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

We prove that every contact metric (κ, μ) -space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κ and μ 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 (κ, μ) -structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of (κ, μ) -spaces related to the existence of Einstein-Weyl and Lorentzian-Sasaki-Einstein structures.