INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

Luis J. Alías; Aldir Brasil; A. Gervasio Colares

doi:10.1017/S0013091502000500

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.

AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Koh, Sung-Eun and Yoo, Myung-Suk 2004. A characterization of totally umbilical hypersurfaces in de Sitter space. Journal of Geometry and Physics, Vol. 51, Issue. 1, p. 34.

Caminha, A. 2006. On spacelike hypersurfaces of constant sectional curvature lorentz manifolds. Journal of Geometry and Physics, Vol. 56, Issue. 7, p. 1144.

ALÍAS, LUIS J. and COLARES, A. GERVASIO 2007. Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 143, Issue. 3, p. 703.

de Lima, Henrique Fernandes 2007. Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space. Journal of Geometry and Physics, Vol. 57, Issue. 3, p. 967.

de Lima, Henrique F. 2008. A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz–Minkowski space and application. Differential Geometry and its Applications, Vol. 26, Issue. 4, p. 445.

Colares, A. Gervasio and de Lima, Henrique F. 2008. Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces. General Relativity and Gravitation, Vol. 40, Issue. 10, p. 2131.

Barros, Abdênago Brasil, Aldir and Caminha, Antonio 2008. Stability of spacelike hypersurfaces in foliated spacetimes. Differential Geometry and its Applications, Vol. 26, Issue. 4, p. 357.

Caminha, A. and de Lima, Henrique F. 2009. Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds. General Relativity and Gravitation, Vol. 41, Issue. 1, p. 173.

de Lima, Henrique F. and de Lima, Joseilson R. 2009. Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space. International Journal of Mathematics and Mathematical Sciences, Vol. 2009, Issue. , p. 1.

Albujer, A.L. Camargo, F.E.C. and de Lima, H.F. 2010. Complete spacelike hypersurfaces with constant mean curvature in −R×Hn. Journal of Mathematical Analysis and Applications, Vol. 368, Issue. 2, p. 650.

Camargo, F. Caminha, A. da Silva, M. and de Lima, H. 2010. On the r-stability of spacelike hypersurfaces. Journal of Geometry and Physics, Vol. 60, Issue. 10, p. 1402.

ALBUJER, ALMA L. CAMARGO, FERNANDA E. C. and DE LIMA, HENRIQUE F. 2011. Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 151, Issue. 2, p. 271.

Gervasio Colares, A. and Palmas, Oscar 2011. On Riemannian manifolds foliated by (n − 1)-umbilical hypersurfaces. Bulletin of the Brazilian Mathematical Society, New Series, Vol. 42, Issue. 1, p. 105.

de Lima, Henrique Fernandes 2012. On Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime. Journal of Geometry, Vol. 103, Issue. 2, p. 219.

Camargo, F. Caminha, A. de Lima, H. and Velásquez, M. 2012. On the geometry of conformally stationary Lorentz spaces. Acta Mathematica Hungarica, Vol. 134, Issue. 3, p. 385.

Impera, Debora 2012. Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces. Journal of Geometry and Physics, Vol. 62, Issue. 2, p. 412.

CAMARGO, FERNANDA CAMINHA, ANTONIO DE LIMA, HENRIQUE and PARENTE, ULISSES 2012. Generalized maximum principles and the rigidity of complete spacelike hypersurfaces. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 153, Issue. 3, p. 541.

ALÍAS, LUIS J. IMPERA, DEBORA and RIGOLI, MARCO 2012. Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 152, Issue. 2, p. 365.

de Lima, Henrique F. and de Lima, Joseílson R. 2012. Complete hypersurfaces immersed in a semi-Riemannian warped product. Differential Geometry and its Applications, Vol. 30, Issue. 1, p. 136.

Alías, Luis J. Colares, Antonio Gervásio and de Lima, Henrique F. 2013. On the rigidity of complete spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime. Bulletin of the Brazilian Mathematical Society, New Series, Vol. 44, Issue. 2, p. 195.

Download full list

Article contents

INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

Abstract

Keywords

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

Abstract

Keywords

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests