Algebraically special, real alpha-geometries

摘要

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of α-planes (algebraically special real α- geometry). In particular, we determine the behaviour of Walker geometry under conformal rescaling and provide a derivation of the hyperheavenly equation from conformal rescaling formulae.