Foliations with rigid Godbillon-Vey class

Abstract.

The general formula for the variation of the Godbillon-Vey class is given in terms of the obstruction to the existence of a projective transversal structure (when a foliation arises by gluing of level sets of local functions with fractional linear transition maps). Using the above formula one obtains (under the technical condition of separability of some topological space of cohomology) that the Godbillon-Vey number of a foliation ${\cal F}$ of codimension one on a compact orientable 3-fold is topologically rigid (i.e. constant under infinitesimal singular deformations) iff ${\cal F}$ admits a projective transversal structure.