Alphabeta Math
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→N be a diffeomorphism of oriented smooth n-manifolds and ω∈Ωcn(N). If F preserves orientation everywhere, ∫MF∗ω=∫Nω; if it reverses orientation everywhere, ∫MF∗ω=−∫Nω. 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→N, pullback sends smooth differential forms on N to smooth differential forms on M, is functorial, and satisfies F∗(α∧β)=F∗α∧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: (ϕiF)−1=F−1ϕi−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

Dependency tree · 2 levels

Sources