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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104801.png" /> be an almost Hermitian manifold (cf. also [[Hermitian structure|Hermitian structure]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104802.png" /> is an [[Almost-complex structure|almost-complex structure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104804.png" /> is a [[Riemannian metric|Riemannian metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104805.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104806.png" /> for any vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104808.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c1104809.png" />. A real submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048011.png" /> is said to be a complex (holomorphic) submanifold if the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048013.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048014.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048015.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048016.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048017.png" /> be the [[Normal bundle|normal bundle]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048018.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048019.png" /> is called a totally real (anti-invariant) submanifold if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048020.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048021.png" />.
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In 1978, A. Bejancu [[#References|[a1]]] introduced the notion of a CR-submanifold as a natural generalization of both complex submanifolds and totally real submanifolds. More precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048022.png" /> is said to be a CR-submanifold if there exists a smooth distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048024.png" /> such that:
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048025.png" /> is a holomorphic distribution, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048026.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048027.png" />;
+
Let  $  ( M,J,g ) $
 +
be an almost Hermitian manifold (cf. also [[Hermitian structure|Hermitian structure]]), where  $  J $
 +
is an [[Almost-complex structure|almost-complex structure]] on  $  M $
 +
and  $  g $
 +
is a [[Riemannian metric|Riemannian metric]] on  $  M $
 +
satisfying  $  g ( JX,JY ) = g ( X,Y ) $
 +
for any vector fields  $  X $
 +
and  $  Y $
 +
on  $  M $.  
 +
A real submanifold  $  N $
 +
of  $  M $
 +
is said to be a complex (holomorphic) submanifold if the tangent bundle  $  TN $
 +
of  $  N $
 +
is invariant under  $  J $,  
 +
i.e. $  J ( T _ {x} N ) = T _ {x} N $
 +
for any $  x \in N $.  
 +
Let  $  TN  ^  \perp  $
 +
be the [[Normal bundle|normal bundle]] of  $  N $.  
 +
Then  $  N $
 +
is called a totally real (anti-invariant) submanifold if  $  J ( T _ {x} N ) \subset  T _ {x} N  ^  \perp  $
 +
for any  $  x \in N $.
  
the complementary orthogonal distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048029.png" /> is a totally real distribution, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048030.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048031.png" />.
+
In 1978, A. Bejancu [[#References|[a1]]] introduced the notion of a CR-submanifold as a natural generalization of both complex submanifolds and totally real submanifolds. More precisely,  $  N $
 +
is said to be a CR-submanifold if there exists a smooth distribution  $  D $
 +
on  $  N $
 +
such that:
 +
 
 +
$  D $
 +
is a holomorphic distribution, that is,  $  J ( D _ {x} ) = D _ {x} $
 +
for any  $  x \in N $;
 +
 
 +
the complementary orthogonal distribution $  D  ^  \perp  $
 +
of $  D $
 +
is a totally real distribution, that is, $  J ( D _ {x}  ^  \perp  ) \subset  T _ {x} N  ^  \perp  $
 +
for any $  x \in N $.
  
 
The above concept has been mainly investigated from the viewpoint of [[Differential geometry|differential geometry]] (cf. [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]]).
 
The above concept has been mainly investigated from the viewpoint of [[Differential geometry|differential geometry]] (cf. [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048032.png" /> be the [[Second fundamental form|second fundamental form]] of the CR-submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048033.png" />. Then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048034.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048036.png" />-geodesic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048038.png" />-geodesic or mixed geodesic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048039.png" /> vanishes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048041.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048042.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048046.png" />, respectively.
+
Let $  h $
 +
be the [[Second fundamental form|second fundamental form]] of the CR-submanifold $  N $.  
 +
Then one says that $  N $
 +
is $  D $-
 +
geodesic, $  D  ^  \perp  $-
 +
geodesic or mixed geodesic if $  h $
 +
vanishes on $  D $
 +
or $  D  ^  \perp  $,  
 +
or $  h ( X,Y ) = 0 $
 +
for any $  X $
 +
in $  D $
 +
and $  Y $
 +
in $  D  ^  \perp  $,  
 +
respectively.
  
