A note on the vertizontal curvature of fat bundles

TL;DR

This paper investigates the curvature properties of fat principal bundles and Riemannian foliations, providing rigidity results under certain curvature constraints and addressing a question posed by W. Ziller about modifying metrics to achieve uniform vertical curvature.

Contribution

It establishes a rigidity theorem for fat Riemannian foliations with bounded holonomy and specific curvature conditions, extending Ziller's question to compact fiber bundles with certain geometric properties.

Findings

Rigidity result for fat Riemannian foliations with bounded holonomy

Conditions under which vertical curvatures can be uniformly set to 1

Addresses Ziller's question for compact fiber bundles with curvature constraints

Abstract

In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits are totally geodesic, can one modify $h$ to render all vertical curvatures equal to $1$? In this note, we establish a rigidity result for fat Riemannian foliations with bounded holonomy and a specific curvature constraint. Our result addresses Ziller's question for fat fiber bundles with compact structure groups, considering connected compact total spaces under a curvature constraint that holds on various examples, such as locally symmetric spaces. Additionally, we assume that all vertizontal curvatures coincide at a point.