---
tag: D83NMTHT7N
kind: concept
name: Projective connection
msc: [53B10]
specializes: [QPR7DCHXPG]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_connection
same_as:
  wikidata: Q7249464
  wikipedia: Projective connection
---

## Summary

A projective connection is the structure a manifold carries when it knows which curves are its straight lines.
It does not know how these lines are parametrized.
In Cartan's language it is a [[Cartan connection|QPR7DCHXPG]] modelled on real projective space, viewed as a homogeneous space of the projective linear group.
Classically it is described by a class of torsion-free affine connections that have the same geodesics up to reparametrization.
Projective connections appear in projective differential geometry and in the study of second-order ordinary differential equations.

## Explanations

### L1 human

In projective geometry there are lines but no distances and no preferred speed along a line.
A projective connection gives a curved space the same kind of structure: a family of curves that play the role of straight lines.
Two different ways of measuring acceleration can produce exactly the same paths with different speeds; they define the same projective connection.
The flat example is projective space itself, or ordinary space with its usual straight lines.