Foliation Preserving Lie Group Actions and Characteristic Classes

The Diffeomorphism Group of a Lie Foliation

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus T, n ≥ 2, namely codimension one foliations, flows, and the so-called nonquadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than ±id and translations. The computation is an app...

Lie Algebroids, Holonomy and Characteristic Classes

We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus pr...

Topology of Lie Groups and Characteristic Classes

Secondary Characteristic Classes of Lie Algebra Extensions

We introduce a notion of secondary characteristic classes of Lie algebra extensions. As a spin-off of our construction we obtain a new proof of Lecomte’s generalization of the Chern–Weil homomorphism.

Characteristic Classes of Lie Algebroid Morphisms

We extend R. Fernandes’ construction of the secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism. In [4] R. Fernandes has constructed a sequence of secondary characteristic classes of a Li...