Twistor bundles, Einstein equations and real structures

Details Twistor bundles, Einstein equations and real structures Twistor bundles, Einstein equations and real structures

Jan 1997

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997cqgra..14a.261n&link_type=abstract

Classical and Quantum Gravity, Volume 14, Issue 1A, pp. A261-A290 (1997).

Computer Science

Scientific paper

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

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