Sasakian manifold

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

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.

A Sasakian manifold is a manifold whose Riemannian cone is Kähler. If, in addition, this cone is Ricci flat, is called Sasaki–Einstein; if it is hyperkähler, 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.