17 v 1 2 7 O ct 1 99 6 TWISTOR BUNDLES , EINSTEIN EQUATIONS AND REAL STRUCTURES ∗

We consider S 2 bundles P and P ′ of totally null planes of maximal dimension over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP ′ of P and P ′ is found to be appropriate for the encoding of both the selfdual 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 PP ′. 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 PP ′ 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 PP ′ .