The transverse geometry of $G$-manifolds and Riemannian foliations

Abstract

Given a compact Riemannian manifold on which a compact Lie group acts by isometries, it is shown that there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension of the orbit may be different from the dimension of the leaf closure. Conversely, given a Riemannian foliation, there is a metric on the basic manifold (an $O(q)$-manifold associated to the foliation) such that the leaf closure space is isometric to the $O(q)$-orbit space of the basic manifold via an isometry that preserves the volume of the leaf closures of maximal dimension. Thus, the orbit space of any Riemannian G-manifold is isometric to the orbit space of a Riemannian $O(q)$-manifold via an isometry that preserves the volumes of orbits of maximal dimension. Consequently, the spectrum of the Laplacian restricted to invariant functions on any $G$-manifold may be identified with the spectrum of the Laplacian restricted to invariant functions on a Riemannian $O(q)$-manifold. Other similar results concerning the spectrum of differential operators on sections of vector bundles over Riemannian foliations and $G$-manifolds are discussed.