Isospectrum And Totally Umbilical Of Submanifold And Remarks Unit Tangent Sphere Bundle

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 M_n1.n2 = Sⁿ₁((n₁/n)^1/2)× Sⁿ₂((n₂/n)^1/2) be a Clifford minimal hypersurface. If S pec^p(M) = S pec^p(M_n1.n2) and S pec^q (M) = S pec^q(M_n1.n2), (0≤p2 + q² - nq - np)≠ 0) then M isometric to M_n1.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.