Browse › Differential geometry › 345 Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
Infinitely many closed geodesic images on every Riemannian sphere
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Abstract
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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Zhiqi Chen, Shaoqiang Deng and Hui Zhang, Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces (arXiv:2610.21054v1, 2026-10-15). Credits the OpenAI sphere manuscript as the source of its finite-cut and doubling strategy and claims infinitude for arbitrary smooth metrics on spheres, complex and quaternionic projective spaces, and the Cayley plane, stating that its proofs do not assume the manuscript's theorem or intermediate lemmas; Codex assistance is acknowledged.
- Kenta Kitamura (GitHub KitaKen1), Infinitely Many Closed Geodesic Images on Reversible Finsler Spheres (GitHub repository, public draft 1, 9 October 2026). Claims the same conclusion for smooth, strongly convex, reversible Finsler metrics on $S^n$, $n\ge3$, with a README proof sketch (degree-uniform filling bound, reflection-equivariant classes, half-translation fixed locus, doubling inequality) that follows this paper's outline without naming it. [AI-generated per source]
- Kenta Kitamura (KitaKen1), Add a conjecture: infinitely many closed geodesic images on reversible Finsler spheres (google-deepmind/formal-conjectures issue #7041, 9 October 2026). Proposes the reversible Finsler analogue of this paper's Theorem 1.1 as a new Formal Conjectures entry, linking the note above; its disclosure says ChatGPT and OpenAI Codex assisted. [AI-generated per source]
Comments and discussion
- Marco Mazzucchelli, Alexandru Oancea and Egor Shelukhin (organizers), Closed geodesics on Riemannian spheres (Symplectic Zoominar working group, first listed talk 2026-10-21). A specialist reading group on this paper, listing Oancea's opening talk "Tools for the closed geodesics problem, and history" followed by talks by Mazzucchelli and Shelukhin.
Related prior literature mentioned by others
- Viktor L. Ginzburg, Başak Z. Gürel and Marco Mazzucchelli, Barcode entropy of geodesic flows (arXiv:2212.00943v2, 2024-12-11). Cited in Section 1.4 of the Chen–Deng–Zhang follow-up as context for the filtered loop-space methods used in this paper and its extension.
Possibly relevant literature (found by the agent, not checked)
- Alexandru Oancea, with an appendix by Umberto Hryniewicz, Morse theory, closed geodesics, and the homology of free loop spaces (arXiv:1406.3107, 2014-06-12). Survey of free-loop-space Morse theory, including Bott iteration, Gromoll–Meyer and SO(2) versus O(2) symmetry, background for the paper's reflection-equivariant argument; the appendix treats the classical two-sphere infinitude result used in Theorem 1.1.
- Keith Burns and Vladimir S. Matveev, Open problems and questions about geodesics (arXiv:1308.5417, Ergodic Theory Dynam. Systems 41 (2021) 641–684). Survey that discusses the infinitude problem as open beyond surfaces and generic metrics, the setting of the paper's main claim for spheres.
Digests and expositions
- Latent Seminar, Infinitely many closed geodesic images (2026-10-07). Exposition of the uniform filling estimate, transient reflection classes, doubling argument and finite-cover consequences, which states that it is not an independent verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 345
Independent sites; their content is not reviewed here.
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}
}Family 345: 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.
This is the only paper in this family.