On Compact Riemannian Manifolds with Harmonic Weyl Curvature

  • Hai-Ping Fu
  • Hui-Ya HeEmail author


We give some rigidity theorems for an n-dimensional (\(n\ge 4\)) compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant \(\sigma _2\) curvature. Moreover, we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant \(\sigma _2\) curvature is isometric to a quotient of the round \(\mathbb {S}^4\).


Einstein manifold harmonic Weyl curvature Schouten tensor 

Mathematics Subject Classification

Primary 53C21 Secondary 53C20 



The authors are very grateful to Professor Haizhong Li for his guidance and constant support.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangPeople’s Republic of China
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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