Abstract: 
In this paper we study the geometry of direct connections in smooth vector
bundles (see N. Teleman \cite{Tn.3});
 we show that the infinitesimal part,
$\nabla^{\tau}$, of a direct connection $\tau$ is a linear connection. We
determine the curvature tensor of the associated linear connection
$\nabla^{\tau}.$

As an application of these results, we present a direct proof of N. Teleman's
Theorem 6.2 \cite{Tn.3}, which shows that it is possible to represent
the Chern character of smooth vector bundles as the periodic cyclic homology
class of a specific periodic cyclic cycle $\Phi_{\ast}^{\tau},$
manufactured from a direct connection $\tau$, rather than from a smooth linear
connection as the Chern-Weil construction does. In addition, we show 
 that the image of the cyclic cycle $\Phi_{\ast}^{\tau}$ into the de Rham
cohomology (through the A. Connes' isomorphism) coincides with the
cycle provided by the Chern-Weil construction applied to the
underlying linear connection $\nabla^{\tau}.$

For more details about these constructions, the reader is referred to
\cite{M}, N. Teleman\break \cite{Tn.1}, \cite{Tn.2}, \cite{Tn.3}, C. Teleman
\cite{Tc}, A. Connes \cite{C.1}, \cite{C.2} and A. Connes and H. Mos-\break covici \cite{C.M}.

1. Institute of Mathematics
   Technical University of £ódŸ
   Wólczañska 215
   93-005 £ódŸ, Poland
   and
   Mathematical Institute of Polish Academy of Sciences
   Œniadeckich 8
   00-950 Warszawa, Poland
   
2. Dipartimento di Scienze Matematiche
   Universita Politecnica delle Marche
   60161 Ancona, Italy