Bundle type sub-Riemannian structures on holonomy bundles

TL;DR

This paper introduces controllable principal connections on principal G-bundles, estimates Gromov-Hausdorff distances, and shows how reductions of structure groups lead to convergence to reductive homogeneous G-spaces, revealing geometric structures via topological invariants.

Contribution

It combines classical theorems to define controllable principal connections and establishes convergence results linking bundle reductions to homogeneous spaces.

Findings

Estimates Gromov-Hausdorff distance between bundles and homogeneous spaces.

Shows reductions induce convergent sequences of Riemannian metrics.

Detects reductive homogeneous spaces in metric space closures using topological invariants.

Abstract

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.