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

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

1 paper

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.

Papers 1

Infinitely many closed geodesic images on every Riemannian sphere

OpenAI · 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.

Read PDF Download Source

Cite

@misc{OAI:Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026,
  author = {{OpenAI}},
  title = {{Infinitely many closed geodesic images on every Riemannian sphere}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf}{OAI:Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026}},
  year = {2026}
}

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

Watch

Papers 1

Infinitely many closed geodesic images on every Riemannian sphere

Formal verification