Annals of Global Analysis and Geometry

, Volume 19, Issue 3, pp 209–234 | Cite as

Transversal Twistor Spaces of Foliations

  • Izu Vaisman


The transversal twistor space of a foliation \(F\) of an even codimension is the bundle \(Z(F)\) of the complex structures of the fibers of the transversalbundle of \(F\). On \(Z(F)\)there exists a foliation \(\hat F\)by covering spaces of the leaves of \(F\), and any Bottconnection of \(F\) produces an ordered pair\((\ell _1 ,\ell _2 )\)of transversal almost complex structures of \(\hat F\). The existence of a Bott connection which yields a structure\(\ell \)1 that is projectable to the space of leaves isequivalent to the fact that \(F\) is a transversallyprojective foliation. A Bott connection which yields a projectablestructure \(\ell \)2 exists iff \(F\) isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, \(\ell \)1is projectable as well. \(\ell \)2 is never integrable.The essential integrability condition of \(\ell \)1 isthe flatness of the transversal projective structure of \(F\).

foliated (projectable) objects foliations transversal twistor spaces 


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  1. 1.
    Atiyah, M. F., Hitchin, N. J. and Singer, I. M.: Self-duality in four-dimensional Riemannian geometry, Proc. Royal Soc. London A 362 (1978), 425-461.Google Scholar
  2. 2.
    Bott, R.: Lectures on Characteristic Classes of Foliations, Lecture Notes in Math. 279, Springer, New York, 1972.Google Scholar
  3. 3.
    Jensen, G. R. and Rigoli, M.: Twistor and Gauss lifts of surfaces in four-manifolds, in: Contemp. Math. 132, Amer. Math. Soc., Providence, RI, 1989, pp. 197-232.Google Scholar
  4. 4.
    Kamber, F. and Tondeur, Ph.: Foliated Bundles and Characteristic Classes, Lecture Notes in Math. 493, Springer, New York, 1975.Google Scholar
  5. 5.
    Kobayashi, S.: Transformation Groups in Differential Geometry, 2nd edn, Springer, New York, 1995.Google Scholar
  6. 6.
    Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry I, II, Interscience, New York, 1963, 1969.Google Scholar
  7. 7.
    Molino, P.: Propriétés cohomologiques et propriétés topologiques des feuilletages à connexion transverse projetable, Topology 12 (1973), 317-325.Google Scholar
  8. 8.
    Molino, P.: Riemannian Foliations, Progr. Math. Series 73, Birkhäuser, Boston, 1988.Google Scholar
  9. 9.
    Nirenberg, L.: A complex Frobenious theorem, in: Seminar on Analytic Functions, Vol. I, Inst. for Adv. Study, Princeton, NJ, 1957, pp. 172-179.Google Scholar
  10. 10.
    Nishikawa, S. and Sato, H.: On characteristic classes of Riemannian, conformal and projective foliations, J. Math. Soc. Japan 28 (1976), 223-241.Google Scholar
  11. 11.
    O'Brian, N. R. and Rawnsley, J. H.: Twistor spaces, Ann. Global Anal. Geom. 3 (1985), 29-58.Google Scholar
  12. 12.
    Penrose, R.: The twistor programme, Rep. Math. Phys. 12 (1977), 65-76.Google Scholar
  13. 13.
    Vaisman, I.: Variétés Riemanniennes feuielletées, Czech. Math. J. 21 (1971), 46-75.Google Scholar
  14. 14.
    Vaisman, I.: Cohomology and Differential Forms, Marcel Dekker, New York, 1973.Google Scholar
  15. 15.
    Vaisman, I.: Conformal foliations, Kodai Math. J. 2 (1979), 26-37.Google Scholar
  16. 16.
    Vaisman, I.: A note on projective foliations, Publ. Univ. Autonoma Barcelona 27 (1983), 109-128.Google Scholar
  17. 17.
    Vaisman, I., On the twistor space of almost Hermitian manifolds, Ann. Global Anal. Geom. 16 (1998), 335-356.Google Scholar

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Izu Vaisman
    • 1
  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael

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