A characterization of isoparametric hypersurfaces of Clifford type (Q1878971)

Let \(M\) be an isoparametric hypersurface in the unit sphere \(\mathbb S\) of Euclidean space \(\mathbb{R}^{2l}\) . Suppose that \(M\) has four distinct principal curvatures and that the two focal manifolds \(M_{+}\) and \(M_{-}\) satisfy \(\dim\,M_+\geq \dim\,M_-\). Let \(U\) be a subspace of the space \(\mathcal{S}_{2l}\) of symmetric \((2l\times 2l)\)-matrices with the property that each non-zero matrix in \(U\) is regular. If \(U\) is generated by a Clifford system and \(M\) is an isoparametric hypersurface of Clifford type associated with the system, then, by a theorem of Ferus, Karcher and Münzner, \(M_{+}=\{x\in \mathbf{S}\mid \langle x, Ax\rangle=0\) for every \(A\in U\}\). The author of the paper under review proves the converse. The author does it by studying the properties of quadratic forms vanishing on a focal manifold and by proving a structure theorem for isoparameric triple systems.