Integrable almost complex structures in principal bundles and holomorphic curves

Abstract

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.