Waning in principal bundles

TL;DR

This paper investigates the limits of tangent bundles of principal bundles under fiber rescaling, revealing a new fiber bundle structure influenced by the holonomy of the metric.

Contribution

It characterizes the Gromov-Hausdorff limits of tangent bundles of principal bundles with rescaled fibers, linking the limit fibers to the holonomy of the metric.

Findings

Limit tangent bundles form a locally trivial fiber bundle.

Limit fibers are determined by the holonomy of the bi-invariant metric.

Berger 3-spheres exemplify fibers remaining of dimension 3.

Abstract

Let $P\to M$ be a principal bundle. Consider a sequence of metrics on $P$ obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to be a locally trivial fiber bundle containing the tangent space to the base as a subbundle in a natural way. Berger 3-spheres provide an example where the limit fibers are still of dimension 3. The fibers are shown to be entirely determined by the riemannian holonomy of the chosen bi-invariant metric.