Projective connection

In differential geometry, a projective connection is a geometric structure on a differentiable manifold that specifies a distinguished class of curves, called geodesics, up to projective reparametrization. Equivalently, in one common formulation, it is given by an equivalence class of torsion-free affine connections having the same unparametrized geodesics.

Projective connections are modeled on the geometry of projective space. In modern terms, they may be described as Cartan connections modeled on projective space; in the normal torsion-free case, this Cartan-geometric description is equivalent to the classical description by projectively equivalent affine connections.

Unlike a Riemannian or pseudo-Riemannian connection, a projective connection does not determine a notion of length, angle, or distance. Its basic geometric datum is instead a notion of straightness: it determines which curves are to be regarded as geodesics, while forgetting the affine parametrization of those curves.