Sasakian manifold

In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold formula_1 equipped with a special kind of Riemannian metric formula_2, called a "Sasakian" metric.
Definition.
A Sasakian metric is defined using the construction of the "Riemannian cone". Given a Riemannian manifold formula_3, its Riemannian cone is the product
of formula_5 with a half-line formula_6,
equipped with the "cone metric"
where formula_8 is the parameter in formula_6.
A manifold formula_5 equipped with a 1-form formula_11
is contact if and only if the 2-form
on its cone is symplectic (this is one of the possible
definitions of a contact structure). A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a Kähler manifold with
Kähler form
Examples.
As an example consider
where the right hand side is a natural Kähler manifold and read as the cone over the sphere (endowed with embedded metric). The contact 1-form on formula_15 is the form associated to the tangent vector formula_16, constructed from the unit-normal vector formula_17 to the sphere (formula_18 being the complex structure on formula_19).
Another non-compact example is formula_20 with coordinates formula_21 endowed with contact-form
formula_22
and the Riemannian metric
formula_23
As a third example consider:
formula_24
where the right hand side has a natural Kähler structure, and the group formula_25 acts by reflection at the origin.
History.
Sasakian manifolds were introduced in 1960 by the Japanese geometer Shigeo Sasaki. There was not much activity in this field after the mid-1970s, until the advent of String theory. Since then Sasakian manifolds have gained prominence in physics and algebraic geometry, mostly due to a string of papers by Charles P. Boyer and Krzysztof Galicki and their co-authors.
The Reeb vector field.
The homothetic vector field on the cone over a Sasakian manifold is defined to be
As the cone is by definition Kähler, there exists a complex structure "J". The "Reeb vector field" on the Sasaskian manifold is defined to be
It is nowhere vanishing. It commutes with all holomorphic Killing vectors on the cone and in particular with all isometries of the Sasakian manifold. If the orbits of the vector field close then the space of orbits is a Kähler orbifold. The Reeb vector field at the Sasakian manifold at unit radius is a unit vector field and tangential to the embedding.
Sasaki–Einstein manifolds.
A Sasakian manifold formula_5 is a manifold whose Riemannian cone is Kähler. If, in addition, this cone is Ricci-flat, formula_5 is called Sasaki–Einstein; if it is hyperkähler, formula_5 is called 3-Sasakian. Any 3-Sasakian manifold is both an Einstein manifold and a spin manifold.
If "M" is positive-scalar-curvature Kahler–Einstein manifold, then, by an observation of Shoshichi Kobayashi, the circle bundle "S" in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection from "S" to "M" into a Riemannian submersion. (For example, it follows that 
there exist Sasaki–Einstein metrics on suitable circle bundles over the 3rd through 8th del Pezzo surfaces.) While this Riemannian submersion construction provides a correct local picture of any Sasaki–Einstein manifold, 
the global structure of such manifolds can be more complicated. For example, one can more generally
construct Sasaki–Einstein manifolds by starting from a Kahler–Einstein orbifold "M." Using this observation, Boyer, Galicki, and János Kollár constructed infinitely many homeotypes of Sasaki-Einstein 5-manifolds. 
The same construction shows that the moduli space of Einstein metrics on the 5-sphere has at least several hundred connected components.