| The study on uniqueness of spacelike hypersurface in semi-Riemannian manifolds is mo-tived by physical and mathematical interests. In recent years, there is a steady growing interest in the study of uniqueness of hypersurface immersed into a semi-Riemannian warped warped product space ?R ×fMn (?= ±1), meanwhile, fruitful works have been obtained.In this paper, we study the uniqueness of spacelike hypersurface immersed in semi-Riemannian warped producted space by applying Omori-Yau maximum principal and a result of the Stokes'theorem on the spacelike hypersurface of semi-Riemannian manifolds, which extended positive higher mean curvature and derivative of warping function to the case of non-vanishing. The main contents of this paper is as follows:Firstly, we give a suitable restriction on the higher order mean curvature and the norm of gradient on the height function of the hypersurface we obtain some rigidity theorems in GRW (generalized Robertson-Walker) spacetimes.Secondly, we consider the ambient space is Lorentzian warped product space-R ×fMn with vanishing and non-vanishing f', respectively. When f' vanishing, we study the uniqueness of hypersurface immersed in -R × Mn, where the sectional curvature of fiber Mn is bounded from below; When f' non-vanishing, by applying Omori-Yau maximum principles and a result of Stokes'theorem, we establish the uniqueness results of Lorentzian warped product-R×f Mn with non-vanishing higher order mean curvatures. Moreover, there are some applications are given on -R ×t Hn and -R ×cosht Sn, etc.Finally, when the mean curvature non-vanishing, we obtain a sign relationship between the derivative of warping function with angle function by applying the Omori-Yau maximum principal, then imposing suitable restrictions on the higher order mean curvatures, we establish uniqueness results for entire graph in a Riemannian warped product space. Furthermore, some applications are given on (-?/2,?/2) ×cost Hn and R ×cosht Hn, etc. |
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