Building New Einstein Spaces by Deforming Symmetric Einstein Spaces

Abstract

We introduce an approach to determine new pseudo-Riemannian Einstein spaces by deforming symmetric pseudo-Riemannian Einstein spaces. The metrics of the spaces we will deform are associated with complex hyperbolic spaces and are (para-)Kähler manifolds. That is, they admit a parallel field of skew-symmetric endomorphisms, called a (para-)complex structure K, such that K2 = −Id (K2 = Id), where Id denotes the identity endomorphism. These metrics also belong to the class of \(\mathcal {I}\)-degenerate metrics and can be deformed to produce new metrics with the same set of scalar polynomial curvature invariants. Using the metric deformations of \(\mathcal {I}\)-degenerate metrics which preserve the set of scalar polynomial curvature invariants we explicitly construct examples of new (para-)Kähler or almost-(para-)Kähler Einstein spaces.