TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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The exterior derivative commutes with pullback

Statement

For every smooth map $F : M \to N$ and every form $ω$ on $N$ , $d (F^{*} ω) = F^{*} (d ω) .$

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that On a chart, if $ω = \sum_{I} ω_{I} d x^{I}$ , then $d ω = \sum_{I} d ω_{I} \land d x^{I} .$ (The local coordinate formula for the exterior derivative).

Proof

technique · direct

1.1given

Fix $p \in M$ , choose target coordinates $(y^{1}, \dots, y^{n})$ near $F (p)$ , and write $ω = \sum_{I} ω_{I} d y^{I}$ there. Pullback sends $ω_{I}$ to $ω_{I} \circ F$ and $d y^{a}$ to $d (y^{a} \circ F)$ .

2.1F1step 1.1∎

On a neighbourhood of $p$ , the coordinate formula and the pullback wedge law give $\begin{aligned} d (F^{*} ω) & = \sum_{I} d (ω_{I} \circ F) \land d (y^{i_{1}} \circ F) \land \dots \land d (y^{i_{k}} \circ F) \\ = F^{*} ⁣ (\sum_{I} d ω_{I} \land d y^{I}) = F^{*} (d ω) . \end{aligned}$ Since $p$ was arbitrary, the identity holds on $M$ .

Depends on

Used by

Closed differential forms are locally exact Corollary
Maximal compact subgroups exist and are conjugate in a connected finite center semisimple Lie group Corollary
Pullback carries closed forms to closed forms and exact forms to exact forms Corollary
Bigraded complex forms and the Dolbeault operators Definition
Averaged pullback Whitney moments and uniform near identity Lemma
Form calculus extends locally across a manifold boundary Lemma
Restriction of a foliation transverse to the boundary Lemma
The de Rham complex and pullback extend to manifolds with boundary Lemma
A graph of a one-form is Lagrangian exactly when the form is closed Proposition
Cartan decomposition gives the invariant metric and curvature of G mod K Proposition
Cotangent lifts are symplectomorphisms Proposition
Lie derivatives are natural for related vector fields Proposition
Products and opposites of symplectic manifolds Proposition
Pullback is a morphism of de rham complexes Proposition
De rham homotopy formula for a smooth homotopy Theorem
Stokes theorem for smooth singular chains 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)