Singular Riemannian foliations with sections

Abstract

A singular foliation on a complete riemannian manifold is said to be riemannian if every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. In this paper, we study singular riemannian foliations with sections. A section is a totally geodesic complete immersed submanifold that meets each leaf orthogonally and whose dimension is the codimension of the regular leaves.

We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidean space. This result provides local information about the singular foliation and in particular about the singular stratification of the foliation. It allows us to describe the plaques of the foliation as level sets of a transnormal map (a generalization of an isoparametric map). We also prove that the regular leaves of a singular riemannian foliation with sections are locally equifocal. We use this property to define a singular holonomy. Then we establish some results about this singular holonomy and illustrate them with a couple of examples.