Shu, Shichang Complete spacelike hypersurfaces in a de Sitter space. (English) Zbl 1098.53051 Bull. Aust. Math. Soc. 73, No. 1, 9-16 (2006). Summary: We characterise the \(n\)-dimensional \((n\geq 3)\) complete space-like hyper-surfaces \(M^n\) in a de Sitter space \(S_1^{n+1}\) with constant scalar curvature and with two distinct principal curvatures. We show that if the multiplicities of such principal curvatures are greater than 1, then \(M^n\) is isometric to \(H^k(\sinh r)\times S^{n-k}(\cosh r)\), \(1< k< n-1\). In particular, when \(M^n\) is the complete space-like hypersurfaces in \(S_1^{n+1}\) with the scalar curvature and the mean curvature being linearly related, we also obtain a characteristic theorem of such hypersurfaces. Cited in 1 ReviewCited in 7 Documents MSC: 53C40 Global submanifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics Keywords:multiplicities; principal curvatures; scalar curvature; mean curvature × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Abe, Yokohama Math. J. 35 pp 123– (1987) [2] DOI: 10.1007/BF01425237 · Zbl 0349.53041 · doi:10.1007/BF01425237 [3] DOI: 10.2307/2160241 · Zbl 0802.53017 · doi:10.2307/2160241 [4] DOI: 10.1007/BF01179263 · Zbl 0611.53047 · doi:10.1007/BF01179263 [5] DOI: 10.1007/BF00773403 · Zbl 0863.53040 · doi:10.1007/BF00773403 [6] DOI: 10.1016/0926-2245(96)00006-X · Zbl 0848.53035 · doi:10.1016/0926-2245(96)00006-X [7] DOI: 10.1512/iumj.1987.36.36020 · Zbl 0626.53041 · doi:10.1512/iumj.1987.36.36020 [8] DOI: 10.2307/2373502 · Zbl 0196.25102 · doi:10.2307/2373502 [9] DOI: 10.2307/2373587 · Zbl 0302.53028 · doi:10.2307/2373587 [10] DOI: 10.2748/tmj/1178225410 · Zbl 0848.53039 · doi:10.2748/tmj/1178225410 [11] Nomizu, Japan. J. Math. 7 pp 217– (1981) [12] DOI: 10.1007/s002290100187 · Zbl 1002.53043 · doi:10.1007/s002290100187 [13] A.B., Illinois J. Maths. 47 pp 847– (2003) [14] DOI: 10.1017/S0305004100054153 · Zbl 0386.53042 · doi:10.1017/S0305004100054153 [15] DOI: 10.2206/kyushumfs.44.67 · Zbl 0732.53044 · doi:10.2206/kyushumfs.44.67 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.