Twistor bundles, Einstein equations and real structures

Abstract

We consider S2 bundles \U0001d4ab and \U0001d4ab' of totally null planes of maximal dimension and opposite self-duality over a four-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product \U0001d4ab\U0001d4ab' of \U0001d4ab and \U0001d4ab' is found to be appropriate for the encoding of both the self-dual and the Einstein - Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on \U0001d4ab\U0001d4ab'. The formulation is suitable for complex-valued metrics and unifies results for all three possible real signatures. In the purely Riemannian positive-definite case it implies the existence of a natural almost Hermitian structure on \U0001d4ab\U0001d4ab' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost Hermitian structure on \U0001d4ab\U0001d4ab'.