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Annals of Global Analysis and Geometry

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

Transversal Twistor Spaces of Foliations

  • Izu Vaisman
Article

Abstract

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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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

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

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