Darboux's theorem
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 -dimensional symplectic manifold can be made to look locally like the linear symplectic space  with its canonical symplectic form. There is also an analogous consequence of the theorem applied to contact geometry. Statement
Suppose that  is a differential 1-form on an -dimensional manifold, such that  has constant rank . Then

 if  everywhere, then there is a local system of coordinates  in which 
 if  everywhere, then there is a local system of coordinates  in which

Darboux's original proof used induction on  and it can be equivalently presented in terms of distributions or of differential ideals. Frobenius' theorem 
Darboux's theorem for  ensures the any 1-form  such that  can be written as  in some coordinate system . This recovers one of the formulation of Frobenius theorem in terms of differential forms: if  is the differential ideal generated by , then  implies the existence of a coordinate system  where  is actually generated by . Darboux's theorem for symplectic manifolds 
Suppose that  is a symplectic 2-form on an -dimensional manifold . In a neighborhood of each point  of , by the Poincaré lemma, there is a 1-form  with . Moreover,  satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart  near  in which

Taking an exterior derivative now shows
 
The chart  is said to be a Darboux chart around . The manifold  can be covered by such charts. To state this differently, identify  with  by letting . If is a Darboux chart, then  can be written as the pullback of the standard symplectic form  on :

A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using Moser's trick. Comparison with Riemannian geometry 
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  can be made to take the standard form in an entire neighborhood around . 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. Darboux's theorem for contact manifolds 
Another particular case is recovered when ; if  everywhere, then  is a contact form. A simpler proof can be given, as in the case of symplectic structures, by using Moser's trick. The Darboux-Weinstein theorem
Alan Weinstein showed that the Darboux's theorem for sympletic manifolds can be strengthened to hold on a neighborhood of a submanifold:
Let  be a smooth manifold endowed with two symplectic forms  and , and let  be a closed submanifold. If , then there is a neighborhood  of  in  and a diffeomorphism  such that . The standard Darboux theorem is recovered when  is a point and  is the standard symplectic structure on a coordinate chart. This theorem also holds for infinite-dimensional Banach manifolds. See also

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

References

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

Differential systems
Symplectic geometry
Coordinate systems in differential geometry
Theorems in differential geometry
Mathematical physics