In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.

It is a foundational result in several fields, the chief among them being symplectic geometry. Indeed, one of its many consequences is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every ${\displaystyle 2n}$-dimensional symplectic manifold can be made to look locally like the linear symplectic space ${\displaystyle \mathbb {C} ^{n}}$ with its canonical symplectic form.

There is also an analogous consequence of the theorem applied to contact geometry.

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

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

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 ${\displaystyle dx_{1}}$.

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