Submanifold With Parallel Second Fundamental Form In A Riemannian Manifold

An important problem in differential geometry is the study of relations between the geometric strtcture and the geometric invariants of Riemannian submanifolds. The submanifold with parallel second fundamental form and submaniflod with parallel mean vector is two kind of special submanifolds, the studys of embodiment nature about these two kind of sub-manifolds is also belongs to this category. Does not flow swiftly aftermany great mathematicians diligently, has already obtained the verygreat achievement to this question research! This article studies inthe predecessor in the foundation continues to this question toconduct the research In this paper .Making use of 1-forms and geometric inequalities ,We prove a intergaral inequality for submanifolds with parallel second fundamental form in a Riemannian manifold .Second,we prove a pinching theorem for Pseudo-umbilical submanifolds with parallel second fundamental in a complete simply connected pinched Riemannian manifold .Finally,we obtain a pinching theorem for the square norm of the second fundamental form of M under the assumption that M is a submanifolds with parallel second fundamental in a Riemannian manifold.