---
tag: TD6VEP3SCE
kind: concept
name: Ehresmann connection
msc: [53C05]
sources:
  - url: https://en.wikipedia.org/wiki/Ehresmann_connection
same_as:
  wikidata: Q2993471
  wikipedia: Ehresmann connection
---

## Summary

An Ehresmann connection on a [[fibre bundle|CX4M8K61AB]] is a choice of horizontal directions at every point of the total space.
These directions are complementary to the directions along the fibres.
It works for any smooth fibre bundle, not only for vector bundles, and needs no linear structure.
It lets one lift curves in the base to horizontal curves in the total space, which gives parallel transport between fibres.
Connections on vector bundles and on principal bundles are Ehresmann connections with extra symmetry.

## Explanations

### L1 human

Picture a bundle as a family of fibres standing over the points of a base.
At each point of the total space some directions go along the fibre; these are vertical and are given by the bundle.
An Ehresmann connection chooses which of the remaining directions count as horizontal.
For the product of a line and a circle, the obvious choice is to call the line directions horizontal.
Then moving in the base lifts uniquely to moving horizontally upstairs.