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The Godbillon-Vey form eta wedge d eta is closed

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ . Then $d η \land ω = 0$ ; the $2$ -form $d η$ is divisible by $ω$ , i.e. $d η = ζ \land ω$ for a smooth $1$ -form $ζ$ ; $d η \land d η = 0$ ; and $d (η \land d η) = 0$ . Consequently $η \land d η$ is a closed $3$ -form on $M$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ of a smooth manifold $M$ with defining one-form $ω$ and a one-form $η$ satisfying $d ω = η \land ω$ , and the standing countable choice assumption.

[F1]

Under ${A C}_{ω}$ , if $ω$ is nowhere vanishing and $θ \land ω = 0$ for a smooth two-form $θ$ , then $θ = β \land ω$ for a smooth one-form $β$ . (Divisibility by a nowhere-vanishing one-form).

[F2]

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

[F3]

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

[F4]

The wedge product is associative and graded-commutative, so $ζ \land ζ = 0$ for a one-form $ζ$ and $ω \land ω = 0$ . (The wedge product is associative and graded commutative).

Proof

technique · direct

1.1F2F3F4given

Differentiating $d ω = η \land ω$ with the Leibniz rule [F2] and $d^{2} = 0$ [F3] gives $0 = d η \land ω - η \land d ω = d η \land ω - η \land η \land ω$ , and $η \land η = 0$ by graded commutativity [F4], so $d η \land ω = 0$ .

2.1F1step 1.1

Since $ω$ is nowhere vanishing and the two-form $d η$ satisfies $d η \land ω = 0$ , the divisibility lemma [F1] gives a smooth one-form $ζ$ with $d η = ζ \land ω$ .

3.1F4step 2.1

Then $d η \land d η = (ζ \land ω) \land (ζ \land ω) = - ζ \land ζ \land ω \land ω = 0$ by associativity and graded commutativity with $ζ \land ζ = 0$ and $ω \land ω = 0$ [F4].

4.1F2F3step 3.1∎

Finally $d (η \land d η) = d η \land d η - η \land d (d η) = 0$ by the graded Leibniz rule [F2] and $d^{2} = 0$ [F3], so $η \land d η$ is a closed three-form. The standing ${A C}_{ω}$ assumption licenses the global divisibility result in step 2.1; the remaining calculations are formal exterior-algebra identities.

Depends on

Used by

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27 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)