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2014-01-23T03:02:10Z

193.52.24.8:

In [[mathematics]], a '''[[Cauchy-Riemann|CR]] manifold''' is a [[differentiable manifold]] together with a geometric structure modeled on that of a real [[hypersurface]] in a [[Real number|complex vector space]], or more generally modeled on an [[Edge-of-the-wedge theorem|edge of a wedge]].

Formally, a '''CR manifold''' is a differentiable manifold ''M'' together with a preferred complex distribution ''L'', or in other words a [[subbundle]] of the [[complexified]] [[tangent bundle]] '''C'''T''M'' = T''M'' ⊗ '''C''' such that
* <math>[L,L]\subseteq L</math> (''L'' is '''formally [[Frobenius integration theorem|integrable]]''')
* <math>L\cap\bar{L}=\{0\}</math> ('''L''' is [[almost Lagrangian]]).
The bundle ''L'' is called a '''CR structure''' on the manifold ''M''.

The abbreviation CR stands for [[Cauchy-Riemann]] or [https://secure.msri.org/calendar/sgw/WorkshopInfo/434/show_sgw Complex-Real].

==Introduction and motivation==
The notion of a CR structure attempts to describe ''intrinsically'' the property of being a hypersurface in complex space by studying the properties of [[holomorphic]] [[vector field]]s which are tangent to the hypersurface.

Suppose for instance that ''M'' is the hypersurface of '''C'''2 given by the equation
:<math>F(z,w) := |z|^2+|w|^2=1,</math>
where ''z'' and ''w'' are the usual complex coordinates on '''C'''2. The ''holomorphic tangent bundle'' of '''C'''2 consists of all linear combinations of the vectors
:<math>\frac{\partial}{\partial z},\quad \frac{\partial}{\partial w}.</math>
The distribution ''L'' on ''M'' consists of all combinations of these vectors which are ''tangent'' to ''M''. In detail, the tangent vectors must annihilate the defining equation for ''M'', so ''L'' consists of complex scalar multiples of
:<math>\bar{w}\frac{\partial}{\partial z}-\bar{z}\frac{\partial}{\partial w}.</math>
Note that ''L'' gives a CR structure on ''M'', for [''L'',''L''] = 0 (since ''L'' is one-dimensional) and <math>L\cap\bar{L}=\{0\}</math> since ∂/∂''z'' and ∂/∂''w'' are linearly independent of their complex conjugates.

More generally, suppose that ''M'' is a real hypersurface in '''C'''n, with defining equation ''F''(''z''1, ..., ''z''n) = 0. Then the CR structure ''L'' consists of those linear combinations of the basic holomorphic vectors on '''C'''n:
:<math>\frac{\partial}{\partial z_1}, ..., \frac{\partial}{\partial z_n}</math>
which annihilate the defining function. In this case, <math>L\cap\bar{L}=\{0\}</math> for the same reason as before. Moreover, [''L'',''L''] ⊂ ''L'' since the commutator of vector fields annihilating ''F'' is again a vector field annihilating ''F''.

===Embedded and abstract CR manifolds===
There is a sharp contrast between the theories of embedded CR manifolds (hypersurface and edges of wedges in complex space) and abstract CR manifolds (those given by the Lagrangian distribution ''L''). Many of the formal geometrical features are similar. These include:
* A notion of [[Convex set|convexity]] (supplied by the '''Levi form''')
* A [[differential operator]], analogous to the [[Dolbeault operator]], and an associated [[cohomology]] (the '''tangential Cauchy-Riemann complex''').

Embedded CR manifolds possess some additional structure, though: a [[Neumann problem|Neumann]] and [[Dirichlet problem]] for the Cauchy-Riemann equations.

This article first treats the geometry of embedded CR manifolds, shows how to defined these structures intrinsically, and then generalizes these to the abstract setting.

==Embedded CR manifolds==

===Preliminaries===
Embedded CR manifolds are, first and foremost, submanifolds of '''C'''n. Define a pair of subbundles of the complexified tangent bundle '''C''' ⊗ T'''C''''n by:
*''T''(1,0)'''C'''n consists of the complex vectors annihilating the [[antiholomorphic]] functions. In the [[complex manifold|holomorphic coordinates]]:
::<math>T^{(1,0)}\mathbb{C}^n = \mathrm{span}\left(\frac{\partial}{\partial z_1},\dots,\frac{\partial}{\partial z_n}\right).</math>
*T(0,1)'''C'''n consists of the complex vectors annihilating the [[holomorphic]] functions. In coordinates:
::<math>T^{(0,1)}\mathbb{C}^n = \mathrm{span}\left(\frac{\partial}{\partial \bar{z}_1},\dots,\frac{\partial}{\partial \bar{z}_n}\right).</math>

