Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics Charles P. Boyer and Krzysztof Galicki

This review article has grown out of notes for the three lectures the second author presented during the XXIV-th Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years while, at the same time, we summarize all the known results. We do not give any technical details other than what is necessary for the clarity of the exposition. In conclusion we would like to argue that Sasakian geometry has emerged as one of the most powerful tools of constructing and proving existence of special Riemannian metrics, such as Einstein metrics or metrics of positive Ricci curvature, on a wide range of odd-dimensional manifolds. The key geometric object in the theory is that of a contact structure (hence, only odd dimensions) together with a Riemannian metric naturally adapted to the contact form. Sasakian metrics in contact geometry are analogous to the Kähler metrics in the symplectic case. We begin with basic facts about contact and Sasakian manifolds after which we focus on exploring the fundamental relation between the Sasakian and the transverse Kähler geometry. In this context positive Sasakian, Sasakian-Einstein, and 3-Sasakian manifolds are introduced. In Section 3 we present all known constructions of 3-Sasakian manifolds. These come as V-bundles over compact quaternion Kähler orbifolds and large families can be explicitly obtained using symmetry reduction. We also discuss Sasakian-Einstein manifolds which are not 3-Sasakian. Here there has been only one effective method of producing examples, namely by representing a Sasakian-Einstein manifold as the total space of an S Seifert bundle over a Kähler-Einstein orbifold. In the smooth case with a trivial orbifold structure, this construction goes back to Kobayashi [Kob56]. Any smooth Fano variety Z which admits a Kähler-Einstein metric can be used for the base of a unique simply connected circle bundle P which is Sasakian-Einstein. Any Sasakian-Einstein manifold obtained this way is automatically regular. It is clear that, in order to get non-regular examples of Sasakian-Einstein structures, one should replace the smooth Fano structure with a Fano orbifold. This was done in [BG00] where we generalized the Kobayashi construction to V-bundles over Fano orbifolds. However, at that time, with the exception of twistor spaces of known 3-Sasakian metrics, compact Fano orbifolds known to admit orbifold Kähler-Einstein metrics were rare. The first examples of non-regular Sasakian-Einstein manifolds which are not 3-Sasakian were obtained in [BG00]. There we observed that Sasakian-Einstein manifolds have the structure of a monoid under a certain “join” operation. A