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Isoparametric submanifolds and their Coxeter groups. (English) Zbl 0615.53047

This is a fundamental paper on the generalization of Cartan’s isoparametric hypersurfaces to higher codimension. For a different attempt see S. Carter and A. West [Proc. Lond. Math. Soc., III. Ser. 51, 520-542 (1985; Zbl 0587.53055)]. A submanifold of a space form (say of \({\mathbb{R}}^{n+k})\) is isoparametric, if its normal bundle is flat, and if the principal curvatures with respect to each parallel normal field are constant. We may then assume that the manifold M is full in \({\mathbb{R}}^{n+k}\), and \(M\subset S^{n+k-1}.\)
M is foliated by families of mutually orthogonal curvature leaves, which are round spheres in \({\mathbb{R}}^{n+k}\). At \(x\in M\) the (linear) reflexions of the affine normal space along the radial directions of the curvature leaves generate a finite Coxeter group W of rank k. W acts freely on M, where the above generators correspond to the antipodal maps on the curvature leaves. The author shows the existence of an associated ”Cartan polynomial” \({\mathbb{R}}^{n+k}\to {\mathbb{R}}^ k\) with M as a level. (The contents of this paper have been considerably extended by the subsequent [W. Y. Hsiang, R. S. Palais and the author, Proc. Natl. Acad. Sci. USA 82, 4863-4865 (1985; Zbl 0573.53033)].
Reviewer: D.Ferus

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C35 Differential geometry of symmetric spaces
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