Also relevant are the characteristic annihilators from the [[Dolbeault complex]]:
*Ω(1,0)'''C'''n = (''T''(0,1)'''C'''n)⊥. In coordinates,
::<math>\Omega^{(1,0)}\mathbb{C}^n = \mathrm{span}(dz_1,\dots,dz_n).</math>
*Ω(0,1)'''C'''n = (''T''(1,0)'''C'''n)⊥. In coordinates,
::<math>\Omega^{(0,1)}\mathbb{C}^n = \mathrm{span}(d\bar{z}_1,\dots,d\bar{z}_n).</math>
The [[exterior product]]s of these are denoted by the self-evident notation Ω(p,q), and the Dolbeault operator and its complex conjugate map between these spaces via
:<math>\partial : \Omega^{(p,q)} \rightarrow \Omega^{(p+1,q)}</math>
:<math>\bar{\partial} : \Omega^{(p,q)} \rightarrow \Omega^{(p,q+1)}</math>
Furthermore, there is a decomposition of the usual [[exterior derivative]] via <math>d = \partial + \bar{\partial}</math>.

===Real submanifolds of complex space===
Let ''M'' ⊂ '''C'''''n'' be a real submanifold, defined locally as the locus of a system of smooth real-valued functions
:''F''1 = 0, ''F''2 = 0, ..., ''F''''k'' = 0.
Suppose that this system has maximal rank, in the sense that the differentials satisfy the following ''independence condition'':

:<math>\partial F_1\wedge\bar{\partial} F_1\wedge\dots \wedge \partial F_k\wedge \bar{\partial} F_k \not= 0.</math>

Note that this condition is strictly stronger than needed to apply the [[implicit function theorem]]: in particular, ''M'' is a manifold of real dimension 2''n'' − ''k''. We say that ''M'' is an embedded CR manifold of '''CR codimension''' ''k''. In most applications, ''k'' = 1, in which case the manifold is said to be of '''hypersurface type'''.

Let ''L'' ⊂ ''T''(1,0)'''C'''''n''|''M'' be the subbundle of vectors annihilating all of the defining functions ''F''1, ..., ''F''''k''. Note that, by the usual considerations for integrable distributions on hypersurfaces, ''L'' is involutive. Moreover, the independence condition implies that ''L'' is a bundle of constant rank ''n'' − ''k''.

Henceforth, suppose that ''k'' = 1 (so that the CR manifold is of hypersurface type), unless otherwise noted.

===The Levi form===
Let ''M'' be a CR manifold of hypersurface type with single defining function ''F'' = 0. The '''Levi form''' of ''M'', named after [[Eugenio Elia Levi]],<ref>See {{harv|Levi|909|p=207}}: the Levi form is the [[differential form]] associated to the [[differential operator]] ''C'', according to Levi's notation.</ref> is the [[Hermitian metric|Hermitian 2-form]]
:<math>h=i\partial\bar{\partial}F|_L.</math>
This determines a metric on ''L''. ''M'' is said to be '''strictly pseudoconvex''' if ''h'' is positive definite (or ''pseudoconvex'' in case ''h'' is positive semidefinite). Many of the analytic existence and uniqueness results in the theory of CR manifolds depend on the strict pseudoconvexity of the Levi form.

This nomenclature comes from the study of [[pseudoconvexity|pseudoconvex domain]]s: ''M'' is the boundary of a (strictly) pseudoconvex domain in '''C'''n if and only if it is (strictly) pseudoconvex as a CR manifold. (See [[plurisubharmonic function]]s and [[Stein manifold]].)

==Abstract CR structures==

An abstract CR structure on a manifold ''M'' of dimension ''n'' consists of a subbundle ''L'' of the complexified tangent bundle which is formally integrable, in the sense that [''L'',''L''] ⊂ ''L'', which is linearly independent of its complex conjugate. The '''CR codimension''' of the CR structure is k = ''n'' - 2 dim ''L''. In case ''k'' = 1, the CR structure is said to be of '''hypersurface type'''. Most examples of abstract CR structures are of hypersurface type, unless otherwise made explicit.

===The Levi form and pseudoconvexity===
Suppose that ''M'' is a CR manifold of hypersurface type. The Levi form is the [[vector valued form]], defined on ''L'', with values in the [[line bundle]]
:<math>V = \frac{TM\otimes{\mathbb C}}{L\oplus\bar{L}}</math>
given by
:<math>h(v,w) = \frac{1}{2i}[v,\bar{w}] \mod L\oplus\bar{L},\quad v,w\in L.</math>
''h'' defines a [[sesquilinear]] form on ''L'' since it does not depend on how ''v'' and ''w'' are extended to sections of ''L'', by the integrability condition. This form extends to a [[hermitian form]] on the bundle <math>L\oplus\bar{L}</math> by the same expression. The extended form is also sometimes referred to as the Levi form.

