Linear direct connections
Volume 76 / 2007
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 \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. Moscovici \cite{C.M}.
