Hyperheavenly spaces and their application in Walker and para-K\"ahler geometries: Part I

TL;DR

This paper explores special classes of spaces with null string congruences, focusing on Walker and para-K"ahler geometries, providing classification criteria and examples of these spaces.

Contribution

It introduces new classification criteria for Walker and para-K"ahler spaces based on null string congruences and presents explicit examples of such geometries.

Findings

Properties of intersections of null string congruences are used for classification.

Spaces in the two-sided Walker and para-K"ahler classes are characterized.

Examples of para-K"ahler and para-K"ahler-Einstein spaces are provided.

Abstract

Spaces equipped with congruences of null strings are considered. A special attention is paid to the spaces which belong to the two-sided Walker class and para-K\"ahler class. Properties of an intersection of self-dual and anti-self-dual congruences of null strings are used as an additional criterion for a classification of such spaces. Finally, a few examples of para-K\"ahler and para-K\"ahler-Einstein spaces are presented.