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

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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.

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Vector-bundle curvature is an endomorphism-valued two-form

Statement

For a connection $\nabla$ on $E \to M$ , the curvature $R^{\nabla} (X, Y) s$ is $C^{\infty} (M)$ -linear separately in $X$ , $Y$ , and $s$ , and is alternating in $X, Y$ . Its pointwise values depend smoothly on the base point and therefore define

$R^{\nabla} \in Ω^{2} (M; End (E)) .$

Facts & Assumptions

[F1]

Bundle curvature is the bracket-corrected commutator of covariant derivatives. Curvature of a vector-bundle connection.

[F2]

A connection is function-linear in its differentiating field and satisfies the section Leibniz rule. Connection laws in directional form.

[F3]

The Lie bracket satisfies $[f X, Y] = f [X, Y] - Y (f) X$ and $[X, f Y] = f [X, Y] + X (f) Y$ . Leibniz rules for the Lie bracket with function multiples.

[F4]

A smooth two-form is a smooth section of the alternating second cotangent power. A smooth differential $k$ -form.

[F5]

The Hom bundle has fibre $Hom (E_{p}, E_{p}) = End (E_{p})$ . Dual and Hom vector bundles.

[F6]

The Hom construction carries a smooth vector-bundle structure, and finite tensor products of smooth vector bundles carry canonical smooth product-frame structures. Dual and Hom transition functions define smooth bundles, Finite tensor products of smooth vector bundles.

Proof

Given: Smooth vector fields $X, Y$ , a smooth function $f$ , a smooth section $s$ of $E$ , and a connection $\nabla$ .

1.1

Expanding $R (f X, Y) s$ with [F1]–[F3] produces $f R (X, Y) s - Y (f) \nabla_{X} s + Y (f) \nabla_{X} s = f R (X, Y) s$ ; expanding $R (X, f Y) s$ produces $f R (X, Y) s + X (f) \nabla_{Y} s - X (f) \nabla_{Y} s = f R (X, Y) s$ . Interchanging $X, Y$ in [F1] and using $[Y, X] = - [X, Y]$ gives $R (Y, X) s = - R (X, Y) s$ .

F1F2F3algebra

2.1

Applying the section Leibniz rule twice gives $R (X, Y) (f s) = f R (X, Y) s + (X (Y f) - Y (X f) - [X, Y] f) s = f R (X, Y) s$ . Thus evaluation at a point depends only on $X_{p}, Y_{p}, s_{p}$ , and the result is alternating in the tangent entries.

F1F2step 1.1algebra

3.1

On a neighborhood with tangent frame $E_{i}$ and bundle frame $e_{a}$ , each $R (E_{i}, E_{j}) e_{a}$ is a smooth section because [F1] combines connection derivatives and a Lie bracket of smooth inputs. Its smooth frame coefficients are alternating in $i, j$ by step 1.1 and define a smooth section of $⋀^{2} T^{*} M \otimes End (E)$ by [F4]–[F6]. Step 2.1 shows that this section acts on arbitrary $X, Y, s$ as the original curvature.

F1F4F5F6step 1.1step 2.1∎

Depends on

Used by

Flat connections have locally path-independent parallel transport on a coordinate ball Proposition
A flat connection admits local parallel frames Theorem
Curvature two-form structure equation Theorem

Dependency tree · two levels

22 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

Will J. Merry, Differential Geometry (2021) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)