3-Sasakian Manifolds

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Abstract
This chapter introduces a special case of Sasaki-Einstein manifolds which have a somewhat richer structure and occur only in dimensions 4m + 3. As any Sasakian manifold they are foliated by 1-dimensional leaves but now the transverse space has additional properties. In addition to a Kähler-Einstein metric, it has a complex contact structure making it into an orbifold twistor space. Furthermore, 3-Sasakian manifolds fiber as Konishi orbibundles over quaternionic Kähler orbifolds. Most of the 3-Sasakian metrics considered are obtained via symmetry reduction similar to hyper Kähler and quaternionic Kähler reduction. Indeed the three reductions are all related, so the manifolds and the metrics obtained are quite often explicit and can be studied as quotients. A detailed study and a classification of toric 3-Sasakian manifolds is presented. Some non-toric examples are also constructed as quotients as well as Konishi orbibundles over Hitchin-Tod self-dual Einstein orbifolds.

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