Abstract
In this paper, we deal with complete spacelike submanifolds \(M^n\) immersed in the de Sitter space \(\mathbb S_p^{n+p}\) of index p with parallel normalized mean curvature vector and constant scalar curvature R. Imposing a suitable restriction on the values of R, we apply a maximum principle for the so-called Cheng–Yau operator L, which enables us to show that either such a submanifold must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only the submanifold is isometric to a hyperbolic cylinder of the ambient space. In particular, when \(n=2\) this provides a nice characterization of the totally umbilical spacelike surfaces of \(\mathbb {S}^{2+p}_p\) with codimension \(p\ge 2\). Furthermore, we also study the case in which these spacelike submanifold are L-parabolic.
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Acknowledgements
The authors would like to thank the referee for reading the manuscript in great detail and giving several valuable suggestions and useful comments which improved the paper. This work is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. The first author was partially supported by MINECO/FEDER project reference MTM2015-65430-P and Fundación Séneca project reference 19901/GERM/15, Spain, and Ciência sem Fronteiras, Programa PVE, project A012/2013, CAPES, Brazil. The second author was partially supported by CNPq, Brazil, Grant 303977/2015-9.
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Alías, L.J., Lima, H.F.d. & Santos, F.R.d. Characterizations of Spacelike Submanifolds with Constant Scalar Curvature in the de Sitter Space. Mediterr. J. Math. 15, 12 (2018). https://doi.org/10.1007/s00009-017-1057-9
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DOI: https://doi.org/10.1007/s00009-017-1057-9
Keywords
- De Sitter space
- parallel normalized mean curvature vector
- constant scalar curvature
- totally umbilical submanifolds
- hyperbolic cylinders