Characterizations Of Spacelike Hypersurfaces In Einstein Spacetimes

The studies on spacelike submanifolds in Riemannian Geometry have been paid more attention to the physicists and geometers, it is convenient as initial surfaces for the Cauchy problem in arbitrary spacetimes and for studying the propagation of gravitational radia-tion, respectively. In this article, we mainly study the spacelike hypersurface Mn in the Einstein spacetime N1n+1. By supposing to obey certain appropriated conditions to cur-vature, we prove that a such spacelike hypersurface must be either totally umbilical or isoparametric.The primary results obtained in this paper include the following three part.1. The complete hypersurface Mn with constant mean curvature in Einstein spacetime N1n+1 will be studied. By assuming the hypersurface has two distinct principle, we prove that such a hypersurface must be isoparametric according to Hpof maximum principle, and meanwhile a pinching conclusion about supM |?|~2 is obtained, where |?|~2=S-nH~2.2. The space-like hypersurface Mn with harmonic Riemannian curvature tensor, namely the Ricci tensor is Codazzi-type tensor, in Einstein spacetime N1n+1 will be s-tudied. In case of complete, if the mean curvature of the hypersurface is constant, we prove that it must be a totally umbilical sphere. In case of compact, if the sectional curva-ture of the hypersurface is non-negative, or the squared norm of the second fundamental form S of the hypersurface satisfies ?~2?S?nH~2+?~2 and c2?0, then we prove that it must be a totally umbilical sphere.3. The linear Weingarten hypersurfaces, that is the hypersurfaces which the normal-ized scalar curvature R and the mean curvature H satisfy R=aH+b, where a,b?R, in Einstein spacetimes will be studied. In case of copmact, if the mean curvature of the hy-persurfaces satisfy 0, we prove that it must be totally umbilical. In case of complete, suppose that the hypersurfaces have two distinct principle curvature, with hen we prove that it must be isoparametric, and meanwhile a pinching conclusion about supM |?|~2 is obtained.