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On \(\eta\)-Einstein Sasakian geometry. (English) Zbl 1103.53022

Summary: A compact quasi-regular Sasakian manifold \(M\) is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold \(\mathcal Z\). In the case when the transverse space \(\mathcal Z\) is also Einstein the corresponding Sasakian manifold \(M\) is said to be Sasakian \(\eta\)-Einstein. In this article we study \(\eta\)-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 \(\eta\)-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.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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