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Math-2401 · mathematics · Geometry

Vanishing of the $hat A$-Genus Under Nonnegative Curvature

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Does the $\hat A$-genus vanish for every spin manifold that admits a metric of nonnegative sectional curvature?

Formulation references: [S1], [S2].

Short English statement

Does the $\hat A$-genus vanish for every spin manifold that admits a metric of nonnegative sectional curvature?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2401-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2401

Zero-plane Einstein rigidity: The explicit nonnegative Einstein fourfold classification gives vanishing hat-A for its spin positive-Einstein subclass. The target allows arbitrary spin dimensions and arbitrary nonnegative-curvature metrics.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Vanishing of the \$hat A\$-Genus Under Nonnegative Curvature

OpenAI's claim

Zero-plane Einstein rigidity: The claim assumes a positive Einstein constant on a closed four-manifold. If sectional curvature is nonnegative and one plane is flat, the universal cover is the product of two round constant-curvature surfaces. The radii are fixed by the Einstein normalization.

Mathematical statement of the claimed result

Mathematical claim 1.
For a connected smooth closed four-manifold with $\operatorname{Ric}_g=3g$ and sectional curvature $\ge0$, the existence of one zero-curvature tangent two-plane is claimed to force its universal Riemannian cover to be \[S^2(1/\sqrt3)\times S^2(1/\sqrt3).\] Its corollary classifies every such universal cover as this product, round $S^4$ of curvature $1$, or Fubini--Study $\mathbb{CP}^2$ with $\operatorname{Ric}=3g$. This is an Einstein four-dimensional assertion, not a classification of all nonnegatively curved manifolds.

Outline of the argument

Zero-plane Einstein rigidity: Bonnet--Myers reduces the analysis to the compact simply connected cover. The two Weyl blocks satisfy a sharp coupled spectral constraint at the boundary of the curvature cone. Extending the weighted comparisons to this boundary and tracking equality forces the product geometry rather than merely a weaker norm bound.

Formal verification

Family 348 describes Lean coverage for the strict-positive-curvature Einstein classification; zero-plane rigidity and the small-energy gap are separate.

Zero-plane Einstein rigidity: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

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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