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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100401.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100402.png" />-dimensional [[Differentiable manifold|differentiable manifold]] of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100403.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100404.png" /> be a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100405.png" /> (cf. also [[Tensor on a vector space|Tensor on a vector space]]), a [[Vector field|vector field]] and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100406.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100407.png" /> (cf. [[Differential form|Differential form]]), respectively, such that
+
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<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/s/s110/s110040/s1100408.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s1100409.png" /> is the identity on the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004011.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004012.png" /> is said to be an almost contact structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004014.png" /> is called an almost contact manifold. If follows that
+
Let  $  M $
 +
be a  $  ( 2m + 1 ) $-
 +
dimensional [[Differentiable manifold|differentiable manifold]] of class  $  C  ^  \infty  $
 +
and let  $  \phi, \xi, \eta $
 +
be a tensor field of type  $  ( 1,1 ) $(
 +
cf. also [[Tensor on a vector space|Tensor on a vector space]]), a [[Vector field|vector field]] and a  $  1 $-
 +
form on  $  M $(
 +
cf. [[Differential form|Differential form]]), respectively, such that
  
<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/s/s110/s110040/s11004015.png" /></td> </tr></table>
+
$$
 +
\phi  ^ {2} = - I + \eta \otimes \xi, \quad \eta ( \xi ) = 1,
 +
$$
  
and therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004016.png" /> has the constant rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004018.png" />. Moreover, there exists a [[Riemannian metric|Riemannian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004020.png" /> such that
+
where  $  I $
 +
is the identity on the [[Tangent bundle|tangent bundle]] $  TM $
 +
of  $  M $.  
 +
Then  $  ( \phi, \xi, \eta ) $
 +
is said to be an almost contact structure on $  M $,
 +
and  $  M $
 +
is called an almost contact manifold. If follows that
  
<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/s/s110/s110040/s11004021.png" /></td> </tr></table>
+
$$
 +
\eta \circ \phi = 0, \quad \phi ( \xi ) = 0,
 +
$$
  
<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/s/s110/s110040/s11004022.png" /></td> </tr></table>
+
and therefore  $  \phi $
 +
has the constant rank  $  2m $
 +
on  $  M $.  
 +
Moreover, there exists a [[Riemannian metric|Riemannian metric]]  $  g $
 +
on  $  M $
 +
such that
  
for any vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004025.png" /> [[#References|[a2]]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004026.png" /> is said to be an almost contact metric structure and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004027.png" /> an almost contact metric manifold. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004028.png" /> one defines the fundamental <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004030.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004031.png" /> by
+
$$
 +
g ( \phi x, \phi Y ) = g ( X,Y ) - \eta ( X ) \eta ( Y ) ,
 +
$$
  
<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/s/s110/s110040/s11004032.png" /></td> </tr></table>
+
$$
 +
\eta ( X ) = g ( X, \xi ) ,
 +
$$
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004033.png" /> is said to be a contact metric structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004034.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004035.png" />.
+
for any vector fields  $  X $,
 +
$  Y $
 +
on  $  M $[[#References|[a2]]]. Then $  ( \phi, \xi, \eta,g ) $
 +
is said to be an almost contact metric structure and  $  M $
 +
an almost contact metric manifold. On  $  M $
 +
one defines the fundamental  $  2 $-
 +
form  $  \Phi $
 +
by
  
The Nijenhuis tensor field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004036.png" /> is the tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004037.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004038.png" /> given by
+
$$
 +
\Phi ( X,Y ) = g ( X, \phi Y ) .
 +
$$
  
<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/s/s110/s110040/s11004039.png" /></td> </tr></table>
+
Then  $  ( \phi, \xi, \eta,g ) $
 +
is said to be a contact metric structure on  $  M $
 +
if  $  \Phi = d \eta $.
  
<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/s/s110/s110040/s11004040.png" /></td> </tr></table>
+
The Nijenhuis tensor field of  $  \phi $
 +
is the tensor field  $  [ \phi, \phi] $
 +
of type  $  ( 1,2 ) $
 +
given by
  
The almost contact structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004041.png" /> is said to be normal if
+
$$
 +
[ \phi, \phi ] ( X,Y ) = \phi  ^ {2} [ X,Y ] - [ \phi X, \phi Y ] +
 +
$$
  
<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/s/s110/s110040/s11004042.png" /></td> </tr></table>
+
$$
 +
- \phi [ \phi X,Y ] - \phi [ X, \phi Y ] .
 +
$$
  
A manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004043.png" /> endowed with a normal contact metric structure is called a Sasakian manifold. To study Sasakian manifolds one often uses the following characterization (cf. [[#References|[a4]]]): An almost contact metric manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004044.png" /> is Sasakian if and only if
+
The almost contact structure ( \phi, \xi, \eta ) $
 +
is said to be normal if
  
<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/s/s110/s110040/s11004045.png" /></td> </tr></table>
+
$$
 +
[ \phi, \phi ] + 2d \eta \otimes \xi = 0.
 +
$$
  
for any vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004049.png" /> is the [[Levi-Civita connection|Levi-Civita connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004050.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004051.png" />.
+
A manifold  $  M $
 +
endowed with a normal contact metric structure is called a Sasakian manifold. To study Sasakian manifolds one often uses the following characterization (cf. [[#References|[a4]]]): An almost contact metric manifold  $  M $
 +
is Sasakian if and only if
  
A plane section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004053.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004055.png" />-section if there exists a unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004057.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004059.png" /> is an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004060.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004062.png" />-sectional curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004063.png" /> with respect to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004064.png" />-section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004065.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004067.png" /> is the curvature tensor field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004068.png" />. When the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004069.png" />-sectional curvature does not depend on both the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004070.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004071.png" />-section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004072.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004073.png" /> has constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004075.png" />-sectional curvature and calls it a Sasakian space form.
+
$$
 +
( \nabla _ {X} \phi ) Y = g ( X,Y ) \xi - \eta ( Y ) X,
 +
$$
 +
 
 +
for any vector fields  $  X $,
 +
$  Y $
 +
on  $  M $,
 +
where  $  \nabla $
 +
is the [[Levi-Civita connection|Levi-Civita connection]] on  $  M $
 +
with respect to  $  g $.
 +
 
 +
A plane section $  \pi $
 +
in $  T _ {x} M $
 +
is called a $  \phi $-
 +
section if there exists a unit vector $  X $
 +
in $  T _ {x} M $
 +
orthogonal to $  \xi $
 +
such that $  \{ X, \phi X \} $
 +
is an orthonormal basis of $  \pi $.  
 +
The $  \phi $-
 +
sectional curvature of $  M $
 +
with respect to a $  \phi $-
 +
section $  \pi $
 +
is defined by $  H ( \pi ) = g ( R ( X, \phi X ) \phi X,X ) $,  
 +
where $  R $
 +
is the curvature tensor field of $  \nabla $.  
 +
When the $  \phi $-
 +
sectional curvature does not depend on both the point $  x \in M $
 +
and the $  \phi $-
 +
section $  \pi $,  
 +
one says that $  M $
 +
has constant $  \phi $-
 +
sectional curvature and calls it a Sasakian space form.
  
 
General references for Sasakian manifolds are [[#References|[a2]]], [[#References|[a3]]], [[#References|[a6]]].
 
General references for Sasakian manifolds are [[#References|[a2]]], [[#References|[a3]]], [[#References|[a6]]].
  
 
==Submanifolds of Sasakian manifolds.==
 
==Submanifolds of Sasakian manifolds.==
Three classes of submanifolds of a Sasakian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004076.png" /> have been studied intensively.
+
Three classes of submanifolds of a Sasakian manifold $  M $
 +
have been studied intensively.
  
First, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004077.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004078.png" />-dimensional submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004079.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004080.png" /> is tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004082.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004083.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004084.png" /> is said to be an invariant submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004085.png" />. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004086.png" /> is a Sasakian manifold too, and, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004087.png" /> inherits the properties of the ambient Sasakian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004088.png" />.
+
First, let $  N $
 +
be a $  ( 2n + 1 ) $-
 +
dimensional submanifold of $  M $
 +
such that $  \xi $
 +
is tangent to $  N $
 +
and $  \phi ( T _ {x} N ) \subset  T _ {x} N $,  
 +
for all $  x \in N $.  
 +
Then $  N $
 +
is said to be an invariant submanifold of $  M $.  
 +
It follows that $  N $
 +
is a Sasakian manifold too, and, in general, $  N $
 +
inherits the properties of the ambient Sasakian manifold $  M $.
  
Next, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004089.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004091.png" /> is an anti-invariant submanifold if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004092.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004094.png" /> is the normal space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004095.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004096.png" />. The most important results on anti-invariant submanifolds have been collected in [[#References|[a5]]].
+
Next, an $  n $-
 +
dimensional submanifold $  N $
 +
of $  M $
 +
is an anti-invariant submanifold if $  \phi ( T _ {x} N ) \subset  T _ {x} N  ^  \perp  $
 +
for all $  x \in N $,  
 +
where $  T _ {x} N  ^  \perp  $
 +
is the normal space of $  N $
 +
at $  x $.  
 +
The most important results on anti-invariant submanifolds have been collected in [[#References|[a5]]].
  
