Let M^n be an n（n≥4）-dimensional compact oriented submanifold in the non- negative space forms N^n＋p（c） with S ≤ S（c,H）. Then M^n is either homeomorphic to a standard n-dimensional sphere S^n or isometric to a Clifford torus. We also prove that a compact oriented submanifold in any N^n＋p （c） is diffeomorphic to a sphere if S≤（n^2H^2）/（n-1）＋2c.
In this paper, we consider existence and uniqueness of positive solutions to three coupled nonlinear SchrSdinger equations which appear in nonlinear optics. We use the behaviors of minimizing sequences for a bound to obtain the existence of positive solutions for three coupled system. To prove the uniqueness of positive solutions, we use the radial symmetry of positive solutions to transform the system into an ordinary differential system, and then integrate the system. In particular, for N = 1, we prove the uniqueness of positive solutions when 0≤ β=μ1 = μ2 =μ3 or β 〉 max{l,μ2,μ3}.
