Connection (vector Bundle) - Parallel Transport and Holonomy

Parallel Transport and Holonomy

A connection ∇ on a vector bundle E → M defines a notion of parallel transport on E along a curve in M. Let γ : → M be a smooth path in M. A section σ of E along γ is said to be parallel if

for all t ∈ . More formally, one can consider the pullback γ*E of E by γ. This is a vector bundle over with fiber E_γ(t) over t ∈ . The connection ∇ on E pulls back to a connection on γ*E. A section σ of γ*E is parallel if and only if γ*∇(σ) = 0.

Suppose γ is a path from x to y in M. The above equation defining parallel sections is a first-order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector v in E_x there exists a unique parallel section σ of γ*E with σ(0) = v. Define a parallel transport map

by τ_γ(v) = σ(1). It can be shown that τ_γ is a linear isomorphism.

Parallel transport can be used to define the holonomy group of the connection ∇ based at a point x in M. This is the subgroup of GL(E_x) consisting of all parallel transport maps coming from loops based at x:

The holonomy group of a connection is intimately related to the curvature of the connection (AmbroseSinger 1953).