Linear direct connections

Jan Kubarski ; Nicolae Teleman

Banach Center Publications, Tome 75 (2007), p. 425-436 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper we study the geometry of direct connections in smooth vector bundles (see N. Teleman [Tn.3]); we show that the infinitesimal part, $∇^{τ}$, of a direct connection τ is a linear connection. We determine the curvature tensor of the associated linear connection $∇^{τ}.$ As an application of these results, we present a direct proof of N. Teleman’s Theorem 6.2 [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 $Φ_{*}^{τ},$ manufactured from a direct connection τ, rather than from a smooth linear connection as the Chern-Weil construction does. In addition, we show that the image of the cyclic cycle $Φ_{*}^{τ}$ 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 $∇^{τ}.$ For more details about these constructions, the reader is referred to [M], N. Teleman [Tn.1], [Tn.2], [Tn.3], C. Teleman [Tc], A. Connes [C.1], [C.2] and A. Connes and H. Moscovici [C.M].

Détail

Publié le : 2007-01-01

Zbl 1125.53018

EUDML-ID : urn:eudml:doc:281881