On Ricci tensor of focal submanifolds of isoparametric hypersurfaces
In [Geom. Dedicata 7, 259--280 (1978; Zbl 0378.53018)], \textit{A. Gray} defined two classes of Riemannian manifolds called \(\mathcal{A}\)-manifolds and \(\mathcal{B}\)-manifolds. An \textit{\(\mathcal{A}\)-manifold} is a Riemannian manifold with cyclic-parallel Ricci curvature tensor, i.e., \[ (\nabla_{X_1} \mathrm{Ric})(X_2,X_3)+(\nabla_{X_2} \mathrm{Ric})(X_3,X_1)+(\nabla_{X_3} \mathrm{Ric})(X_1,X_2)=0. \] A \textit{\(\mathcal{B}\)-manifold} is a Riemannian manifold with Ricci curvature tensor satisfying the condition \((\nabla_{X_1} \mathrm{Ric})(X_2,X_3)=(\nabla_{X_2} \mathrm{Ric})(X_1,X_3)\). A Riemannian manifold is Ricci parallel if it is at the same time an \(\mathcal{A}\)-manifold and a \(\mathcal{B}\)-manifold. In [Adv. Math. 285, 1970--2000 (2015; Zbl 1327.53060)], \textit{Z. Tang} and \textit{W. Yan} used a case-by-case analysis to show that all focal submanifolds of any isoparametric hypersurface in the unit sphere with \(4\) distinct principal curvatures are \(\mathcal{A}\)-manifolds, except possibly for one case with multiplicities \((m_1,m_2)=(7,8)\). Here the authors give a short proof of the general theorem, including the remaining case. They also show that for each family of isoparametric hypersurfaces in the unit sphere with \(6\) distinct principal curvatures, only one focal submanifold is an \(\mathcal{A}\)-manifold and none of them is a \(\mathcal{B}\)-manifold.
scientific article; zbMATH DE number 6485628
scientific article