Let $E \to M$ be a vector bundle with a connection $\nabla$ . For any smooth path $\gamma : [0,1] \to M$ we will define a linear isomorphism $T_\gamma : E_{\gamma(0)} \to E_{\gamma(1)}$ called the parallel transport along $\gamma$ .

The construction [local-0]

More precisely, we construct a family of linear isomorphisms:

T_t : E_{\gamma(0)} \to E_{\gamma(t)}

for all $t \in [0, 1]$ . Consider arbitrary $t \in [0, 1]$ , let $u_0 \in E_{\gamma(0)}$ be a vector, then define $u_t := T_t(u_0)$ , we know $u_t \in E_{\gamma(t)}$ . What we are searching is a "constant" path, in the sense that derivative is $0$ , hence we want

\nabla_{\frac{d}{dt}} u_t = 0, \quad \text{where } \frac{d}{dt} = \dot{\gamma}

so this suggests a way of defining $T_t$ : For any $u_0 \in E_{\gamma(0)}$ and any $t \in [0,1]$ , define $T_t(u_0)$ as the value at $t$ of the solution of the initial value problem:

\begin{cases} \nabla_{\frac{d}{dt}} u(t) = 0 \\ u(0) = u_0 \end{cases}

And this is a system of linear ordinary differential equations in disguise.