The Rigidity Characterizations Of Some Special Riemannian Manifolds

It is well known that the study of rigidity theorem on Riemannian manifolds is an important subject in submanifold geometry.This paper aims to study rigidity theorems on some special Riemannian manifolds.The main structure arrangement is shown as follows:In the second chapter,?-Bach tensor Bij? is defined on the basis of Bach tensor Bij.Under the conditions of the manifold compact and the manifold complete,we explore under which conditions the manifold will be Einstein manifold,and obtain the relevant rigidity theorems on ?-Bach flat manifolds.In the third chapter,based on the results of Catino's study on the rigidity theorems of manifolds with normal quantitative curvature,the condition that the quantitative curvature is constant is removed and some rigidity theorems of manifolds with positive quantitative curvature are obtained.Finally,in the fourth chapter,based on the Schouten tensor P,a generalized Schouten tensor P? is defined.And then the k-curvature ?k(P?)and k-curvature functional defined by PT are constructed as follows:and then we can deduce the condition that satisfies ?k(P?)is constant if Riemannian manifold Mn is compact and g is critical metric of Mn,and obtained the first and second variational formula of Fk?(g).