Characterizations of real hypersurfaces in complex space forms in terms of curvature tensors
type: Item
instance of: scholarly article
title: Characterizations of real hypersurfaces in complex space forms in terms of curvature tensors
zbMATH Open document ID: 0837.53023
author: Young Jin Suh
author: Yong-Soo Pyo
published in: Tsukuba Journal of Mathematics
publication date: 1996-05-13T00:00:00Z
review text: The authors provide characterizations of certain real hypersurfaces in a complex projective space \(P_n \mathbb{C}\) or a complex hyperbolic space \(H_n\mathbb{C}\) in terms of Riemannian curvature. Let \(M\) be such a hypersurface with induced Riemannian metric \(g\), Riemannian curvature tensor \(R\) and Riemannian connection \(\nabla\). Moreover, let \(\xi\) be the vector field on \(M\) obtained by applying the Kähler structure of the ambient space to a unit normal field of \(M\), and \(A\) the shape operator of \(M\) with respect to \(\xi\). The authors obtain the following results: (1) If (a) \(n\geq 3\), \(\nabla_\xi R= 0\) and \(g(A\xi, \xi)\neq 0\), or if (b) \(n\geq 2\) and the Lie derivative of \(R\) in direction of \(\xi\) vanishes, then \(M\) is an open part of a tube around a totally geodesic complex subspace of \(P_n \mathbb{C}\) or \(H_n\mathbb{C}\), or of a horosphere in \(H_n \mathbb{C}\). (2) If \(n\geq 3\), \(\nabla_\xi R= 0\), and if the ambient space is \(P_n \mathbb{C}\), then \(M\) is a non-homogeneous real hypersurface lying on a tube of a certain radius around a certain Kähler submanifold in \(P_n\mathbb{C}\), or is an open part of a tube around a totally geodesic complex subspace of \(P_n \mathbb{C}\).
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53B35
zbMATH DE Number: 828344
zbMATH Keywords: complex space form
zbMATH Keywords: tubes
zbMATH Keywords: real hypersurface
zbMATH Keywords: Riemannian curvature tensor
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162805
OpenAlex ID: W1608527140
DOI: 10.21099/TKBJM/1496162805
description: scientific article; zbMATH DE number 828344