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 Space-like submanifolds with parallel mean curvature in de Sitter spaces

2000 ◽  
Vol 69 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Chongzhen Ouyang ◽  
Zhenqi Li
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Parallel Mean Curvature Vector ◽  
Complete Space ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature

AbstractThis paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

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.

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