Darboux's theorem

Suppose that ${\displaystyle \theta }$  is a differential 1-form on an ${\displaystyle n}$ -dimensional manifold, such that ${\displaystyle \mathrm {d} \theta }$  has constant rank ${\displaystyle p}$ . Then

• if ${\displaystyle \theta \wedge \left(\mathrm {d} \theta \right)^{p}=0}$  everywhere, then there is a local system of coordinates ${\displaystyle (x_{1},\ldots ,x_{n-p},y_{1},\ldots ,y_{p})}$  in which $${\displaystyle \theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{p}\,\mathrm {d} y_{p};}$$ 
• if ${\displaystyle \theta \wedge \left(\mathrm {d} \theta \right)^{p}\neq 0}$  everywhere, then there is a local system of coordinates ${\displaystyle (x_{1},\ldots ,x_{n-p},y_{1},\ldots ,y_{p})}$  in which$${\displaystyle \theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{p}\,\mathrm {d} y_{p}+\mathrm {d} x_{p+1}.}$$ 

Darboux's original proof used induction on ${\displaystyle p}$  and it can be equivalently presented in terms of distributions^[3] or of differential ideals.^[4]

Darboux's theorem for ${\displaystyle p=0}$  ensures that any 1-form ${\displaystyle \theta \neq 0}$  such that ${\displaystyle \theta \wedge d\theta =0}$  can be written as ${\displaystyle \theta =dx_{1}}$  in some coordinate system ${\displaystyle (x_{1},\ldots ,x_{n})}$ .

This recovers one of the formulation of Frobenius theorem in terms of differential forms: if ${\displaystyle {\mathcal {I}}\subset \Omega ^{*}(M)}$  is the differential ideal generated by ${\displaystyle \theta }$ , then ${\displaystyle \theta \wedge d\theta =0}$  implies the existence of a coordinate system ${\displaystyle (x_{1},\ldots ,x_{n})}$  where ${\displaystyle {\mathcal {I}}\subset \Omega ^{*}(M)}$  is actually generated by ${\displaystyle dx_{1}}$ .^[4]

Suppose that ${\displaystyle \omega }$  is a symplectic 2-form on an ${\displaystyle n=2m}$ -dimensional manifold ${\displaystyle M}$ . In a neighborhood of each point ${\displaystyle p}$  of ${\displaystyle M}$ , by the Poincaré lemma, there is a 1-form ${\displaystyle \theta }$  with ${\displaystyle \mathrm {d} \theta =\omega }$ . Moreover, ${\displaystyle \theta }$  satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart ${\displaystyle U}$  near ${\displaystyle p}$  in which$${\displaystyle \theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{m}\,\mathrm {d} y_{m}.}$$ 

Taking an exterior derivative now shows

$${\displaystyle \omega =\mathrm {d} \theta =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\ldots +\mathrm {d} x_{m}\wedge \mathrm {d} y_{m}.}$$ 

The chart ${\displaystyle U}$  is said to be a Darboux chart around ${\displaystyle p}$ .^[5] The manifold ${\displaystyle M}$  can be covered by such charts.

To state this differently, identify ${\displaystyle \mathbb {R} ^{2m}}$  with ${\displaystyle \mathbb {C} ^{m}}$  by letting ${\displaystyle z_{j}=x_{j}+{\textit {i}}\,y_{j}}$ . If ${\displaystyle \varphi :U\to \mathbb {C} ^{n}}$  is a Darboux chart, then ${\displaystyle \omega }$  can be written as the pullback of the standard symplectic form ${\displaystyle \omega _{0}}$  on ${\displaystyle \mathbb {C} ^{n}}$ :

${\displaystyle \omega =\varphi ^{*}\omega _{0}.\,}$ 

A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using Moser's trick.^[5][6]

Darboux's theorem for symplectic manifolds implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares of coordinate differentials.

The difference is that Darboux's theorem states that ${\displaystyle \omega }$  can be made to take the standard form in an entire neighborhood around ${\displaystyle p}$ . In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.

Another particular case is recovered when ${\displaystyle n=2p+1}$ ; if ${\displaystyle \theta \wedge \left(\mathrm {d} \theta \right)^{p}\neq 0}$  everywhere, then ${\displaystyle \theta }$  is a contact form. A simpler proof can be given, as in the case of symplectic structures, by using Moser's trick.^[7]

Alan Weinstein showed that the Darboux's theorem for sympletic manifolds can be strengthened to hold on a neighborhood of a submanifold:^[8]

Let ${\displaystyle M}$  be a smooth manifold endowed with two symplectic forms ${\displaystyle \omega _{1}}$  and ${\displaystyle \omega _{2}}$ , and let ${\displaystyle N\subset M}$  be a closed submanifold. If ${\displaystyle \left.\omega _{1}\right|_{N}=\left.\omega _{2}\right|_{N}}$ , then there is a neighborhood ${\displaystyle U}$  of ${\displaystyle N}$  in ${\displaystyle M}$  and a diffeomorphism ${\displaystyle f:U\to U}$  such that ${\displaystyle f^{*}\omega _{2}=\omega _{1}}$ .

The standard Darboux theorem is recovered when ${\displaystyle N}$  is a point and ${\displaystyle \omega _{2}}$  is the standard symplectic structure on a coordinate chart.

This theorem also holds for infinite-dimensional Banach manifolds.

• Carathéodory–Jacobi–Lie theorem, a generalization of this theorem.
• Moser's trick
• Symplectic basis

• G. Darboux, "On the Pfaff Problem", transl. by D. H. Delphenich
• G. Darboux, "On the Pfaff Problem (cont.)", transl. by D. H. Delphenich