Collapsing regular Riemannian foliations with flat leaves

Abstract

In this manuscript, we present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that in the case when the manifold is compact and simply connected, the foliation is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions.