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ON THE UMBILICITY OF HYPERSURFACES IN THE HYPERBOLIC SPACE

Part of: Global differential geometry Applications to physics - Differential geometry General relativity Local differential geometry

Published online by Cambridge University Press:  16 November 2016

C. P. AQUINO ,
M. BATISTA  and
H. F. DE LIMA
C. P. AQUINO
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí, 64049-550 Teresina, Piauí, Brazil email cicero.aquino@ufpi.edu.br
M. BATISTA*
Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, 57072-970 Maceió, Alagoas, Brazil email mhbs@mat.ufal.br
H. F. DE LIMA
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58429-970 Campina Grande, Paraíba, Brazil email henrique@mat.ufcg.edu.br
*
✉mhbs@mat.ufal.br
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Abstract

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In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space $\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for $\mathbb{H}^{n+1}$ and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in $\mathbb{H}^{n+1}$.

Keywords

hyperbolic spacecomplete hypersurfacestotally umbilical hypersurfaceshigher order mean curvaturesLorentzian Gauss map

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics 53Z05: Applications to physics 83C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 1 , August 2017 , pp. 45 - 58
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author is partially supported by CNPq, Brazil, grant number 302738/2014-2. The second author is partially supported by CNPq, Brazil, grant number 456755/2014-4. The third author is partially supported by CNPq, Brazil, grant number 303977/2015-9.

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