CLOSED CONVEX SPACELIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACES
CLOSED CONVEX SPACELIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACES
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发表时间：
2017
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通讯作者：
Zhongyang Sun
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作者：
Zhongyang Sun
1997年，H. Li [12]提出了一个猜想：如果Mn（n > 3）是de Sitter空间S1（1）中的完备类空超曲面，其常标准化数量曲率R满足n−2 n 6 R 6 1，那么Mn是全脐的吗？最近，F. E. C. Camargo等人（[5]）部分证明了这个猜想。本文从另一个角度研究了局部对称Lorentz空间L1中的闭凸类空超曲面Mn，并证明了当闭凸类空超曲面Mn的第二基本形式的长度平方为常数时Mn是全脐的，即定理1.另一方面，我们得到了若局部对称Lorentz空间L1中闭凸类空超曲面Mn的截面曲率满足K（Mn）> 0，则Mn是全脐的，即定理2.
In 1997, H. Li [12] proposed a conjecture: if Mn(n > 3) is a complete spacelike hypersurface in de Sitter space S 1 (1) with constant normalized scalar curvature R satisfying n−2 n 6 R 6 1, then is Mn totally umbilical? Recently, F. E. C. Camargo et al. ([5]) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface Mn in locally symmetric Lorentz space L 1 and also prove that M n is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface Mn is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface Mn in locally symmetric Lorentz space L 1 satisfies K(M n) > 0, then Mn is totally umbilical, i.e., Theorem 2.