Annals of Global Analysis and Geometry

, Volume 42, Issue 2, pp 147–170 | Cite as

Homogeneous nearly Kähler manifolds

  • J. C. González Dávila
  • F. Martín CabreraEmail author


The structure of nearly Kähler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233–248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over quaternionic Kähler manifolds and six-dimensional (6D) nearly Kähler manifolds, where the homogeneous nearly Kähler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly Kähler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly Kähler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.


Nearly Kähler Homogeneous space 3-Symmetric space Intrinsic torsion Minimal connection 

Mathematics Subject Classification (2000)

53C20 53C30 53C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekseevskii D.V.: Compact quaternion spaces. Funk. Anal. i Prilozen. 2(2), 11–20 (1968)MathSciNetGoogle Scholar
  2. 2.
    Besse A.L.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge. Springer, Berlin (1987)Google Scholar
  3. 3.
    Burstall F.E., Gutt S., Rawnsley J.H.: Twistor spaces for Riemannian symmetric spaces. Math. Ann. 295, 729–743 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Butruille J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Glob. Anal. Geom. 27, 201–225 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Butruille J.-B.: Twistors and 3-symmetric spaces. Proc. Lond. Math. Soc. (3) 96, 738–766 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    González-Dávila J.C., Martín Cabrera F.: Harmonic G-structures. Math. Proc. Camb. Philos. Soc. 146, 435–459 (2008)CrossRefGoogle Scholar
  7. 7.
    Gray A.: Riemannian manifolds with geodesic symmetries of order 3. J. Differ. Geom. 7, 343–369 (1972)zbMATHGoogle Scholar
  8. 8.
    Gray A.: The structure of nearly Kähler manifolds. Math. Ann. 223, 233–248 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Helgason S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York (1978)zbMATHGoogle Scholar
  10. 10.
    Hitchin N.: Kählerian twistor spaces. Proc. Lond. Math. Soc. 43(3), 133–150 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kirichenko V.F.: K-spaces of maximal rank. Mat. Zametki 22, 465–476 (1977)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Nagy P.A.: On nearly Kähler geometry. Ann. Glob. Anal. Geom. 22, 167–178 (2002a)zbMATHCrossRefGoogle Scholar
  13. 13.
    Nagy P.A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6, 481–504 (2002b)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Nomizu K.: Invariant affine connections on homogeneous spaces. Am. J. Math. 76, 33–65 (1954)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    O’Brian N.R., Rawnsley J.H.: Twistor spaces. Ann. Glob. Anal. Geom. 3(1), 29–58 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tricerri F., Vanhecke L.: Homogeneous Structures on Riemannian Manifolds, London Math. Soc. Lect. Note Series 83. Cambridge University Press, Cambridge (1983)Google Scholar
  17. 17.
    Watanabe Y., Takamatsu K.: On a K-space of constant holomorphic sectional curvature. Kodai Math. Semin. Rep. 25, 297–306 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wolf J.A., Gray A.: Homogeneous spaces defined by Lie group automorphisms, I, II. J. Differ. Geom. 2, 77–159 (1968)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Yano K.: Differential Geometry on Complex and Almost Complex Spaces. International Series of Monographs in Pure and Applied Mathematics, 49. Pergamon Press, New York (1965)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • J. C. González Dávila
    • 1
  • F. Martín Cabrera
    • 2
    Email author
  1. 1.Department of Fundamental MathematicsUniversity of La LagunaLa Laguna, TenerifeSpain
  2. 2.Department of Fundamental Mathematics, CSIC Associated UnityUniversity of La LagunaLa Laguna, TenerifeSpain

Personalised recommendations