Closed convex spacelike hypersurfaces in locally symmetric Lorentz spaces
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In 1997, H. Li {\cite{Li97}} proposed a conjecture: if $M^{n} (n\geqslant3)$ is a complete spacelike hypersurface in de Sitter space $S_{1}^{n+1}(1)$ with constant normalized scalar curvature $R$ satisfying $\frac{n-2}{n}\leqslant R\leqslant1$, then is $M^{n}$ totally umbilical? Recently, F.~E.~C.~Camargo et al.~(\cite{Camargo08}) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface $M^{n}$ in locally symmetric Lorentz space $L^{n+1}_1$ and also prove that $M^{n}$ is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface $M^{n}$ is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface $M^{n}$ in locally symmetric Lorentz space $L_{1}^{n+1}$ satisfies $K(M^{n})>0$, then $M^{n}$ is totally umbilical, i.e., Theorem 2.
Abstract
In 1997, H. Li {\cite{Li97}} proposed a conjecture: if $M^{n} (n\geqslant3)$ is a complete spacelike hypersurface in de Sitter space $S_{1}^{n+1}(1)$ with constant normalized scalar curvature $R$ satisfying $\frac{n-2}{n}\leqslant R\leqslant1$, then is $M^{n}$ totally umbilical? Recently, F.~E.~C.~Camargo et al.~(\cite{Camargo08}) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface $M^{n}$ in locally symmetric Lorentz space $L^{n+1}_1$ and also prove that $M^{n}$ is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface $M^{n}$ is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface $M^{n}$ in locally symmetric Lorentz space $L_{1}^{n+1}$ satisfies $K(M^{n})>0$, then $M^{n}$ is totally umbilical, i.e., Theorem 2.
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