Proposition: Gauge-covariant parallel transport

The matrix ${T_A}^{\gamma }$ is unitary. Under $A^{\prime }=U^{-1}AU+U^{-1}dU$,

In particular, transport between endpoints on $\Gamma$ is unchanged when $U{|}_{\Gamma }=I$.

Proof. Skew-Hermiticity of $A(\dot{\gamma })$ gives $\frac{d}{dt}(T^{\ast }T)=0$. Differentiate $U(\gamma (t){)}^{-1}{T_A}^{\gamma {|}_{[0,t]}}U(\gamma (0))$. It obeys the transformed differential equation and initial condition.