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A Sasakian manifold is a contact manifold $(M,\theta)$ equipped with a special kind of Riemannian metric $g$, called a Sasakian metric.
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Paper about Sasaki-Einstein manifolds
Well, now there is a great textbook for Sasaki-Einstein geometry, by Boyer-Galicki: Sasakian Geometry.
Here is a link to the book at Oxford University Press, DOI: 10.1093/acprof:oso/9780198564959.001. …