Some Rigidity Theorems On Riemannian Manifolds

The classification of Riemannian manifolds is always an important problem in Dif-ferential Geometry.In this paper,we study rigidity problems of some special Riemannian manifolds.The main results are the following five parts:1.Let(Mn,g)(n?4)be an n-dimensional compact locally conformally flat Riemannian manifold with constant scalar curvature and constant squared norm of Ricci curvature.Applying moving frame method,we prove that such kind of Riemannian manifold does not exist if its Ricci curvature tensor has three distinct eigenvalues.2.We prove that an n-dimensional(n?4)compact Bach-flat manifold with positive scalar curvature and positive constant ?2 is an Einstein manifold,provided that its Weyl curvature satisfies a suitable pinching condition.3.We give some rigidity theorems for an n-dimensional(n?4)compact Riemannian manifold with harmonic Weyl tensor,positive scalar curvature and positive constant ?2.Moreover,when n=4,we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant ?2 is isometric to a quotient of the round S4.4.A 3-dimensional compact,oriented,connected Miao-Tam critical metric(M3,g,f)with smooth boundary(?)M and non-negative Ricci curvature is isometric to a geodesic ball in a simply connected space form R3 or S3.5.We give a new proof of n-dimensional(n?3)CPE conjecture,if divC=0 and 3-dimensional CPE conjecture,if div3C=0.Moreover,we prove 3-dimensional CPE conjecture with non-negative Ricci curvature.