Almost Kähler-Einstein Structures on 8-Dimensional Walker Manifolds

We study almost Kähler-Einstein structures on 8-dimensional Walker manifolds, i.e., pseudo-Riemannian 8-manifolds admitting a field of parallel null 4-planes, whence the metric is of neutral signature. We construct on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Bbb R}^8$\end{document} explicit almost Kähler-Einstein Walker metrics which are not Kähler. An appropriate restriction induces examples of such metrics on the 8-torus, thereby producing a counterexample to Goldberg’s conjecture in the case of neutral signature.