Bangert–Franks theorem on closed geodesics on the sphere

Remarks

The proof combines two results. Franks treated, with a theorem on area-preserving homeomorphisms of the open annulus, the metrics for which a simple closed geodesic gives an annulus of section for the geodesic flow. Bangert treated the remaining metrics by variational methods on the loop space. The theorem of the three geodesics gives a simple closed geodesic to start from.

The Gromoll–Meyer theorem does not apply here, because the Betti numbers of the free loop space of $S^2$ are bounded. The hypothesis that the metric is Riemannian matters: Katok constructed non-reversible Finsler metrics on $S^2$ with only finitely many closed geodesics.

Going deeper

For a student

The result answers a long-standing question for the simplest closed surface. It mixes two methods: dynamics of the geodesic flow, through a surface of section and a return map of an annulus, and Morse theory of the energy on loop spaces. For higher-dimensional spheres the corresponding question for Riemannian metrics is open.