An example of algebraically general para-Kähler Einstein space
Adam Chudecki, Michał Dobrski
TL;DR
This work constructs and analyzes algebraically general para-Kähler Einstein spaces with a 3D algebra of infinitesimal symmetries, showing that the presence of a 2D trivial subalgebra reduces the vacuum Einstein equations with cosmological constant to a first-order Abel equation and presenting a fully explicit example. The approach relies on the second Plebański formalism and hyperheavenly spaces to formulate a single key function $\Theta$ whose derivatives encode the curvature; symmetry reduction via Killing vectors yields tractable ODEs or Abel equations in multiple cases. The results demonstrate that algebraically general para-Kähler Einstein spaces exist and can realize all Petrov-Penrose types for the ASD Weyl tensor, thereby answering a longstanding question and enriching the landscape of exact solutions. The explicit example and discriminant analyses provide a concrete foundation for further classification of gpKE-spaces under larger symmetry algebras and in complex versus real settings.
Abstract
Algebraically general para-Kähler Einstein spaces equipped with 3D algebras of infinitesimal symmetries are considered. It is shown that if the algebra contains 2D trivial subalgebra then vacuum Einstein field equations with cosmological constant can be reduced to a single, first-order differential equation. One of the cases is solved explicitly. Hence, a first example of an algebraically general para-Kähler Einstein space is given.