Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds

Proves that every smooth Riemannian metric on Sⁿ, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.

Manuscript

Infinitely many closed geodesic images on every Riemannian sphere

September 24, 2026 · source · cite as \cite{OAI:Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026}

We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere Sⁿ, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.

Result 345 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.