  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.
