Cohomogeneity one disk bundles with normal homogeneous collars

TL;DR

This paper classifies cohomogeneity one disk bundles with normal homogeneous collars that admit nonnegative curvature, focusing on ranks 6 and 8 completely and providing partial results for rank 3.

Contribution

It provides a complete classification of such bundles for ranks 6 and 8, and partial classification for rank 3, advancing understanding of geometric structures with nonnegative curvature.

Findings

Ranks 2, 3, 4, 6, 8 are necessary for nontrivial bundles with the specified properties.

Complete classification achieved for ranks 6 and 8.

Partial classification provided for rank 3.

Abstract

We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in $\{2,3,4,6,8\}$. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.