TheoremStatement: 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.

Gauss’s Theorema Egregium

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 $M^{2} \subseteq R^{3}$ be a surface with its induced metric. For every $p \in M$ and either smooth local unit normal $ν$ near $p$ ,

$\det S_{ν, p} = K (T_{p} M) .$

The left side is independent of the sign of $ν$ and equals the intrinsic sectional curvature, so it is determined by the induced Riemannian metric. The choice hypothesis is inherited through both the smooth submanifold projection constructions and the supplied sectional-curvature interface.

Facts & Assumptions

Given: Countable choice, the Euclidean surface, a point $p$ , and a supplied local unit normal $ν$ .

[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]

The Gauss equation expresses intrinsic curvature as ambient curvature plus the two ordered quadratic II terms. Gauss equation for a Riemannian submanifold.

[F2]

The shape operator satisfies $g (S_{ν} X, Y) = ⟨ I I (X, Y), ν ⟩$ and is self-adjoint. Weingarten equation and adjointness of the shape operator.

[F3]

Euclidean space is locally isometric to itself and hence flat. A Riemannian manifold is flat iff it is locally isometric to Euclidean space.

[F4]

For an orthonormal tangent pair $(e_{1}, e_{2})$ , $K (T_{p} M) = {Rm}^{M} (e_{1}, e_{2}, e_{2}, e_{1})$ . Sectional curvature.

[F5]

The determinant of an endomorphism is basis independent. The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space.

[F6]

The induced metric determines a unique Levi–Civita connection. Fundamental theorem of riemannian geometry.

Proof

technique · direct

1.1

Choose one orthonormal basis $(e_{1}, e_{2})$ of the supplied tangent plane. Since the normal fibre is spanned by $ν$ , [F2] gives $I I (e_{i}, e_{j}) = g (S_{ν} e_{i}, e_{j}) ν .$ Write $h_{i j} = g (S_{ν} e_{i}, e_{j})$ . In this basis [F5] gives $\det S_{ν, p} = h_{11} h_{22} - h_{12} h_{21}$ .

F2F5choosealgebra

2.1

Apply [F1] with $X = W = e_{1}$ and $Y = Z = e_{2}$ . The ambient term vanishes by [F3], and step 1.1 yields ${Rm}^{M} (e_{1}, e_{2}, e_{2}, e_{1}) = h_{11} h_{22} - h_{12} h_{21} = \det S_{ν, p} .$ By [F4] this is $K (T_{p} M)$ .

A1F1F3F4step 1.1algebra

3.1

Replacing $ν$ by $- ν$ replaces $S_{ν}$ by $- S_{ν}$ ; in dimension two, $\det (- S_{ν}) = (- 1)^{2} \det S_{ν} = \det S_{ν}$ . Thus the value is independent of the local normal sign. By [F6], the curvature tensor and therefore [F4]'s sectional curvature are determined solely by the induced metric, proving the intrinsic conclusion.

F5F6step 2.1algebra

4.1

The assertion is vacuous for the empty surface and is specifically two-dimensional, so zero- and one-dimensional cases are outside its hypothesis. The fibrewise calculation applies at boundary points and positive definiteness supplies the orthonormal basis. Only one finite basis and one supplied local normal are used. The stated ${A C}_{ω}$ is inherited through [F1]–[F2] and [F4], with no new family selection.

F1F2F4F5step 1.1step 2.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

31 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)