Torsions of connections on some natural bundles. (English) Zbl 0783.53021
The importance of connections in principal bundles and associated geometric objects in the study of gauge fields is now well established. It has opened up a new area of study, namely, gauge theoretic topology and geometry. It is, therefore, natural to study connections in more general settings such as fibered manifolds and consider its applications. The Frölicher-Nijenhuis bracket plays a fundamental role in this study. If \(\Gamma\) is a connection on a fibered manifold \(E\) over base \(M\) and \(Q\) is a vector-valued 1-form on \(E\), then \(\tau_ Q\), the torsion of \(\Gamma\) of type \(Q\) is defined by \(\tau_ Q:=[\Gamma,Q]\). In this paper the authors define and classify natural vector-valued 1-forms \(Q\) on \(T^*M\) and on a family of natural bundles over \(M\) associated with product preserving functors. These forms are then used to study the torsion \(\tau_ Q\). Detailed expressions are given in several particular cases. These are used to compare \(\tau_ Q\) with the classical torsion of linear connections as well as with other concepts of torsion such as the torsion associated with the Liouville form on the cotangent bundle \(T^*M\).
Reviewer: K.Marathe (Brooklyn)
MSC:
| 53C05 | Connections (general theory) |
| 53B15 | Other connections |
| 55R15 | Classification of fiber spaces or bundles in algebraic topology |
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