Integrable almost complex structures in principal bundles and holomorphic curves

We consider almost complex structures that arise naturally in a particular class of principal fibre bundles, where the choice of a connection can be used to determine equivariant isomorphisms between the vertical and horizontal tangent bundles of the total space. For instance, such data always exist on the frame bundle of a 3-manifold, but also in many other situations. We study the integrability condition to a complex structure, obtaining a system of gauge invariant coupled first order partial differential equations. This yields to a few correspondences between complex-geometric properties on the total space and metric properties on the base.