Sasaki-Einstein and paraSasaki-Einstein metrics from (κ, μ)-structures

We prove that every contact metric (kappa, mu)-space admits a canonical eta-Einstein Sasakian or eta-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of kappa and mu for which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (kappa, mu)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of (kappa, mu)-spaces related to the existence of Einstein-Weyl and Lorentzian-Sasaki-Einstein structures. (C) 2013 Elsevier B.V. All rights reserved.