Foliations with rigid Godbillon-Vey class (Q1297994)

The paper studies smooth codimension one foliations on smooth manifolds. To these foliations a very natural De Rham three-dimensional cohomology class is associated called the Godbillon-Vey class. A general form for the variation of this class is given in terms of the obstruction to the existence of a real projective transverse structure in the foliation. Using the above formula, the author obtains (under the technical condition of separability of some topological space of cohomology) that the Godbillon-Vey number of a codimension one foliation on a compact orientable 3-manifold is topologically rigid. This means, that it remains constant under infinitesimal singular deformations iff the foliation admits a real projective transverse structure. Here an infinitesimal singular deformation, roughly speaking, is a passage from smooth leaves to continuous but rough leaves.