On a certain class of para-Hermite Einstein spaces

TL;DR

This paper classifies a special class of complex para-Hermite Einstein spaces with degenerate anti-self-dual Weyl tensor, providing explicit metrics and analyzing their geometric properties.

Contribution

It introduces a classification of para-Hermite Einstein spaces with algebraically degenerate anti-self-dual Weyl tensor and derives explicit metric solutions.

Findings

Explicit metrics for the classified spaces are obtained.

The spaces are shown to have a non-parallel null congruence.

The anti-self-dual Weyl tensor is algebraically degenerate in these spaces.

Abstract

A special class of (complex) para-Hermite Einstein spaces is analyzed. It is well-known that the self-dual Weyl tensor in para-Hermite Einstein spaces is of the Petrov-Penrose type [D]. In what follows we assume that the anti-self-dual Weyl tensor is algebraically degenerate. It is equivalent to the existence of an anti-self-dual congruence of null strings which is assumed not to be parallely propagated. Hence, spaces analyzed here are not Walker spaces. A classification of such spaces is given and the explicit metrics are found.