From the viewpoint of complex analysis, a CR-submanifold is an imbedded [[CR-manifold|CR-manifold]] in a complex manifold. In this case a real hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048047.png" /> of a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110480/c11048048.png" /> is a CR-submanifold (cf. [[#References|[a4]]]).
+
From the viewpoint of complex analysis, a CR-submanifold is an imbedded [[CR-manifold|CR-manifold]] in a complex manifold. In this case a real hypersurface $  N $
 +
of a complex manifold $  ( M,J ) $
 +
is a CR-submanifold (cf. [[#References|[a4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Bejancu,  "CR submanifolds of a Kaehler manifold I"  ''Proc. Amer. Math. Soc.'' , '''69'''  (1978)  pp. 134–142</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bejancu,  "Geometry of CR submanifolds" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.E. Blair,  B.Y. Chen,  "On CR submanifolds of Hermitian manifolds"  ''Israel J. Math.'' , '''34'''  (1979)  pp. 353–363</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Boggess,  "CR manifolds and tangential Cauchy–Riemann complex" , CRC  (1991)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.Y. Chen,  "Geometry of submanifolds and its applications" , Tokyo Sci. Univ.  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Yano,  M. Kon,  "CR submanifolds of Kaehlerian and Sasakian manifolds" , Birkhäuser  (1983)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Yano,  M. Kon,  "Structures on manifolds" , World Sci.  (1984)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Bejancu,  "CR submanifolds of a Kaehler manifold I"  ''Proc. Amer. Math. Soc.'' , '''69'''  (1978)  pp. 134–142</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bejancu,  "Geometry of CR submanifolds" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.E. Blair,  B.Y. Chen,  "On CR submanifolds of Hermitian manifolds"  ''Israel J. Math.'' , '''34'''  (1979)  pp. 353–363</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Boggess,  "CR manifolds and tangential Cauchy–Riemann complex" , CRC  (1991)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.Y. Chen,  "Geometry of submanifolds and its applications" , Tokyo Sci. Univ.  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Yano,  M. Kon,  "CR submanifolds of Kaehlerian and Sasakian manifolds" , Birkhäuser  (1983)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Yano,  M. Kon,  "Structures on manifolds" , World Sci.  (1984)</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


Let $ ( M,J,g ) $ be an almost Hermitian manifold (cf. also Hermitian structure), where $ J $ is an almost-complex structure on $ M $ and $ g $ is a Riemannian metric on $ M $ satisfying $ g ( JX,JY ) = g ( X,Y ) $ for any vector fields $ X $ and $ Y $ on $ M $. A real submanifold $ N $ of $ M $ is said to be a complex (holomorphic) submanifold if the tangent bundle $ TN $ of $ N $ is invariant under $ J $, i.e. $ J ( T _ {x} N ) = T _ {x} N $ for any $ x \in N $. Let $ TN ^ \perp $ be the normal bundle of $ N $. Then $ N $ is called a totally real (anti-invariant) submanifold if $ J ( T _ {x} N ) \subset T _ {x} N ^ \perp $ for any $ x \in N $.

In 1978, A. Bejancu [a1] introduced the notion of a CR-submanifold as a natural generalization of both complex submanifolds and totally real submanifolds. More precisely, $ N $ is said to be a CR-submanifold if there exists a smooth distribution $ D $ on $ N $ such that:

$ D $ is a holomorphic distribution, that is, $ J ( D _ {x} ) = D _ {x} $ for any $ x \in N $;

the complementary orthogonal distribution $ D ^ \perp $ of $ D $ is a totally real distribution, that is, $ J ( D _ {x} ^ \perp ) \subset T _ {x} N ^ \perp $ for any $ x \in N $.

The above concept has been mainly investigated from the viewpoint of differential geometry (cf. [a2], [a3], [a5], [a6], [a7]).

Let $ h $ be the second fundamental form of the CR-submanifold $ N $. Then one says that $ N $ is $ D $- geodesic, $ D ^ \perp $- geodesic or mixed geodesic if $ h $ vanishes on $ D $ or $ D ^ \perp $, or $ h ( X,Y ) = 0 $ for any $ X $ in $ D $ and $ Y $ in $ D ^ \perp $, respectively.

From the viewpoint of complex analysis, a CR-submanifold is an imbedded CR-manifold in a complex manifold. In this case a real hypersurface $ N $ of a complex manifold $ ( M,J ) $ is a CR-submanifold (cf. [a4]).

References

[a1] A. Bejancu, "CR submanifolds of a Kaehler manifold I" Proc. Amer. Math. Soc. , 69 (1978) pp. 134–142
[a2] A. Bejancu, "Geometry of CR submanifolds" , Reidel (1986)
[a3] D.E. Blair, B.Y. Chen, "On CR submanifolds of Hermitian manifolds" Israel J. Math. , 34 (1979) pp. 353–363
[a4] A. Boggess, "CR manifolds and tangential Cauchy–Riemann complex" , CRC (1991)
[a5] B.Y. Chen, "Geometry of submanifolds and its applications" , Tokyo Sci. Univ. (1981)
[a6] K. Yano, M. Kon, "CR submanifolds of Kaehlerian and Sasakian manifolds" , Birkhäuser (1983)
[a7] K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984)
How to Cite This Entry:
CR-submanifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CR-submanifold&oldid=46184
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article