| Firstly, we calculate the Laplace of the square of the length of the second fundamental form for submanifold and introduce a self-adjoint second order operator in de Sitter space. Secondly, we define the higher mean curvature and use Hopf Theorem, the method of J.Simons' proof of his pinching theorem. Finally, we obtain some integral inequalities and rigidity theorems for hypersurfaces and submanifolds in de Sitter space. It mainly concludes the following:1. We obtain the rigidity theorem for hypersurfaces with constant mean curvature and nonnegative sectional curvature in de Sitter space. Also, the inequality for sectional curvature and Ricci curvature is got.2. A integral inequality of constant scalar curvature in de Sitter space is obtained. At the same time, we get the pinching theorem about the square of the length of the second fundamental form.3. We obtain some integral inequalities and rigidity theorems on condition that it uses the higher mean curvature in de Sitter space.4. The submanifolds get the pinching theorems, the integral inequality and some rigidity theorems with the sectional curvature and Ricci curvature, which have parallel mean curvature in de Sitter space.5. We obtain a rigidity theorem for submanifold with parallel the second fundamental form in de Sitter space. |
