The Blaschke conjecture: Blaschke manifolds are compact rank-one symmetric spaces
A connected closed Riemannian manifold of positive dimension is a Blaschke manifold if its injectivity radius equals its diameter, so every unit-speed geodesic minimizes up to the diameter. Round spheres, the real, complex and quaternionic projective spaces and the Cayley plane with their standard metrics are Blaschke. Blaschke posed the problem for surfaces (settled by Green, 1963); the sphere and real projective cases were settled by Berger, Kazdan, Weinstein and Yang, and Bott-Samelson theory fixes the cohomology type, but the metric question was open for the complex, quaternionic and Cayley types. Is every Blaschke manifold isometric, up to scaling, to a compact rank-one symmetric space?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Riemannian geometry; manifolds all of whose geodesics are closed
- Posed by
- Wilhelm Blaschke (surfaces); general metric form in Arthur Besse's monograph (1978)
- Year posed
- 1921
- Years open
- 105y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if is connected, closed, smooth, of positive dimension, with , then is isometric up to scale to a round sphere, a standard real, complex or quaternionic projective space, or the Cayley plane. Theorem 1.2 gives against the model at equal diameter, with equality only for isometry; the upper bound in quaternionic dimension , , is proved from characteristic classes and the geodesic space. It does not treat the weaker Besse-type questions on manifolds all of whose geodesics are closed without the Blaschke condition (SC or P-manifolds in general).
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it covers every connected closed smooth Blaschke manifold of positive dimension, including all quaternionic dimensions and the Cayley plane, with conclusion isometry up to scale to a compact rank-one symmetric space. The argument is a sharp volume lower bound with rigidity (via Szabo's classification of harmonic manifolds) plus a matching upper bound from topology; it cites results of Yang, Kramer-Stolz and a theorem attributed to Reznikov via Wilking, whose hypotheses it says it verifies. Not refereed. No Lean formalization in the release.