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.

Weingarten equation and adjointness of the shape operator

Statement

Assume ${A C}_{ω}$ . For tangent fields $X, Y$ and a normal field $ν$ along an embedded Riemannian submanifold,

${\overline{\nabla}}_{X} ν = - S_{ν} X + \nabla_{X}^{⊥} ν$

and

$g (S_{ν} X, Y) = \overline{g} (I I (X, Y), ν) .$

Consequently every shape operator $S_{ν}$ is self-adjoint. The choice hypothesis is inherited exactly through the smooth normal-bundle projections.

Facts & Assumptions

Given: Countable choice, the embedded Riemannian submanifold, tangent fields $X, Y$ , and a normal field $ν$ .

[F1]

The shape operator is $S_{ν} X = - ({\overline{\nabla}}_{X} ν)^{⊤}$ . Shape operator.

[F2]

The normal connection is $\nabla_{X}^{⊥} ν = ({\overline{\nabla}}_{X} ν)^{⊥}$ . Normal connection.

[F3]

The Gauss decomposition is ${\overline{\nabla}}_{X} Y = \nabla_{X}^{M} Y + I I (X, Y)$ . Induced connection and second fundamental form.

[F4]

The ambient Levi–Civita connection is compatible with $\overline{g}$ . Levi civita connection.

[F5]

The second fundamental form is symmetric. The second fundamental form is a symmetric normal-bundle-valued two-tensor.

Proof

technique · direct

1.1

Split ${\overline{\nabla}}_{X} ν$ into its tangential and normal components. By [F1] its tangential component is $- S_{ν} X$ , and by [F2] its normal component is $\nabla_{X}^{⊥} ν$ . This proves the first displayed identity.

F1F2algebra

2.1

Since $\overline{g} (ν, Y) = 0$ along $M$ , differentiation in the tangent direction $X$ and [F4] give $0 = X \overline{g} (ν, Y) = \overline{g} ({\overline{\nabla}}_{X} ν, Y) + \overline{g} (ν, {\overline{\nabla}}_{X} Y) .$ By step 1.1 the first inner product is $- g (S_{ν} X, Y)$ ; by [F3] the second is $\overline{g} (ν, I I (X, Y))$ , since $ν$ is normal and $\nabla_{X}^{M} Y$ is tangent. Rearranging proves the second displayed identity.

F3F4step 1.1algebra

3.1

Using [F5] and the symmetry of the metric, $g (S_{ν} X, Y) = \overline{g} (I I (X, Y), ν) = \overline{g} (I I (Y, X), ν) = g (S_{ν} Y, X) = g (X, S_{ν} Y) .$ Thus $S_{ν}$ is self-adjoint.

F5step 2.1algebra

4.1

All identities are vacuous on the empty submanifold and reduce to the unique zero maps when tangent or normal rank is zero. They apply unchanged in rank one and at boundary points. Positive definiteness is used for the orthogonal splitting and for the usual self-adjoint interpretation. The stated ${A C}_{ω}$ is inherited through [F1]–[F3], and no new selection occurs.

F1F2F3step 1.1step 2.1step 3.1∎

Depends on

Used by

Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface Definition
A great sphere is totally geodesic Example
The catenoid has zero mean curvature but is not totally geodesic Example
The cylinder has zero Gaussian curvature but nonzero second fundamental form Example
The round sphere has positive constant sectional curvature Example
Euclidean hypersurface sectional curvature from principal curvatures Proposition
First variation of volume for a normal variation Proposition
Mean curvature and minimal submanifolds Remark
Gauss equation for a Riemannian submanifold Theorem
Gauss’s Theorema Egregium Theorem
Ricci equation for the normal connection Theorem

Dependency tree · two levels

12 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

Chuu-Lian Terng, Lecture Notes on Curves and Surfaces in R^3 and Riemannian Geometry (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)
John M. Lee, Riemannian Manifolds: An Introduction to Curvature (standard reference, not scraped)