Parallel transport

Remarks

Existence and uniqueness hold because, in a local frame, $D_t V = 0$ is a linear system of ordinary differential equations, whose solutions exist on the whole interval. $P_\gamma$ is a linear isomorphism, its inverse is the transport along the reversed curve, and the transport along a concatenation is the composition. Example: for the flat connection on $T\R^n$, $D_t V = \dot V$, so parallel sections are constant and $P_\gamma$ is the identity of $\R^n$ for every curve $\gamma$. Non-example of path independence: take the unit sphere $S^2$ with its Levi-Civita connection. Transport around the boundary of an octant, a geodesic triangle with three right angles, rotates tangent vectors by $\pi/2$, the area of the octant. For the Levi-Civita connection of a Riemannian metric, $P_\gamma$ is an isometry between tangent spaces. The connection is recovered by $\nabla_{\gamma'(0)} s = \frac{d}{dt}\big|_{t = 0} P_{\gamma|_{[0, t]}}^{-1} s(\gamma(t))$.