Editing Darboux's theorem

{{short description|Foundational result in symplectic geometry}}
{{About|Darboux's theorem in differential geometry|Darboux's theorem related to the intermediate value theorem|Darboux's theorem (analysis)}}

In [[differential geometry]], a field in [[mathematics]], '''Darboux's theorem''' is a [[theorem]] providing a normal form for special classes of [[differential forms|differential 1-forms]], partially generalizing the [[Frobenius integration theorem]]. It is named after [[Jean Gaston Darboux]]<ref>{{cite journal |last=Darboux |first=Gaston |author-link=Jean Gaston Darboux |year=1882 |title=Sur le problème de Pfaff |trans-title=On the Pfaff's problem |url=http://gallica.bnf.fr/ark:/12148/bpt6k68005v |journal=Bull. Sci. Math. |language=fr |volume=6 |pages=14–36, 49–68 |jfm=05.0196.01}}</ref> who established it as the solution of the [[Johann Friedrich Pfaff|Pfaff]] problem.<ref>{{cite journal |last=Pfaff |first=Johann Friedrich |author-link=Johann Friedrich Pfaff |year=1814–1815 |title=Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi |trans-title=A general method to completely integrate partial differential equations, as well as ordinary differential equations, of order higher than one, with any number of variables |url=https://archive.org/details/abhandlungenderp14akad/page/76/mode/1up?view=theater |journal=Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin |language=la |pages=76–136}}</ref>

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 manifold]]s of the same dimension are locally [[symplectomorphism|symplectomorphic]] to one another. That is, every <math>2n </math>-dimensional symplectic manifold can be made to look locally like the [[linear symplectic space]] <math>\mathbb{C}^n </math> with its canonical symplectic form.

There is also an analogous consequence of the theorem applied to [[contact geometry]].

==Statement==
Suppose that <math>\theta </math> is a differential 1-form on an ''<math>n </math>''-dimensional manifold, such that <math>\mathrm{d} \theta </math> has constant [[Exterior algebra#Rank of a k-vector|rank]] ''<math>p </math>''. Then

* if <math> \theta \wedge \left(\mathrm{d}\theta\right)^p = 0 </math> everywhere, then there is a local system of coordinates <math> (x_1,\ldots,x_{n-p},y_1,\ldots, y_p) </math> in which <math display="block"> \theta=x_1\,\mathrm{d}y_1+\ldots + x_p\,\mathrm{d}y_p; </math>
* if <math> \theta \wedge \left( \mathrm{d} \theta \right)^p \ne 0 </math> everywhere, then there is a local system of coordinates <math> (x_1,\ldots,x_{n-p},y_1,\ldots, y_p) </math> in which<math display="block"> \theta=x_1\,\mathrm{d}y_1+\ldots + x_p\,\mathrm{d}y_p + \mathrm{d}x_{p+1}.</math>

Darboux's original proof used [[Mathematical induction|induction]] on ''<math>p </math>'' and it can be equivalently presented in terms of [[Distribution (differential geometry)|distributions]]<ref>{{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}}</ref> or of [[Differential ideal|differential ideals]].<ref name=":0">{{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 }}</ref>

=== Frobenius' theorem ===
Darboux's theorem for ''<math>p=0 </math>'' ensures that any 1-form ''<math>\theta \neq 0 </math>'' such that ''<math>\theta \wedge d\theta = 0 </math>'' can be written as ''<math>\theta = dx_1 </math>'' in some coordinate system <math> (x_1,\ldots,x_n) </math>.

