CHARACTERIZATIONS OF COMPLETE HYPERSURFACES IN LOCALLY SYMMETRIC RIEMANNIAN MANIFOLDS

被引：0

作者：

Araujo, Jogli G. ^{[1
]}

Batista, Marcio ^{[2
]}

De Lima, Henrique F. ^{[1
]}

机构：

[1] Univ Fed Campina Grande, Dept Matemat, BR-58429970 Campina Grande, Paraiba, Brazil

[2] Univ Fed Alagoas, Inst Matemat, BR-57072970 Maceio, Alagoas, Brazil

来源：

COLLOQUIUM MATHEMATICUM | 2018年 / 153卷 / 02期

关键词：

locally symmetric Riemannian manifolds; Einstein manifolds; complete linear Weingarten hypersurfaces; totally umbilical hypersurfaces; isoparametric hypersurfaces; LINEAR WEINGARTEN HYPERSURFACES; CONSTANT SCALAR CURVATURE; ORDER MEAN-CURVATURE; SPACE-FORMS; GEOMETRY; THEOREM; SPHERE;

D O I：

10.4064/cm7047-5-2017

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We deal with complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold whose sectional curvature obeys certain standard constraints. Under an appropriate restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Our approach is based on the use of a suitable Simons type formula combined with applications of some generalized maximum principles.

引用

页码：149 / 162

页数：14

共 30 条

[1] HYPERSURFACES WITH CONSTANT MEAN-CURVATURE IN SPHERES [J].

ALENCAR, H ;

DOCARMO, M .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1994, 120 (04) :1223-1229

[2] Constant higher-order mean curvature hypersurfaces in Riemannian spaces [J].

Alias, Luis J. ;

De Lira, Jorge H. S. ;

Malacarne, J. Miguel .

JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, 2006, 5 (04) :527-562

[3] Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds [J].

Alias, Luis J. ;

de Lima, Henrique F. ;

Melendez, Josue ;

dos Santos, Fabio R. .

MATHEMATISCHE NACHRICHTEN, 2016, 289 (11-12) :1309-1324

[4]

Alías LJ, 2013, T AM MATH SOC, V365, P591

[5] A maximum principle for hypersurfaces with constant scalar curvature and applications [J].

Alias, Luis J. ;

Carolina Garcia-Martinez, S. ;

Rigoli, Marco .

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, 2012, 41 (03) :307-320

[6] On the scalar curvature of constant mean curvature hypersurfaces in space forms [J].

Alias, Luis J. ;

Carolina Garcia-Martinez, S. .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2010, 363 (02) :579-587

[7] Generalized Maximum Principles and the Characterization of Linear Weingarten Hypersurfaces in Space Forms [J].

Aquino, Cicero P. ;

de Lima, Henrique F. ;

Velasquez, Marco Antonio L. .

MICHIGAN MATHEMATICAL JOURNAL, 2014, 63 (01) :27-40

[8] On the geometry of linear Weingarten hypersurfaces in the hyperbolic space [J].

Aquino, Cicero P. ;

de Lima, Henrique F. .

MONATSHEFTE FUR MATHEMATIK, 2013, 171 (3-4) :259-268

[9] A NEW CHARACTERIZATION OF COMPLETE LINEAR WEINGARTEN HYPERSURFACES IN REAL SPACE FORMS [J].

Aquino, Cicero P. ;

de Lima, Henrique F. ;

Velasquez, Marco A. L. .

PACIFIC JOURNAL OF MATHEMATICS, 2013, 261 (01) :33-43

[10]

Berard P., 1994, AN ACAD BRAS CIENC, V66, P397

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