*(English)*Zbl 1102.53044

The paper deals with $n$-dimensional space-like hypersurfaces of constant sectional curvature Lorentz manifolds given by an immersion of the form $x:{M}^{n}\to {\tilde{M}}^{n+1}\xb7$ The author first computes ${L}_{r}\left({S}_{r}\right)$ for such immersions where ${L}_{r}:\mathcal{D}\left(M\right)\to \mathcal{D}\left(M\right)$ is the second order differential operator defined as

where ${P}_{r}$ is the $r$-th Newton transformation on $M\xb7$ Then the author makes use of the formula not only to consider $r$-maximal space-like hypersurfaces of $\tilde{M}$ but also to obtain conditions for the sectional curvature of $M$ to make it umbilic in the case of a constant higher order mean curvature. One of the main results of the paper in terms of applications of the formula for ${L}_{r}\left({S}_{r}\right)$ is Theorem 1, i.e, if $x:{M}^{n}\to {\tilde{M}}_{c}^{n+1}$ is a closed space-like hypersurface of a time-oriented Lorentz manifold of constant sectional curvature $c\ge 0$ and $M$ is of constant scalar curvature $R$ such that

(a) $c\left(\frac{n-2}{n}\right)<R\le c$, then $M$ is totally umbilical,

(b) $c\left(\frac{n-2}{n}\right)\le R\le c$ and ${S}_{3}\ne 0,$ then $M$ is totally umbilical,

(c) $c\left(\frac{n-2}{n}\right)\le R<c$, then $M$ has constant mean curvature.

Moreover, the case of $c\left(\frac{n-2}{n}\right)<R<c$ is considered under some assumptions on the mean curvature $H$ of $M$ in Theorem 2.