Connection theory on differentiable fibre bundles: A concise introduction

Bozhidar Z Iliev

doi:10.1155/ijmms/2006/56397

Abstract

The paper contains a partial review on the general connection theory on differentiable fibre bundles. Particular attention is paid on (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic and arbitrary frames are considered.

Highlights

This is a partial review of the connection theory on differentiable fibre bundles
We have presented a short review of the connection theory on bundles whose base and bundle spaces are (C2) differentiable manifolds
Special attention was paid to connections, in particular linear ones, on vector bundles, which find wide applications in physics [7, 24]

Summary

Introduction

This is a partial review of the connection theory on differentiable fibre bundles. From different view points, this theory can be found in many works, like [2,3,4,5,6, 9, 13,14,15, 18, 19, 24, 26, 27, 30, 31, 35,36,37,38,39,40, 42]. The general results are specified on the (co)tangent bundle over a manifold in Section 4.2; Section 4.3 is devoted to linear connections on vector bundles, that is, connections such that the parallel transport assigned to them is a linear mapping. They are defined via the Lie derivatives and a mapping realizing an isomorphism between the vertical vector fields on the bundle space and the sections of the bundle.

Preliminaries

Connections on bundles

Connections on vector bundles

Conclusion