Theorem: Parallel transport defines a G-equivariant map between fibers

Let $\pi:P\to M$ be a principal G-bundle with a principal connection .

Let $\gamma:[a,b]\to M$ be a smooth curve with $\gamma(a)=x$ and $\gamma(b)=y$.

Theorem

Define $\tau_\gamma:P_x\to P_y$ by the rule: for $p\in P_x$, let $\widetilde\gamma_p$ be the unique horizontal lift of $\gamma$ with $\widetilde\gamma_p(a)=p$ (existence/uniqueness is in horizontal lift existence/uniqueness ), and set

Then:

1. $\tau_\gamma$ is a smooth bijection (indeed a diffeomorphism) from $P_x$ to $P_y$.
2. $\tau_\gamma$ is right $G$-equivariant: $\tau_\gamma(p\cdot g)=\tau_\gamma(p)\cdot g\qquad \forall\,p\in P_x,\ g\in G.$
3. $\tau_\gamma$ depends smoothly on $p$ and on the curve $\gamma$ under smooth variations.

This map is the principal-bundle version of parallel transport .

Examples

1. Trivial bundle. For $P=M\times G$ with connection form $A$, parallel transport along $\gamma$ acts by $(x,g_0)\mapsto (y,g(b))$, where $g(t)$ solves the ODE determined by $A$ along $\gamma$.

2. Flat connection. If the connection is flat (zero curvature), then $\tau_\gamma$ depends only on the homotopy class of $\gamma$ with fixed endpoints; the resulting representation of $\pi_1(M,x)$ is the holonomy representation.

3. Circle bundle phases. For $G=U(1)$, $\tau_\gamma$ is multiplication by a phase factor in $U(1)$; on associated complex line bundles this is the usual phase change of a transported vector.