The Levi form can alternatively be characterized in terms of duality. Consider the line subbundle of the complex [[cotangent bundle]] annihilating ''V''
:<math>H_0M = V^* = (L\oplus\bar{L})^\perp\sub T^*M\otimes{\mathbb C}.</math>
For each local section α∈Γ(''H''0''M''), let
:<math>h_\alpha(v,w) = d\alpha(v,\bar{w}) = -\alpha([v,\bar{w}]),\quad v,w\in L\oplus\bar{L}.</math>
The form ''h''α is a complex-valued hermitian form associated to α.

Generalizations of the Levi form exist when the manifold is not of hypersurface type, in which case the form no longer assumes values in a line bundle, but rather in a vector bundle. One may then speak, not of a Levi form, but of a collection of Levi forms for the structure.

===Characteristic ideals===

===The tangential Cauchy–Riemann complex===

==Examples==
The canonical example of a CR manifold is the real <math>2n+1</math> sphere as a submanifold of <math>\mathbb{C}^{n+1}</math>. The bundle <math>L</math> described above is given by
:<math>L = \mathbb{C}TS^{2n+1} \cap T^{1,0}\mathbb{C}^{n+1}</math>
where <math>T^{1,0}\mathbb{C}^{n+1}</math> is the bundle of holomorphic vectors. The real form of this is given by <math>P=\Re (L\oplus \bar{L})</math>, the bundle given at a point <math>p\in S^{2n+1}</math> concretely in terms of the complex structure, <math>I</math>, on <math>\mathbb{C}^{n+1}</math> by
:<math>P_p = \{ X\in T_pS^{2n+1} : IX \in T_pS^{2n+1}\subset T_p\mathbb{C}^{n+1}\},</math>
and the almost complex structure on <math>P</math> is just the restriction of <math>I</math>.

The [[Heisenberg group]] also is an example of a CR manifold.<ref> {{cite book | last1 = Capogna | first1 = Luca | last2 = Danielli | first2 = Donatella | last3 = Pauls | first3 = Scott | last4 = Tyson | first4 = Jeremy | title = An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem | chapter = Applications of Heisenberg Geometry | volume = 259 | work = Progress in Mathematics | publisher = Birkhauser | year = 2007 | location = Berlin | pages = 45-48 | accessdate = 2013-08-06}}</ref>

==See also==
*[[Eugenio Elia Levi]]
*[[Pseudoconvexity]]

==Notes==
{{reflist|30em}}

==Bibliography==
*{{Citation
| last = Levi
| first = Eugenio Elia
| author-link = Eugenio Elia Levi
| title = Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse
| language = [[Italian language|Italian]]
| journal = [http://www.springerlink.com/content/108198/?p=723d441f3aee4d4bb65e370e90b9c567&pi=0 Annali di Matematica Pura e Applicata]
| series = s. III,
| volume = XVII
| issue = 1
| pages = 61–87
| year = 1910
| url = http://www.springerlink.com/content/yr0150m4tq64j465/
| doi = 10.1007/BF02419336
| id =
| jfm = 41.0487.01
}}. An important paper in the [[Several complex variables|theory of functions of several complex variables]]. An English translation of the title reads as:-"''studies on essential singular points of analytic functions of two or more complex variables''".

==References==
* {{cite book|first=Albert|last=Boggess|title=CR Manifolds and the Tangential Cauchy Riemann Complex|publisher=CRC Press|year=1991}}
* {{cite journal|author=Hill, D. and Nacinovich, M.|url=http://www.numdam.org/numdam-bin/fitem?id=ASNSP_1995_4_22_2_315_0|title=Duality and distribution cohomology of CR manifolds|journal=Ann. Scuola Norm. Sup. Pisa|volume=22|issue=2|year=1995|pages=315–339}}
* {{cite journal|author=Chern S. S. and Moser, J.K.|title=Real hypersurfaces in complex manifolds|journal=Acta Math.|volume=133|pages=219–271|year=1974|doi=10.1007/BF02392146}}
* {{cite journal|author=Harvey, F.R. and Lawson, H.B., Jr.|title=On boundaries of complex analytic varieties|journal=Ann. Math.|year=1978|pages=223–290|volume=102|doi=10.2307/1971032|issue=2|jstor=1971032}}

{{DEFAULTSORT:Cr Manifold}}
[[Category:Smooth manifolds]]

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