Complete Manifolds with Harmonic Curvature and Finite $L^p$Norm Curvature 
Received:January 25, 2016 Revised:February 27, 2017 
Key Word:
Harmonic curvature tracefree curvature tensor constant curvature space

Fund ProjectL:Supported by the National Natural Science Foundations of China (Grant Nos.11261038; 11361041) and the Natural Science Foundation of Jiangxi Province (Grant No.20132BAB201005). 

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Abstract: 
Let $(M^n, g)~(n\geq3)$ be an $n$dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by $R$ and $\mathring{Rm}$ the scalar curvature and the tracefree Riemannian curvature tensor of $M$, respectively. The main result of this paper states that $\mathring{Rm}$ goes to zero uniformly at infinity if for $p\geq n$, the $L^{p}$norm of $\mathring{Rm}$ is finite. As applications, we prove that $(M^n, g)$ is compact if the $L^{p}$norm of $\mathring{Rm}$ is finite and $R$ is positive, and $(M^n, g)$ is scalar flat if $(M^n, g)$ is a complete noncompact manifold with nonnegative scalar curvature and finite $L^{p}$norm of $\mathring{Rm}$. We prove that $(M^n, g)$ is isometric to a spherical space form if for $p\geq \frac n2$, the $L^{p}$norm of $\mathring{Rm}$ is sufficiently small and $R$ is positive. In particular, we prove that $(M^n, g)$ is isometric to a spherical space form if for $p\geq n$, $R$ is positive and the $L^{p}$norm of $\mathring{Rm}$ is pinched in $[0,C)$, where $C$ is an explicit positive constant depending only on $n, p$, $R$ and the Yamabe constant. 
Citation: 
DOI:10.3770/j.issn:20952651.2017.03.011 
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