In this thesis, we mainly study the relations between compact minimal hypersurfaces of Sn+1(1) and the Clifford minimal hypersurface of Sn+1(1) by their spectrums, and the relations between unit tangent sphere bundle T1 M and its base manifold M. Besides,we also study the totally umbilical space-like hpersurfaces in De Sitter.In chapter one, we introduce some studies of submanifolds.In chapter two, we obtain the following result: Let M be a compact minimal hypersurface of Sn+1(1) and Mn1.n2 = Sn1((n1/n)1/2)× Sn2((n2/n)1/2) be a Clifford minimal hypersurface. If S pecp(M) = S pecp(Mn1.n2) and S pecq (M) = S pecq(Mn1.n2), (0≤p2 + q2 - nq - np)≠0) then M isometric to Mn1.n2 .In chapter three, we study the relation between a Riemannian manifold and its unit tangent sphere bundle equipped with a standard contact metric structure. We find there exists no unit tangent sphere bundle which has constant curvature for a Riemannian manifold Mn+1(n ≥ 2) and a necessary and sufficient condition for the base manifold of a unit tangent sphere to have constant curvature. Futher, we also derive some other results about them.In chapter four, we consider compact spacelike hypersurfaces in De Sitter space with constant nonnegative r-mean curvature. We prove that compact spacelike hypersurfaces are totally umbilical under appopariate hypothesis.
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