This recovers one of the formulation of [[Frobenius theorem (differential topology)|Frobenius theorem]] in terms of differential forms: if <math> \mathcal{I} \subset \Omega^*(M) </math> is the differential ideal generated by <math> \theta </math>, then ''<math>\theta \wedge d\theta = 0 </math>'' implies the existence of a coordinate system <math> (x_1,\ldots,x_n) </math> where <math> \mathcal{I} \subset \Omega^*(M) </math> is actually generated by <math> d x_1 </math>.<ref name=":0" />

== Darboux's theorem for symplectic manifolds ==
Suppose that <math>\omega </math> is a [[Symplectic form|symplectic 2-form]] on an <math>n=2m </math>-dimensional manifold ''<math>M </math>''. In a neighborhood of each point ''<math>p </math>'' of ''<math>M </math>'', by the [[Poincaré lemma]], there is a 1-form <math>\theta </math> with <math>\mathrm{d} \theta = \omega</math>. Moreover, <math>\theta </math> satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a [[coordinate chart]] ''<math>U </math>'' near ''<math>p </math>'' in which<math display="block"> \theta=x_1\,\mathrm{d}y_1+\ldots + x_m\,\mathrm{d}y_m. </math>

Taking an [[exterior derivative]] now shows
: <math display="block"> \omega = \mathrm{d} \theta = \mathrm{d}x_1 \wedge \mathrm{d}y_1 + \ldots + \mathrm{d}x_m \wedge \mathrm{d}y_m.</math>
The chart ''<math>U </math>'' is said to be a '''Darboux chart''' around ''<math>p </math>''.<ref name=":32">{{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}}</ref> The manifold ''<math>M </math>'' can be [[cover (topology)|covered]] by such charts.

To state this differently, identify <math>\mathbb{R}^{2m}</math> with <math>\mathbb{C}^{m}</math> by letting <math>z_j=x_j+\textit{i}\,y_j</math>. If <math>\varphi: U \to \mathbb{C}^n</math> is a Darboux chart, then <math> \omega </math> can be written as the [[pullback (differential geometry)|pullback]] of the standard symplectic form <math>\omega_0</math> on <math>\mathbb{C}^{n}</math>:
:<math>\omega = \varphi^{*}\omega_0.\,</math>
A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using [[Moser's trick]].<ref name=":32" /><ref name=":02">{{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 Science+Business Media|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}}</ref>

=== Comparison with Riemannian geometry ===
Darboux's theorem for symplectic manifolds implies that there are no local invariants in symplectic geometry: a [[Darboux frame|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 of Riemannian manifolds|curvature]] is a local invariant, an obstruction to the [[metric tensor|metric]] being locally a sum of squares of coordinate differentials.

The difference is that Darboux's theorem states that <math> \omega </math> can be made to take the standard form in an ''entire neighborhood'' around ''<math>p </math>''. 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.

=== Darboux's theorem for contact manifolds ===
Another particular case is recovered when <math> n=2p+1 </math>; if <math> \theta \wedge \left( \mathrm{d} \theta \right)^p \ne 0 </math> everywhere, then <math> \theta </math> is a [[Contact geometry#Contact forms and structures|contact form]]. A simpler proof can be given, as in the case of symplectic structures, by using [[Moser's trick]].<ref>{{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}}</ref>

==The Darboux-Weinstein theorem==
[[Alan Weinstein]] showed that the Darboux's theorem for sympletic manifolds can be strengthened to hold on a [[Neighbourhood (mathematics)|neighborhood]] of a [[submanifold]]:<ref>{{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}}</ref>
<blockquote>''Let <math>M</math> be a smooth manifold endowed with two symplectic forms <math>\omega_1</math> and <math>\omega_2</math>, and let <math>N \subset M</math> be a closed submanifold. If <math> \left.\omega_1\right|_N = \left.\omega_2\right|_N </math>, then there is a neighborhood <math> U </math> of <math>N</math> in <math>M</math> and a diffeomorphism <math>f : U \to U</math> such that <math>f^*\omega_2 = \omega_1</math>.''</blockquote>

The standard Darboux theorem is recovered when <math>N</math> is a point and <math>\omega_2</math> is the standard symplectic structure on a coordinate chart.

This theorem also holds for infinite-dimensional [[Banach manifold]]s.

==See also==

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

==References==
<references/>
==External links==
* [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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