Sphere With Flat Normal Plexus Sub-manifold Outside Of Yun Geometric Study

In this paper, we discuss the square of the norm of the second foundmental form and the mean curvature of submanifolds in the Riemannian manifold N^n+p(c) with constant curvature. Besides, we study the submanifolds of S^n+p(1) with parallel mean curvature vector field and flat normal bundle, and obtain the following theorem:Main Theorem: Let Mⁿ be a compact submanifold of S^n+p(1) with mean curvature vector field and flat normal bundle. If S < α(n,H), then Mⁿ must be the umbilical hypersurface in Sⁿ⁺¹(1), whereOur theorem is a generalization of the rigidity theorem due to X. H. Mo[16].