String homology, and closed geodesics on manifolds which are elliptic spaces

TL;DR

This paper proves that on certain elliptic manifolds, if their cohomology algebra isn't generated by a single element, then any Riemannian metric admits infinitely many geometrically distinct closed geodesics, using string homology techniques.

Contribution

It establishes a new link between the algebraic structure of cohomology and the existence of infinitely many closed geodesics on elliptic spaces.

Findings

Manifolds with non-single-generator cohomology have infinite closed geodesics.

Uses spectral sequences and Hopf algebra structure to prove the result.

Extends classical geodesic existence results to elliptic spaces.

Abstract

Let $M$ be a closed simply connected smooth manifold. Let $\F_p$ be the finite field with $p$ elements where $p> 0$ is a prime integer. Suppose that $M$ is an $\F_p$-elliptic space in the sense of [FHT91]. We prove that if the cohomology algebra $H^*(M, \F_p)$ cannot be generated (as an algebra) by one element, then any Riemannian metric on $M$ has an infinite number of geometrically distinct closed geodesics. The starting point is a classical theorem of Gromoll and Meyer [GM69]. The proof uses string homology, in particular the spectral sequence of [CJY04], the main theorem of [McC87], and the structure theorem for elliptic Hopf algebras over $\F_p$ from [FHT91].