TWISTOR BUNDLES, EINSTEIN EQUATIONS AND REAL STRUCTURES ∗

We consider S2 bundles and ' 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 ' of and ' 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 '. 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 ' 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 '.