Annals of Global Analysis and Geometry

, Volume 37, Issue 2, pp 185–219 | Cite as

Invariant Einstein metrics on flag manifolds with four isotropy summands

  • Andreas Arvanitoyeorgos
  • Ioannis ChrysikosEmail author
Original Paper


A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands by the use of \({\mathfrak{t}}\)-roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.


Homogeneous manifold Einstein metric Generalized flag manifold Isotropy representation \({\mathfrak{t}}\)-roots 

Mathematics Subject Classification (2000)

Primary 53C25 Secondary 53C30 


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  1. 1.
    Akhiezer D.N.: Lie Group Actions in Complex Analysis, Aspects of Mathematics, vol. E27. Vieweg, Braunschweig (1995)Google Scholar
  2. 2.
    Alekseevsky, D.V.: Homogeneous Einstein metrics. In: Differential Geometry and its Applications (Proccedings of the Conference), pp. 1–21. Univ. of. J. E. Purkyne, Chechoslovakia (1987)Google Scholar
  3. 3.
    Alekseevsky D.V., Arvanitoyeorgos A.: Riemannian flag manifolds with homogeneous geodesics. Trans. Am. Math. Soc. 359(8), 3769–3789 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alekseevsky D.V., Perelomov A.M.: Invariant Kähler-Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl. 20(3), 171–182 (1986)CrossRefGoogle Scholar
  5. 5.
    Alekseevsky, D.V., Spiro, A.F.: Flag Manifolds and Homogeneous CR Structures. Lecture notes given by the first author at the First Colloquium on Lie Theory and Applications in Spain (2000)Google Scholar
  6. 6.
    Arvanitoyeorgos A.: New invariant Einstein metrics on generalized flag manifolds. Trans. Am. Math. Soc. 337(2), 981–995 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Arvanitoyeorgos A.: Geometry of flag manifolds. Int. J. Geom. Methods Mod. Phys. 3(5-6), 1–18 (2006)MathSciNetGoogle Scholar
  8. 8.
    Arvanitoyeorgos, A., Chrysikos, I.: Invariant Einstein metrics on generalized flag manifolds with two isotropy summands, Preprint (2008), arXiv:0902.1826v1Google Scholar
  9. 9.
    Arvanitoyeorgos, A., Chrysikos, I.: Motion of charged particles and homogeneous geodesics in Kähler C-spaces with two isotropy summands, Tokyo J. Math. (to appear)Google Scholar
  10. 10.
    Besse A.L.: Einstein Manifolds. Springer-Verlag, Berlin (1986)Google Scholar
  11. 11.
    Bordeman M., Forger M., Römer H.: Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models. Comm. Math. Phys. 102, 604–647 (1986)CrossRefGoogle Scholar
  12. 12.
    Borel A.: Kahlerian coset spaces of semisimple Lie groups. Proc. Natl Acad. Sci. USA 40, 1147–1151 (1954)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Borel A., Hirzebruch F.: Characteristics classes and homogeneous spaces I. Am. J. Math. 80, 458–538 (1958)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Böhm C., Kerr M.: Low-dimensional homogeneous Einstein manifolds. Trans. Amer. Math. Soc. 358(4), 1455–1468 (2005)CrossRefGoogle Scholar
  15. 15.
    Böhm C., Wang M., Ziller W.: A variational approach for homogeneous Einstein metrics. Geom. Funct. Anal. 14(4), 681–733 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bourbaki N.: Éléments De Mathématique: Groupes et algèbres De Lie, Chapitres 4, 5 et 6. Masson Publishing, Paris (1981)Google Scholar
  17. 17.
    Boyer C.P., Galicki K.: On Sasakian-Einstein geometry. Int. J. Math. 11, 873–909 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Burstall F.E.: Riemannian twistor spaces and holonomy groups. In: Bailey, T.N., Baston, R.J. (eds) Twistors in Mathematics and Physics, pp. 53–70. Cambridge University Press, Cambridge (1990)Google Scholar
  19. 19.
    Burstall, F.E., Rawnsley, J.H.: Twistor Theory for Riemannian Symmetric Spaces. Lectures Notes in Mathematics, Springer-Verlag (1990)Google Scholar
  20. 20.
    Dickinson W., Kerr M.: The geometry of compact homogeneous spaces with two isotropy summands. Ann. Glob. Anal. Geom. 34, 329–350 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dos Santos E.C.F., Negreiros C.J.C.: Einstein metrics on flag manifolds. Revista Della, Unión Mathemática Argetina 47(2), 77–84 (2006)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Gorbatzevich, V.V., Onishchik, A.L., Vinberg, E.B.: Structure of Lie Groups and Lie Algebras, Encycl. Math. Sci. v41, Lie Groups and Lie Algebras-3, Springer–VerlagGoogle Scholar
  23. 23.
    Graev M.M.: On the number of invariant Einstein metrics on a compact homogeneous space, Newton polytopes and contraction of Lie algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1047–1075 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Heber J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978)zbMATHGoogle Scholar
  26. 26.
    Itoh M.: On curvature properties of Kähler C-spaces. J. Math. Soc. Jpn 30(1), 39–71 (1978)CrossRefGoogle Scholar
  27. 27.
    Kimura M.: Homogeneous Einstein metrics on certain Kähler C-spaces. Adv. Stud. Pure Math. 18(I), 303–320 (1990)Google Scholar
  28. 28.
    LeBrun, C., Wang, M. (eds): Surveys in Differential Geometry, Volume VI: Essays on Einstein Manifolds. International Press, Boston (1999)Google Scholar
  29. 29.
    Lomshakov, A., Nikonorov, Yu.G., Firsov, E.: Invariant Einstein Metrics on Three-Locally-Symmetric spaces. (Russian) Mat. Tr. 6(2), 80–101 (2003), (engl. transl. in Siberian Adv. Math. 14(3), 43–62 (2004))Google Scholar
  30. 30.
    Nikonorov Yu.G.: Compact homogeneous Einstein 7-manifolds. Geom. Dedic. 109, 7–30 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Nikonorov Yu.G., Rodionov E.D., Slavskii V.V.: Geometry of homogeneous Riemannian manifolds. J. Math. Sci. 146(6), 6313–6390 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Nishiyama M.: Classification of invariant complex structures on irreducible compact simply connected coset spaces. Osaka J. Math. 21, 39–58 (1984)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Park J.-S., Sakane Y.: Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math. 20(1), 51–61 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Sakane Y.: Homogeneous Einstein metrics on flag manifolds. Lobachevskii J. Math. 4, 71–87 (1999)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Samelson H.: Notes on Lie Algebras. Springer-Verlag, New York (1990)zbMATHGoogle Scholar
  36. 36.
    Siebenthal J.: Sur certains modules dans une algébre de Lie semisimple. Comment. Math. Helv. 44(1), 1–44 (1964)CrossRefGoogle Scholar
  37. 37.
    Tian G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Wang H.C.: Closed manifolds with homogeneous complex structures. Am. J. Math. 76(1), 1–32 (1954)zbMATHCrossRefGoogle Scholar
  39. 39.
    Wang M., Ziller W.: On normal homogeneous Einstein manifolds. Ann. Sci. Éc. Norm. Sup. 18(4), 563–633 (1985)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Wang M., Ziller W.: Existence and non-excistence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PatrasRionGreece

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