The metric Blaschke conjecture

Claims the metric Blaschke conjecture: injectivity radius equal to diameter forces a round sphere or projective space.

1 paper September 23, 2026 Explained by AI

On a round sphere, every straightest path stays shortest until it reaches the opposite point. A few other symmetric spaces, the projective spaces and the Cayley plane, share this property. The metric Blaschke conjecture says no other closed smooth space has it. Surfaces and some other cases were settled earlier. This paper claims the full classification in every dimension.

First step

1921

Wilhelm Blaschke, who asked the original question for surfaces

Significance

Landmark

It claims a long-standing conjecture in Riemannian geometry, studied for about a century, in every dimension.

Trust

Claimed, not checked

Field

Differential geometry

1 paper

The explanation

Take a round sphere of radius 1. The farthest two points can be is π, half a great circle, so the diameter is π. Start at the north pole and walk along any straight path. It stays the shortest route until it hits the south pole, at distance exactly π. So every straightest path from every point stays shortest for exactly the diameter. Now glue each point of the sphere to its opposite point. This gives the real projective plane. Its diameter is π/2, and its closed straightest paths have length π, twice the diameter. Again every straightest path stays shortest for exactly the diameter. A flat square torus with side 1 is different. Some straight paths stop being shortest after length 1/2. But its diameter is the half-diagonal, √2/2, about 0.707. The two numbers differ.

The injectivity radius at a point is how far you can go in every direction before straight paths stop being unique. The smallest value over all points is the global injectivity radius. It is never larger than the diameter. A Blaschke manifold is one where the two are equalSource 1. The round spheres, the real, complex and quaternionic projective spaces, and the Cayley plane are all Blaschke. These are the compact rank-one symmetric spaces, the most symmetric curved spaces of their kind. The metric Blaschke conjecture asks whether they are the only onesSource 2.

Green settled the surface caseSource 3. Berger, Kazdan, Weinstein and Yang settled the sphere and real projective casesSource 4. Yang showed a Blaschke manifold shaped like complex projective space has the model's volumeSource 5. The paper claims the full result. Every Blaschke manifold is, up to scale, one of these symmetric spacesSource 6.

The plan has two volume bounds. First, the paper claims every Blaschke manifold has at least the volume of its model at the same diameterSource 7. Equality holds only for the model itselfSource 8. So it is enough to prove the volume is at most the model'sSource 9. The lower bound uses Jacobi fields, which measure how nearby straight paths spread apart. For the quaternionic spaces, the upper bound was the main obstacle. The paper proves it with topology, using characteristic classes and the space of all closed straight pathsSource 10.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof combines delicate analysis with algebraic topology. No specialist review of the paper has been published.

What it doesn't show. This does not cover spaces that are not smooth or not closed. It has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark-level claim if the proof holds: it would settle the metric Blaschke conjecture in every dimension. The proof is long and has no Lean check or published specialist review.

Words used

Geodesic: A straightest possible path on a curved space.
Diameter: The largest distance between any two points of the space.
Injectivity radius: How far one can go from a point, in every direction, before straight paths stop being unique.
Blaschke manifold: A closed space whose global injectivity radius equals its diameter.
Projective space: A space of lines through the origin; the real projective plane is a sphere with opposite points glued.
Compact rank-one symmetric space: A round sphere, a real, complex or quaternionic projective space, or the Cayley plane.
Jacobi field: A measure of how a family of nearby straight paths spreads apart.
Isometric: Matched by a map that keeps all distances.

The original paper

The metric Blaschke theorem (PDF, opens in a new tab)
September 23, 2026

All quoted lines (10)

We call $M$ a Blaschke manifold if it has positive dimension and $\inj(M,g)=\diam(M,g)$ .
The metric Blaschke theorem Back to text
The metric Blaschke conjecture asks whether this property characterizes these compact rank-one symmetric spaces.
The metric Blaschke theorem Back to text
Green settled the surface problem; the later geometric formulation and its development are described in and in the corrected account of.
The metric Blaschke theorem Back to text
The sphere and real-projective metric cases were established through the work of Berger, Kazdan, Weinstein and Yang; see and.
The metric Blaschke theorem Back to text
Yang proved that a Blaschke manifold of complex projective homotopy type has the model volume at the same prime geodesic period.
The metric Blaschke theorem Back to text
If $\inj(M,g)=\diam(M,g)$ , then $(M,g)$ 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.
The metric Blaschke theorem Back to text
Every Blaschke manifold and its model at the same diameter satisfy $\Vol(M,g)\ge\Vol(M_0,g_0).$
The metric Blaschke theorem Back to text
Equality holds if and only if $(M,g)$ is isometric to $(M_0,g_0)$ .
The metric Blaschke theorem Back to text
If a Blaschke manifold has volume at most that of its model at the same diameter, then it is isometric to that model.
The metric Blaschke theorem Back to text
Characteristic-class and signature theory enter here in their usual integral and oriented forms.
The metric Blaschke theorem Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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339Katok's entropy rigidity conjectureDifferential geometry
349The Solomon–Yau least-volume conjectureDifferential geometry

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