---
tag: JX65KSVHSJ
kind: concept
name: Affine connection
msc: [53B05]
specializes: [5ZYRRVE8BW]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_connection
same_as:
  wikidata: Q1371213
  wikipedia: Affine connection
---

## Summary

An affine connection on a smooth manifold is a [[connection|5ZYRRVE8BW]] on its tangent bundle.
It lets one differentiate vector fields along vector fields, and so defines parallel transport of tangent vectors, geodesics, torsion and curvature.
On a Riemannian manifold there is a preferred affine connection, the Levi-Civita connection.
The name comes from the idea that a connection identifies nearby tangent spaces as pieces of one affine space.

## Explanations

### L1 human

In the plane, comparing arrows at two points is easy: slide one arrow to the other point without turning it.
On a curved surface no such sliding is given in advance.
An affine connection is a rule that tells how to differentiate one field of arrows along another, and so how to slide arrows along curves.
In the plane the usual rule takes the directional derivative of each component of the vector field.
Straight lines are then the curves whose velocity does not change, and on a curved surface the same idea gives geodesics.