Transversal Twistor Spaces of Foliations

The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost complex structures of F'. The existence of a Bott connection which yields a structure I that is projectable to the space of leaves is equivalent to the fact that F is a transversally projective foliation. A Bott connection which yields a projectable structure J exists iff F is a transversally projective foliation which satisfies a supplementary cohomological condition, and, in this case, I is projectable as well. J is never integrable. The essential integrability condition of I is the flatness of the transversal projective structure of F.