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A Spinor Approach to Walker Geometry

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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 [11] and Plebañski [30] in which Walker geometry is implicit. In addition to studying local Walker geometry, we address a global question raised by the use of spinors.

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Authors and Affiliations

  1. 4 Mack Place, Monroe, NY, 10950, USA

    Peter R. Law

  2. Section of Mathematics, School of Engineering, University of Shiga Prefecture, Hikone, 522-8533, Japan

    Yasuo Matsushita

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  1. Peter R. Law
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  2. Yasuo Matsushita
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Correspondence to Peter R. Law.

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Communicated by G.W. Gibbons

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Law, P.R., Matsushita, Y. A Spinor Approach to Walker Geometry. Commun. Math. Phys. 282, 577–623 (2008). https://doi.org/10.1007/s00220-008-0561-y

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  • DOI: https://doi.org/10.1007/s00220-008-0561-y

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