Metrisability of Painlev\'e equations

TL;DR

This paper determines which Painlevé equations and related second-order ODEs have solution curves that can be interpreted as geodesics of a (pseudo) Riemannian metric, solving a specific metrisability problem.

Contribution

It solves the metrisability problem for all six Painlevé equations and related ODEs with Painlevé property, identifying when their solutions are geodesics of a metric.

Findings

Identified which Painlevé equations are metrizable.

Characterized the metric structures for these equations.

Extended results to all 2nd order ODEs with Painlevé property.

Abstract

We solve the metrisability problem for the six Painlev\'e equations, and more generally for all 2nd order ODEs with Painlev\'e property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.