Indifferential geometry, aSasakian manifold (named afterShigeo Sasaki) is acontact manifold${\displaystyle (M,\theta )}$ equipped with a special kind ofRiemannian metric${\displaystyle g}$, called aSasakian metric. They are studied as a natural odd-dimensional counterpart ofKähler manifolds (which are necessarily even-dimensional).

A Sasakian metric is defined using the construction of theRiemannian cone. Given aRiemannian manifold${\displaystyle (M,g)}$, its Riemannian cone is the product

${\displaystyle (M\times {\mathbb{R}}^{>0})}$

of${\displaystyle M}$ with a half-line${\displaystyle {\mathbb{R}}^{>0}}$,equipped with thecone metric

${\displaystyle t^{2}g+d{t}^2,}$

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

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

${\displaystyle d(t^2\theta )=t^{2}d\theta +2tdt\wedge \theta }$

on its cone is symplectic (this is one of the possibledefinitions of a contact structure). A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is aKähler manifold with Kähler form

${\displaystyle t^{2}d\theta +2tdt\wedge \theta .}$

As an example consider

${\displaystyle S^{2n-1}\hookrightarrow {\mathbb{R}}^{2n}={\mathbb{C}}^n}$

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${\displaystyle S^{2n-1}}$ is the form associated to the tangent vector${\displaystyle i\overrightarrow{N}}$, constructed from the unit-normal vector${\displaystyle \overrightarrow{N}}$ to the sphere (${\displaystyle i}$ being the complex structure on${\displaystyle {\mathbb{C}}^n}$).

Another non-compact example is${\displaystyle {\mathbb{R}}^{2n+1}}$ with coordinates${\displaystyle (\overrightarrow{x},\overrightarrow{y},z)}$ endowed with contact-form

${\displaystyle \theta =\frac{1}{2}dz+\sum _i{y}_{i}d{x}_i}$

and the Riemannian metric

${\displaystyle g=\sum _i(d{x}_i{)}^2+(d{y}_i{)}^2+{\theta }^2.}$

As a third example consider:

${\displaystyle {\mathbb{P}}^{2n-1}\mathbb{R}\hookrightarrow {\mathbb{C}}^n/{\mathbb{Z}}_2}$

where the right hand side has a natural Kähler structure, and the group${\displaystyle {\mathbb{Z}}_2}$ acts by reflection at the origin.

Sasakian manifolds were introduced in 1960 by the Japanese geometerShigeo Sasaki.^[1] There was not much activity in this field after the mid-1970s, until the advent ofString theory. Since then Sasakian manifolds have gained prominence in physics and algebraic geometry, mostly due to a string of papers byCharles P. Boyer and Krzysztof Galicki and their co-authors.

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

${\displaystyle t\mathrm{\partial }/\mathrm{\partial }t.}$

As the cone is by definition Kähler, there exists a complex structureJ. TheReeb vector field on the Sasaskian manifold is defined to be

${\displaystyle \xi =-J(t\mathrm{\partial }/\mathrm{\partial }t).}$

It is nowhere vanishing. It commutes with all holomorphicKilling vectors on the cone and in particular with allisometries 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.

A Sasakian manifold${\displaystyle M}$ is a manifold whose Riemannian cone is Kähler. If, in addition, this cone isRicci-flat,${\displaystyle M}$ is calledSasaki–Einstein; if it ishyperkähler,${\displaystyle M}$ is called3-Sasakian. Any 3-Sasakian manifold is both anEinstein manifold and aspin manifold.

IfM is positive-scalar-curvature Kähler–Einstein manifold, then, by an observation ofShoshichi Kobayashi, the circle bundleS in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection fromS toM into a Riemannian submersion. (For example, it follows that there exist Sasaki–Einstein metrics on suitablecircle bundles over the 3rd through 8thdel 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 generallyconstruct Sasaki–Einstein manifolds by starting from a Kähler–EinsteinorbifoldM. Using this observation, Boyer, Galicki, andJános 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.

• Shigeo 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 andShing-Tung Yau, "Sasaki-Einstein Manifolds and Volume Minimization",ArXiv hep-th/0603021

• EoM page,Sasakian manifold

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• Symplectic geometry
• Structures on manifolds

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