Hypersurfaces in space forms satisfying some curvature conditions
type: Item
instance of: scholarly article
title: Hypersurfaces in space forms satisfying some curvature conditions
zbMATH Open document ID: 1343.53019
author: Ryszard Deszcz
author: M. Głogowska
author: Marian Hotloś
author: Georges Zafindratafa
published in: Journal of Geometry and Physics
publication date: 2015-12-18T00:00:00Z
review text: In [\textit{B. E. Abdalla} and \textit{F. Dillen}, Proc. Am. Math. Soc. 130, No. 6, 1805--1808 (2002; Zbl 1008.53021)] an example of a non-semisymmetric Ricci-symmetric quasi-Einstein austere hypersurface \(M^n\) isometrically immersed in a Euclidean space \(\mathbb R^{n+1}\), \(n>4\), was constructed. In the paper the authors state that, at every point of the hypersurface \(M^n\), the following condition \((\ast)\) is satisfied: \((n-2)(R\cdot C-C\cdot R)=Q(S,C)\). Here \(R\) is the Riemann curvature tensor and \(C\) is the Weyl conformal curvature tensor. Due to [\textit{R. Deszcz} et al., Kragujevac J. Math. 35, No. 2, 223--247 (2011; Zbl 1289.53037)], given a symmetric \((0, 2)\)-tensor \(A\) and a \((0, k)\)-tensor field \(B\), the Tachibana tensor \(Q(A,B)\) of \(A\) and \(B\) is a \((0, k+2)\)-tensor \(Q(A,B)(X_1,\dots, X_k,X,Y)= -T((X\wedge_A Y)X_1,\dots, X_k)-\dots -T(X_1,\dots, (X\wedge_A Y)X_k)\), where \((X\wedge_A Y)Z=A(Y,Z)X-A(X,Z)Y\). It is shown that non-conformally flat and non-Einstein hypersurfaces with vanishing scalar curvature having at every point two distinct principal curvatures, as well as some hypersurfaces having at every point three distinct principal curvatures, satisfy \((\ast)\), and certain examples of hypersurfaces are presented.
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53B50
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 6522228
zbMATH Keywords: generalized Einstein metric condition
zbMATH Keywords: Roter-type manifold
zbMATH Keywords: Tachibana tensor
zbMATH Keywords: austere hypersurface
zbMATH Keywords: quasi-Einstein hypersurface
zbMATH Keywords: 2-quasi-Einstein hypersurface
reviewed by: Vladimir Yu. Rovenskij
MaRDI profile type: Publication
full work available at URL: j.geomphys.2015.10.010
OpenAlex ID: W1890520051
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Wikidata QID: Q125896687
DOI: 10.1016/J.GEOMPHYS.2015.10.010
description: scientific article; zbMATH DE number 6522228