Advertisement

Annals of Global Analysis and Geometry

, Volume 19, Issue 4, pp 307–319 | Cite as

Nearly Kähler 6-Manifolds with Reduced Holonomy

  • Florin Belgun
  • Andrei Moroianu
Article

Abstract

We consider a complete six-dimensional nearly Kählermanifold together with the first canonical Hermitian connection. We showthat if the holonomy of this connection is reducible, then the manifoldendowed with a modified metric and almost complex structure is aKählerian twistor space. This result was conjectured byReyes-Carrión.

nearly Kähler manifolds twistor spaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M. F., Hitchin, N. J. and Singer, I. M.: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London A 362 (1978), 425-461.Google Scholar
  2. 2.
    Bär, C.: Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993), 509-521.Google Scholar
  3. 3.
    Besse, A. L.: Einstein Manifolds, Springer, New York, 1987.Google Scholar
  4. 4.
    Eells, J. and Salamon, S.: Twistorial construction of harmonic maps of surfaces into fourmanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), 589-640.Google Scholar
  5. 5.
    Friedrich, Th.: A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds, Math. Nachr. 102 (1981), 53-56.Google Scholar
  6. 6.
    Friedrich, Th. and Grunewald, R.: On Einstein metrics on the twistor space of a fourdimensional Riemannian manifold, Math. Nachr. 123 (1985), 55-60.Google Scholar
  7. 7.
    Friedrich, Th. and Grunewald, R.: On the first eigenvalue of the Dirac operator on 6-dimensional manifolds, Ann. Global Anal. Geom. 3 (1985), 265-273.Google Scholar
  8. 8.
    Grunewald, R.: Six-dimensional Riemannian manifolds with real Killing spinors, Ann. Global Anal. Geom. 8 (1990), 43-59.Google Scholar
  9. 9.
    Gray, A.: The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233-248.Google Scholar
  10. 10.
    Gray, A. and Hervella, L. M.: The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35-58.Google Scholar
  11. 11.
    Hermann, R.: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236-242.Google Scholar
  12. 12.
    Kirchberg, K.-D.: Compact six-dimensional Kähler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator, Math. Ann. 282 (1988), 157-176.Google Scholar
  13. 13.
    O'Neill, B.: The fundamental equations of a submersion, Michigan Math J. 13 (1966), 459-469.Google Scholar
  14. 14.
    Reeb, G.: Stabilité des feuilles compactes à groupe de Poincaré fini, C.R. Acad. Sci. Paris 228 (1949), 47-48.Google Scholar
  15. 15.
    Reyes-Carrión, R.: Some special geometries defined by Lie groups, PhD Thesis, Oxford, 1993.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Florin Belgun
    • 1
  • Andrei Moroianu
    • 2
  1. 1.Institut für Reine MathematikHumboldt Universität zu BerlinBerlinGermany
  2. 2.Centre de Mathématiques, École Polytechnique (UMR 7640 du CNRS)PalaiseauFrance

Personalised recommendations