LP RICCI CURVATURE PINCHING THEOREMS FOR CONFORMALLY FLAT RIEMANNIAN MANIFOLDS

摘要

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" /2 norm of the Ricci curvature tensor of M is pinched in [0, C1 (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 0, and the L"/2 norm of the trace-free Ricci curvature tensor of M is pinched in [0, C2 (n)), then M is isometric to a space form. Finally, we prove an L" trace-free Ricci curvature pinching theorem for complete locally conformally flat Riemannian manifolds with constant nonzero scalar curvature. Here C 1 (n) and C2 (n) are explicit positive constants depending only on n.