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.We call a pseudo-Riemannian 4-manifold, which admits a field of parallel null 2-planes, a Walker 4-manifold. A pseudo-Riemannian metric of a Walker 4-manifold is necessarily of neutral signature, and it admits an orthogonal almost complex structure. We show that such a Walker 4-manifold can carry various structures with respect to a certain kind of almost complex structure, e.g., symplectic structures,...
. 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 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...
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