{"id":"openai-math/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere","pdf":"https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026","tags":["riemannian-geometry","closed-geodesics","differential-geometry"],"score":0,"title":"Infinitely many closed geodesic images on every Riemannian sphere","domain":"Differential geometry","family":"345","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds"}