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Math-2667 · mathematics · Geometry

Metric classification of Blaschke manifolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If a connected closed smooth Riemannian manifold of positive dimension has injectivity radius equal to diameter, must some positive scaling of its metric be isometric to a standard round sphere, real/complex/quaternionic projective space, or Cayley plane?

Short English statement

If a connected closed smooth Riemannian manifold of positive dimension has injectivity radius equal to diameter, must some positive scaling of its metric be isometric to a standard round sphere, real/complex/quaternionic projective space, or Cayley plane?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2667-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2667

Metric Blaschke rigidity: The manuscript claims that a closed connected manifold with injectivity radius equal to diameter is a scaled standard compact rank-one symmetric space. The assertion includes all positive dimensions and model types. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Metric classification of Blaschke manifolds

OpenAI's claim

Metric Blaschke rigidity: The manuscript claims that a closed connected manifold with injectivity radius equal to diameter is a scaled standard compact rank-one symmetric space. The assertion includes all positive dimensions and model types.

Mathematical statement of the claimed result

Mathematical claim 1.
For a smooth closed manifold with a Riemannian metric, assume connectedness and positive dimension, equality of global injectivity radius and diameter, $\operatorname{inj}(M,g)=\operatorname{diam}(M,g)$, is claimed to imply that after a positive constant scaling the metric is isometric to one of these standard models: a round sphere, projective space over $\mathbb R$, $\mathbb C$ or $\mathbb H$, or a Cayley plane. The conclusion classifies the metric, rather than only the diffeomorphism type.

Outline of the argument

Metric Blaschke rigidity: Normal Jacobi fields along a closed geodesic are encoded as Lagrangian planes. A fixed graph chart factors the endpoint determinant using a nonsingular block and its Schur complement, with the latter controlled by the cut-time intersection data. Sharp volume comparison and equality analysis combine with the corresponding upper bound, forcing the metric to equal its model.

Formal verification

Metric Blaschke rigidity: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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