Sasaki but not Einstein - MathOverflow

Sasaki but not Einstein

2010-11-19T15:18:44Z

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

Thank you and best regards!

Answer by Anirbit for Sasaki but not Einstein

2010-11-19T15:26:21Z

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.

Answer by José Figueroa-O'Farrill for Sasaki but not Einstein

2010-11-19T18:26:58Z

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.

Answer by Tarun Chitra for Sasaki but not Einstein

2011-04-05T20:03:39Z

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