## Sasaki but not Einstein

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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 2010 at 2:24

## 3 Answers

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 2010 at 8:38

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This is not an answer to your question, but rather an extension of José Figueroa-O'Farrill's answer. Roughly a year ago, a quick Google search led me to a preprint of Boyer & Galicki's wonderful book; it can be found here

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I am not very sure of everything here but I wonder if the example given in Appendix A on Page 21 of this paper by one of my professors meets your criteria.

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Are you sure that is the right paper? Can you give me the construction of the manifold? Best regards! – Differentialgeometer Nov 19 2010 at 17:39

1
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 2010 at 18:37
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