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Characterizations of complete spacelike submanifolds in the \(\mathbf{(n+p)}\)-dimensional anti-de Sitter space of index \(\mathbf{q}\)

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Our purpose in this paper is to study the geometry of n-dimensional complete spacelike submanifolds immersed in the \((n+p)\)-dimensional anti-de Sitter space \(\mathbb {H}^{n+p}_{q}\) of index q, with \(1\le q\le p\). Under suitable constraints on the Ricci curvature and the second fundamental form, we show that a complete maximal spacelike submanifold of \(\mathbb {H}^{n+p}_{q}\) must be totally geodesic. Furthermore, we establish sufficient conditions to guarantee that a complete spacelike submanifold with nonzero parallel mean curvature vector in \(\mathbb {H}^{n+p}_{p}\) must be pseudo-umbilical, which means that its mean curvature vector is an umbilical direction.

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Acknowledgments

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. The authors would like to thank the referee for giving some valuable suggestions and comments which improved the paper.

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  1. Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58429-970, Brazil

    Henrique F. de Lima, Fábio R. dos Santos & Marco Antonio L. Velásquez

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  1. Henrique F. de Lima
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  2. Fábio R. dos Santos
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  3. Marco Antonio L. Velásquez
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Correspondence to Henrique F. de Lima.

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de Lima, H.F., dos Santos, F.R. & Velásquez, M.A.L. Characterizations of complete spacelike submanifolds in the \(\mathbf{(n+p)}\)-dimensional anti-de Sitter space of index \(\mathbf{q}\) . RACSAM 111, 921–930 (2017). https://doi.org/10.1007/s13398-016-0330-2

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  • DOI: https://doi.org/10.1007/s13398-016-0330-2

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