Gromoll–Meyer theorem on closed geodesics

Remarks

The idea of the proof is this. Suppose there were only finitely many geometrically distinct closed geodesics. Each critical orbit of the energy on $\Lambda M$, with all its iterates, contributes to the homology of $\Lambda M$ only in a controlled way. The indices of iterates grow linearly, and the local homology of iterates takes only finitely many forms. This forces the Betti numbers of $\Lambda M$ to be bounded, a contradiction.

By a theorem of Vigué-Poirrier and Sullivan, for $F = \Q$ the Betti numbers of $\Lambda M$ are unbounded exactly when the rational cohomology ring $H^*(M; \Q)$ needs at least two generators. So the hypothesis holds, for example, for $M = S^2 \times S^2$. The hypothesis fails for spheres and for compact rank one symmetric spaces, whose rational cohomology is generated by one element. For $S^2$ the conclusion still holds by the theorem of Bangert and Franks.

Going deeper

For a student

The theorem is a basic result of Morse theory on the free loop space. There closed geodesics are critical orbits of the energy, for the circle action that rotates the parameter. Its main technical inputs are the growth of the Morse index under iteration and a finiteness result for local homology of degenerate critical orbits. It is a well-known open question whether every compact simply connected Riemannian manifold of dimension at least $2$ has infinitely many closed geodesics. The Gromoll–Meyer theorem settles it when the loop space homology grows.