Real AlphaBeta-geometries and Walker geometry

By a real αβαβ-geometry we mean a four-dimensional manifold MM equipped with a neutral metric hh such that (M,h)(M,h) admits both an integrable distribution of αα-planes and an integrable distribution of ββ-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 αα- and ββ-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 αβαβ-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.