On Walker and para-Hermite Einstein spaces

A special class of (complex) para-Hermite Einstein spaces is analyzed. For this class of spaces the self-dual Weyl tensor is type-[D] in the Petrov-Penrose classification. The anti-self-dual Weyl tensor is algebraically degenerate, equivalently, there exists an anti-self-dual congruence of null strings. It is assumed that this congruence is parallely propagated. Thus, the spaces are not only para-Hermite but also Walker. A classification of the spaces according to three criteria is given. Finally, explicit metrics of all admitted Petrov-Penrose types are found.

💡 Research Summary

This paper investigates a special class of four‑dimensional complex (or real neutral) Einstein spaces equipped with a para‑Hermitian structure and a Walker structure. The defining feature of the class is that the self‑dual (SD) part of the Weyl tensor is of Petrov‑Penrose type