Abstract: Let Mⁿ be an n（n≥4）-dimensional compact oriented submanifold in the nonnegative space forms N^n+p（c） with S ≤ S（c,H）.Then Mⁿ is either homeomorphic to a standard n-dimensional sphere Sⁿ or isometric to a Clifford torus.We also prove that a2 xt-2compact oriented submanifold in any N^n+p（c） is diffeomorphic to a sphere if S ≤（n²H²）/（n-1）+2c. 

Key words: submanifolds, principle curvature, Ricci curvature