On the multiplicity of isometry-invariant geodesics on product manifolds

TL;DR

This paper proves that on certain product manifolds with specific topological and geometric conditions, there are infinitely many geodesics invariant under isometries homotopic to the identity.

Contribution

It establishes the existence of infinitely many isometry-invariant geodesics on product manifolds under new topological and geometric conditions.

Findings

Infinitely many isometry-invariant geodesics exist on the specified manifolds.

The result applies to manifolds with nontrivial first homology and dimension of at least 2.

The proof leverages topological and geometric properties of the manifolds.

Abstract

We prove that on any closed Riemannian manifold $(M_1\times M_2,g)$, with $\rank\Hom_1(M_1)\neq0$ and $\dim(M_2)\geq2$, every isometry homotopic to the identity admits infinitely many isometry-invariant geodesics.