  We define and study isoparametric submanifolds of general ambient spaces and
of arbitrary codimension. In particular we study their behaviour with respect
to Riemannian submersions and their lift into a Hilbert space. These results
are used to prove a Chevalley type restriction theorem which relates by
restriction eigenfunctions of the Laplacian on a compact Riemannian manifold
which contains an isoparametric submanifold with flat sections to
eigenfunctions of the Laplacian of a section. A simple example of such an
isoparametric foliation is given by the conjugacy classes of a compact Lie
group and in that case the restriction theorem is a (well known) fundamental
result in representation theory. As an application of the restriction theorem
we show that isoparametric submanifolds with flat sections in compact symmetric
spaces are level sets of eigenfunctions of the Laplacian and are hence related
to representation theory. In addition we also get the following results.
Isoparametric submanifolds in Hilbert space have globally flat normal bundle,
and a general result about Riemannian submersions which says that focal
distances do not change if a submanifold of the base is lifted to the total
space.
