Some Rigidity Results Of Riemannian Manifolds

In this thesis,we studied the rigidity problems of two kinds Riemannian manifolds with parallel Cotton tensor Riemannian manifolds and Bach-flat Riemannian manifolds.The main content and the structure of this thesis are as follows:In section 1,the research backgrounds of parallel Cotton tensor Riemannian manifolds and Bach-flat Riemannian manifolds were introduced.In addition,the main results were given.In section 2,we gave some sufficient conditions for stochastically complete Riemannian manifolds with parallel Cotton tensor to be either Einstein or constant sectional curvature.In particular,this chapter proposed a complete classification for a parabolic complete four-dimensional Riemannian manifolds with parallel Cotton tensor and positive constant scalar curvature.In section 3,under certain curvature pinching conditions,we analyzed the rigidity problems of the stochastically complete Bach-flat Riemannian manifolds.Moreover,we obtained an optimal rigidity theorem for an n-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature.