On Parallel Transport and Curvature —— Graduate Project

Given connection ∇ on a smooth vector bundle E → M , with connected base space M , the set of the parallel transport maps (associated to ∇) along closed loops based at x ∈M form a subgroup, Hol(∇), of the general linear group on the fibre Ex, GL(Ex). The group Hol(∇) is known as the holonomy group of the connection and it is independent of the base point x under conjugation of elements of the general linear group. It therefore defines a global invariant for the connection. If M is simply connected, then Hol(∇) is a Lie subgroup of GL(k,R). Restricting Hol(∇) to nullhomotopic loops gives rise the restricted holonomy group, Hol(∇), which is exactly the identity component of Hol(∇). The Lie algebra associated to Hol(∇) and Hol(∇) is called the holonomy algebra, hol(∇). The holonomy algebra is a linear subspace of End(Ex) and it coincides with the the subspace of End(Ex) generated by a special class of endomorphism obtained through the curvature tensor R(∇) of the connection. This important result is captured by the Ambrose-Singer holonomy theorem. In this report we investigate the dependence of parallel transport maps on the curvature, building the necessary tools to prove the Ambrose-Singer holonomy theorem.