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[1] who established it as the solution of the Pfaff problem.
[2] 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.
-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.
Then Darboux's original proof used induction on
and it can be equivalently presented in terms of distributions[3] or of differential ideals.
[4] Darboux's theorem for
This recovers one of the formulation of Frobenius theorem in terms of differential forms: if
is the differential ideal generated by
implies the existence of a coordinate system
satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart
Taking an exterior derivative now shows The chart
To state this differently, identify
can be written as the pullback of the standard symplectic form
: 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
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.
Another particular case is recovered when
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
be a smooth manifold endowed with two symplectic forms
.The standard Darboux theorem is recovered when
is the standard symplectic structure on a coordinate chart.
This theorem also holds for infinite-dimensional Banach manifolds.