Abstract
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 \({\mathbb{R}^{n+1}_{1}}\).
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The first author is partially supported by CNPq, Brazil, grant 303977/2015-9.
The second author was partially supported by PNPD/UFCG/CAPES, Brazil.
The third author is partially supported by CNPq, Brazil, grant 308757/2015-7.
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De Lima, H.F., Dos Santos, F.R. & Velásquez, M.A.L. Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space. Acta Math. Hungar. 150, 217–227 (2016). https://doi.org/10.1007/s10474-016-0646-6
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DOI: https://doi.org/10.1007/s10474-016-0646-6
Key words and phrases
- semi-Euclidean space
- Lorentz–Minkowski space
- complete spacelike submanifold
- totally geodesic submanifold
- parallel mean curvature vector
- normalized scalar curvature
- constant mean curvature spacelike hypersurface