Twistor Bundles, Einstein Equations and Real Structures

Details Twistor Bundles, Einstein Equations and Real Structures Twistor Bundles, Einstein Equations and Real Structures

1996-10-27

arxiv.org/abs/dg-ga/9610017v1

Mathematics

Differential Geometry

Paper accepted in Classical and Quantum Gravity, Special issue in honour of Professor Andrzej Trautman

Scientific paper

10.1088/0264-9381/14/1A/021

We consider sphere bundles P and P' of totally null planes of maximal dimension and opposite self-duality 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 both complex- and real-valued metrics. It 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'.

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