Rigidity of submanifolds in the Euclidean and spherical spaces

被引：0
作者：
Jonatan F. da Silva
Henrique F. de Lima
Fábio R. dos Santos
Marco Antonio L. Velásquez
机构：
[1] Universidade Federal do Ceará,Departamento de Matemática
[2] Universidade Federal de Campina Grande,Departamento de Matemática
来源：
Rendiconti del Circolo Matematico di Palermo Series 2 | 2020年 / 69卷
关键词：
Euclidean and spherical spaces; Complete submanifolds; Pseudo-umbilical submanifolds; Parallel mean curvature vector field; Primary 53C42;
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摘要：
We deal with n-dimensional complete submanifolds Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^n$$\end{document} immersed with nonzero parallel mean curvature vector field H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H}$$\end{document} either in the Euclidean space Rn+p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n+p}$$\end{document} or in the Euclidean sphere Sn+p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^{n+p}$$\end{document}. In this setting, we establish sufficient conditions to guarantee that such a submanifold Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^n$$\end{document} must be pseudo-umbilical, which means that H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H}$$\end{document} is an umbilical direction. Moreover, assuming a suitable lower bound for the Ricci curvature, we conclude that Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^n$$\end{document} must be isometric to Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^{n}$$\end{document}, up to scaling.
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页码：159 / 165
页数：6
相关论文
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