Linear direct connections

• Autorzy: Jan Kubarski , Nicolae Teleman
• Języki publikacji: EN

Abstrakty

EN

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].

Kategorie tematyczne

• 53C05: Connections, general theory
• 57R20: Characteristic classes and numbers
• 58A05: Differentiable manifolds, foundations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2007
• Tom: 76
• Numer: 1
• Strony: 425-436

Daty

wydano

2007

Twórcy

autor

Jan Kubarski

• Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
• Mathematical Institute of Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

autor

Nicolae Teleman

• Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, 60161 Ancona, Italy

Identyfikatory

DOI

10.4064/bc76-0-20