Rigidity of Secondary Characteristic Classes

In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the Leibniz cohomology of the Lie algebra vanishes, then all secondary characteristic classes are rigid. In the case of codimension one foliations (with trivial normal bundle), we show that a characteristic map from the Leibniz cohomology of formal vector fields to the de Rham cohomology of the manifold detects a one-parameter variation of the Godbillon-Vey class. Recall that Leibniz cohomology groups are invariants of Lie algebras defined by J-L. Loday.