LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rescaling the defining form changes the Godbillon-Vey form by an exact form

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ . For $ω^{'} = e^{f} ω$ with $f \in C^{\infty} (M)$ one has $d ω^{'} = η^{'} \land ω^{'}$ with $η^{'} = η + d f$ , and $η^{'} \land d η^{'} = η \land d η + d (f d η)$ ; in particular $[η^{'} \land d η^{'}] = [η \land d η]$ in $H_{d R}^{3} (M; R)$ . The same conclusion holds for any nowhere-vanishing smooth multiple $ω^{'} = g ω$ , since on each connected component $g$ is $e^{f}$ or $- e^{f}$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ , and a smooth function $f$ with $ω^{'} = e^{f} ω$ .

[F1]

With $d ω = η \land ω$ one has $d η \land ω = 0$ and $η \land d η$ is closed. (The Godbillon-Vey form eta wedge d eta is closed).

[F2]

For homogeneous smooth forms $α, β$ of degrees $p, q$ one has $d (α \land β) = d α \land β + (- 1)^{p} α \land d β$ . (The exterior derivative is a graded derivation).

[F3]

For every differential form $ω$ , $d (d ω) = 0$ . (The exterior derivative squares to zero).

Proof

technique · direct

1.1F2given

Compute $d ω^{'} = d (e^{f} ω) = e^{f} d f \land ω + e^{f} d ω = e^{f} (d f \land ω + η \land ω) = (η + d f) \land ω^{'}$ by [F2], so $η^{'} = η + d f$ is admissible for $ω^{'}$ .

2.1F1F2F3step 1.1

Then $d η^{'} = d η + d (d f) = d η$ by [F3], so $η^{'} \land d η^{'} = (η + d f) \land d η = η \land d η + d (f d η)$ because $d (f d η) = d f \land d η$ by [F2]; the difference is exact and $η \land d η$ is closed by [F1], so the two forms define the same class in $H_{d R}^{3} (M; R)$ .

3.1step 2.1∎

For a general nowhere-vanishing smooth multiple $ω^{'} = g ω$ the same computation applies on each connected component with $g = \pm e^{f}$ : the sign choice leaves $d ω = η \land ω$ and the form $η \land d η$ unchanged, so the class is independent of the rescaling; no choice principle is used.

Depends on

Used by

The Godbillon-Vey class of a codimension-one foliation Definition
Explicit Godbillon-Vey rescaling calculation Example

Dependency tree · two levels

19 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

Steven Hurder and Remi Langevin, Dynamics and the Godbillon-Vey Class of C1 Foliations (complete author-hosted PDF) (standard reference, not scraped)