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  • SPACE-LIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACE

  • Part of
  • SHICHANG SHU
  • Journal:
    The ANZIAM Journal / Volume 50 / Issue 1 / July 2008
    Published online by Cambridge University Press:
    01 July 2008, pp. 1-11
    • Article
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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.