## On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds.(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 $$x: M^n \to \widetilde M^{n+1}.$$ The author first computes $$L_r(S_r)$$ for such immersions where $$L_r: \mathcal D(M) \to \mathcal D(M)$$ is the second order differential operator defined as $L_r(f)=\text{tr}(P_r \text{Hess} f)$ where $$P_r$$ is the $$r$$-th Newton transformation on $$M.$$ Then the author makes use of the formula not only to consider $$r$$-maximal space-like hypersurfaces of $$\widetilde M$$ 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 $$L_r(S_r)$$ is Theorem 1, i.e, if $$x: M^n \to \widetilde M^{n+1}_c$$ is a closed space-like hypersurface of a time-oriented Lorentz manifold of constant sectional curvature $$c \geq 0$$ and $$M$$ is of constant scalar curvature $$R$$ such that
(a) $$c(\frac{n-2}{n})< R \leq c$$, then $$M$$ is totally umbilical,
(b) $$c(\frac{n-2}{n})\leq R \leq c$$ and $$S_3 \neq 0,$$ then $$M$$ is totally umbilical,
(c) $$c(\frac{n-2}{n})\leq R < c$$, then $$M$$ has constant mean curvature.
Moreover, the case of $$c(\frac{n-2}{n})< R < c$$ is considered under some assumptions on the mean curvature $$H$$ of $$M$$ in Theorem 2.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C99 Global differential geometry
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### References:

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