How the Curvature Generates the Holonomy of a Connection in an Arbitrary Fibre Bundle

Abstract.

For an arbitrary fibre bundle with a connection, the holonomy group of which is a Lie transformation group, it is shown how the parallel displacement along a null-homotopic loop can be obtained from the curvature by integration. The result also sheds some new light on the situation for vector bundles and principal fibre bundles. The Theorem of Ambrose–Singer is derived as a corollary in our general setting. The curvature of the connection is interpreted as a differential 2-form with values in the holonomy algebra bundle, the elements of which are special vector fields on the fibres of the given bundle.