Examples of self-dual, Einstein metrics of $(2,2)$-signature

Abstract

In this paper we use para-quaternionic reduction to construct a family of examples of self-dual Einstain metrics of neutral signature, which are not Ricci flat, nor locally homogenous. The curvature of these manifolds is studied in detail and these examples are compared with the orbifolds $\mathcal O_{p,q}(1)$ given by Galicki and Lawson. Particular attention is given to the sign and to the pinching of the sectional curvatures.