| Denote by S the square norm of the second fundamental form. In the second section, we fist prove a generalized Simons integral inequality. Second, we prove a pinching theorem on S. The results generalize the results obtained by Hong wei Xu. In the third second, we study a closed oriented submanifold with parallel mean curvature vector field in a sphere. Let S = S-nH2. we fist obtain a pinching theorem on 5. which generalizes the results obtained by H. Alencar and M. do Carrno. Second, we prove a pinching theorem on the Ricci curvature of the subrnanifold. |
