Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space. (English) Zbl 1399.53093
Summary: Our aim in this article is to study the geometry of \(n\)-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \(\mathbb{R}^{n+p}_q\) of index \(q\), with \(1 \leq q \leq p\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \(\mathbb{R}^{n+p}_q\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \(\mathbb{R}^{n+p}_p\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz-Minkowski space \(\mathbb{R}^{n+1}_1\).
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C20 | Global Riemannian geometry, including pinching |
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
Keywords:
semi-Euclidean space; Lorentz-Minkowski space; complete space-like submanifold; totally geodesic submanifold; parallel mean curvature vector; normalized scalar curvature; constant mean curvature; space-like hypersurfaceReferences:
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