A characterization of isoparametric hypersurfaces of Clifford type

Stefan Immervoll

Abstract: Let $M$ be an isoparametric hypersurface with four distinct principal curvatures in the unit sphere $\bf S \subseteq {\bf R}^{2l}$ with focal manifolds $M_+$ and $M_-$. Let $\cal U$ be a vector space of symmetric $(2l\times 2l)$-matrices such that each matrix in $\cal U\backslash \set{0}$ is regular, and assume that $M_+$ is the intersection of $\bf S$ with the quadrics ${x \in \bf R^{2l}}{\mid {x,Ax}=0}$, $A\in \cal U$. Then $\cal U$ is generated by a Clifford system and $M$ is an isoparametric hypersurface of Clifford type provided that $\dim M_+ \geq \dim M_-$. The proof of this theorem is based on properties of quadratic forms vanishing on $M_+$ and on a structure theorem for isoparametric triple systems, which we prove in this paper.

Keywords: isoparametric hypersurface, triple system, Peirce decomposition, Clifford system/sphere

Classification (MSC2000): 53C40; 17A40, 15A66

Full text of the article:

• Compressed PostScript file (64 kilobytes)
• PDF file (152 kilobytes)