A spinor approach to Walker geometry

Abstract

A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker geometry, which we exploit to draw together several themes in recent explicit studies of Walker geometry and in other work of Dunajski (2002) and Plebanski (1975) in which Walker geometry is implicit. In addition to studying local Walker geometry, we address a global question raised by the use of spinors.

41 pages. Typos which persisted into published version corrected, notably at (2.15)

References in corpus (1)

• Anti-self-dual four-manifolds with a parallel real spinor

Cited by in corpus (8)

• The geometry of modified Riemannian extensions
• Spin coefficients for four-dimensional neutral metrics, and null geometry
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• Algebraically Special, Real Alpha Geometries
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• 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