Finally, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004097.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s11004099.png" /> is said to be a semi-invariant submanifold (a contact CR-submanifold; cf. also [[CR-submanifold|CR-submanifold]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040100.png" /> is tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040101.png" /> and there exist two distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040103.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040105.png" /> has the orthogonal decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040106.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040108.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040110.png" /> denotes the distribution spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040111.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040112.png" />. For the geometry of semi-invariant submanifolds, see [[#References|[a1]]].
+
Finally, an $  n $-
 +
dimensional submanifold $  N $
 +
of $  M $
 +
is said to be a semi-invariant submanifold (a contact CR-submanifold; cf. also [[CR-submanifold|CR-submanifold]]) if $  \xi $
 +
is tangent to $  N $
 +
and there exist two distributions $  D $
 +
and $  D  ^  \perp  $
 +
on $  N $
 +
such that $  TN $
 +
has the orthogonal decomposition $  TN = D \oplus D  ^  \perp  \oplus \{ \xi \} $,  
 +
with $  \phi ( D _ {x} ) = D _ {x} $
 +
and $  \phi ( D _ {x}  ^  \perp  ) \subset  T _ {x} N  ^  \perp  $
 +
for all $  x \in N $,  
 +
where $  \{ \xi \} $
 +
denotes the distribution spanned by $  \xi $
 +
on $  N $.  
 +
For the geometry of semi-invariant submanifolds, see [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Bejancu,  "Geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040113.png" /> submanifolds" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.E. Blair,  "Contact manifolds in Riemannian geometry" , ''Lecture Notes in Mathematics'' , '''509''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Sasaki,  "Almost contact manifolds" , ''Lecture Notes'' , '''1–3''' , Math. Inst. Tôhoku Univ.  (1965–1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Sasaki,  Y. Hatakeyama,  "On differentiable manifolds with contact metric strctures"  ''J. Math. Soc. Japan'' , '''14'''  (1962)  pp. 249–271</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Yano,  M. Kon,  "Anti-invariant submanifolds" , M. Dekker  (1976)</TD></TR><TR><TD valign="top">[a6]</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,  "Geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110040/s110040113.png" /> submanifolds" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.E. Blair,  "Contact manifolds in Riemannian geometry" , ''Lecture Notes in Mathematics'' , '''509''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Sasaki,  "Almost contact manifolds" , ''Lecture Notes'' , '''1–3''' , Math. Inst. Tôhoku Univ.  (1965–1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Sasaki,  Y. Hatakeyama,  "On differentiable manifolds with contact metric strctures"  ''J. Math. Soc. Japan'' , '''14'''  (1962)  pp. 249–271</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Yano,  M. Kon,  "Anti-invariant submanifolds" , M. Dekker  (1976)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Yano,  M. Kon,  "Structures on manifolds" , World Sci.  (1984)</TD></TR></table>

Revision as of 08:12, 6 June 2020


Let $ M $ be a $ ( 2m + 1 ) $- dimensional differentiable manifold of class $ C ^ \infty $ and let $ \phi, \xi, \eta $ be a tensor field of type $ ( 1,1 ) $( cf. also Tensor on a vector space), a vector field and a $ 1 $- form on $ M $( cf. Differential form), respectively, such that

$$ \phi ^ {2} = - I + \eta \otimes \xi, \quad \eta ( \xi ) = 1, $$

where $ I $ is the identity on the tangent bundle $ TM $ of $ M $. Then $ ( \phi, \xi, \eta ) $ is said to be an almost contact structure on $ M $, and $ M $ is called an almost contact manifold. If follows that

$$ \eta \circ \phi = 0, \quad \phi ( \xi ) = 0, $$

and therefore $ \phi $ has the constant rank $ 2m $ on $ M $. Moreover, there exists a Riemannian metric $ g $ on $ M $ such that

$$ g ( \phi x, \phi Y ) = g ( X,Y ) - \eta ( X ) \eta ( Y ) , $$

$$ \eta ( X ) = g ( X, \xi ) , $$

for any vector fields $ X $, $ Y $ on $ M $[a2]. Then $ ( \phi, \xi, \eta,g ) $ is said to be an almost contact metric structure and $ M $ an almost contact metric manifold. On $ M $ one defines the fundamental $ 2 $- form $ \Phi $ by

$$ \Phi ( X,Y ) = g ( X, \phi Y ) . $$

Then $ ( \phi, \xi, \eta,g ) $ is said to be a contact metric structure on $ M $ if $ \Phi = d \eta $.

