Linear connections and secondary characteristic classes of Lie algebroids

Abstract

Lie algebroids appear in many structures related to geometry. Although the motivations for defining the concept of Lie algebroid come from Lie groupoids [74], on the one hand, we can view them as some generalizations of a tangent bundle or integrable distribution on a differential manifold, and on the other hand, as a generalization of Lie algebra. Further, the structures of the Lie algebroid can be generalized to, for example, structures in which the Lie bracket does not satisfy the Jacobi condition or at all the structure without the Lie bracket, however, equipped with a morphism acting from a given vector bundle into a tangent bundle (called an anchor). A vector bundle equipped with an anchor allows us to introduce the concept of connection. Our considerations focus on linear connections and their properties, and on the existence of a connection in a given vector bundle compatible with an existing metric structure.[...]