# 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 that is a differential 1-form on an *n* dimensional manifold, such that has constant rank *p*. If

- everywhere,

then there is a local system of coordinates in which

- .

If, on the other hand,

- everywhere,

then there is a local system of coordinates ' in which

- .

In particular, suppose that is a symplectic 2-form on an *n*=2*m* dimensional manifold *M*. In a neighborhood of each point *p* of *M*, 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 *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, identify with by letting . If is a Darboux chart, then is the pullback of the standard symplectic form on :

## 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

## Notes

- ↑ Darboux (1882).
- ↑ Pfaff (1814–1815).
- ↑ Sternberg (1964) p. 140–141.
- ↑ Cf. with McDuff and Salamon (1998) p. 96.

## References

- Darboux, Gaston (1882). "Sur le problème de Pfaff".
*Bull. Sci. Math*.**6**: 14–36, 49–68. - Pfaff, Johann Friedrich (1814–1815). "Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi".
*Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin*: 76–136. - Sternberg, Shlomo (1964).
*Lectures on Differential Geometry*. Prentice Hall. - McDuff, D.; Salamon, D. (1998).
*Introduction to Symplectic Topology*. Oxford University Press. ISBN 0-19-850451-9.

## External links

- "Proof of Darboux's Theorem".
*PlanetMath*. - G. Darboux, "On the Pfaff Problem," transl. by D. H. Delphenich
- G. Darboux, "On the Pfaff Problem (cont.)," transl. by D. H. Delphenich