Twisted submersions in nonnegative sectional curvature

In 16], 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 Riemannian submersions from nonnegatively curved spaces even without the strict positive curvature assumption and irrespective of the particular metric.