Lp RICCI CURVATURE PINCHING THEOREMS FOR CONFORMALLY FLAT RIEMANNIAN MANIFOLDS

Let M be an n-dimensional complete locally conformally flat Riemannian manifold with constant scalar curvature R and n >= 3. We first prove that if R = 0 and the L-n/2 norm of the Ricci curvature tensor of M is pinched in [0, C-1(n)), then M is isometric to a complete flat Riemannian manifold, which improves Pigola, Rigoli, and Setti's pinching theorem. Next, we prove that if n >= 6, R not equal 0, and the L-n/2 norm of the trace-free Ricci curvature tensor of M is pinched in [0; C-2(n)), then M is isometric to a space form. Finally, we prove an L-n trace-free Ricci curvature pinching theorem for complete locally conformally flat Riemannian manifolds with constant nonzero scalar curvature. Here C-1(n) and C-2(n) are explicit positive constants depending only on n.