Annals of Global Analysis and Geometry

, Volume 50, Issue 1, pp 65–84 | Cite as

The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

  • Nurlan Abievich Abiev
  • Yuriĭ Gennadievich Nikonorov


This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces \(\mathrm{SU}(3)/T_{\max }\), \(\mathrm{Sp}(3)/\mathrm{Sp}(1)\times \mathrm{Sp}(1)\times \mathrm{Sp}(1)\), and \(F_4/\mathrm{Spin}(8)\). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces \(\mathrm{Sp}(3)/\mathrm{Sp}(1)\times \mathrm{Sp}(1)\times \mathrm{Sp}(1)\) and \(F_4/\mathrm{Spin}(8)\), the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.


Wallach space Generalized Wallach space Riemannian metric Ricci curvature Ricci flow Scalar curvature Sectional curvature Planar dynamical system Singular point 

Mathematics Subject Classification

Primary 53C30 Secondary 53C44 37C10 34C05 



The authors are grateful to the anonymous referee for helpful comments and suggestions that improved the presentation of this paper. The authors are indebted to Prof. Christoph Böhm, to Prof. Nolan R. Wallach, and to Prof. Wolfgang Ziller for helpful discussions concerning this paper. The project was supported by Grant 1452/GF4 of Ministry of Education and Sciences of the Republic of Kazakhstan for 2015–2017.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.M. Kh. Dulaty Taraz State UniversityTarazKazakhstan
  2. 2.Southern Mathematical Institute of VSC RASVladikavkazRussia

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