Integrable almost complex structures in principal bundles and holomorphic curves

Consider a K-principal fibre bundle P π −→ M and fix a connection A on P . Suppose an ad(P )-valued 1-form α defines a bundle isomorphism TM ∼= −→ ad(P ). We will associate to any such triple (π,A, α) a K-invariant almost complex structure Jα on P . In the first section we give explicitely the integrability condition of Jα. This condition is equivalent to a system on non-linear differential equations of the pair (A, α). In the second section we make the further assumption that K is a compact Lie group and we fix an ad-invariant inner product on its Lie algebra k. This induces the structure of an Euclidean vector bundle on ad(P ) and therefore we get an induced metric gα on the base M . We show that the pull-back of the linear connection ∇A on ad(P ) via α is the Levi-Civita-Connection of (M, gα) if and only if the first of the of our integrability equations is satisfied. The second integrability equation implies that the sectional curvature of gα is non-positive in general. Moreover in the case K = SU(2) or K = SO(3) the metric gα will be hyperbolic. Therefore we get a relation between our integrability problem and hyperbolic geometry. In particular the bundle of orthonormal oriented frames of every hyperbolic oriented 3-manifold is naturally a complex manifold. In the third section we get an equivalence between the complete integrability of Jαholomorphic vector fields and the geodesic completeness of (M, gα), if the first integrability equation is satisfied. In the case where (M, gα) is complete and Jα is integrable we use this equivalence in the fourth section to obtain a holomorphic action of the complexified Lie group G = K on the complex manifold P . This action is locally free and transitive