Projective connection

Remarks

Classical description: two torsion-free affine connections $\nabla, \nabla'$ on $M$ are projectively equivalent if they have the same geodesics 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 $\nabla$.