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In 1907, H. Poincaré wrote a seminal paper, [[#References|[a6]]], in which he showed that two real hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104701.png" /> are, in general, biholomorphically inequivalent (cf. [[Biholomorphic mapping|Biholomorphic mapping]]; [[Hypersurface|Hypersurface]]). Later, E. Cartan [[#References|[a10]]], [[#References|[a11]]] found all the invariants that distinguish one real hypersurface from another. The general solution for complex dimensions greater than two was given by S.S. Chern and J. Moser [[#References|[a3]]] and N. Tanaka [[#References|[a8]]], [[#References|[a7]]].
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The concept of a CR-manifold (CR-structure) has been defined having in mind the geometric structure induced on a real hypersurface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104703.png" />.
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 +
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104704.png" /> be a real [[Differentiable manifold|differentiable manifold]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104705.png" /> the [[Tangent bundle|tangent bundle]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104706.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104707.png" /> is a CR-manifold if there exists a complex subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104708.png" /> of the complexified tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c1104709.png" /> satisfying the conditions:
+
In 1907, H. Poincaré wrote a seminal paper, [[#References|[a6]]], in which he showed that two real hypersurfaces in  $  \mathbf C  ^ {2} $
 +
are, in general, biholomorphically inequivalent (cf. [[Biholomorphic mapping|Biholomorphic mapping]]; [[Hypersurface|Hypersurface]]). Later, E. Cartan [[#References|[a10]]], [[#References|[a11]]] found all the invariants that distinguish one real hypersurface from another. The general solution for complex dimensions greater than two was given by S.S. Chern and J. Moser [[#References|[a3]]] and N. Tanaka [[#References|[a8]]], [[#References|[a7]]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047010.png" />;
+
The concept of a CR-manifold (CR-structure) has been defined having in mind the geometric structure induced on a real hypersurface of  $  \mathbf C  ^ {n} $,
 +
$  n \geq  2 $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047011.png" /> is involutive, i.e., for any complex vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047014.png" /> the [[Lie bracket|Lie bracket]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047015.png" /> is also in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047016.png" />.
+
Let  $  M $
 +
be a real [[Differentiable manifold|differentiable manifold]] and $  TM $
 +
the [[Tangent bundle|tangent bundle]] of  $  M $.  
 +
One says that  $  M $
 +
is a CR-manifold if there exists a complex subbundle  $  H $
 +
of the complexified tangent bundle  $  \mathbf C \otimes TM $
 +
satisfying the conditions:
  
Alternatively, by using real vector bundles it can be proved (cf. [[#References|[a1]]]) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047017.png" /> is a CR-manifold if and only if there exists an almost-complex distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047019.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047020.png" /> is a vector subbundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047022.png" /> is an [[Almost-complex structure|almost-complex structure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047023.png" />) such that
+
$  H \cap {\overline{H}\; } = \{ 0 \} $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047024.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047025.png" />;
+
$  H $
 +
is involutive, i.e., for any complex vector fields  $  U $
 +
and  $  V $
 +
in  $  H $
 +
the [[Lie bracket|Lie bracket]]  $  [ U,V ] $
 +
is also in $  H $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047026.png" /> for any real vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047029.png" />.
+
Alternatively, by using real vector bundles it can be proved (cf. [[#References|[a1]]]) that  $  M $
 +
is a CR-manifold if and only if there exists an almost-complex distribution  $  ( D,J ) $
 +
on  $  M $(
 +
i.e., $  D $
 +
is a vector subbundle of  $  TM $
 +
and  $  J $
 +
is an [[Almost-complex structure|almost-complex structure]] on  $  D $)
 +
such that
  
