Product twistor spaces and Weyl geometry

TL;DR

This paper explores the integrability conditions of certain almost complex structures on the product of twistor spaces over a 4-manifold with a metric connection having skew-symmetric torsion, linking them to Weyl geometry.

Contribution

It introduces a novel interpretation of integrability conditions for almost complex structures on twistor space products via Weyl geometry, providing new examples.

Findings

Integrability conditions are characterized in terms of Weyl geometry.

Examples satisfying the integrability conditions are constructed.

Connections with skew-symmetric torsion influence complex structure integrability.

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.