Projective connections associated with second order ODEs

Abstract

We show that every 2nd order ODE defines a 4-parameter family of projective connections on its 2-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of connections always has a preferred representative. This preferred representative turns out to be identical to the projective connection described in Cartan's classic paper "Sur les Varietes a Connection Projective".