# Darboux's theorem

# Darboux's theorem

**Darboux's theorem** is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux^{[1]} who established it as the solution of the Pfaff problem.^{[2]}

One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2*n*-dimensional symplectic manifold can be made to look locally like the linear symplectic space **C***n* with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry.

Statement and first consequences

`The precise statement is as follows.`

^{[3]}Suppose thatis a differential 1-form on an*n*dimensional manifold, such thathas constantrank*p*. If- everywhere,

`then there is a local system of coordinatesin which`

- .

If, on the other hand,

- everywhere,

`then there is a local system of coordinates 'in which`

- .

`Note that ifeverywhere andthenis acontact form.`

`In particular, suppose thatis a symplectic 2-form on an`

*n*=2*m*dimensional manifold*M*. In a neighborhood of each point*p*of*M*, by thePoincaré lemma, there is a 1-formwith. Moreover,satisfies the first set of hypotheses in Darboux's theorem, and so locally there is acoordinate chart*U*near*p*in which- .

Taking an exterior derivative now shows

The chart *U* is said to be a **Darboux chart** around *p*.^{[4]} The manifold *M* can be covered by such charts.

`To state this differently, identifywithby letting. Ifis a Darboux chart, thenis thepullbackof the standard symplectic formon:`

Comparison with Riemannian geometry

This result 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 ω can be made to take the standard form in an *entire neighborhood* around *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.

See also

Carathéodory-Jacobi-Lie theorem, a generalization of this theorem.

Symplectic basis