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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: shushichang@126.com)
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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

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