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