View source for CR-submanifold

<!--
c1104801.png
$#A+1 = 46 n = 0
$#C+1 = 46 : ~/encyclopedia/old_files/data/C110/C.1100480 CR\AAhsubmanifold
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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 $.

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]]]).

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  $  N $
of a complex manifold  $  ( M,J ) $
is a CR-submanifold (cf. [[#References|[a4]]]).

====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>