Darboux's theorem

Suppose that $\backslash theta$ is a differential 1-form on an $n$-dimensional manifold, such that $\backslash mathrm\{d\}\; \backslash theta$ has constant rank $p$. Then

• if $\backslash theta\; \backslash wedge\; \backslash left(\backslash mathrm\{d\}\backslash theta\backslash right)^p\; =\; 0$ everywhere, then there is a local system of coordinates $(x\_1,\backslash ldots,x\_\{n-p\},y\_1,\backslash ldots,\; y\_p)$ in which $$\backslash theta=x\_1\backslash ,\backslash mathrm\{d\}y\_1+\backslash ldots\; +\; x\_p\backslash ,\backslash mathrm\{d\}y\_p;$$
• if $\backslash theta\; \backslash wedge\; \backslash left(\; \backslash mathrm\{d\}\; \backslash theta\; \backslash right)^p\; \backslash ne\; 0$ everywhere, then there is a local system of coordinates $(x\_1,\backslash ldots,x\_\{n-p\},y\_1,\backslash ldots,\; y\_p)$ in which$$\backslash theta=x\_1\backslash ,\backslash mathrm\{d\}y\_1+\backslash ldots\; +\; x\_p\backslash ,\backslash mathrm\{d\}y\_p\; +\; \backslash mathrm\{d\}x\_\{p+1\}.$$

Darboux's original proof used induction on $p$ and it can be equivalently presented in terms of distributions{{Cite book |last=Sternberg |first=Shlomo |url=https://archive.org/details/lecturesondiffer0000ster |title=Lectures on Differential Geometry |publisher=Prentice Hall |year=1964 |isbn=9780828403160 |pages=140-141 |author-link=Shlomo Sternberg}} or of differential ideals.{{Cite journal |last=Bryant |first=Robert L. |author-link=Robert Bryant (mathematician) |last2=Chern |first2=S. S. |author-link2=Shiing-Shen Chern |last3=Gardner |first3=Robert B. |author-link3=Robert Brown Gardner |last4=Goldschmidt |first4=Hubert L. |last5=Griffiths |first5=P. A. |author-link5=Phillip Griffiths |date=1991 |title=Exterior Differential Systems |url=https://doi.org/10.1007/978-1-4613-9714-4 |journal=Mathematical Sciences Research Institute Publications |language=en |doi=10.1007/978-1-4613-9714-4 |issn=0940-4740|url-access=subscription }}

Darboux's theorem for $p=0$ ensures that any 1-form $\backslash theta\; \backslash neq\; 0$ such that $\backslash theta\; \backslash wedge\; d\backslash theta\; =\; 0$ can be written as $\backslash theta\; =\; dx\_1$ in some coordinate system $(x\_1,\backslash ldots,x\_n)$.

This recovers one of the formulation of Frobenius theorem in terms of differential forms: if $\backslash mathcal\{I\}\; \backslash subset\; \backslash Omega^*(M)$ is the differential ideal generated by $\backslash theta$, then $\backslash theta\; \backslash wedge\; d\backslash theta\; =\; 0$ implies the existence of a coordinate system $(x\_1,\backslash ldots,x\_n)$ where $\backslash mathcal\{I\}\; \backslash subset\; \backslash Omega^*(M)$ is actually generated by $d\; x\_1$.

Suppose that $\backslash omega$ is a symplectic 2-form on an $n=2m$-dimensional manifold $M$. In a neighborhood of each point $p$ of $M$, by the Poincaré lemma, there is a 1-form $\backslash theta$ with $\backslash mathrm\{d\}\; \backslash theta\; =\; \backslash omega$. Moreover, $\backslash theta$ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart $U$ near $p$ in which$$\backslash theta=x\_1\backslash ,\backslash mathrm\{d\}y\_1+\backslash ldots\; +\; x\_m\backslash ,\backslash mathrm\{d\}y\_m.$$

Taking an exterior derivative now shows

: $$\backslash omega\; =\; \backslash mathrm\{d\}\; \backslash theta\; =\; \backslash mathrm\{d\}x\_1\; \backslash wedge\; \backslash mathrm\{d\}y\_1\; +\; \backslash ldots\; +\; \backslash mathrm\{d\}x\_m\; \backslash wedge\; \backslash mathrm\{d\}y\_m.$$

