Hypersurfaces in space forms satisfying some curvature conditions

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• 作者：Ryszard Deszcz^a ; ^{Ryszard.Deszcz@up.wroc.pl" class="auth_mail" title="E-mail the corresponding author} ; Małgorzata Głogowska^a ; ^{Malgorzata.Glogowska@up.wroc.pl" class="auth_mail" title="E-mail the corresponding author} ; Marian Hotloś^b ; ^{Marian.Hotlos@pwr.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Georges Zafindratafa^c ; ^{Georges.Zafindratafa@univ-valenciennes.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 53B20 ; 53B25 ; 53B30 ; 53B50 ; secondary ; 53C25 ; 53C40
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：99
• 期：Complete
• 页码：218-231
• 全文大小：483 K

文摘

In Abdalla and Dillen (2002) an example of a non-semisymmetric Ricci-symmetric quasi-Einstein austere hypersurface dd1a4577c8554ebda3f855c797fd3c4e" title="Click to view the MathML source">M$M$ isometrically immersed in an Euclidean space was constructed. In this paper we state that, at every point of the hypersurface dd1a4577c8554ebda3f855c797fd3c4e" title="Click to view the MathML source">M$M$, the following generalized Einstein metric curvature condition is satisfied: (∗$\ast$) the difference tensor cdd6da4c6cc221e3302abdc9e0149" title="Click to view the MathML source">R⋅C−C⋅R$R\cdot C-C\cdot R$ and the Tachibana tensor Q(S,C)$Q(S,C)$ are linearly dependent. Precisely, $(n-2)(R\cdot C-C\cdot R)=Q(S,C)$ on dd1a4577c8554ebda3f855c797fd3c4e" title="Click to view the MathML source">M$M$. We also prove 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 (∗$\ast$). We present examples of hypersurfaces satisfying (∗$\ast$).