Twisted submersions in nonnegative sectional curvature

TL;DR

This paper investigates the properties of dual foliations in nonnegatively curved Riemannian manifolds, demonstrating that the phenomenon of a single leaf in the dual foliation often persists even without positive curvature.

Contribution

It extends Wilking's results by showing that the single leaf property of dual foliations holds in broader nonnegative curvature settings without requiring strict positivity.

Findings

Dual foliation often contains a single leaf in nonnegative curvature.

The single leaf property holds regardless of specific metric choices.

The phenomenon occurs even without strict positive curvature conditions.

Abstract

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for nonnegatively curved Riemannian submersions even without the strict positive curvature condition, and irrespective of the particular metric.