---
tag: TVAGN3D6W5
kind: concept
name: Parallel transport
msc: [53C05, 53B05]
sources:
  - url: https://en.wikipedia.org/wiki/Parallel_transport
same_as:
  wikidata: Q1814838
  wikipedia: Parallel transport
---

## Summary

Parallel transport moves vectors along a curve in a manifold while keeping them "constant" in the sense of a given [[connection|5ZYRRVE8BW]].
It gives, for each curve, a linear isomorphism between the fibres over its endpoints.
The result usually depends on the curve, not only on its endpoints, and this dependence is measured by curvature and holonomy.
A connection can be recovered from its parallel transports, so the two describe the same data.

## Explanations

### L1 human

Walk along a path on a surface carrying a spear, and never turn the spear relative to the surface as you go.
Where the spear ends up pointing is the parallel transport of its starting direction.
On a flat plane the spear always points the same way, whatever path you take.
On a sphere, walk from the north pole down to the equator, a quarter of the way around the equator, and back to the pole.
The spear comes back turned by a right angle, which shows that the sphere is curved.