Introduction To Connections, Curvature And Holonomy Groups

Abstract In this chapter we will introduce the theory of connections, focussing in particular on two topics, the curvature and the holonomy group of a connection. Connections can be defined in two different sorts of bundle, that is, vector bundles and principal bundles. Both definitions will be given in 2.1. Sections 2.2–2.4 define the holonomy group of a connection on a vector or principal bundle, and explain some of its basic properties, including its relationship with the curvature of the connection. The curvature is a local invariant of the connection, since it varies from point to point on the manifold, whereas the holonomy group is a global invariant, as it is independent of any base point in the manifold.