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# 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 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cn 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 constantrankp. 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=2m 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 chartU 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.

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

• Symplectic basis

## References

[1]
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[2]
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[3]
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[4]
Citation Linkopenlibrary.orgCf. with McDuff and Salamon (1998) p. 96.
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[5]
Citation Linkgallica.bnf.fr"Sur le problème de Pfaff"
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[6]
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[7]
Citation Linkneo-classical-physics.infoG. Darboux, "On the Pfaff Problem," transl. by D. H. Delphenich
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[8]
Citation Linkneo-classical-physics.infoG. Darboux, "On the Pfaff Problem (cont.)," transl. by D. H. Delphenich
Sep 20, 2019, 6:31 PM
[9]
Citation Linkgallica.bnf.fr"Sur le problème de Pfaff"
Sep 20, 2019, 6:31 PM
[10]