The Rigidity Results Of Bach-flat Riemannian Manifold With Constant Scalar Curvature Under The L~p Curvature Pinching Condition

The curvature pinching problems play an important role in global differential geometry.This paper will study the rigidity problems of the complete Bach-flat Riemannian manifolds with constant scalar curvature under theL~p curvature pinching condition,and obtain the following main results: under someL~p curvature pinching conditions,the compact Bach-flat Riemannian manifolds with the positive constant scalar curvature and the complete,simply connected,locally conformally flat Riemannian manifolds with constant scalar curvature are isometric to the constant curvature space.This paper's structure is as follows:In section 1,the historical backgrounds of Bach-flat Riemannian manifolds and locally conformally flat Riemannian manifolds and the paper's main results are introduced.In section 2,the necessary formulas,definitions and lemmas of this paper are introduced.In section 3,We study the rigidity problems of the compact Bach-flat Riemannian manifolds with the positive constant scalar curvature under the L~p curvature pinching condition.In section 4,We study the rigidity problems of the complete,simply connected,locally conformally flat Riemannian manifolds with constant scalar curvature under theL~p curvature pinching condition.