Generalizations of isoparametric foliations

Abstract

Isoparametric submanifolds and hypersurfaces in space forms are geometric objects that have been studied since E. Cartan. Another important class of geometric objects is the orbits of polar actions on a Riemannian manifold, e.g., the orbits of the adjoint action of a compact Lie group on itself. These two classes of submanifolds share some common properties. For example, they are leaves of singular Riemannian foliations with section (s.r.f.s. for short). A singular foliation on a complete Riemannian manifold is said to be Riemannian if every geodesic that is perpendicular at one point to a given leaf remains perpendicular to every leaf it meets. Moreover, the singular foliation admits sections if, for each regular point p, there is a totally geodesic complete immersed submanifold through p that meets each leaf orthogonally and whose dimension is the codimension of the regular leaves. The purpose of this paper is to review some results of the theory of s.r.f.s., introduced in [1] and developed in [2],[3] and [4]. This paper is organized as follows. In Section 2 we recall the definitions of isoparametric submanifolds, polar actions and equifocal submanifolds (the last one introduced by Terng and