Isoparametric submanifolds and a Chevalley-type restriction theorem

TL;DR

This paper introduces a new class of isoparametric submanifolds in various ambient spaces, establishes a Chevalley-type restriction theorem linking eigenfunctions on manifolds and sections, and explores their applications in symmetric spaces and representation theory.

Contribution

It defines isoparametric submanifolds in general spaces, proves a Chevalley-type restriction theorem, and applies these results to symmetric spaces and representation theory.

Findings

Isoparametric submanifolds with flat sections have flat normal bundles.

Restriction theorem relates eigenfunctions on manifolds and sections.

Focal distances are invariant under Riemannian submersions.

Abstract

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.