Projective connections associated with second order ODEs

TL;DR

This paper explores how second order ordinary differential equations naturally induce a family of projective connections on their solution spaces, identifying special cases with a canonical representative linked to Cartan's classical work.

Contribution

It establishes a correspondence between second order ODEs and projective connections, highlighting a special case with a canonical representative related to Cartan's theory.

Findings

Every second order ODE defines a 4-parameter family of projective connections.

A special class of ODEs admits a unique preferred projective connection.

The preferred connection coincides with Cartan's classical projective connection.

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".