Let (M n , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R andRm the scalar curvature and the trace-free Riemannian curvature tensor of M , respectively. The main result of this paper states thatRm goes to zero uniformly at infinity if for p ≥ n 2 , the L p -norm ofRm is finite. Moreover, If R is positive, then (M n , g) is compact. As applications, we prove that (M n , g) is isometric to a spherical space form if for p ≥ n 2 , R is positive and the L p -norm ofRm is pinched in [0, C 1 ), where C 1 is an explicit positive constant depending only on n, p, R and the Yamabe constant.In particular, we prove an L p ( n 2 ≤ p < n−2 2 (1+ 1 − 4 n ))-norm ofRic pinching theorem for complete, simply connected, locally conformally flat Riemannian n(n ≥ 6)-manifolds with constant negative scalar curvature.We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which improves Thereom 1.1 and Corollary 1 of E. Hebey and M. Vaugon [18]. This rsult is sharped, and we can precisely characterize the case of equality.

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We prove that an n-dimensional, n ≥ 4, compact gradient shrinking Ricci soliton satisfying a L n 2 -pinching condition is isometric to a quotient of the round S n , which improves the rigidity theorem given by G. Catino [5]

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