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 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.