Direct Connections and Chern Character

We show how the Chern character of the tangent bundle of a smooth manifold may be extracted from the geodesic distance function by means of cyclic homology. In Sect.6 we introduce the notion of linear direct connection in vector bundles, notion which generalizes the notion of linear connection. The basic difference between a linear direct connection and a linear connection consists of the fact that while a linear connection provides a transport of fibers along curves, a linear direct connection provides a direct transport of fibres from point to point. For this reason, linear direct connections could be defined in contexts where differentiability is not available. We show next that the algebraic procedure for constructing the Chern character, discussed in Sect. 6, applies also in the case of linear direct connections. This paper provides a geometric interpretation of the non commutative Chern character due to A. Connes [1,2]. We show that this interpretation goes along the same lines as those presented by N. Teleman [11] and C. Teleman [10]. The arguments discussed here may be extended to the language of groupoids. In a subsequent paper we are going to improve some of the considerations presented here and extend their field of application to more singular situations.