## Complete space-like submanifolds with parallel mean curvature vector of an indefinite space form.(English)Zbl 0804.53089

The authors prove the following theorems:
Theorem 1. Let $$M$$ be an $$n$$-dimensional complete space-like submanifold with parallel mean curvature vector of an indefinite space form $$M^{n+p}_ p(c)$$. If one of the following conditions is satisfied: (1) $$c\leq 0$$, (2) $$c>0$$ and $$n^ 2 H^ 2\geq 4(n-1)c$$ then $$S\leq S_ +(p)+ k(p)$$, where $$k(p)$$ is a constant defined by $k(p)= (p-1) H\bigl\{nH+ \sqrt{n(n-1)\{S_ +(1)- n^ 2 H^ 2\}}\bigr\}.$ [$$S$$ is the squared norm of the second fundamental form.]
Theorem 2. The hyperbolic cylinder $$H^ 1(c_ 1)\times \mathbb{R}^{n-1}$$ in $$\mathbb{R}^{n+1}_ 1$$ is the only complete connected space-like $$n$$- dimensional submanifold with parallel mean curvature vector of $$\mathbb{R}^{n+p}_ p$$ satisfying $$S= S_ +(p)+ k(p)$$.
Theorem 3. The hyperbolic cylinder $$H^ 1(c_ 1)\times H^{n-1}(c_ 2)$$ of $$H^{n-1}_ 1(c)$$ and the maximal submanifolds $$H^{n_ 1}(c_ 1)\times\cdots \times H^{n_{p+1}}(c_{p+1})$$ of $$H^{n+p}_ p(c)$$ are the only complete connected space-like $$n$$- dimensional submanifolds with parallel mean curvature vector satisfying $$S= S_ +(p)+ k(p)$$, where $$c_ r= (n/n_ r)c$$ and $$\sum^{p+1}_{r=1} n_ r= n$$ in the latter case.
Reviewer: B.Rouxel (Quimper)

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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