Real AlphaBeta-Geometries

Abstract

By a real alphabeta-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 consider real αβ-geometries for which the corresponding spinors are both multiple Weyl principal spinors.

14 pages

References in corpus (3)

• A spinor approach to Walker geometry
• Spin coefficients for four-dimensional neutral metrics, and null geometry
• Algebraically Special, Real Alpha Geometries

Cited by in corpus (6)

• On some examples of para-Hermite and para-Kähler Einstein spaces with $\Lambda \ne 0$
• Null Killing vectors and geometry of null strings in Einstein spaces
• On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces
• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part I
• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: part II
• On Walker and para-Hermite Einstein spaces