A new look at condition A
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.
scientific article; zbMATH DE number 6029031