Isoparametric submanifolds and a Chevalley-type restriction theorem

By E. Heintze, Xiaobo Liu, C. Olmos. 2000

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We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Rie- mannian 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 Lapla- cian 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d64adc46b2dd2823d9271097fd77cadf2a49a750

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On Constructing Biharmonic Maps and Metrics (2003, background): “As a consequence of [ 10 ], this result holds more generally for functions defined on a simply connected symmetric space of compact type.”
Singular Riemannian foliations with sections (2003, background): “Isoparametric submanifolds in space forms and equifocal submanifolds with flat sections in simply connected symmetric spaces of compact type can always be described as regular level sets of transnormal analytic maps, see R.Palais and C.L.Terng [7] and E. Heintzte, X.Liu and C.Olmos [5].”