For complete noncompact Riemannian manifolds with Bach-flat tensor andpositive constant scalar curvature, we provide a theorem which states that the$L^2$-norm of traceless Ricci curvature is unbounded under a pointwiseinequality involving the Weyl curvature and the traceless Ricci curvature.Furthermore, we prove some rigidity results under an inequality involving$L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature andthe Sobolev constant.
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机译：对于具有Bach平坦张量和正常数标量曲率的完全非紧黎曼流形，我们提供了一个定理，指出无痕Ricci曲率的$ L ^ 2 $范数在涉及Weyl曲率和无痕Ricci曲率的点向不等式下是无界的。 ，我们证明了在涉及Weyl曲率，无迹Ricci曲率和Sobolev常数的$ L ^ {\ frac {n} {2}} $-范数的不等式下的一些刚性结果。
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