Characterization of totally \(\eta\)-umbilic real hypersurfaces in nonflat complex space forms by some inequality
type: Item
instance of: scholarly article
title: Characterization of totally \(\eta\)-umbilic real hypersurfaces in nonflat complex space forms by some inequality
zbMATH Open document ID: 1074.53011
author: Takehiro Itoh
author: Sadahiro Maeda
published in: Proceedings of the Japan Academy. Series A
publication date: 2005-02-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1116442245
review text: Let \(\widetilde{M}_n(c)\) be an \(n\)-dimensional nonflat complex space form of constant holomorphic curvature \(c\) which is either a complex projective space \(\mathbb{C}P^n(c)\) or a complex hyperbolic space \(\mathbb{C}H^n(c)\). It is known that there is no totally umbilic real hypersurface in \(\widetilde{M}_n(c)\), but there exist totally \(\eta\)-umbilic real hypersurfaces. A real hypersurface \(M\) of \(\widetilde{M}_n(c)\) is called totally \(\eta\)-umbilic if its shape operator \(A\) is of the form \(AX=\alpha X\) for each vector \(X\) on \(M\) which is orthogonal to the characteristic vector \(\xi\) of \(M\), where \(\alpha\) is a smooth function on \(M\). The main result of the paper under review is the following: Theorem. Let \(M\) be a real hypersurface in a nonflat complex space form \(\widetilde{M}_n(c)\) (\(n \geq 2\)). Then the following inequality holds: \((\operatorname{trace} A -\langle A\xi,\xi\rangle)^2 \leq 2(n-1)(\operatorname{trace} A^2 -\| A\xi\| ^2)\), where \(A\) is the shape operator of \(M\) in the ambient space \(\widetilde{M}_n(c)\). Moreover, the equality holds on \(M\) if and only if \(M\) is totally \(\eta\)-umbilic in \(\widetilde{M}_n(c)\).
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 2138338
zbMATH Keywords: complex projective space
zbMATH Keywords: complex hyperbolic space
zbMATH Keywords: real hypersurfaces
zbMATH Keywords: \(\eta\)-umbilic real hypersurfaces
reviewed by: Yurii G. Nikonorov
MaRDI profile type: Publication
OpenAlex ID: W1981704501
cites work: Real hypersurfaces some of whose geodesics are plane curves in nonflat complex space forms
cites work: Characterizations of geodesic hyperspheres in a projective space by observing the extrinsic shape of geodesics
cites work: Q4394855
DOI: 10.3792/PJAA.80.61
description: scientific article; zbMATH DE number 2138338