Rigidity of Riemannian manifolds with positive scalar curvature



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.


Yamabe invariant Rigidity Bach-flat Harmonic curvature 

Mathematics Subject Classification

Primary 53C24 Secondary 53C21 



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.


  1. 1.
    Besse, A.L.: Einstein Manifolds. Springer-Verlag, Berlin (2008)MATHGoogle Scholar
  2. 2.
    Bach, R.: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. 9, 110–135 (1921)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barros, A., Diógenes, R., Ribeiro Jr., E.: Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary. J. Geom. Anal. 25, 2698–2715 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bour, V.: Fourth order curvature flows and geometric applications. arXiv:1012.0342 [math.DG] (2010)
  5. 5.
    Bourguignon, J.-P.: The “magic” of Weitzenböck formulas, Variational methods (Paris, 1988), pp.. 251–271, Progress in Nonlinear Differential Equations and their Applications, vol. 4, Birkhäuser Boston, Boston (1990)Google Scholar
  6. 6.
    Calderbank, D., Gauduchon, P., Herzlich, M.: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173, 214–255 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao, H.-D., Chen, Q.: On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162, 1149–1169 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Catino, G.: Integral pinched shrinking Ricci solitons. Adv. Math. 303, 279–294 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Derdzinski, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compos. Math. 49, 405–433 (1983)MATHGoogle Scholar
  10. 10.
    do Carmo, M.P.: Riemannian Geometry, Mathematics: Theory & Applications. Birkhäuser, Boston (1992). (translated from the second Portuguese edition by Francis Flaherty) CrossRefGoogle Scholar
  11. 11.
    Fang, Y., Yuan, W.: A sphere theorem for Bach-flat manifolds with positive constant scalar curvature. arXiv:1704.06633
  12. 12.
    Fu, H.-P., Xiao, L.-Q.: Rigidity theorem for integral pinched shrinking Ricci solitons. arXiv:1510.07121
  13. 13.
    Hebey, E., Vaugon, M.: Effective \(L_p\) pinching for the concircular curvature. J. Geom. Anal. 6, 531–553 (1996)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Huang, G.Y.: Integral pinched gradient shrinking \(\rho \)-Einstein solitons. J. Math. Anal. Appl. 451, 1045–1055 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Yuan, W.: Volume comparision with respect to scalar curvature. arXiv:1609.08849

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© 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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