Abstract
By a real αβ-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of α-planes and an integrable distribution of β-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 α- and β-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 αβ-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.
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