---
tag: P7YZFF8851
kind: definition
name: Parallel transport
msc: [53C05, 53B05]
defines: TVAGN3D6W5
uses: [5ZYRRVE8BW, 5FZDQQHQKM]
sources:
  - url: https://en.wikipedia.org/wiki/Parallel_transport
---

## Definition

Let $M$ be a smooth manifold, let $E \to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\nabla$, and let $\gamma : [a, b] \to M$ be a piecewise smooth curve.
Let $D_t$ be the [[covariant derivative along $\gamma$|5FZDQQHQKM]].
A section $V$ of $E$ along $\gamma$ is *parallel* if $D_t V = 0$.
For each $v \in E_{\gamma(a)}$ there is a unique parallel section $V$ along $\gamma$ with $V(a) = v$.
The *parallel transport* along $\gamma$ is the map
$$P_\gamma : E_{\gamma(a)} \to E_{\gamma(b)}, \quad v \mapsto V(b).$$

## 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 $\nabla_{\gamma'(0)} s = \frac{d}{dt}\big|_{t = 0} P_{\gamma|_{[0, t]}}^{-1} s(\gamma(t))$.