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Abstract
It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the kth mean curvature function Hk is a linear combination of Hk+1,…, Hn. This is a new angle to characterize round spheres.
Highlights
Let R1n+2 be the (n + 2)-dimensional Lorentz-Minkowski space, namely, the real vector space Rn+2 endowed with the Lorentzian inner product, given by n +1∑ ( ) ( ) v, w =vi wi − vn+2wn+2, v = v1, vn+2, w = w1, wn+2 ∈ Rn+2. i =1{ } the n-dimensional de Sitter space is defined by S1n+1 = x ∈ R1n+2 x, x = 1
In [2], Montiel showed that the only compact spacelike hypersurfaces in S1n+1 with constant mean curvature H1 were the totally umbilical round spheres
Theorem 1([4], Theorem 7) Let ψ : M n → S1n+1 ⊂ R1n+2 be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future of an equator of S1n+1
Summary
Let R1n+2 be the (n + 2)-dimensional Lorentz-Minkowski space, namely, the real vector space Rn+2 endowed with the Lorentzian inner product , given by n +1. Cheng and Ishikawa [3] have shown that the totally umbilical round spheres are the only compact spacelike hypersurfaces in de Sitter space with constant scalar curvature S < n(n −1). In [6] the authors considered that Hk is the linear combination of H1, , Hk−1 , and proved: Theorem 3 ([6]) Let ψ : M n → S1n+1 ⊂ R1n+2 be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of S1n+1. Theorem 4 Let ψ : M n → S1n+1 ⊂ R1n+2 be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of S1n+1. The corresponding theorem characterizes ellipsoids holds in affine differential geometry
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