Singular Riemannian Foliations

Abstract

The global geometry of Riemannian foliations that we depicted in the last chapter leads us naturally to consider “singular” Riemannian foliations. Indeed, if (M, F, g T ) is a Riemannian foliation on a compact connected manifold, then the partition \(\overline F\) of M by the leaf closures defines, outside the singular closures, a Riemannian foliation. Moreover, transverse to the leaves, this partition is locally defined by the orbits of a Lie algebra of Killing vector fields. This is a typical example of a singular Riemannian foliation.