The metric Blaschke conjecture
From OpenAI Aftermath
The metric Blaschke conjecture asks whether a closed Riemannian manifold on which every geodesic minimizes up to the same distance, equal to the diameter, must be one of the compact rank-one symmetric spaces: a round sphere, a real, complex or quaternionic projective space, or the Cayley plane. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026 and listed as its result 344, claims the conjecture in full. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Blaschke's original question of 1921 concerned surfaces, and Green settled the surface problem in 1963. The later geometric formulation is described in Besse's 1978 book and in McKay's corrected account, and the Bott–Samelson restrictions explain why the compact rank-one symmetric models appear.
- The sphere and real projective cases were established through the work of Berger, Kazdan, Weinstein and Yang; Berger's argument combines a volume problem with Kazdan's integral inequality, and Berger's later sharp isoembolic inequality uses a weighted comparison of Jacobi determinants.
- For the projective models the manuscript distinguishes topology, volume and isometry: Yang proved the model volume for Blaschke manifolds of complex projective homotopy type, Kramer and Stolz identified the projective planes diffeomorphically, and Wilking records an equal-volume theorem attributed to Reznikov.
- Rigidity was known under additional hypotheses: an upper curvature bound (Shankar, Spatzier and Wilking), a normalized lower curvature bound (Su, Sun and Wang), and curvature bounds with agreement with a model away from a convex set (Connell, Islam, Nguyen and Spatzier). A preprint of K. Song, Blaschke conjecture and complex geometry (arXiv:2609.23822, 20 September 2026), proves metric rigidity assuming an adapted complex structure on the entire tangent bundle, a hypothesis additional to the Blaschke condition; the OpenAI manuscript cites it.
The original result
- Claim. The main theorem of The metric Blaschke theorem states that a connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is isometric, up to multiplication of the metric by a positive constant, to a round sphere, a standard real, complex or quaternionic projective space, or the Cayley plane, in every quaternionic dimension.
- Method, as described: the manuscript proves a sharp lower bound for the volume in terms of the model with the same diameter, by splitting the determinant of the Jacobi-field endpoint pairing along each closed geodesic into a regular factor and a Schur-complement factor and combining the resulting scalar estimates with a logarithmic form of Kazdan's weighted cancellation; equality forces every radial volume density to agree with the model, and Szabó's classification of compact simply connected harmonic manifolds then gives isometry. A matching upper bound comes from Yang's theorem for the complex projective type, from the Kramer–Stolz identification and the equal-volume theorem recorded by Wilking for the projective planes, and, in quaternionic dimension 4r with r ≥ 2, from an intrinsic argument through the tangent Pontryagin class of an incidence manifold, the signature theorem, integral Gysin sequences of the space of geodesics and the first negative bundle of the free-loop energy.
- Formalization. The repository's catalogue lists no Lean formalization for this family.
- Source (openai/math repository, commit
7ef5c0c): The metric Blaschke theorem, dated 23 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | The metric Blaschke theorem, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the claim that a closed Riemannian manifold on which every geodesic minimizes up to the diameter is a standard compact rank-one symmetric space up to scale, and the two volume comparisons of the proof: the lower bound from Jacobi fields and a weighted convexity identity, and the quaternionic upper bound from the geometry and integral topology of the space of closed geodesics. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_409 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap and one minor note: the upper volume bound for the Cayley plane rests on a secondary attribution, Wilking's crediting of an equal-volume theorem to Reznikov, without a primary source covering that case, so a reader would need to confirm that the cited statement applies. The reviewer states that it read every section and checked the determinant splitting, the weighted scalar identity and the global volume comparison line by line, with a numerical sanity check of the comparison. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The review is published under CC BY 4.0. |
| 2026-10-19 | AI Proved What?, episode 344 Only five shapes?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim and its history. The episode states that the key new step is the sharp lower volume bound with equality only for the model, and that statements about manifolds where only some points have the property, or where geodesics all close but need not minimize that long, are not claimed. It notes that the paper relies on deep earlier theorems, including Szabó's on harmonic manifolds and Yang's volume theorem, and cites Reznikov's theorem only through a later paper, and that Song's September 2026 proof under an extra hypothesis is cited. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 41 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that it found no specialist comment and no reported error. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- See also Infinitely many closed geodesics on Riemannian spheres for another result of the collection on geodesics of closed manifolds.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
