Convexity theorem for isoparametric submanifolds
type: Item
instance of: scholarly article
title: Convexity theorem for isoparametric submanifolds
zbMATH Open document ID: 0615.53048
DOI: 10.1007/BF01390326
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143378
review text: In a previous publication [J. Differ. Geom. 21, 79-107 (1985; see the preceding review)], the author has associated a Weyl group with each isoparametric submanifold \(M^ n\) in \({\mathbb{R}}^{n+k}\). The main theorem of this paper - generalizing results of Schur and Kostant - is: Let \(M^ n\) be a full isoparametric submanifold of \({\mathbb{R}}^{n+k}\), W its associated Weyl group, x in \(M^ n\), and P the orthogonal projection of \({\mathbb{R}}^{n+k}\) onto the normal plane \(x+\gamma_ x\) through x. Let \(M^ n_ q\) be the parallel manifold of \(M^ n\) through a point q in \(x+\gamma_ x.\) Then \(P(M^ n_ q)=convex\) hull (W.q). The proof of this theorem uses previous results of the author on the height function of M and ideas of Atiyah, Guillemin and Sternberg.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 3996574
zbMATH Keywords: convexity theorem
zbMATH Keywords: Coxeter group
zbMATH Keywords: Weyl group
zbMATH Keywords: isoparametric submanifold
zbMATH Keywords: parallel manifold
author: Chuu-Lian Terng
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3996574