes conditions under which a closed manifold admits infinitely many geometrically distinct closed geodesics, linking homotopy group properties and fundamental group conjugacy classes to geodesic multiplicity.

Contribution

It proves that certain non-trivial actions on higher homotopy groups guarantee infinitely many closed geodesics for generic metrics, extending to all metrics if the fundamental group has infinitely many conjugacy classes.

Findings

Existence of infinitely many closed geodesics under specific homotopy group conditions.

Generic Riemannian metrics satisfy the geodesic multiplicity condition.

All metrics have infinitely many geodesics if the fundamental group has infinitely many conjugacy classes.

Abstract

We prove that if the $m$-th homotopy group for $m \geq 2$ of a closed manifold has non-trivial invariants or coinvariants under the action of the fundamental group, then there exist infinitely many geometrically distinct closed geodesics for a $C^4$-generic Riemannian metric. If moreover there are infinitely many conjugacy classes in the fundamental group, then the same holds for every Riemannian metric.