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 Rigidity of complete Riemannian manifolds with vanishing Bach tensor Bull. Korean Math. Soc.Published online August 6, 2019 Guangyue Huang and Bingqing Ma Henan Normal University Abstract : For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant. Keywords : Sobolev constant, rigidity, vanishing Bach tensor MSC numbers : Primary 53C24, Secondary 53C21 Full-Text :