Linear direct connections

Jan Kubarski, and Nicolae Teleman. "Linear direct connections." Banach Center Publications 76.1 (2007): 425-436. <http://eudml.org/doc/281881>.

@article{JanKubarski2007,
abstract = {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].},
author = {Jan Kubarski, Nicolae Teleman},
journal = {Banach Center Publications},
keywords = {direct connection; Chern character; periodic cyclic homology},
language = {eng},
number = {1},
pages = {425-436},
title = {Linear direct connections},
url = {http://eudml.org/doc/281881},
volume = {76},
year = {2007},
}

TY - JOUR
AU - Jan Kubarski
AU - Nicolae Teleman
TI - Linear direct connections
JO - Banach Center Publications
PY - 2007
VL - 76
IS - 1
SP - 425
EP - 436
AB - 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].
LA - eng
KW - direct connection; Chern character; periodic cyclic homology
UR - http://eudml.org/doc/281881
ER -