TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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Change of variables on oriented manifolds

Statement

Let $F : M \to N$ be a diffeomorphism of oriented smooth $n$ -manifolds and $ω \in Ω_{c}^{n} (N)$ . If $F$ preserves orientation everywhere, $\int_{M} F^{*} ω = \int_{N} ω$ ; if it reverses orientation everywhere, $\int_{M} F^{*} ω = - \int_{N} ω$ . If the sign varies between components, apply the appropriate signed equality on each component and add. The compact-support integrals are the choice-free finite-chart values, equal to the global partition integrals whenever those are available.

Facts & Assumptions

[F1]

Finite chart localization of compactly supported top forms, including boundary constructs the finite-chart compact-support integral in every dimension, including genuine boundary; proves signed diffeomorphism change of variables componentwise; and identifies its value with the global partition integral under countable choice.

[F2]

Pullback of forms is smooth functorial and preserves wedges: 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 $F^{*} (α \land β) = F^{*} α \land F^{*} β .$

Proof

Given: The objects and hypotheses in the statement above.

1.1F1F2given

The support of $F^{*} ω$ is $F^{- 1} (supp ω)$ because the tangent maps are isomorphisms. It is compact, being the continuous image of the compact support under the inverse homeomorphism. Take the finite target localization from [F1] and pull its weights and charts back through $F$ .

2.1F1F2step 1.1

If orientation is preserved, the corresponding source and target charts have the same sign and the same localized coefficient: $(ϕ_{i} F)^{- 1} = F^{- 1} ϕ_{i}^{- 1}$ and functoriality cancels the pullbacks. The finite sums therefore agree and [F1] makes both intrinsic.

3.1F1step 2.1∎

For global reversal the chart signs are opposite, giving the negative finite sum. More generally the sign is locally constant because a continuous nonzero determinant has constant sign on each connected component; compact support meets finitely many components, so add their signed equalities. In dimension zero the diffeomorphism is a bijection of finite supports with the corresponding point signs; empty support and zero forms give zero.

Depends on

Used by

A cohomology class obstructs a global symplectomorphism Counterexample
Bochner–Martinelli on the unit ball Example
Change of variables on an oriented circle Example
Reflection reverses the signed form integral Example
K-finite vectors detect nonzero closed invariant subspaces Lemma
The invariant pairing between opposite principal-series parameters Lemma
Top de Rham integration equals fundamental-class evaluation Lemma
Degree of an orientation-preserving or reversing diffeomorphism Proposition
Integration on an oriented embedded submanifold Proposition
Iwasawa decomposition and Haar integration formula for SL2(R) Theorem
The general Stokes theorem Theorem

Dependency tree · two levels

18 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

Lee Proposition 16.6(d), pp.407–408 (standard reference, not scraped)