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

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

Statement

Assume ${A C}_{ω}$ . The second fundamental form of an embedded Riemannian submanifold is $C^{\infty} (M)$ -bilinear and symmetric in its tangent arguments. Hence it is a smooth section $I I \in Γ (S^{2} T^{*} M \otimes ν M)$ .

Facts & Assumptions

Given: Countable choice, an embedded Riemannian submanifold, and tangent fields $X, Y$ .

[F1]

The second fundamental form is the normal projection $I I (X, Y) = ({\overline{\nabla}}_{X} Y)^{⊥}$ of a well-defined smooth field along $M$ . Induced connection and second fundamental form.

[F2]

Covariant differentiation is function-linear in its direction and obeys the section Leibniz rule. Connection laws in directional form.

[F3]

The induced connection is torsion free and equals the Levi–Civita connection of the induced metric. The induced connection is Levi–Civita.

Proof

technique · direct

1.1

For $f \in C^{\infty} (M)$ , function-linearity in the first slot and fibrewise linearity of the normal projection give $I I (f X, Y) = ({\overline{\nabla}}_{f X} Y)^{⊥} = f ({\overline{\nabla}}_{X} Y)^{⊥} = f I I (X, Y) .$ Real linearity follows identically.

F1F2algebra

1.2

The Leibniz rule in the second slot gives $I I (X, f Y) = (X (f) Y + f {\overline{\nabla}}_{X} Y)^{⊥} = f I I (X, Y),$ because $X (f) Y$ is tangent and has zero normal projection. Thus $I I$ is $C^{\infty} (M)$ -bilinear.

F1F2algebra

1.3

Subtract the two Gauss decompositions from [F1]. Ambient torsion freeness gives $0 = {\overline{\nabla}}_{X} Y - {\overline{\nabla}}_{Y} X - [X, Y] = (\nabla_{X}^{M} Y - \nabla_{Y}^{M} X - [X, Y]) + I I (X, Y) - I I (Y, X) .$ The parenthesized tangent term is zero by [F3], so the remaining normal term proves $I I (X, Y) = I I (Y, X)$ .

F1F3algebra

2.1

Smoothness was supplied in [F1], while steps 1.1–1.3 give tensoriality and symmetry; this is exactly a section of $S^{2} T^{*} M \otimes ν M$ . For an empty or zero-dimensional $M$ it is the unique zero section, and the formulas apply unchanged in dimension one, codimension zero, and at boundary points. Degenerate ambient forms are excluded by the Riemannian hypothesis. The stated ${A C}_{ω}$ is inherited exactly through [F1] and [F3]; no new selection occurs.

F1F3step 1.1step 1.2step 1.3∎

Depends on

Used by

Mean curvature vector Definition
Totally geodesic submanifold Definition
Codazzi equation for a Riemannian submanifold Theorem
Equivalent characterizations of a totally geodesic submanifold Theorem
Weingarten equation and adjointness of the shape operator 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)