  We consider principal fibre bundles with a given connection and construct
almost complex structures on the total space if the adjoint bundle is
isomorphic to the tangent bundle of the base. We derive the integrability
condition. If the structure group is compact, then a choice of an ad-invariant
inner product on its Lie algebra gives naturally the structure of a Riemannian
manifold to the base. The integrability condition is then expressed in
geometric terms. In particular we get a relation to hyperbolic geometry if the
structure group is SU(2) or SO(3). The bundle of orthonormal frames of a
hyperbolic oriented 3-manifold is naturally a complex manifold. If the base is
geodesically complete and connected, then we can endow the total space with a
locally free transitive holomorphic action of the complexified structure group.
We then get some restrictions for holomorphic maps from Riemann surfaces to the
total space. If the pull-back of a canonical Lie algebra valued 1-form on the
total space is of scalar form, then the holomorphic map factorises through an
elliptic curve. If the induced map to the base is conformal, then the
associated holomorphic map with value in $P^2 \mathbb{C}$ factorises through a
smooth quadric.
