Convexity theorem for isoparametric submanifolds
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.
scientific article; zbMATH DE number 3996574
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