Abstract
Let M n be a spacelike linear Weingarten submanifold in a de Sitter space \({S^{n+p}_{p}(1)}\) with R = aH + b, where R and H are the normalized scalar curvature and the length of the mean curvature vector respectively. In this paper, we give intrinsic and extrinsic conditions for M n to be totally umbilical, respectively.
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Abe N., Koike N., Yamaguchi S.: Congruence theorems for proper semi-Riemannian hypersurfaces in a real space form. Yokohama Math. J. 35, 123–136 (1987)
Aiyama R.: Compact spacelike m-submanifolds in a pseudo-Riemannian sphere \({S^{m+p}_{p_{}}(c)}\). Tokyo J. Math. 18, 81–90 (1995)
Akutagawa K.: On spacelike hypersurfaces with constant mean curvature in the de Sitter space. Math. Z 196, 13–19 (1987)
Alias L.J.: On the Ricci curvature of compact spacelike hypersurfaces in de Sitter space. Geom. Dedicata. 77, 297–304 (1999)
Brasil A. Jr, Colares A.G., Palmas O.: Complete hypersurfaces with constant scalar curvature in spheres. Monatshefte für Math. 161, 369–380 (2010)
Camargo F.E.C., Chaves R.M.B., Sousa L.A.M. Jr: Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in de Sitter space. Differ. Geom. Appl. 26, 592–599 (2008)
Caminha A.: A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds. Differ. Geom. Appl. 24, 652–659 (2006)
Cheng Q.M.: Complete space-like submanifolds in de Sitter space with parallel mean curvature vector. Math. Z 206, 333–339 (1991)
Cheng Q.M.: Complete space-like hypersurfaces of a de Sitter space with r = kH. Mem. Fac. Sci. Kyushu Univ. 44, 67–77 (1990)
Cheng Q.M., Ishikawa S.: Space-like hypersurfaces with constant scalar curvature. Manuscr. Math. 95, 499–505 (1998)
Cheng S.Y., Yau S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–240 (1977)
Goddard A.J.: Some remarks on the existence of spacelike hypersurfaces of constant mean curvature. Math. Proc. Camb. Phil. Soc. 82, 489–495 (1977)
Hou Z.H., Yang D.: Linear Weingarten spacelike hypersurfaces in de Sitter space. Bull. Belg. Math. Soc. Simon Stevin. 17, 769–780 (2010)
Kashani S.M.B.: On some compact spacelike submanifolds of pseudo-sphere. Geom. Dedicata 108, 125–130 (2004)
Li H.: Global rigidity theorems of hypersurfaces. Ark. Mat. 35, 327–351 (1997)
Liu X.M.: Space-like submanifolds in de Sitter spaces. J. Phys. A Math. Gen. 34, 5463–5468 (2001)
Montiel S.: An integral inequality for compact spacelike hypersurfaces in the de Sitter space and application to case of constant mean curvature. Indiana Univ. Math. J. 37, 909–917 (1988)
Okumura M.: Hypersurfaces and a pinching problem on the second fundamental tensor. Am. J. Math. 96, 207–213 (1874)
Ramanathan J.: Complete space-like hypersurfaces of constant mean curvature in the de Sitter spaces. Indiana Univ. Math. J. 36, 349–359 (1987)
Zhang, J.F.: Submanifolds with constant scalar curvature and the harmonic function of Finsler manifold. Ph.D. dissertation, Zhejiang university (2005)
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Yang, D., Hou, Z. Linear Weingarten spacelike submanifolds in de Sitter space. J. Geom. 103, 177–190 (2012). https://doi.org/10.1007/s00022-012-0110-x
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DOI: https://doi.org/10.1007/s00022-012-0110-x