The two-or-infinity conjecture for prime closed geodesics on the two-sphere
Let be a Finsler metric on , and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Two-or-infinity conjecture for Finsler metrics. Every Finsler metric on has either two or infinitely many prime closed geodesics. This is the Finsler-geometric analogue of the two-or-infinity prediction for Reeb flows: Katok constructed a metric with exactly two closed geodesics, while the corresponding general assertion is described in the source as a longstanding conjecture.
References
Primary source
Dan Cristofaro-Gardiner, “Low-dimensional topology and symplectic dynamics”, arXiv:2510.07680 (2025).
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI. Claims infinitely many prime closed geodesics with distinct images for every smooth Riemannian sphere of dimension at least two, including degenerate metrics, and for every nonempty closed smooth three-manifold.See full solution
Claimed by OpenAI. Claims infinitely many prime closed geodesics with distinct images for every smooth Riemannian sphere of dimension at least two, including degenerate metrics, and for every nonempty closed smooth three-manifold.
Scope relative to this problem: This is progress only for the Riemannian subclass of the target Finsler metrics on S^2. The source asserts infinitely many distinct prime geodesic images there; it does not settle the two-or-infinity assertion for arbitrary, possibly irreversible, Finsler metrics. Its additional higher-dimensional sphere and three-manifold results do not enlarge that target subclass.
GitHub repository: https://github.com/openai/math
- OpenAI-345-01-Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere.pdfOpen