Real AlphaBeta-geometries and Walker geometry

By a real alpha beta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M, h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i.e., is a Walker geometry) and the three-dimensional distribution spanned by the alpha- and beta-distributions is integrable. The case when both distributions are parallel, which has been called two-sided Walker geometry, is obtained as a special case. We also study real alpha beta-geometries for which the corresponding spinors are both multiple Weyl principal spinors. All these results have natural analogues in the context of the hyperheavens of complex general relativity. (c) 2012 Elsevier B.V. All rights reserved.