Proofs of Conjectures about singular riemannian foliations

Abstract

We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section) which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem inspired by a remark of Palais and Terng and a work of Szenthe about polar actions. We also study the singular holonomy of a singular riemannian foliation with sections (s.r.f.s for short) and in particular the transverse orbit of the closure of each leaf. Furthermore we prove that the closure of the leaves of a s.r.f.s. on M form a partition of M which is a singular riemannian foliation. This result proves partially a conjecture of Molino.