Bundle type sub-Riemannian structures on holonomy bundles

In this paper, combining the Rashevsky-Chow-Sussmann (orbit) theorem with the Ambrose-Singer theorem, we introduce the notion of controllable principal connections on principal G-bundles. Using this concept, under a mild assumption of compactness, we estimate the Gromov-Hausdorff distance between principal G-bundles and certain reductive homogeneous G-spaces. In addition, we prove that every reduction of the structure group G to a closed connected subgroup gives rise to a sequence of Riemannian metrics on the total space for which the underlying sequence of metric spaces converges, in the Gromov-Housdorff topology, to a normal reductive homogeneous G-space. This last finding allows one to detect the presence of certain reductive homogeneous G-spaces in the Gromov-Housdorff closure of the moduli space of Riemannian metrics of the total space of the bundle through topological invariants provided by obstruction theory.