A new look at condition A
type: Item
instance of: scholarly article
title: A new look at condition A
zbMATH Open document ID: 1246.53078
author: Quo-Shin Chi
published in: Osaka Journal of Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.0377
full work available at URL: https://projecteuclid.org/euclid.ojm/1332337242
review text: An isoparametric hypersurface in \(S^n\) can be obtained as the intersection of \(S^n\) with the level set of a homogeneous polynomial on \(\mathbb {R}^{n+1}\), of degree \(g=1\), \(2\), \(3\), \(4\) or \(6\). These isoparametric hypersurfaces have \(g\) principal curvatures, where only \(2\) different multiplicities \(m_i\), \(i=1,2\), of the principal curvatures occur. For the construction of inhomogeneous isoparametric hypersurfaces with \(4\) different principal curvatures, \textit{H. Ozeki} and \textit{M. Takeuchi} [Tohoku Math. J., II. Ser. 27, 515--559 (1975; Zbl 0359.53011)] imposed conditions (``A'' and ``B'') on the focal submanifolds of the isoparametric hypersurface, the first of which appeared to be crucial in subsequent work. In particular, \textit{J. Dorfmeister} and \textit{E. Neher} [Osaka J. Math. 20, 145--175 (1983; Zbl 0543.53046)] showed that the first of these conditions suffices to characterize a particular class of isoparametric hypersurfaces [\textit{D. Ferus}, \textit{H. Karcher} and \textit{H.-F. Münzner}, Math. Z. 177, 479--502 (1981; Zbl 0443.53037)]. The author provides a simpler and more geometric proof of this result, aiming to shed light on the geometric meaning of the condition for the construction of classes of examples, paying particular attention to the exceptional multiplicity pairs \((3,4)\) and \((7,8)\). In particular, the author analyses a construction of isoparametric hypersurfaces using octonians in order to provide a better understanding of the relation between the algebraic and geometric structures involved.
reviewed by: Udo Hertrich-Jeromin
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 6029031
zbMATH Keywords: isoparametric hypersurface
zbMATH Keywords: inhomogeneous hypersurface
zbMATH Keywords: octonians
zbMATH Keywords: Clifford algebra
MaRDI profile type: Publication
arXiv ID: 0907.0377
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description: scientific article; zbMATH DE number 6029031