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.