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SPACE-LIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACE

Part of: Local differential geometry

Published online by Cambridge University Press:  01 July 2008

SHICHANG SHU
SHICHANG SHU*
Affiliation:
Department of Mathematics, Xianyang Normal University, Xianyang, 712000 Shaanxi, People’s Republic of China (email: [email protected])
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Abstract

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Let M be an n-dimensional space-like hypersurface in a locally symmetric Lorentz space, with n(n−1)R=κH(κ>0) and satisfying certain additional conditions on the sectional curvature. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M, respectively. We show that if the mean curvature is nonnegative and attains its maximum on M, then:

  1. (1) if H2<4(n−1)c/n2, M is totally umbilical;

  2. (2) if H2=4(n−1)c/n2, M is totally umbilical or is an isoparametric hypersurface;

  3. (3) if H2>4(n−1)c/n2 and S satisfies some pinching conditions, M is totally umbilical or is an isoparametric hypersurface.

Keywords

locally symmetricLorentz spacehypersurfacestotally umbilical

MSC classification

Secondary: 53B20: Local Riemannian geometry
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 1 , July 2008 , pp. 1 - 11
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Akutagawa, K., “On space-like hypersurfaces with constant mean curvature in a de Sitter space”, Math. Z. 196 (1987) 13–19.CrossRefGoogle Scholar
[2]Alencar, H. and do Carmo, M. P., “Hypersurfaces with constant mean curvature in spheres”, Proc. Amer. Math. Soc. 120 (1994) 1223–1229.Google Scholar
[3]Baek, J. D., Cheng, Q.-M. and Suh, Y. J., “Complete space-like hypersurfaces in locally symmetric Lorentz spaces”, J. Geom. Phys. 49 (2004) 231–247.CrossRefGoogle Scholar
[4]Cheng, Q.-M., “Complete space-like hypersurfaces of a de Sitter space with r=kH”, Mem. Fac. Sci. Kyushu Univ. 44 (1990) 67–77.Google Scholar
[5]Cheng, Q.-M. and Ishikawa, S., “Space-like hypersurfaces with constant scalar curvature”, Manuscripta Math. 95 (1998) 499–505.Google Scholar
[6]Cheng, S. Y. and Yau, S. T., “Hypersurfaces with constant scalar curvature”, Math. Ann. 225 (1977) 195–204.CrossRefGoogle Scholar
[7]Chouque-Bruhat, Y., Fisher, A. E. and Marsdan, J. E., “Maximal hypersurfaces and positivity mass”, in Proc. E. Fermi Summer School of the Italian Physical Society (ed. J. Ehlers), (North-Holland, Amsterdam, 1979).Google Scholar
[8]Goddard, A. J., “Some remarks on the existence of space-like hypersurfaces of constant mean curvature”, Math. Proc. Cambridge Philos. Soc. 82 (1977) 489–495.CrossRefGoogle Scholar
[9]Li, H., “Global rigidity theorems of hypersurfaces”, Ark. Mat. 35 (1997) 327–351.Google Scholar
[10]Marsden, J. and Tipler, F., “Maximal hypersurfaces and foliations of constant mean curvature in general relativity”, Bull. Am. Phys. Soc. 23 (1978) 84–90.Google Scholar
[11]Montiel, S., “A characterization of hyperbolic cylinders in the de Sitter space”, Tôhoku Math. J. 48 (1996) 23–31.CrossRefGoogle Scholar
[12]Okumura, M., “Hypersurfaces and a pinching problem on the second fundamental tensor”, Amer. J. Math. 96 (1974) 207–213.Google Scholar
[13]Ramanathan, J., “Complete space-like hypersurfaces of constant mean curvature in the de Sitter space”, Indiana Univ. Math. J. 36 (1987) 349–359.CrossRefGoogle Scholar
[14]Shu, S. C., “Complete space-like hypersurfaces in a de Sitter space”, Bull. Austral. Math. Soc. 73 (2006) 9–16.Google Scholar
[15]Stumbles, S., “Hypersurfaces of constant mean extrinsic curvature”, Ann. of Phys. 133 (1981) 28–56.Google Scholar
[16]Suh, Y. J., Choi, Y. S. and Yang, H. Y., “On space-like hypersurfaces with constant mean curvature in a Lorentz manifold”, Houston J. Math. 28 (2002) 47–70.Google Scholar