Submanifolds with constant scalar curvature in a space form

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• 作者：Jogli G. Araú ; jo ^{jogli@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Henrique F. de Lima ; ^{henrique@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Fá ; bio R. dos Santos ^{fabio@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Marco Antonio L. Velá ; squez ^{marco.velasquez@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Riemannian space forms ; Complete submanifolds ; Parallel normalized mean curvature vector field ; Constant normalized scalar curvature ; Clifford torus ; Circular and hyperbolic cylinders
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：488-498
• 全文大小：313 K

文摘

We deal with complete submanifolds Mⁿ$M^n$ having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form ${\mathbb{Q}}_c^{n+p}$ of constant sectional curvature c∈{1,0,−1}$c\in \{1,0,-1\}$. In this setting, we show that such a submanifold Mⁿ$M^n$ must be either totally umbilical or isometric to a Clifford torus ${\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\times {\mathbb{S}}^{n-1}(r)$, when c=1$c=1$, a circular cylinder R×Sⁿ⁻¹(r)$\mathbb{R}\times {\mathbb{S}}^{n-1}(r)$, when c=0$c=0$, or a hyperbolic cylinder ${\mathbb{H}}^1(-\sqrt{1+r^2})\times {\mathbb{S}}^{n-1}(r)$, when c=−1$c=-1$. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli [2], Cheng [4] and Guo and Li [6].