LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Closedness of invariant curvature forms

Statement

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $E \to M$ be a smooth finite-rank real or complex vector bundle with a supplied $G$ -frame atlas and a connection compatible with that reduction, and let $P_{k}$ be a homogeneous degree- $k$ $G$ -invariant polynomial as in Evaluation of an invariant polynomial on curvature. If $Ω$ is the curvature matrix in a supplied $G$ -frame, then its global evaluation $P_{k} (Ω, \dots, Ω)$ is closed. This includes $k = 0$ , where the evaluation is the constant $0$ -form. The assertion applies to $G = {GL}_{r} (C)$ , ${GL}_{r} (R)$ , $U (r)$ , and $SO (2 m)$ within their stated invariant-polynomial scopes. Finite sums are closed degree by degree.

Facts & Assumptions

Given: The manifold, bundle, compatible connection, curvature, and invariant polynomial in the Statement.

[F1]

For the curvature $Ω$ of a bundle connection, the covariant exterior derivative satisfies $d^{\nabla} Ω = 0$ (Second Bianchi identity for a bundle connection).

[F2]

For homogeneous $g$ -valued forms, the differential of the invariant-polynomial extension is the signed sum obtained by applying $D^{\nabla}$ in each slot; the identity also holds in boundary charts (Invariant polynomials cancel connection commutators).

[F3]

The compatible connection and invariant polynomial define the global curvature evaluation used in the Statement (Evaluation of an invariant polynomial on curvature).

Proof

Proof technique: apply the covariant Leibniz identity to repeated copies of the curvature and use the second Bianchi identity.

1.1F2F3givenalgebra

If $k = 0$ , the evaluation is a constant $0$ -form and its exterior derivative is zero. Suppose $k \geq 1$ . On each supplied $G$ -frame chart use [F2] with $A_{1} = \dots = A_{k} = Ω$ . Since every $A_{j}$ has degree $2$ , each prefix sign is $(- 1)^{2 (j - 1)} = 1$ , and [F2] gives $d P_{k} (Ω, \dots, Ω) = \sum_{j = 1}^{k} P_{k} (Ω, \dots, D^{\nabla} Ω, \dots, Ω)$ . By [F3] this local expression is the differential of the global form in the Statement.

2.1F1F2step 1.1algebra

In the supplied frame, $D^{\nabla}$ from [F2] is the local expression for the covariant exterior derivative $d^{\nabla}$ in [F1]. Hence $D^{\nabla} Ω = 0$ , so every summand in step 1.1 vanishes. Thus the differential is zero on every chart and therefore is zero globally. The calculation is coefficientwise and also restricts to boundary charts as specified in [F2]. The conclusion for a finite sum follows by linearity. $□$

Depends on

Used by

Chern–Weil map for a chosen connection Definition
Chern, Pontryagin, and Euler characteristic forms Definition

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

Raoul Bott, Lectures on Characteristic Classes and Foliations (standard reference, not scraped)
Stefan Haller, The Atiyah–Singer Index Theorem, Vienna lecture notes (2013) (standard reference, not scraped)