On some examples of para-Hermite and para-Kähler Einstein spaces with $\Lambda \ne 0$

Abstract

Spaces equipped with two complementary (distinct) congruences of self-dual null strings and at least one congruence of anti-self-dual null strings are considered. It is shown that if such spaces are Einsteinian then the vacuum Einstein field equations can be reduced to a single nonlinear partial differential equation of the second order. Different forms of these equations are analyzed. Finally, several new explicit metrics of the para-Hermite and para-Kähler Einstein spaces with $\Lambda \ne 0$ are presented. Some relation of that metrics to a modern approach to mechanical issues is discussed.

References in corpus (1)

• The dancing metric, ${\mathrm{G}}_2$-symmetry and projective rolling

Cited by in corpus (6)

• The Geometry of Almost Einstein (2,3,5) Distributions
• Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions
• On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces
• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part I
• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: part II
• On Walker and para-Hermite Einstein spaces