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In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold $$(M,\theta)$$ equipped with a special kind of Riemannian metric g, called a Sasakian metric.

Definition

A Sasakian metric is defined using the construction of the Riemannian cone. Given a Riemannian manifold (M,g), its Riemannian cone is a product

$$(M\times {\Bbb R}^{>0})\,$$

of M with a half-line $${\Bbb R}^{>0}$$ , equipped with the cone metric

$$t^2 g + dt^2,\,$$

where t is the parameter in $${\Bbb R}^{>0}.$$

A manifold M equipped with a 1-form $$\theta$$ is contact if and only if the 2-form

$$t^2\,d\theta + 2t\, dt \cdot \theta\,$$

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

$$t^2\,d\theta + 2t\,dt \cdot \theta.\,$$

Examples

As an example consider

$$S^{2n-1}\hookrightarrow {\Bbb R}^{2n, *}={\Bbb C}^{n, *}$$

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 $$S^{2n+1}$$ is the form associated to the tangent vector i\vec{n}, constructued from the unit-normal vector $$\vec{n}$$ to the sphere (i being the complex structure on $${\Bbb C}^n)$$ .

Another non-compact example is $${{\Bbb R}^{2n+1}}$$ with coordinates $$(\vec{x},\vec{y},z)$$ endowed with

contact-form $$\theta=\frac12 dz+\sum_i y_i\,dx_i$$ and

the Riemannian metric $$g=\sum_i (dx_i)^2+(dy_i)^2+\theta^2.$$

As a third example consider

$${\Bbb P}^{2n-1}{\Bbb R}\hookrightarrow {\Bbb C}^{n,*}/{\Bbb Z}_2$$

where the right hand side has a natural Kähler structure (and the $${\Bbb Z}_2$$ acts by reflection at the origin).
History

Sasakian manifolds were introduced in 1960 by the Japanese geometer Shigeo Sasaki[1]. 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 Boyer, Galicki and their co-authors.
The Reeb vector field

The homothetic vector field on the cone over a Sasakian manifold is defined to be

$$t\partial/\partial t.$$

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

$$\xi =-J(t\partial/\partial t).$$

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 M is one with the Riemannian cone Kähler. If the cone is, in addition, Ricci-flat, M is called Sasaki–Einstein; if it is hyperkähler, M is called 3-Sasakian. Any 3-Sasakian manifold is an Einstein manifold and a spin manifold.

Examples include all round odd-dimensional spheres, and the product of a 2-sphere and a 3-sphere with a homogeneous metric. The cones are respectively complex vector spaces without the origin, and the conifold.

It is also known that there exist Sasaki-Einstein metrics on some circle bundles over the 3rd through 8th del Pezzo surfaces.

In 2005 an infinite family of 5-dimensional Sasaki-Einstein metrics was constructed. These are denoted

$$L^{a,b,c}\$$

where a, b and c are three integral parameters. A 2-parameter family had been constructed the previous year, before which only a finite number of 5-dimensional examples were known.
Notes

^ Sasaki biography

References

S. Sasaki, "On differentiable manifolds with certain structures which are closely related to almost contact structure", Tohoku Math. J. 2 (1960), 459-476.
Charles P. Boyer, Krzysztof Galicki, Sasakian geometry
Charles P. Boyer, Krzysztof Galicki, "3-Sasakian Manifolds", Surveys Diff. Geom. 7 (1999) 123-184
Dario Martelli, James Sparks and Shing-Tung Yau, "Sasaki-Einstein Manifolds and Volume Minimization", ArXiv hep-th/0603021