On Eta-einstein Sasakian Geometry

A compact quasi-regular Sasakian manifold M is foliated by onedimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold Z. In the case when the transverse space Z is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.