Thus the CR-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047030.png" /> is determined either by the complex vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047031.png" /> or by the almost-complex distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047032.png" />. The abbreviation CR refers to A.L. Cauchy and B. Riemann, because, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047035.png" /> consists of the induced Cauchy–Riemann operators (cf. [[Cauchy–Riemann conditions|Cauchy–Riemann conditions]]).
+
$  [ JX,JY ] - [ X,Y ] $
 +
lies in $  D $;
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047036.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047037.png" /> is called a CR-function if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047038.png" /> for all complex vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047040.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047041.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047042.png" /> is said to be a CR-mapping if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047044.png" /> is the tangent mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047045.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047046.png" /> is a [[Diffeomorphism|diffeomorphism]], one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047047.png" /> is a pseudo-conformal mapping and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047049.png" /> are CR-diffeomorphic or, briefly, that they are equivalent. A CR-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047050.png" /> is said to be realizable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047051.png" /> is equivalent to some real hypersurface of a complex Euclidean space.
+
$  [ JX,JY ] - [ X,Y ] - J ( [ JX,Y ] + [ X,JY ] ) = 0 $
 +
for any real vector fields  $  X $,
 +
$  Y $
 +
in  $  D $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047052.png" /> be the natural projection mapping. Then the Levi form for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047053.png" /> is the mapping
+
Thus the CR-structure on  $  M $
 +
is determined either by the complex vector bundle  $  H $
 +
or by the almost-complex distribution  $  ( D,J ) $.  
 +
The abbreviation CR refers to A.L. Cauchy and B. Riemann, because, for  $  M $
 +
in  $  \mathbf C  ^ {n} $,
 +
$  H $
 +
consists of the induced Cauchy–Riemann operators (cf. [[Cauchy-Riemann equations]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047054.png" /></td> </tr></table>
+
A  $  C  ^ {1} $-
 +
function  $  f : {( M,H ) } \rightarrow \mathbf C $
 +
is called a CR-function if  $  Lf = 0 $
 +
for all complex vector fields  $  L $
 +
in  $  H $.
 +
A  $  C  ^ {1} $-
 +
mapping  $  F : {( M,H ) } \rightarrow {( {\widetilde{M}  } , {\widetilde{H}  } ) } $
 +
is said to be a CR-mapping if  $  F _ {*} H \subset  {\widetilde{H}  } $,
 +
where  $  F _ {*} $
 +
is the tangent mapping of  $  F $.  
 +
In particular, if  $  F $
 +
is a [[Diffeomorphism|diffeomorphism]], one says that  $  F $
 +
is a pseudo-conformal mapping and that  $  M $
 +
and  $  {\widetilde{M}  } $
 +
are CR-diffeomorphic or, briefly, that they are equivalent. A CR-structure on  $  M $
 +
is said to be realizable if  $  M $
 +
is equivalent to some real hypersurface of a complex Euclidean space.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047055.png" /></td> </tr></table>
+
Let  $  \pi : {\mathbf C \otimes TM } \rightarrow {( \mathbf C \otimes TM ) / ( H \oplus {\overline{H}\; } ) } $
 +
be the natural projection mapping. Then the Levi form for  $  M $
 +
is the mapping
  
for any complex vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047057.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047058.png" /> is the real hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047059.png" /> given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047061.png" /> is smooth, then the Levi form for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047062.png" /> is identified with the restriction of the complex Hessian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047063.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047064.png" /> (cf. also [[Hessian matrix|Hessian matrix]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047065.png" /> is positive- or negative-definite on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047066.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110470/c11047067.png" /> is strictly pseudo-convex.
+
$$
 +
h : H \rightarrow {( \mathbf C \otimes TM ) / ( H \oplus {\overline{H}\; } ) } ,
 +
$$
 +
 
 +
$$
 +
h ( U ) = {
 +
\frac{1}{2i }
 +
} \pi ( [ U, {\overline{U}\; } ] ) ,
 +
$$
 +
 
 +
for any complex vector field $  U $
 +
in $  H $.  
 +
If $  M $
 +
is the real hypersurface in $  \mathbf C  ^ {n} $
 +
given by the equation $  g ( z ) = 0 $,  
 +
where $  g : {\mathbf C  ^ {n} } \rightarrow \mathbf R $
 +
is smooth, then the Levi form for $  M $
 +
is identified with the restriction of the complex Hessian of $  g $
 +
to $  H $(
 +
cf. also [[Hessian matrix|Hessian matrix]]). When $  h $
 +
is positive- or negative-definite on $  M $,  
 +
one says that $  M $
 +
is strictly pseudo-convex.
  
 
The differential geometry of CR-manifolds (cf. [[#References|[a4]]]) has potential applications to both partial differential equations (cf. [[#References|[a2]]]) and mathematical physics (cf. [[#References|[a5]]] and [[#References|[a9]]]).
 
The differential geometry of CR-manifolds (cf. [[#References|[a4]]]) has potential applications to both partial differential equations (cf. [[#References|[a2]]]) and mathematical physics (cf. [[#References|[a5]]] and [[#References|[a9]]]).

Latest revision as of 06:29, 30 May 2020


In 1907, H. Poincaré wrote a seminal paper, [a6], in which he showed that two real hypersurfaces in $ \mathbf C ^ {2} $ are, in general, biholomorphically inequivalent (cf. Biholomorphic mapping; Hypersurface). Later, E. Cartan [a10], [a11] found all the invariants that distinguish one real hypersurface from another. The general solution for complex dimensions greater than two was given by S.S. Chern and J. Moser [a3] and N. Tanaka [a8], [a7].

The concept of a CR-manifold (CR-structure) has been defined having in mind the geometric structure induced on a real hypersurface of $ \mathbf C ^ {n} $, $ n \geq 2 $.

Let $ M $ be a real differentiable manifold and $ TM $ the tangent bundle of $ M $. One says that $ M $ is a CR-manifold if there exists a complex subbundle $ H $ of the complexified tangent bundle $ \mathbf C \otimes TM $ satisfying the conditions:

$ H \cap {\overline{H}\; } = \{ 0 \} $;

$ H $ is involutive, i.e., for any complex vector fields $ U $ and $ V $ in $ H $ the Lie bracket $ [ U,V ] $ is also in $ H $.

