scholarly journals On the umbilicity of linear Weingarten spacelike submanifolds immersed in the de Sitter space

2020 ◽  
pp. 1-12
Author(s):  
Weiller F. C. Barboza ◽  
Eudes L. de Lima ◽  
Henrique F. de Lima ◽  
Marco Antonio L. Velásquez
Keyword(s):  
Mean Curvature ◽  
Curvature Vector ◽  
De Sitter Space ◽  
Index Formula ◽  
Curvature Function ◽  
De Sitter ◽  
Totally Umbilical ◽  
Spacelike Submanifold ◽  
Totally Umbilical Submanifolds ◽  
Spacelike Submanifolds

We investigate the umbilicity of [Formula: see text]-dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field in the de Sitter space [Formula: see text] of index [Formula: see text]. We recall that a spacelike submanifold is said to be linear Weingarten when its mean curvature function [Formula: see text] and its normalized scalar curvature [Formula: see text] satisfy a linear relation of the type [Formula: see text], for some constants [Formula: see text]. Under suitable constraints on the values of [Formula: see text] and [Formula: see text], we apply a generalized maximum principle for a modified Cheng–Yau operator [Formula: see text] in order to show that such a spacelike submanifold must be either totally umbilical or isometric to a product [Formula: see text], where the factors [Formula: see text] are totally umbilical submanifolds of [Formula: see text] which are mutually perpendicular along their intersections. Moreover, we also study the case in which these spacelike submanifolds are [Formula: see text]-parabolic.

Complete linear Weingarten spacelike submanifolds with higher codimension in the de Sitter space

2019 ◽  
Vol 16 (04) ◽  
pp. 1950050 ◽  
Author(s):  
Jogli Gidel da Silva Araújo ◽  
Henrique Fernandes de Lima ◽  
Fábio Reis dos Santos ◽  
Marco Antonio Lázaro Velásquez
Keyword(s):  
Mean Curvature ◽  
Curvature Vector ◽  
De Sitter Space ◽  
Second Fundamental Form ◽  
Index Formula ◽  
Curvature Function ◽  
De Sitter ◽  
Spacelike Submanifold ◽  
Spacelike Submanifolds ◽  
Complete Riemannian Manifolds

We study complete linear Weingarten spacelike submanifolds with arbitrary high codimension [Formula: see text] in the de Sitter space [Formula: see text] of index [Formula: see text] and whose normalized mean curvature vector is parallel. Under suitable restrictions on the values of the mean curvature function and on the norm of the traceless part of the second fundamental form, we prove that such a spacelike submanifold must be either totally umbilical or isometric to a certain hyperbolic cylinder of [Formula: see text]. Our approach is based on the use of a Simons type formula related to an appropriate Cheng–Yau modified operator jointly with an extension of Hopf’s maximum principle for complete Riemannian manifolds.

Complete spacelike submanifolds in a locally symmetric semi-Riemannian space

2020 ◽  
Vol 31 (04) ◽  
pp. 2050033
Author(s):  
Shicheng Zhang ◽  
Yuntao Zhang
Keyword(s):  
Maximum Principle ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Curvature Vector ◽  
Isoparametric Hypersurface ◽  
Mean Curvature Vector ◽  
Totally Umbilical ◽  
Type Formula ◽  
Spacelike Submanifolds ◽  
Generalized Maximum Principle

In this paper, complete spacelike submanifolds with parallel normalized mean curvature vector are investigated in semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and apply it jointly with the well-known generalized maximum principle of Omori–Yau to show that it must be totally umbilical submanifold or isometric to an isoparametric hypersurface in a submanifold [Formula: see text] of [Formula: see text].

scholarly journals SUBMANIFOLDS OF EUCLIDEAN SPACES SATISFYING $\Delta H =AH$

1995 ◽  
Vol 25 (1) ◽  
pp. 71-81
Author(s):  
BANG-YEN CHEN
Keyword(s):  
General Condition ◽  
Mean Curvature ◽  
Curvature Vector ◽  
De Sitter Space ◽  
Hyperbolic Spaces ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Euclidean Spaces

In [5] the author initiated the study of submanifolds whose mean curvature vector $H$ satisfying the condition $\Delta H =\lambda H$ for some constant $\lambda$ and proved that such submanifolds are either biharmonic or of 1-type or of null 2-type. Submanifolds of hyperbolic spaces and of de Sitter space-times satisfy this condition have been investigated and classified in [6,7]. In this article, we study submanifolds of $E^m$ whose mean curvature vector $H$ satisfies a more general condition; namely, $\Delta H =AH$ for some $m \times m$ matrix $A$.

scholarly journals COMPLETE SPACELIKE SUBMANIFOLDS IN DE SITTER SPACES WITH

2013 ◽  
Vol 87 (3) ◽  
pp. 386-399 ◽  
Author(s):  
JIANCHENG LIU ◽  
JINGJING ZHANG
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Linear Relation ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Spacelike Submanifolds ◽  
The Mean

AbstractIn this paper, we give a classification of spacelike submanifolds with parallel normalised mean curvature vector field and linear relation$R= aH+ b$of the normalised scalar curvature$R$and the mean curvature$H$in the de Sitter space${ S}_{p}^{n+ p} (c)$.

scholarly journals Compact space-like hypersurfaces in de Sitter space

2005 ◽  
Vol 2005 (13) ◽  
pp. 2053-2069 ◽  
Author(s):  
Jinchi Lv
Keyword(s):  
Compact Space ◽  
Mean Curvature ◽  
De Sitter Space ◽  
Higher Order ◽  
De Sitter ◽  
Totally Umbilical ◽  
Higher Order Mean Curvatures ◽  
Integral Formulas

We present some integral formulas for compact space-like hypersurfaces in de Sitter space and some equivalent characterizations for totally umbilical compact space-like hypersurfaces in de Sitter space in terms of mean curvature and higher-order mean curvatures.

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