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

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

2011-09-28

arxiv.org/abs/1109.6248v1

Mathematics

Differential Geometry

Scientific paper

We prove that any non-Sasakian 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 topological and geometrical properties of (\kappa,\mu)-spaces related to the existence of Eistein-Weil and Lorentzian Sasakian Einstein structures.

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