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  "body": "\n## Definition\n\nLet $n \\ge 2$, 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 $G / H \\cong \\R P^n$.\nLet $M$ be a smooth manifold of dimension $n$.\nA *projective connection* on $M$ is a [[Cartan connection|QPR7DCHXPG]] on $M$ modelled on $(G, H)$.\n\n## Remarks\n\nClassical description: two torsion-free [[affine connections|JX65KSVHSJ]] $\\nabla, \\nabla'$ on $M$ are *projectively equivalent* if they have the same [[geodesics|NSYMQMHPDN]] up to reparametrization.\nThis holds if and only if $\\nabla'_X Y = \\nabla_X Y + \\alpha(X) Y + \\alpha(Y) X$ for some $1$-form $\\alpha$ on $M$.\nA projective equivalence class of torsion-free affine connections is called a projective structure.\nCartan showed that it determines a projective connection in the sense above, unique once a normalization of the curvature is imposed.\nExample: 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.\nFor 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.\nNon-example: let $\\nabla$ be the flat connection of $\\R^n$, let $g$ be the Euclidean metric, and let $V$ be a fixed nonzero vector.\nPut $\\nabla'_X Y = \\nabla_X Y + g(X, Y) V$.\nThis changes the geodesics as curves, so $\\nabla'$ is not projectively equivalent to $\\nabla$.\n",
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