Foliate partial holomorphic structures on principal bundles

The foliate partial holomorphic (f.p.h.) pseudogroup is the pseudogroup of the local diffeomorphisms of ℝ^m which preserve a distribution of the form span {Mathematical expression}, where y^a are real coordinates, z^α are complex coordinates, and ℝ^m may have also some other real coordinates. The f.p.h. structures on manifolds are described geometrically by the Nirenberg-Frobenius theorem [N], and occur in many interesting situations [R], [FW], [DK1], [V3], etc. The present paper discusses f.p.h. structures on principal bundles, and associates with such structures adapted connections, and forms with values in an associated bundle of Lie algebras.