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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

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

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

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

and therefore  $  \phi $
has the constant rank  $  2m $
on  $  M $. 
Moreover, there exists a [[Riemannian metric|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 $[[#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

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

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. [[#References|[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|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]]].

==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 [[#References|[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|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====
<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>