Abstract
The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.
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Acknowledgements
The first author would like to express his sincere thanks to Professor Zejun Hu and Professor Xi Zhang for their consistent assistance and encouragements. The second author would like to thank Professor Kefeng Liu and Professor Hongyu Wang for constant support and help.
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Yawei Chu was supported by grants of NSFC (No. 11371330), NSF of Anhui Provincial Education Department (No. KJ2014A196). Shouwen Fang was supported by grant of NSFC (No. 11401514).
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Chu, Y., Fang, S. Rigidity of complete manifolds with parallel Cotton tensor. Arch. Math. 109, 179–189 (2017). https://doi.org/10.1007/s00013-017-1047-y
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DOI: https://doi.org/10.1007/s00013-017-1047-y