Parallel transport along a piecewise smooth curve

For a piecewise smooth curve $\gamma :[a,b]\to M$ and $v\in E_{\gamma (a)}$, let $V_v$ be the unique continuous piecewise parallel section with $V_v(a)=v$, supplied by Existence and uniqueness of parallel sections. Its parallel transport is $P_{\gamma }:E_{\gamma (a)}\to E_{\gamma (b)},P_{\gamma }(v)=V_v(b)\mathrm{.}$ More generally, $P_{\gamma ;s,t}$ sends initial data at $s$ to its value at $t$, allowing either order of the parameters. The same theorem gives independence from frames and subdivisions. On a singleton, $P_{\gamma }$ is the identity of its fibre. An empty curve has no endpoint fibres and is not assigned such a map. Zero initial data maps to zero; the transport depends on the supplied connection as well as the curve. Linearity and invertibility are proved next.