The Ricci Flow on Some Generalized Wallach Spaces

  • N. A. Abiev
  • A. Arvanitoyeorgos
  • Yu. G. Nikonorov
  • P. Siasos
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 72)


We study the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as a special planar dynamical system. All nonsymmetric generalized Wallach spaces can be naturally parametrized by three positive numbers \(a_{1},a_{2},a_{3}\). Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points \((a_{1},a_{2},a_{3})\) in the cube \((0,1/2] \times (0,1/2] \times (0,1/2]\). We also consider in detail some important partial cases.


Riemannian metric Einstein metric Generalized Wallach space Ricci flow Ricci curvature Planar dynamical system Real algebraic surface 

Mathematics Subject Classification (2010):

201053C30 53C44 37C10 34C05 14P05 



The authors are indebted to Prof. Yusuke Sakane for useful discussions concerning computational aspects of this project and to Tanya Nikonorova for her help with the graphics. The project was supported in part by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (grant NSh-921.2012.1) and by Federal Target Grant “Scientific and educational personnel of innovative Russia” for 2009–2013 (agreement no. 8206, application no. 2012-1.1-12-000-1003-014).


  1. 1.
    Abiev N.A., Arvanitoyeorgos A., Nikonorov Yu.G., Siasos P. The dynamics of the Ricci flow on generalized Wallach spaces // Differential Geom. Appl. (2014),
  2. 2.
    Anastassiou S., Chrysikos I. The Ricci flow approach to homogeneous Einstein metrics on flag manifolds // J. Geom. Phys. (2011), V. 61, No. 8, P. 1587–1600.Google Scholar
  3. 3.
    Arvanitoyeorgos A. New invariant Einstein metrics on generalized flag manifolds // Trans. Amer. Math. Soc. (1993), V. 337, No. 2, P. 981–995.Google Scholar
  4. 4.
    Arnold V. I., Gusein-Zade S. M. and Varchenko A. N. Singularities of differentiable maps. Vol. I, Monogr. Math. 82, Birkhäuser Boston, Inc., Boston, MA, 1986.Google Scholar
  5. 5.
    Besse A. L. Einstein Manifolds. Springer-Verlag. Berlin, etc., 1987.Google Scholar
  6. 6.
    Böhm C., Wilking B. Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature // GAFA Geom. Func. Anal. (2007), v. 17, P. 665–681.Google Scholar
  7. 7.
    Buzano M. Ricci flow on homogeneous spaces with two isotropy summands // Ann. Glob. Anal. Geom. (2014), V. 45, No. 1, P. 25–45.Google Scholar
  8. 8.
    Chow B., Knopf D. The Ricci Flow: an Introduction. Mathematical Surveys and Monographs, AMS, Providence, RI, 2004.Google Scholar
  9. 9.
    D’Atri J. E., Nickerson N. Geodesic symmetries in space with special curvature tensors // J. Different. Geom. (1974), V. 9, P. 251–262.Google Scholar
  10. 10.
    D’Atri J. E., Ziller W. Naturally reductive metrics and Einstein metrics on compact Lie groups // Mem. Amer. Math. Soc. (1979), V. 215, P. 1–72.Google Scholar
  11. 11.
    Darboux G. Lecȩons sur la Théorie générale des Surfaces, IV, Gauthier-Villars, Paris, 1896.Google Scholar
  12. 12.
    Dumortier F., Llibre J., Artes J. Qualitative theory of planar differential systems. Universitext. Springer-Verlag, Berlin, 2006. xvi+298 pp.Google Scholar
  13. 13.
    Glickenstein D. Payne T. L. Ricci flow on three-dimensional, unimodular metric Lie algebras // Commun. Anal. Geom. (2010), V. 18,  No. 5, P. 927–961.Google Scholar
  14. 14.
    Hamilton R. S. Three-manifolds with positive Ricci curvature // J. Differential Geom. (1982), V. 17, P. 255–306.Google Scholar
  15. 15.
    Isenberg J., Jackson M., Lu P. Ricci flow on locally homogeneous closed 4-manifolds // Comm. Anal. Geom. (2006), V. 14, No. 2, P. 345–386.Google Scholar
  16. 16.
    Jiang Q., Llibre J. Qualitative classification of singular points // Qual. Theory Dyn. Syst. (2005), V. 6, No. 1, P. 87–167.Google Scholar
  17. 17.
    Kimura M. Homogeneous Einstein metrics on certain Kähler C-spaces // Adv. Stud. Pure Math. (1990), V. 18, No. 1, P. 303–320.Google Scholar
  18. 18.
    Lauret J. Ricci flow of homogeneous manifolds // Math. Z. (2013), V. 274, No. 1–2, P. 373–403.Google Scholar
  19. 19.
    Lomshakov A. M., Nikonorov Yu. G., Firsov E. V. On invariant Einstein metrics on three-locally-symmetric spaces // Doklady Mathematics (2002), V. 66, No. 2, P. 224–227.Google Scholar
  20. 20.
    Lomshakov A. M., Nikonorov Yu. G., Firsov E. V. Invariant Einstein Metrics on Three-Locally-Symmetric Spaces // Matem. tr. (2003), V. 6, No. 2. P. 80–101 (Russian); English translation in: Siberian Adv. Math. (2004), V. 14, No. 3, P. 43–62.Google Scholar
  21. 21.
    Nikonorov Yu. G., Rodionov E. D., Slavskii V. V. Geometry of homogeneous Riemannian manifolds // Journal of Mathematical Sciences (New York) (2007), V. 146, No. 7, P. 6313–6390.Google Scholar
  22. 22.
    Nikonorov Yu. G. On a class of homogeneous compact Einstein manifolds // Sibirsk. Mat. Zh. (2000), V. 41, No. 1, P. 200–205 (Russian); English translation in: Siberian Math. J. (2000), V. 41, No. 1, P. 168–172.Google Scholar
  23. 23.
    Payne T. L. The Ricci flow for nilmanifolds // J. Mod. Dyn. (2010), V. 4, No. 1, P. 65–90.Google Scholar
  24. 24.
    Rodionov E. D. Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature // Sibirsk. Mat. Zh. (1991), V. 32, No. 3, P. 126–131 (Russian); English translation in: Siberian Math. J. (1991), V. 32, No. 3, P. 455–459.Google Scholar
  25. 25.
    Thom R. Topological models in biology // Topology (1969), V. 8, P. 313–335.Google Scholar
  26. 26.
    Topping P. Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, vol. 325, Cambridge University Press, Cambridge, 2006.Google Scholar
  27. 27.
    Wallach N. Compact homogeneous Riemannian manifolds with strictly positive curvature // Annals of Mathematics, Second Series. (1972), V. 96, P. 277–295.Google Scholar
  28. 28.
    Woodcock A. E. R., Poston T. A geometrical study of the elementary catastrophes, Lecture Notes in Mathematics, vol. 373, Springer-Verlag, Berlin-New York, 1974.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • N. A. Abiev
    • 1
  • A. Arvanitoyeorgos
    • 2
  • Yu. G. Nikonorov
    • 3
  • P. Siasos
    • 2
  1. 1.Taraz State University named after M.Kh. DulatyTarazKazakhstan
  2. 2.Department of MathematicsUniversity of PatrasRionGreece
  3. 3.South Mathematical Institute of Vladikavkaz Scientific Centre of the Russian Academy of SciencesVladikavkazRussia

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