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

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Curvature two-form structure equation

Statement

Let $e = (e_{1}, \dots, e_{r})$ be a local frame, let $ω = (ω^{i}_{j})$ be its connection matrix, and let $Ω = (Ω^{i}_{j})$ be the matrix of the curvature two-form, defined by $R^{\nabla} (X, Y) e_{j} = Ω^{i}_{j} (X, Y) e_{i}$ . Then

$Ω = d ω + ω \land ω,$

where multiplication order is fixed by

$(ω \land ω)^{i}_{j} = \sum_{k} ω^{i}_{k} \land ω^{k}_{j} .$

Facts & Assumptions

[F1]

Bundle curvature is an $End (E)$ -valued two-form. Vector-bundle curvature is an endomorphism-valued two-form.

[F2]

In the supplied frame, $\nabla_{X} e_{j} = ω^{i}_{j} (X) e_{i}$ . Connection one form in a local frame.

[F3]

For $s = e u$ , $\nabla_{X} s = e (X u + ω (X) u)$ . Local coordinate formula for a bundle connection.

[F4]

The exterior derivative of a local coordinate expansion differentiates its scalar coefficients. The local coordinate formula for the exterior derivative.

[F5]

The wedge product is the pointwise alternating product of forms. The wedge product of differential forms.

Proof

Given: A local frame $e$ , a coordinate chart on its domain, and coordinate fields $\partial_{a}, \partial_{b}$ .

1.1

Since coordinate fields commute, expand the defining curvature commutator on $e_{j}$ with [F2]–[F3]. The coefficient of $e_{i}$ is $\partial_{a} (ω^{i}_{j} (\partial_{b})) - \partial_{b} (ω^{i}_{j} (\partial_{a})) + \sum_{k} (ω^{i}_{k} (\partial_{a}) ω^{k}_{j} (\partial_{b}) - ω^{i}_{k} (\partial_{b}) ω^{k}_{j} (\partial_{a}))$ .

F1F2F3algebra

2.1

By [F4], the first two terms in step 1.1 are $d ω^{i}_{j} (\partial_{a}, \partial_{b})$ ; by [F5], the sum is $\sum_{k} (ω^{i}_{k} \land ω^{k}_{j}) (\partial_{a}, \partial_{b})$ in precisely the stated matrix order. Both sides are two-forms by [F1], so equality on every coordinate-frame pair proves $Ω^{i}_{j} = d ω^{i}_{j} + \sum_{k} ω^{i}_{k} \land ω^{k}_{j}$ .

F1F4F5step 1.1algebra∎

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Sources

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