The chart $U$ is said to be a Darboux chart around $p$.{{Cite book |last=McDuff |first=Dusa |url=https://academic.oup.com/book/43512 |title=Introduction to Symplectic Topology |last2=Salamon |first2=Dietmar |date=2017-06-22 |publisher=Oxford University Press |isbn=978-0-19-879489-9 |volume=1 |language=en |doi=10.1093/oso/9780198794899.001.0001 |author-link=Dusa McDuff |author-link2=Dietmar Salamon}} The manifold $M$ can be covered by such charts.

To state this differently, identify $\backslash mathbb\{R\}^\{2m\}$ with $\backslash mathbb\{C\}^\{m\}$ by letting $z\_j=x\_j+\backslash textit\{i\}\backslash ,y\_j$. If $\backslash varphi:\; U\; \backslash to\; \backslash mathbb\{C\}^n$ is a Darboux chart, then $\backslash omega$ can be written as the pullback of the standard symplectic form $\backslash omega\_0$ on $\backslash mathbb\{C\}^\{n\}$:

:$\backslash omega\; =\; \backslash varphi^\{*\}\backslash omega\_0.\backslash ,$

A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using Moser's trick.{{Cite book |last=Cannas Silva |first=Ana |url=https://link.springer.com/book/10.1007/978-3-540-45330-7 |title=Lectures on Symplectic Geometry |publisher=Springer |year=2008 |isbn=978-3-540-42195-5 |language=en |doi=10.1007/978-3-540-45330-7 |author-link=Ana Cannas da Silva}}

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 $\backslash omega$ 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.

Another particular case is recovered when $n=2p+1$; if $\backslash theta\; \backslash wedge\; \backslash left(\; \backslash mathrm\{d\}\; \backslash theta\; \backslash right)^p\; \backslash ne\; 0$ everywhere, then $\backslash theta$ is a contact form. A simpler proof can be given, as in the case of symplectic structures, by using Moser's trick.{{Cite book |last=Geiges |first=Hansjörg |url=https://www.cambridge.org/core/books/an-introduction-to-contact-topology/F851B2A2E7E78C6B9967A18A6641B40C |title=An Introduction to Contact Topology |date=2008 |publisher=Cambridge University Press |isbn=978-0-521-86585-2 |series=Cambridge Studies in Advanced Mathematics |location=Cambridge |pages=67-68 |doi=10.1017/cbo9780511611438}}

Alan Weinstein showed that the Darboux's theorem for sympletic manifolds can be strengthened to hold on a neighborhood of a submanifold:{{cite journal|last = Weinstein|first = Alan|authorlink=Alan Weinstein|title = Symplectic manifolds and their Lagrangian submanifolds|journal=Advances in Mathematics|volume=6|year=1971| issue=3 |pages=329–346|doi=10.1016/0001-8708(71)90020-X|doi-access=free}}

Let $M$ be a smooth manifold endowed with two symplectic forms $\backslash omega\_1$ and $\backslash omega\_2$, and let $N\; \backslash subset\; M$ be a closed submanifold. If $\backslash left.\backslash omega\_1\backslash right|\_N\; =\; \backslash left.\backslash omega\_2\backslash right|\_N$, then there is a neighborhood $U$ of $N$ in $M$ and a diffeomorphism $f\; :\; U\; \backslash to\; U$ such that $f^*\backslash omega\_2\; =\; \backslash omega\_1$.

The standard Darboux theorem is recovered when $N$ is a point and $\backslash omega\_2$ is the standard symplectic structure on a coordinate chart.

This theorem also holds for infinite-dimensional Banach manifolds.

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

• [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/darboux_-_pfaff_problem_i.pdf G. Darboux, "On the Pfaff Problem", transl. by D. H. Delphenich]
• [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/darboux_-_pfaff_problem_ii.pdf G. Darboux, "On the Pfaff Problem (cont.)", transl. by D. H. Delphenich]

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Category:Differential systems

Category:Symplectic geometry

Category:Coordinate systems in differential geometry

Category:Theorems in differential geometry

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