Principal Bundles and Connections

Abstract

The notion of principal bundle as introduced in Section 1.3 is recalled. Invariant vector fields are then introduced as well as the bundle of vertical automorphisms and the bundle of infinitesimal generators of vertical principal automorphisms. Lie derivatives for principal automorphisms and infinitesimal generators of vertical automorphisms are considered in detail. Bundle connections and principal connections are then introduced in many equivalent ways and Lie derivatives of principal connections are calculated. The induced connections on bundles which are associated to a principal bundle are introduced. As examples and applications, the explicit expression for principal connections is obtained on the frame bundle and on higher order frame bundles. The induced connection on the bundle of automorphisms and infinitesimal generators of vertical automorphisms are also calculated. Finally, we introduce the connection-dependent structures on bundles, such as parallel transport, covariant derivatives of a section and the curvature of a principal connection.