A characterization of a real hypersurface of type \(B\)
type: Item
instance of: scholarly article
title: A characterization of a real hypersurface of type \(B\)
zbMATH Open document ID: 0726.53018
author: U-Hang Ki
author: Hyang Sook Kim
author: Hisao Nakagawa
published in: Tsukuba Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let \(N\) be the \(n\)-dimensional complex projective space \(P_n\mathbb{C}\) or the \(n\)-dimensional complex hyperbolic space \(H_ n\mathbb{C}\) of constant holomorphic curvature \(4\) resp. \(-4\). Distinguished examples of real hypersurfaces of \(N\) are the six classes \(A_1\), \(A_2\), \(B\)-\(E\) of homogeneous hypersurfaces of \(P_n\mathbb{C}\) [see \textit{R. Takagi}, Osaka J. Math. 10, 495--506 (1973; Zbl 0274.53062)] and the four classes \(A_0\)-\(A_1\) and \(B\) of hypersurfaces with constant principal curvatures for which the normal bundle is mapped into a principal direction by the complex structure of \(N\) [see \textit{J. Berndt}, J. Reine Angew. Math. 395, 132--141 (1989; Zbl 0655.53046)]. The hypersurfaces of type \(A_1\), \(A_2\) and \(B\) are the tubes around the totally geodesic complete submanifolds of \(N\); the horospheres of \(H_n\mathbb{C}\) form the class \(A_0\). The authors derive characterizations for the classes \(A_0\)-\(A_2\) and \(B\) by special properties of the shape operator (pseudo-parallelity, cyclic parallelity, and special formulas for the norm of the covariant derivative of the shape operator).
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53B35
zbMATH DE Number: 4198946
zbMATH Keywords: complex projective space
zbMATH Keywords: complex hyperbolic space
zbMATH Keywords: real hypersurfaces
zbMATH Keywords: constant principal curvatures
zbMATH Keywords: shape operator
reviewed by: Helmut Reckziegel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496161315
OpenAlex ID: W1607045162
DOI: 10.21099/TKBJM/1496161315
description: scientific article; zbMATH DE number 4198946