A characterization of totally umbilical hypersurfaces in de Sitter space

2004 ◽  
Vol 51 (1) ◽  
pp. 34-39 ◽  
Author(s):  
Sung-Eun Koh ◽  
Myung-Suk Yoo
Keyword(s):  
De Sitter Space ◽  
De Sitter ◽  
Totally Umbilical ◽  
Totally Umbilical Hypersurfaces

scholarly journals Weingarten spacelike hypersurfaces in a de Sitter space

2012 ◽  
Vol 20 (1) ◽  
pp. 387-406
Author(s):  
Junfeng Chen ◽  
Shichang Shu
Keyword(s):  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Riemannian Product ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Cylinder

Abstract We study some Weingarten spacelike hypersurfaces in a de Sitter space S1n+1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product Hk(1−coth2 ϱ)× Sn−k(1 − tanh2 ϱ), 1 < k < n − 1 in S1n+1(1), the hyperbolic cylinder H1(1 − coth2 ϱ) × Sn-1(1 − tanh2 ϱ) or spherical cylinder Hn−1(1 − coth2 ϱ) × S1(1 − tanh2 ϱ) in S1n+1 (1)

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.

Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space

2019 ◽  
Vol 69 (6) ◽  
pp. 1437-1446
Author(s):  
Rong Mi
Keyword(s):  
Lorentz Space ◽  
De Sitter Space ◽  
Sao Paulo ◽  
Maximum Principles ◽  
Isoparametric Hypersurface ◽  
De Sitter ◽  
Principal Curvatures ◽  
Totally Umbilical ◽  
Type Formula ◽  
Spacelike Hypersurfaces

Abstract Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19].

Strong (r, s)-stability in the de Sitter space

2015 ◽  
Vol 145 (1) ◽  
pp. 91-104
Author(s):  
Henrique F. de Lima ◽  
Antonio F. de Sousa ◽  
Marco Antonio L. Velásquez
Keyword(s):  
De Sitter Space ◽  
Higher Order ◽  
Closed Space ◽  
De Sitter ◽  
Round Sphere ◽  
Totally Umbilical ◽  
Higher Order Mean Curvatures ◽  
Geometric Conditions ◽  
Stable Space

In this paper, we establish the notion of strong (r, s)-stability concerning closed space-like hypersurfaces immersed with higher-order mean curvatures linearly related in the de Sitter space . In this setting, we prove that totally umbilical round spheres of are strongly (r, s)-stable. Afterwards, we obtain sufficient geometric conditions that guarantee that a closed strongly (r, s)-stable space-like hypersurface in must be a totally umbilical round sphere.

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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