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Rigidity of Riemannian manifolds with positive scalar curvature

Article

Abstract

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the \(L^{\frac{n}{2}}\)-norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.

Keywords

Yamabe invariant Rigidity Bach-flat Harmonic curvature 

Mathematics Subject Classification

Primary 53C24 Secondary 53C21 

Notes

Acknowledgements

The author want to thank the referee for helpful suggestions which make the paper more readable. He also want to thank Professor Xingxiao Li for stimulating discussions on this subject.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHenan Normal UniversityXinxiangPeople’s Republic of China

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