= Global continuation of linear parallel transport
{title2=$P_\gamma:E_{\gamma(0)}\xrightarrow{\sim}E_{\gamma(1)}$}
Along a smooth curve $\gamma:I\to M$, a <horizontal lift> of a linear <connection on a vector bundle> has fibre equation $v'=-\Gamma(\dot\gamma)v$. On each compact time interval the coefficient matrix is bounded, and the <Gronwall inequality> prevents finite-time blowup. A finite cover by <vector bundle trivializations> and ODE uniqueness patch local solutions. Exhausting the open interval gives a unique lift on all of $I$. Its endpoint map is linear, and reversal of the path gives its inverse; no completeness of the base manifold is needed.
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