Product twistor spaces and Weyl geometry

Abstract

Motivated by generalized geometry (\`a la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle ${\mathcal Z}\times {\mathcal Z}\to M$, where ${\mathcal Z}$ is the twistor space of a Riemannian 4-manifold $M$ endowed with a metric connection $D$ with skew-symmetric torsion. These structures are defined by means of the connection $D$ and four (K\"ahler) complex structures on the fibres of this bundle. Their integrability conditions are interpreted in terms of Weyl geometry and this is used to supply examples satisfying the conditions.