Affine geometry of second order ODEs

TL;DR

This paper uses Cartan's method to classify second order ODEs under area-preserving transformations, providing a characterization and constructing an affine Cartan connection related to their invariants.

Contribution

It introduces a new characterization of second order ODEs equivalent to the trivial equation under area-preserving diffeomorphisms and constructs an associated affine Cartan connection.

Findings

Characterization of ODEs equivalent to y''=0 under area-preserving transformations

Construction of an affine normal Cartan connection for certain ODEs

Curvature encodes all area-preserving invariants

Abstract

We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to $y^{\prime\prime}=0$ under such transformations. Moreover we associate to certain of these second order ODEs which satisfy an invariant condition given by the vanishing of a relative differential invariant, an affine normal Cartan connection on the first jet space whose curvature contains all the area-preserving relative differential invariants.