Family 345Differential geometryNo LeanNot yet digested
Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
Proves that every smooth Riemannian metric on S, 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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Infinitely many closed geodesic images on every Riemannian sphere
Sep 24, 2026·paper.pdf
We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere S, 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.