The Existence of Infinitely Many Geometrically Distinct Non-Constant Prime Closed Geodesics on Riemannian Manifolds

TL;DR

This paper establishes a necessary condition for a Riemannian manifold to have infinitely many geometrically distinct prime closed geodesics, linking the existence of such geodesics to the topological structure of the free loop space.

Contribution

It provides a topological criterion involving handle decompositions and cellular homology for the existence of infinitely many prime closed geodesics on any closed Riemannian manifold.

Findings

Infinitely many prime closed geodesics correspond to infinitely many non-degenerate critical points of the energy functional.

Handle decomposition of free loop space is used to analyze topological invariants related to geodesic existence.

The result applies universally to any closed Riemannian manifold, regardless of specific metric properties.

Abstract

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits infinitely many prime closed geodesics such that the energy functional $E:\Lambda M\to\mathbb{R}$ has infinitely many non-degenerate critical points on the free loop space $\Lambda M$ of Sobolev class $H^1=W^{1,2}$. This result is obtained by invoking a handle decomposition of free loop space and using methods of cellular homology to study its topological invariants.