Science China Mathematics

, Volume 62, Issue 3, pp 569–584 | Cite as

Invariant Einstein metrics on generalized Wallach spaces

  • Zhiqi ChenEmail author
  • Yuriĭ Gennadievich Nikonorov


Invariant Einstein metrics on generalized Wallach spaces have been classified except SO(k + l + m)/SO(k) × SO(l) × SO(m). In this paper, we first give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type SO(k+l+m)/SO(k) × SO(l) × SO(m) admitting exactly two, three, or four invariant Einstein metrics up to a homothety.


compact homogeneous space generalized Wallach space symmetric space homogeneous Riemannian metric Einstein metric Ricci flow 


53C20 53C25 53C30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by Ministry of Education and Sciences of the Republic of Kazakhstan for 2015–2017 (Agreement N 299, February 12, 2015) (Grant No. 1452/GF4). The authors are grateful to the referees for helpful comments and suggestions that improved the presentation of this paper.


  1. 1.
    Abiev N-A. On topological structure of some sets related to the normalized Ricci flow on generalized Wallach spaces. Vladikavkaz Math J, 2015, 17: 5–13MathSciNetGoogle Scholar
  2. 2.
    Abiev N-A, Arvanitoyeorgos A, Nikonorov Y-G, et al. The dynamics of the Ricci flow on generalized Wallach spaces. Differential Geom Appl, 2014, 35: 26–43MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abiev N-A, Arvanitoyeorgos A, Nikonorov Y-G, et al. The Ricci flow on some generalizedWallach spaces. In: Geometry and Its Applications. Springer Proceedings in Mathematics Statistics, vol. 72. Switzerland: Springer, 2014, 3–37CrossRefGoogle Scholar
  4. 4.
    Arvanitoyeorgos A. New invariant Einstein metrics on generalized flag manifolds. Trans Amer Math Soc, 1993, 337: 981–995MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Batkhin A-B. A real variety with boundary and its global parameterization. Program Comput Softw, 2017, 43: 75–83MathSciNetCrossRefGoogle Scholar
  6. 6.
    Batkhin A-B, Bruno A-D. Investigation of a real algebraic surface. Program Comput Softw, 2015, 41: 74–83MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Besse A-L. Einstein Manifolds. Berlin: Springer-Verlag, 1987CrossRefzbMATHGoogle Scholar
  8. 8.
    Chen Z, Kang Y, Liang K. Invariant Einstein metrics on three-locally-symmetric spaces. Comm Anal Geom, 2016, 24: 769–792MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen Z, Nikonorov Y-G, Nikonorova Y-V. Invariant Einstein metrics on Ledger-Obata spaces. Differential Geom Appl, 2017, 50: 71–87MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D’Atri J-E, Nickerson N. Geodesic symmetries in space with special curvature tensors. J Differential Geom, 1974, 9: 251–262MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kimura M. Homogeneous Einstein metrics on certain K¨ahler C-spaces. Adv Stud Pure Math, 1990, 18: 303–320CrossRefzbMATHGoogle Scholar
  12. 12.
    Ledger A-J, Obata M. Affine and Riemannian s-manifolds. J Differential Geom, 1968, 2: 451–459MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lomshakov A-M, Nikonorov Y-G, Firsov E-V. Invariant Einstein metrics on three-locally-symmetric spaces. Siberian Adv Math, 2004, 14: 43–62MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nikonorov Y-G. On a class of homogeneous compact Einstein manifolds. Sib Math J, 2000, 41: 168–172MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nikonorov Y-G. Invariant Einstein metrics on the Ledger-Obata spaces. St Petersburg Math J, 2003, 14: 487–497MathSciNetzbMATHGoogle Scholar
  16. 16.
    Nikonorov Y-G. Classification of generalized Wallach spaces. Geom Dedicata, 2016, 181: 193–212MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nikonorov Y-G, Rodionov E-D, Slavskii V-V. Geometry of homogeneous Riemannian manifolds. J Math Sci (NY), 2007, 146: 6313–6390MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rodionov E-D. Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature. Sib Math J, 1991, 32: 455–459CrossRefzbMATHGoogle Scholar
  19. 19.
    Wallach N-R. Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann of Math (2), 1972, 96: 277–295MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang M-Y. Einstein metrics from symmetry and bundle constructions. In: Surveys in Differential Geometry: Essays on Einstein Manifolds. Surveys in Differential Geometry, vol. 6, Boston: Int Press, 1999, 287–325Google Scholar
  21. 21.
    Wang M-Y. Einstein metrics from symmetry and bundle constructions: A sequel. In: Differential Geometry. Advanced Lectures in Mathematics, vol. 22. Somerville: Int Press, 2012, 253–309Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina
  2. 2.Southern Mathematical Institute of Vladikavkaz Scientific Centre of the Russian Academy of SciencesVladikavkazRussia

Personalised recommendations