Twistor Bundles, Einstein Equations and Real Structures

TL;DR

This paper introduces a geometric framework using sphere bundles and their fiber products to encode self-dual and Einstein-Weyl equations for 4-dimensional metrics, unifying various signatures and linking to almost hermitian structures.

Contribution

It develops a unified geometric approach to encode Einstein and self-dual equations via properties of bundles and structures on their fiber product, applicable to all metric signatures.

Findings

Encodes Einstein and self-dual equations using geometrical objects on fiber product of bundles.

Unifies results across different metric signatures, including Riemannian and complex cases.

Links Einstein equations to integrability conditions of almost hermitian structures.

Abstract

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'.