On Ricci tensor of focal submanifolds of isoparametric hypersurfaces
type: Item
instance of: scholarly article
title: On Ricci tensor of focal submanifolds of isoparametric hypersurfaces
zbMATH Open document ID: 1360.53064
published in: Science China. Mathematics
publication date: 2015-09-25T00:00:00Z
review text: 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.
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 6485628
zbMATH Keywords: \(\mathcal{A}\)-manifold
zbMATH Keywords: \(\mathcal{B}\)-manifold
zbMATH Keywords: isoparametric hypersurface
zbMATH Keywords: focal submanifold
author: Qichao Li
author: Wenjiao Yan
MaRDI profile type: Publication
full work available at URL: s11425-014-4928-6
OpenAlex ID: W2253724632
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DOI: 10.1007/S11425-014-4928-6
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description: scientific article; zbMATH DE number 6485628