The paper deals with -dimensional space-like hypersurfaces of constant sectional curvature Lorentz manifolds given by an immersion of the form The author first computes for such immersions where is the second order differential operator defined as
where is the -th Newton transformation on Then the author makes use of the formula not only to consider -maximal space-like hypersurfaces of but also to obtain conditions for the sectional curvature of 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 is Theorem 1, i.e, if is a closed space-like hypersurface of a time-oriented Lorentz manifold of constant sectional curvature and is of constant scalar curvature such that
(a) , then is totally umbilical,
(b) and then is totally umbilical,
(c) , then has constant mean curvature.
Moreover, the case of is considered under some assumptions on the mean curvature of in Theorem 2.