PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Euclidean hypersurface sectional curvature from principal curvatures

Statement

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Sectional curvature; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

Assume ${A C}_{ω}$ . Let $M^{m} \subseteq R^{m + 1}$ be a Euclidean hypersurface with $m \geq 2$ and a supplied smooth unit normal. If $e_{i}, e_{j} \in T_{p} M$ are orthonormal principal directions with $i \neq j$ and principal curvatures $κ_{i}, κ_{j}$ , then

$K (span {e_{i}, e_{j}}) = κ_{i} κ_{j} .$

The choice hypothesis is inherited through both the smooth hypersurface shape/projection constructions and the supplied sectional-curvature interface.

Facts & Assumptions

Given: Countable choice, the Euclidean hypersurface, a point $p$ , and the two supplied orthonormal principal directions.

[A1]

${A C}_{ω}$ is countable choice and is required here through Sectional curvature; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

Principal directions satisfy $S_{ν} e_{a} = κ_{a} e_{a}$ . Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface.

[F2]

For tangent vectors, $g (S_{ν} X, Y) = ⟨ I I (X, Y), ν ⟩$ . Weingarten equation and adjointness of the shape operator.

[F3]

The Gauss equation has quadratic terms in the order stated on this page. Gauss equation for a Riemannian submanifold.

[F4]

Euclidean space is locally isometric to itself and therefore has zero Riemann curvature. A Riemannian manifold is flat iff it is locally isometric to Euclidean space.

[F5]

On an orthonormal pair, sectional curvature is $Rm (e_{i}, e_{j}, e_{j}, e_{i})$ . Sectional curvature.

Proof

technique · direct

1.1

The normal bundle is spanned by the unit field $ν$ . By [F1]–[F2], $I I (e_{i}, e_{i}) = κ_{i} ν, I I (e_{j}, e_{j}) = κ_{j} ν, I I (e_{i}, e_{j}) = g (S_{ν} e_{i}, e_{j}) ν = 0,$ because $e_{i}, e_{j}$ are orthogonal eigenvectors. Symmetry gives the same mixed value in the reversed order.

F1F2algebra

2.1

Substitute $X = e_{i}$ , $Y = Z = e_{j}$ , and $W = e_{i}$ into [F3]. The ambient term is zero by [F4]; step 1.1 makes the first quadratic term $κ_{i} κ_{j}$ and the mixed term zero. Thus ${Rm}^{M} (e_{i}, e_{j}, e_{j}, e_{i}) = κ_{i} κ_{j} .$ Since the pair is orthonormal, [F5] identifies the left side with the asserted sectional curvature.

A1F3F4F5step 1.1algebra

3.1

The assertion is vacuous on an empty hypersurface. Dimensions zero and one are excluded by $m \geq 2$ , exactly because no tangent two-plane exists there. The calculation applies at a boundary point and uses positive definiteness for orthonormality. The directions are supplied, so no eigenbasis is selected; ${A C}_{ω}$ is inherited through [F1]–[F3] and [F5], and no new choice is made.

F1F2F3F5step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)
John M. Lee, Riemannian Manifolds: An Introduction to Curvature (standard reference, not scraped)