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 On totally umbilicalCR-submanifolds of a Kaehler manifold

1993 ◽  
Vol 16 (2) ◽  
pp. 405-408
Author(s):  
M. A. Bashir
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Lens Space ◽  
Holomorphic Sectional Curvature ◽  
Mean Curvature Vector ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
The Mean

LetMbe a compact3-dimensional totally umbilicalCR-submanifold of a Kaehler manifold of positive holomorphic sectional curvature. We prove that if the length of the mean curvature vector ofMdoes not vanish, thenMis either diffeomorphic toS3orRP3or a lens spaceLp,q3.

scholarly journals On pinching theorems for compact pseudo-umbilical submanifold

2012 ◽  
Vol 45 (3) ◽  
pp. 645-654
Author(s):  
Jing Mao ◽  
Shaodong Qin
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold ◽  
Parallel Mean Curvature

AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

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.

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${ S}_{p}^{n+ p} (c)$.

Sign in / Sign up

Export Citation Format

Share Document