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

We prove that every contact metric (κ,μ)(κ,μ)-space admits a canonical ηη-Einstein Sasakian or ηη-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κκ and μμ 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)(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)(κ,μ)-spaces related to the existence of Einstein–Weyl and Lorentzian–Sasaki–Einstein structures.