Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves that every smooth Riemannian metric on S^(n), 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.
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MANUSCRIPT 01
Infinitely many closed geodesic images on every Riemannian sphere
We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere S^(n), 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…
Full source abstract
We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere S^(n), 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.
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