A Spinor Approach to Walker Geometry

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.