Alternatively, by using real vector bundles it can be proved (cf. [a1]) that $ M $ is a CR-manifold if and only if there exists an almost-complex distribution $ ( D,J ) $ on $ M $( i.e., $ D $ is a vector subbundle of $ TM $ and $ J $ is an almost-complex structure on $ D $) such that

$ [ JX,JY ] - [ X,Y ] $ lies in $ D $;

$ [ JX,JY ] - [ X,Y ] - J ( [ JX,Y ] + [ X,JY ] ) = 0 $ for any real vector fields $ X $, $ Y $ in $ D $.

Thus the CR-structure on $ M $ is determined either by the complex vector bundle $ H $ or by the almost-complex distribution $ ( D,J ) $. The abbreviation CR refers to A.L. Cauchy and B. Riemann, because, for $ M $ in $ \mathbf C ^ {n} $, $ H $ consists of the induced Cauchy–Riemann operators (cf. Cauchy-Riemann equations).

A $ C ^ {1} $- function $ f : {( M,H ) } \rightarrow \mathbf C $ is called a CR-function if $ Lf = 0 $ for all complex vector fields $ L $ in $ H $. A $ C ^ {1} $- mapping $ F : {( M,H ) } \rightarrow {( {\widetilde{M} } , {\widetilde{H} } ) } $ is said to be a CR-mapping if $ F _ {*} H \subset {\widetilde{H} } $, where $ F _ {*} $ is the tangent mapping of $ F $. In particular, if $ F $ is a diffeomorphism, one says that $ F $ is a pseudo-conformal mapping and that $ M $ and $ {\widetilde{M} } $ are CR-diffeomorphic or, briefly, that they are equivalent. A CR-structure on $ M $ is said to be realizable if $ M $ is equivalent to some real hypersurface of a complex Euclidean space.

Let $ \pi : {\mathbf C \otimes TM } \rightarrow {( \mathbf C \otimes TM ) / ( H \oplus {\overline{H}\; } ) } $ be the natural projection mapping. Then the Levi form for $ M $ is the mapping

$$ h : H \rightarrow {( \mathbf C \otimes TM ) / ( H \oplus {\overline{H}\; } ) } , $$

$$ h ( U ) = { \frac{1}{2i } } \pi ( [ U, {\overline{U}\; } ] ) , $$

for any complex vector field $ U $ in $ H $. If $ M $ is the real hypersurface in $ \mathbf C ^ {n} $ given by the equation $ g ( z ) = 0 $, where $ g : {\mathbf C ^ {n} } \rightarrow \mathbf R $ is smooth, then the Levi form for $ M $ is identified with the restriction of the complex Hessian of $ g $ to $ H $( cf. also Hessian matrix). When $ h $ is positive- or negative-definite on $ M $, one says that $ M $ is strictly pseudo-convex.

The differential geometry of CR-manifolds (cf. [a4]) has potential applications to both partial differential equations (cf. [a2]) and mathematical physics (cf. [a5] and [a9]).

References

[a1] A. Bejancu, "Geometry of CR submanifolds" , Reidel (1986)
[a2] A. Boggess, "CR manifolds and tangential Cauchy–Riemann complex" , CRC (1991)
[a3] S.S. Chern, J. Moser, "Real hypersurfaces in complex manifolds" Acta Math. , 133 (1974) pp. 219–271
[a4] H. Jacobowitz, "An introduction to CR structures" , Math. Surveys and Monographs , 32 , Amer. Math. Soc. (1990)
[a5] R. Penrose, "Physical space-time and non-realizable CR structures" , Proc. Symp. Pure Math. , 39 , Amer. Math. Soc. (1983) pp. 401–422
[a6] H. Poincaré, "Les functions analytiques de deux variables et la représentation conforme" Rend. Circ. Mat. Palermo , 23 (1907) pp. 185–220
[a7] N. Tanaka, "On the pseudo-conformal geometry of hypersurfaces of the space of complex variables" J. Math. Soc. Japan , 14 (1962) pp. 397–429
[a8] N. Tanaka, "On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections" Japan J. Math. (N.S.) , 2 (1976) pp. 131–190
[a9] J.R. Wells, Jr., "Complex manifolds and mathematical physics" Bull. Amer. Math. Soc. (N.S.) , 1 (1979) pp. 296–336
[a10] É. Cartan, "Sur l'équivalence pseudo-conforme des hypersurfaces de l'espace de deux variables complexes I." Ann. Mathém. , 11 (1932) pp. 17–90
[a11] É. Cartan, "Sur l'équivalence pseudo-conforme des hypersurfaces de l'espace de deux variable complexes II." Ann. Scuola Norm. Sup. Pisa , 1 (1932) pp. 333–354
How to Cite This Entry:
CR-manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CR-manifold&oldid=11362
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article