"Linear direct connections in Lie groupoids, curvature and characteristic classes "

In the paper J.Kubarski and N.Teleman, "Linear Direct Connections", Banach Center Publications (in printing) to each linear direct connections t:U→GL(E) in a vector bundle E [ΔÌUÌM×M, t(x,y):E_y →E_x, t(x,x)=id ] is associated infinitesimally an usual linear connection Ñ^t  in E. Next, it is proved that image of the periodic cyclic cocycle F^t_2k through the Connes' isomorphism coincides with the differential form provided by the classical Chern - Weil theory applied upon the underlying linear connection Ñ^t

W_2k(F^t_2k)=¹/_(2k)! ¹/₂^k  Tr R^k

 where R=(Ñ^t)² is the curvature of the underlying linear connection Ñ^t. We recall that the Connes' isomorphism associates with any periodic cyclic cycle f an even/odd non-homogeneous closed differential form W(f) on M. As an application it is obtained a direct proof of Theorem 6.2 from the paper Teleman N.: Direct Connections and Chern Character, Proceedings of the International Conference in Honor of Jean-Paul Brasselet, Luminy, May 2005, which shows how we can extract the Chern-character of a smooth vector bundle E from any direct linear connection t in E.

    The aim of this talk it is to give a concept of the linear direct connections t in arbitrary (transitive) Lie groupoid F and to show haw we can extract the Chern-Weil homomorphism of F from t.