On spacelike hypersurfaces of constant sectional curvature lorentz manifolds

Let $x:M^n\to {\overline{M}}^{n+1}$ be an n-dimensional spacelike hypersurface of a constant sectional curvature Lorentz manifold $\overline{M}$. Based on previous work of S. Montiel, L. Alías, A. Brasil and G. Colares studied what can be said about the geometry of M when $\overline{M}$ is a conformally stationary spacetime, with timelike conformal vector field K. For example, if $M^n$ has constant higher order mean curvatures $H_r$ and $H_{r+1}$, they concluded that $M^n$ is totally umbilical, provided $H_{r+1}\ne 0$ on it. If div($K$) does not vanish on $M^n$ they also proved that $M^n$ is totally umbilical, provided it has, a priori, just one constant higher order mean curvature.In this paper, we compute $L_r(S_r)$ for such an immersion, and use the resulting formula to study both r-maximal spacelike hypersurfaces of $\overline{M}$, as well as, in the presence of a constant higher order mean curvature, constraints on the sectional curvature of M that also suffice to guarantee the umbilicity of M. Here, by $L_r$ we mean the linearization of the second order differential operator associated to the r-th elementary symmetric function $S_r$ on the eigenvalues of the second fundamental form of x.