Sasakian manifold

Let be a -dimensional differentiable manifold of class and let be a tensor field of type (cf. also Tensor on a vector space), a vector field and a -form on (cf. Differential form), respectively, such that

where is the identity on the tangent bundle of . Then is said to be an almost contact structure on , and is called an almost contact manifold. If follows that

and therefore has the constant rank on . Moreover, there exists a Riemannian metric on such that

for any vector fields , on [a2]. Then is said to be an almost contact metric structure and an almost contact metric manifold. On one defines the fundamental -form by

Then is said to be a contact metric structure on if .

The Nijenhuis tensor field of is the tensor field of type given by

The almost contact structure is said to be normal if

A manifold 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 is Sasakian if and only if

for any vector fields , on , where is the Levi-Civita connection on with respect to .

A plane section in is called a -section if there exists a unit vector in orthogonal to such that is an orthonormal basis of . The -sectional curvature of with respect to a -section is defined by , where is the curvature tensor field of . When the -sectional curvature does not depend on both the point and the -section , one says that has constant -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 have been studied intensively.

First, let be a -dimensional submanifold of such that is tangent to and , for all . Then is said to be an invariant submanifold of . It follows that is a Sasakian manifold too, and, in general, inherits the properties of the ambient Sasakian manifold .

Next, an -dimensional submanifold of is an anti-invariant submanifold if for all , where is the normal space of at . The most important results on anti-invariant submanifolds have been collected in [a5].

Finally, an -dimensional submanifold of is said to be a semi-invariant submanifold (a contact CR-submanifold; cf. also CR-submanifold) if is tangent to and there exist two distributions and on such that has the orthogonal decomposition , with and for all , where denotes the distribution spanned by on . 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)