Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Geometric Flows

Ed. by Carfora, Mauro / Mantegazza, Carlo

1 Issue per year

Open Access
See all formats and pricing
More options …

The Cotton Tensor and the Ricci Flow

Carlo Mantegazza / Samuele Mongodi / Michele Rimoldi
  • Corresponding author
  • Dipartimento di Scienze Matematiche "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-11-30 | DOI: https://doi.org/10.1515/geofl-2017-0001


We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.

Keywords: Ricci flow; Cotton tensor; Bach tensor; Ricci solitons

MSC 2010: 53C21; 53C25


  • [1] A. L. Besse, Einstein manifolds, Springer-Verlag, Berlin, 2008.Google Scholar

  • [2] H.-D. Cao, G. Catino, Q. Chen, C. Mantegazza, and L. Mazzieri, Bach-flat gradient steady Ricci solitons, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 125-138.Google Scholar

  • [3] H.-D. Cao, B.-L. Chen, and X.-P. Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, vol. 12, Int. Press, Somerville, MA, 2008, pp. 47-112.Google Scholar

  • [4] H.-D. Cao and Q. Chen, On locally conformally flat gradient steady Ricci solitons, Trans. Amer. Math. Soc. 364 (2012), 2377-2391.Google Scholar

  • [5] G. Catino and C. Mantegazza, Evolution of the Weyl tensor under the Ricci flow, Ann. Inst. Fourier (2011), 1407-1435.Google Scholar

  • [6] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Springer-Verlag, 1990.Google Scholar

  • [7] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), no. 2, 255-306.Google Scholar

About the article

Received: 2017-04-20

Accepted: 2017-10-23

Published Online: 2017-11-30

Published in Print: 2017-01-27

Citation Information: Geometric Flows, Volume 2, Issue 1, Pages 49–71, ISSN (Online) 2353-3382, DOI: https://doi.org/10.1515/geofl-2017-0001.

Export Citation

© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in