The Nijenhuis tensor field of $ \phi $ is the tensor field $ [ \phi, \phi] $ of type $ ( 1,2 ) $ given by

$$ [ \phi, \phi ] ( X,Y ) = \phi ^ {2} [ X,Y ] - [ \phi X, \phi Y ] + $$

$$ - \phi [ \phi X,Y ] - \phi [ X, \phi Y ] . $$

The almost contact structure $ ( \phi, \xi, \eta ) $ is said to be normal if

$$ [ \phi, \phi ] + 2d \eta \otimes \xi = 0. $$

A manifold $ M $ endowed with a normal contact metric structure is called a Sasakian manifold. To study Sasakian manifolds one often uses the following characterization (cf. [a4]): An almost contact metric manifold $ M $ is Sasakian if and only if

$$ ( \nabla _ {X} \phi ) Y = g ( X,Y ) \xi - \eta ( Y ) X, $$

for any vector fields $ X $, $ Y $ on $ M $, where $ \nabla $ is the Levi-Civita connection on $ M $ with respect to $ g $.

A plane section $ \pi $ in $ T _ {x} M $ is called a $ \phi $- section if there exists a unit vector $ X $ in $ T _ {x} M $ orthogonal to $ \xi $ such that $ \{ X, \phi X \} $ is an orthonormal basis of $ \pi $. The $ \phi $- sectional curvature of $ M $ with respect to a $ \phi $- section $ \pi $ is defined by $ H ( \pi ) = g ( R ( X, \phi X ) \phi X,X ) $, where $ R $ is the curvature tensor field of $ \nabla $. When the $ \phi $- sectional curvature does not depend on both the point $ x \in M $ and the $ \phi $- section $ \pi $, one says that $ M $ has constant $ \phi $- sectional curvature and calls it a Sasakian space form.

General references for Sasakian manifolds are [a2], [a3], [a6].

Submanifolds of Sasakian manifolds.

Three classes of submanifolds of a Sasakian manifold $ M $ have been studied intensively.

First, let $ N $ be a $ ( 2n + 1 ) $- dimensional submanifold of $ M $ such that $ \xi $ is tangent to $ N $ and $ \phi ( T _ {x} N ) \subset T _ {x} N $, for all $ x \in N $. Then $ N $ is said to be an invariant submanifold of $ M $. It follows that $ N $ is a Sasakian manifold too, and, in general, $ N $ inherits the properties of the ambient Sasakian manifold $ M $.

Next, an $ n $- dimensional submanifold $ N $ of $ M $ is an anti-invariant submanifold if $ \phi ( T _ {x} N ) \subset T _ {x} N ^ \perp $ for all $ x \in N $, where $ T _ {x} N ^ \perp $ is the normal space of $ N $ at $ x $. The most important results on anti-invariant submanifolds have been collected in [a5].

Finally, an $ n $- dimensional submanifold $ N $ of $ M $ is said to be a semi-invariant submanifold (a contact CR-submanifold; cf. also CR-submanifold) if $ \xi $ is tangent to $ N $ and there exist two distributions $ D $ and $ D ^ \perp $ on $ N $ such that $ TN $ has the orthogonal decomposition $ TN = D \oplus D ^ \perp \oplus \{ \xi \} $, with $ \phi ( D _ {x} ) = D _ {x} $ and $ \phi ( D _ {x} ^ \perp ) \subset T _ {x} N ^ \perp $ for all $ x \in N $, where $ \{ \xi \} $ denotes the distribution spanned by $ \xi $ on $ N $. For the geometry of semi-invariant submanifolds, see [a1].

References

[a1] A. Bejancu, "Geometry of submanifolds" , Reidel (1986)
[a2] D.E. Blair, "Contact manifolds in Riemannian geometry" , Lecture Notes in Mathematics , 509 , Springer (1976)
[a3] S. Sasaki, "Almost contact manifolds" , Lecture Notes , 1–3 , Math. Inst. Tôhoku Univ. (1965–1968)
[a4] S. Sasaki, Y. Hatakeyama, "On differentiable manifolds with contact metric strctures" J. Math. Soc. Japan , 14 (1962) pp. 249–271
[a5] K. Yano, M. Kon, "Anti-invariant submanifolds" , M. Dekker (1976)
[a6] K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984)
How to Cite This Entry:
Sasakian manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sasakian_manifold&oldid=48611
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article