The \(^{*}\)-Ricci tensor for hypersurfaces in \(\mathbb{C}\mathbb{P}^n\) and \(\mathbb{C}\mathrm{H}^n\)
type: Item
instance of: scholarly article
title: The \(^{*}\)-Ricci tensor for hypersurfaces in \(\mathbb{C}\mathbb{P}^n\) and \(\mathbb{C}\mathrm{H}^n\)
zbMATH Open document ID: 1242.53062
author: Thomas A. Ivey
author: Patrick J. Ryan
published in: Tokyo Journal of Mathematics
publication date: 2012-04-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.0902
reviewed by: Weihuan Chen
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 6020931
zbMATH Keywords: \({}^\ast\)-Einstein hypersurface
zbMATH Keywords: Hopf hypersurface
zbMATH Keywords: pseudo-Ryan hypersurface
MaRDI profile type: Publication
OpenAlex ID: W2068473456
arXiv ID: 1011.0902
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DOI: 10.3836/TJM/1327931396
review text: The authors refine, clarify and extend some of \textit{T. Hamada}'s work see [Tokyo J. Math. 25, No. 2, 473--483 (2002; Zbl 1046.53037)], especially the classification of \({}^\ast\)-Einstein hypersurfaces in complex space forms. The result isNEWLINENEWLINETheorem 5. The \({}^\ast\)-Einstein Hopf hypersurfaces in \(\mathbb{CP}^n\) and \(\mathbb{CH}^n\), where \(n\geq 2\), are preciselyNEWLINENEWLINE(i) the Hopf hypersurfaces whose Hopf principal curvature \(\alpha\) vanishes, andNEWLINENEWLINE(ii) the open connected subsets of homogeneous Hopf hypersurfaces of type A0, A1 and B (using the standard terminology, see \textit{R. Niebergall} and \textit{P. J. Ryan}, [Tight and taut submanifolds. Based on the workshop on differential systems, submanifolds and control theory, Berkeley, CA, USA, March 1-4, 1994. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 32, 233--305 (1997; Zbl 0904.53005)].NEWLINENEWLINEThe authors establish the existence of a family of non-Hopf pseudo-Ryan hypersurfaces in \(\mathbb{CP}^2\) and \(\mathbb{CH}^2\), and proveNEWLINENEWLINETheorem 13. In \(\mathbb{CP}^2\) and \(\mathbb{CH}^2\) the set of non-Hopf \({}^\ast\)-Einstein hypersurfaces is disjoint from the set of non-Hopf pseudo-Ryan hypersurfaces.NEWLINENEWLINEThen the authors construct a family of Hopf hypersurfaces that are not \({}^\ast\)-Einstein, but satisfy a weakened form of the \({}^\ast\)-Einstein condition. They provide a new construction for the non-Hopf hypersurfaces in \(\mathbb{CH}^2\) with constant principal curvatures which were classified by \textit{J. Berndt} and \textit{J. C. Diaz-Ramos} [Proc. Am. Math. Soc. 135, No. 10, 3349--3357 (2007; Zbl 1129.53031)].
description: scientific article; zbMATH DE number 6020931