# Sasakian manifold

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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=52320
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article