In this paper we study the geometry of direct
connections in smooth vector bundles;
we show that the infinitesimal part, Ñt, of a direct connection t is a linear connection. We determine the
curvature tensor of the associated linear connection Ñt.
As an application of these results, we present
a direct proof of N. Teleman's Theorem 6.2. (Direct Connections and Chern
Character, Proceedings of the International Conference in Honor of
Jean-Paul Brasselet, Luminy, May 2005), which had shown that it was possible to
represent the Chern character of smooth vector bundles as the periodic cyclic
homology class of a specific periodic
cyclic cycle Ft* , manufactured from a direct connection t, rather than from a smooth linear connection
as the Chern - Weil construction does. In addition, we show in this paper that
the image of the cyclic cycle Ft* into
the de Rham cohomology (through the A. Connes' isomorphism) coincides with the
cycle provided by the Chern - Weil construction applied upon the underlying
linear connection Ñt.