Para-Kähler–Einstein structures on Walker 4-manifolds

Abstract

A 4-dimensional Walker manifold [Formula: see text] is a semi-Riemannian manifold [Formula: see text] of signature (++––) (or neutral), which admits a field of null 2-plane. The goal of this paper is to study certain almost paracomplex structures [Formula: see text] on 4-dimensional Walker manifolds. We discuss when these structures are integrable and when the para-Kähler forms are symplectic. We show that such a Walker 4-manifold can carry a class of indefinite para-Kähler–Einstein 4-manifolds, examples of indefinite para-Kähler 4-manifolds, and also almost indefinite para-Hermitian–Einstein 4-manifold. Finally, we give a counterexample for the almost para-Hemitian version of Goldberg conjecture.