# Sasaki but not Einstein

Hi, I search for an example of a Sasaki-manifold which is not Einstein. Can you give one?

Thank you and best regards!

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Since you are having difficulty getting hold of the book mentioned in my answer, feel free to get in touch with me by email (my address should be easy to find from my user page). –  José Figueroa-O'Farrill Nov 21 '10 at 2:24

Chapter 11 in Boyer-Galicki's Sasakian Geometry discusses two obstructions to the existence of Sasaki-Einstein structures on Sasakian manifolds. They go on to discuss many such examples, obtained as links of conical singularities of projective varieties in weighted projective spaces.

They are not hard to find! The cone of a Sasakian manifold is Kähler, whereas the cone of a Sasaki-Einstein manifold is Calabi-Yau. Hence your question is the analogue of asking for a Kähler manifold which is not Calabi-Yau. This is the generic situation.

As in my answer to your first question on Sasakian Geometry, there is no excuse for these questions given the wealth of information and the clarity of style of the book by Boyer and Galicki.

Go read the book! It really is good.

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Thanks for the answer! I want to read the book, but the problem is that I am not able to get the book.... –  Differentialgeometer Nov 20 '10 at 8:38
The geometries in that paper are orbifolds $\mathbb{C}^2/\mathbb{Z}_n$. The $\mathbb{Z}_n$ lies inside an $\mathrm{SU}(2)$ subgroup and hence the orbifold will be Calabi-Yau, whence the link of the conical singularity at the origin will be Sasaki-Einstein. –  José Figueroa-O'Farrill Nov 19 '10 at 18:37