Research ArticleOpen AccessGoogle Scholar indexed
A New Characterization of Totally Umbilical Hypersurfaces in de Sitter Space
College of Mathematics and Information Science, Henan Normal University, Xinxiang, China
School of Mathematics and Statistics, Nangyang Normal University, Nangyang, China
Department of Computer Science, Henan Normal University, Xinxiang, China
- 1 College of Mathematics and Information Science, Henan Normal University, Xinxiang, China
- 2 School of Mathematics and Statistics, Nangyang Normal University, Nangyang, China
- 3 Department of Computer Science, Henan Normal University, Xinxiang, China
Advances in Pure Mathematics·Volume 04 (2014)·Pages 42–46·Published 24 February 2014·DOI10.4236/apm.2014.42006
Copy link · social · email
Abstract
It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the k th mean curvature function H k is a linear combination of H k +1 , … , H n . This is a new angle to characterize round spheres.
Keywordsde Sitter SpaceSpacelike HypersurfaceHigher Order Mean Curvatures
- A. J. Goddard, “Some Remarks on the Existence of Spacelike Hypersurfaces of Constant Mean Curvature,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 82, 1977, pp. 489-495. http://dx.doi.org/10.1017/S0305004100054153
- S. Montiel, “An Integral Inequality for Compact Spacelike Hypersurfaces in de Sitter Space and Applications to the Case of Constant Mean Curvature,” Indiana University Mathematics Journal, Vol. 37, No. 4, 1988, pp. 909-917. http://dx.doi.org/10.1017/S0305004100054153
- Q.-M. Cheng and S. Ishikawa, “Spacelike Hypersurfaces with Constant Scalar Curvature,” Manuscripta Mathematica, Vol. 95, No. 4, 1998, pp. 499-505. http://dx.doi.org/10.1007/s002290050043
- J. A. Aledo, L. J. Alias and A. Romero, “Integral Formulas for Compact Space-Like Hypersurfaces in de Sitter Space: Applications to the Case of Constant Higher Mean Curvature,” Journal of Geometry and Physics, Vol. 31, No. 2-3, 1999, pp. 195-208. http://dx.doi.org/10.1016/S0393-0440(99)00008-X
- L. J. Alias and S.-E. Koh, “Remarks on Compact Spacelike Hypersurfaces in de Sitter Space with Constant Higher Order Mean Curvature,” Journal of Geometry and Physics, Vol. 39, No. 1, 2001, pp. 45-49. http://dx.doi.org/10.1016/S0393-0440(00)00073-5
- S.-E. Koh and M. S. Yoo, “A Characterization of Totally Umbilical Hypersurfaces in de Sitter Space,” Journal of Geometry and Physics, Vol. 51, No.1, 2004, pp. 34-39. http://dx.doi.org/10.1016/j.geomphys.2003.09.006
- S. Montiel and A. Ros, “Compact Hypersurfaces: The Alexandrov Theorem for Higher Order Mean Curvatures,” In: B. Lawson and K. Tenenblat, Eds., Differential Geometry, Longman, Essex, 1991, pp. 279-296.
- K. Akutagawa, “On Spacelike Hypersurfaces with Constant Mean Curvature in the de Sitter Space,” Mathematische Zeitschrift, Vol. 196, No. 1, 1987, pp. 13-19. http://dx.doi.org/10.1007/BF01179263
- L. J. Alias and A. G. Colares, “A Further Characterization of Ellipsoids,” Results in Mathematics, Vol. 48, No.1-2, 2005, pp. 1-8. http://dx.doi.org/10.1007/BF03322891
- H. Li, “Global Rigidity Theorems of Hypersurfaces,” Arkiv for Matematik, Vol. 35, No. 2, 1997, pp. 327-351.
- J. Ramanathan, “Complete Spacelike Hypersurfaces of Constant Mean Curvature in de Sitter Space,” Indiana University Mathematics Journal, Vol. 36, No. 2, 1987, pp. 349-359. http://dx.doi.org/10.1512/iumj.1987.36.36020
- Y. Zheng, “On Space-Like Hypersurfaces in the de Sitter Space,” Annals of Global Analysis and Geometry, Vol.13, No. 4, 1995, pp. 317-321. http://dx.doi.org/10.1007/BF00773403