Geometriae Dedicata

, Volume 101, Issue 1, pp 203–216 | Cite as

Symmetries of Contact Metric Manifolds

  • Florin Belgun
  • Andrei Moroianu
  • Uwe Semmelmann


We study the Lie algebra of infinitesimal isometries on compact Sasakian and K-contact manifolds. On a Sasakian manifold which is not a space form or 3-Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K-contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo-Riemannian quaternion-Kähler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that nonregular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds.

K-contact structure infinitesimal automorphism Killing vector field 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekseevsky, D. and Cortes, V.: Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p,q), Comm. Math. Phys. 183(1997), 477–510.Google Scholar
  2. 2.
    . Bar, C.: Real Killing spinors and holonomy, Comm. Math. Phys. 154(1993), 509–521.Google Scholar
  3. 3.
    Baum, H., Friedrich, Th., Grunewald, R. and Kath, I.: Twistor and Killing Spinors on Riemannian Manifolds, Teubner-Verlag, Stuttgart, 1991.Google Scholar
  4. 4.
    Besse, A.: Einstein Manifolds, Springer-Verlag, New York 1987.Google Scholar
  5. 5.
    Boothby, W. M. and Wang, H. C.: On contact manifolds, Ann. Math. 68(1958), 721–734.Google Scholar
  6. 6.
    Boyer, C. and Galicki, K.: 3-Sasakian manifolds, In: Essays on Einstein Manifolds, Surveys Differential Geom. VI, International Press, Boston, MA, 1999, pp. 123–184.Google Scholar
  7. 7.
    Boyer, C. and Galicki, K.: On Sasakian–Einstein geometry, Internat. J. Math. 11(7) (2000), 873–909.Google Scholar
  8. 8.
    Boyer, C. and Galicki, K.: Einstein manifolds and contact geometry, Proc. Amer. Math. Soc. 129(2001), 2419–2430.Google Scholar
  9. 9.
    Gallot, S.: Équations différentielles caract#x00E9;ristiques de la sph#x00E8;re, Ann. Sci. Ec. Norm. Sup. Paris 12(1979), 235–267.Google Scholar
  10. 10.
    Hitchin, N. J.: The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55(1987), 59–126.Google Scholar
  11. 11.
    Kakolewski, T.: The geometry of some Kähler-Einstein manifolds, Internat. J. Math. 2(1991), 287–309.Google Scholar
  12. 12.
    Kashiwada, T.: On a contact 3-structure, Math. Z. 238(2001), 829–832.Google Scholar
  13. 13.
    Konishi, M.: On manifolds with Sasakian 3-structure over quaternion Kähler manifolds, Kodai Math. Sem. Rep. 26(1974/75), 194–200.Google Scholar
  14. 14.
    Moroianu, A.: On the Infinitesimal Isometries of Manifolds with Killing Spinors, J. Geom. Phys. 35(2000), 63–74.Google Scholar
  15. 15.
    Tanno, S.: On the isometry groups of Sasakian manifolds, J. Math. Soc. Japan 22(1970), 579–590.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Florin Belgun
    • 1
  • Andrei Moroianu
    • 2
  • Uwe Semmelmann
    • 3
  1. 1.Institut für MathematikUniversität LeipzigLeipzigGermany
  2. 2.CMAT, École Polytechnique, UMRPalaiseauFrance
  3. 3.Mathematisches InstitutUniversität MünchenMunichGermany

Personalised recommendations