---
tag: AV8Q85BH70
kind: definition
name: Projective connection
msc: [53B10]
defines: D83NMTHT7N
uses: [QPR7DCHXPG, JX65KSVHSJ, NSYMQMHPDN, JZNQAN00ZJ]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_connection
---

## Definition

Let $n \ge 2$, let $G = \mathrm{PGL}(n + 1, \R)$ act on the real projective space $\R P^n$, and let $H \subset G$ be the stabilizer of a point $o \in \R P^n$, so that $G / H \cong \R P^n$.
Let $M$ be a smooth manifold of dimension $n$.
A *projective connection* on $M$ is a [[Cartan connection|QPR7DCHXPG]] on $M$ modelled on $(G, H)$.

## Remarks

Classical description: two torsion-free [[affine connections|JX65KSVHSJ]] $\nabla, \nabla'$ on $M$ are *projectively equivalent* if they have the same [[geodesics|NSYMQMHPDN]] up to reparametrization.
This holds if and only if $\nabla'_X Y = \nabla_X Y + \alpha(X) Y + \alpha(Y) X$ for some $1$-form $\alpha$ on $M$.
A projective equivalence class of torsion-free affine connections is called a projective structure.
Cartan showed that it determines a projective connection in the sense above, unique once a normalization of the curvature is imposed.
Example: the model $G \to \R P^n$ with the Maurer–Cartan form of $G$ is a projective connection, the flat one; its unparametrized geodesics are the projective lines.
For the flat connection of $\R^n$ and $\alpha = dx^1$, the connection $\nabla'$ above has new geodesics such as $t \mapsto (f(t), 0, \dots, 0)$ with $f'' + 2 (f')^2 = 0$, which are straight lines traversed at a non-constant speed.
Non-example: let $\nabla$ be the flat connection of $\R^n$, let $g$ be the Euclidean metric, and let $V$ be a fixed nonzero vector.
Put $\nabla'_X Y = \nabla_X Y + g(X, Y) V$.
This changes the geodesics as curves, so $\nabla'$ is not projectively equivalent to $\nabla$.