Deformations of singular Riemannian foliations while controlling the curvature

In this talk we are going to present two collapsing procedures for singular Riemannian foliations, while maintaining some control on the sectional curvature and diameter of the ambient manifold. In the first one we consider a regular Riemannian foliation by flat tori on a compact manifold, and show that we can collapse the foliated manifold to the orbit space while preserving sectional curvature and diameter bounds. From this construction we obtain a rigidity result, proving that such a foliation on a simply-connected manifold is given by a smooth torus action.

We also present a deformation procedure for foliations induced by Lie groupoids, which extends the classical construction of Cheeger deformations. We also give an explicit description of the sectional curvature of the deformed metrics in the case of this generalized Cheeger deformation proceedure.