Collapse of unit horizontal bundles equipped with a metric of Cheeger-Gromoll type

TL;DR

This paper investigates the geometric and convergence properties of unit horizontal bundles with Cheeger-Gromoll type metrics, including a collapse theorem related to warped Riemannian submersions.

Contribution

It provides new insights into the metric properties and Gromov-Hausdorff convergence behavior of unit horizontal bundles under Cheeger-Gromoll type metrics.

Findings

Analysis of metric properties of unit horizontal bundles

Gromov-Hausdorff collapse theorem for warped Riemannian submersions

Conditions under which collapse occurs

Abstract

We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse theorem for unit horizontal bundles associated with a sequence of warped Riemannian submersions.