Let M be a complete hypersurface with constant mean curvature in a n+1-dimensional conformally flat Lorentzian manifold Ln+11.Let S stands for the square of the length of the second fundamental form of M and C=[2nr-(n+1)R]/[n(n-1)],where R and r is the supremum and infimum of Ricci curvature of M respectively.Assume that the normal direction of M be the Ricci principal direction of Ln+11.(1) If S<2n-1C,then M is a totally umbilical hypersurface;(2) If S=2n-1C,then M is a totally umbilical hypersurface when ...