PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Pullback of forms is smooth functorial and preserves wedges

Statement

For a smooth map $F : M \to N$ , pullback sends smooth differential forms on $N$ to smooth differential forms on $M$ , is functorial, and satisfies

F^{*} (α \land β) = F^{*} α \land F^{*} β .

Facts & Assumptions

Given: A smooth map $F : M \to N$ , a smooth map $G : N \to P$ , and forms $α, β$ on the target.

[F1]

A form pullback is the covariant tensor pullback restricted to alternating tensors (The pullback of a differential form).

[L1]

Covariant tensor pullback is smooth and functorial (Pullback of covariant tensors is smooth and functorial).

[L2]

Fibrewise linear pullback respects tensor products and permutations, hence wedge products (Linear pullback respects tensor products and permutations).

Proof

technique · direct

1.1

By [F1], $F^{*} α$ is obtained from the covariant tensor pullback. Because [L1] sends smooth covariant tensors to smooth covariant tensors and preserves composition, the same is true for forms.

F1L1given

1.2

At each point $p \in M$ , [F1] and [L2] give $(F^{*} (α \land β))_{p} = d F_{p}^{*} (α_{F (p)} \land β_{F (p)}) = d F_{p}^{*} (α_{F (p)}) \land d F_{p}^{*} (β_{F (p)}),$ which is exactly $(F^{*} α \land F^{*} β)_{p}$ .

F1L2givenalgebra

2.1

The identity and composition laws are inherited from [L1], and step 1.2 gives wedge preservation. Therefore pullback of forms is smooth, functorial, and wedge-preserving.

L1step 1.2∎

Depends on

Used by

The volume coordinate expression changes sign under a reflection Counterexample
Pullback of the circle angular form along a parametrized curve Example
The area form in polar coordinates Example
A diffeomorphism pulls back tensor fields and forms isomorphically Proposition

Dependency tree · two levels

14 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 (standard reference, not scraped)