we-solve (beta)

Math-2661 · mathematics · Geometry

Gigli distributional sectional-curvature characterization

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a complete separable metric space (X,d), integer $n\geq 2$ and $\kappa \in R$, is being an n-dimensional Alexandrov space of curvature $\geq \kappa $ equivalent to full-support $RCD((n-1)\kappa ,n)$ with measure $H^{n}_d$ and distributional curvature satisfying $R(A,B,B,A)(f)\geq \kappa \int f(|A|^{2}|B|^{2}-\langle A,B\rangle ^{2})dH^{n}$ for every nonnegative test function f and test vector fields A,B? Use the unreduced RCD and test classes in the cited theorem.

Short English statement

For a complete separable metric space (X,d), integer $n\geq 2$ and $\kappa \in R$, is being an n-dimensional Alexandrov space of curvature $\geq \kappa $ equivalent to full-support $RCD((n-1)\kappa ,n)$ with measure $H^{n}_d$ and distributional curvature satisfying $R(A,B,B,A)(f)\geq \kappa \int f(|A|^{2}|B|^{2}-\langle A,B\rangle ^{2})dH^{n}$ for every nonnegative test function f and test vector fields A,B? Use the unreduced RCD and test classes in the cited theorem.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2661-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2661

The main equivalence uses the exact $n$-dimensional Hausdorff reference measure, full support and the unreduced $\operatorname{RCD}((n-1)\kappa,n)$ condition together with the global test-class curvature inequality. It claims precisely equivalence with $n$-dimensional Alexandrov curvature bounded below by $\kappa$.

Gigli distributional sectional-curvature characterization

OpenAI's claim

Distributional characterization of Alexandrov curvature: The paper claims equivalence between lower Alexandrov curvature and the corresponding noncollapsed RCD bound together with Gigli’s distributional sectional inequality. It uses the original global test classes for every integer dimension at least two.

Mathematical statement of the claimed result

Mathematical claim 1.
For a complete separable metric space $(M,d)$, integer $n\ge2$ and $\kappa\in\mathbb R$, the claim equates being an $n$-dimensional Alexandrov space of curvature $\ge\kappa$ with the following analytic conditions: $m=\mathcal H_d^n$ has full support, $(M,d,m)$ is unreduced $\operatorname{RCD}((n-1)\kappa,n)$, and \[R(X,Y,Y,X)(f)\ge\kappa\int f(|X|^2|Y|^2-\langle X,Y\rangle^2)\,dm\] for every test field pair $X,Y$ and nonnegative test $f$. The test functions lie in the $L^2$ Laplacian domain, have bounded globally Lipschitz representatives and $\Delta f\in W^{1,2}$; test fields are finite sums $a\nabla b$ of such functions. With $[X,Y]=\nabla_XY-\nabla_YX$, the distributional tensor is \[\begin{aligned}R(X,Y,Z,W)(f)=\int\bigl(&-\langle\nabla_X(fW),\nabla_YZ\rangle-f\langle\nabla_YZ,W\rangle\operatorname{div}X\\&+\langle\nabla_Y(fW),\nabla_XZ\rangle+f\langle\nabla_XZ,W\rangle\operatorname{div}Y-f\langle\nabla_{[X,Y]}Z,W\rangle\bigr)\,dm.\end{aligned}\] The curvature sign is $\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$. RCD here combines the unreduced curvature-dimension condition with quadratic Cheeger energy; the reference measure is exactly Hausdorff measure.

Outline of the argument

Distributional characterization of Alexandrov curvature: RCD calculus supplies the starting transport and differential structure. Additional index, frame, and distance-comparison arguments turn the distributional sectional condition into geometric triangle comparison. The companion reweighting theorem propagates weak Hessian bounds to prescribed geodesics, enabling the curvature implication without replacing the original test classes by stronger ones.

Formal verification

Family 356 describes Lean coverage for the weak-Hessian inequality on every geodesic; the full sectional-curvature characterization is separate.

Distributional characterization of Alexandrov curvature: 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

Go to contribution →