Sasaki-Einstein and paraSasaki-Einstein metrics from (\kappa,\mu)-structures

TL;DR

This paper demonstrates that non-Sasakian contact metric (,)-spaces can be equipped with canonical -Einstein Sasakian or paraSasakian metrics, providing explicit curvature formulas and exploring their geometric and topological implications.

Contribution

It introduces a method to construct -Einstein Sasakian and paraSasakian metrics from (,)-spaces and characterizes conditions for their existence, extending the understanding of contact metric geometry.

Findings

Explicit curvature tensor expressions for the new metrics

Identification of ,) parameters for Einstein conditions

Application to tangent sphere bundles and Einstein-Weyl structures

Abstract

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-Weyl and Lorentzian Sasakian Einstein structures.