Abstract: A "singular Riemannian foliation with sections" (SRFS) on a complete Riemannian manifold is a singular Riemannian foliation in the sense of Molino which admits a transversal complete immersed manifold that meets all the leaves and meets them always orthogonally. They were introduced by Boualem and then by Alexandrino as a simultaneous generalization of orbital foliations of polar actions of Lie groups, isoparametric foliations in simply-connected space forms, and foliations by parallel submanifolds of an equifocal submanifold with flat sections in a simply-connected compact symmetric space.

A smooth map from a complete Riemannian manifold to an Euclidean space is called "transnormal" if it is an integrable Riemannian submersion in a neighborhood of any regular level set. We prove that the leaves of a given SRFS coincide with the level sets of some transnormal map in the case in which the ambient manifold is simply-connected, the sections are flat, and the leaves are compact. This extends previous results due to Carter-West, Terng and Heintze-Liu-Olmos. (Joint work with M. Alexandrino.)