Twistor geometry of a pair of second order ODEs

TL;DR

This paper explores the twistor correspondence linking certain third-dimensional path geometries with vanishing invariants to anti-self-dual conformal structures of signature (2, 2), providing new methods and examples for reconstructing these structures from ODE systems.

Contribution

It introduces a new framework for reconstructing conformal structures from ODEs with vanishing invariants and presents novel examples with high symmetry, including analogues of plane wave spacetimes.

Findings

Reconstruction of conformal structures from ODEs is possible using twistor methods.

New examples of highly symmetric ODE systems leading to anti-self-dual structures.

A variational principle for twistor curves related to Finsler structures with scalar flag curvature.

Abstract

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature $(2, 2)$. We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti--self--dual structures with conformal symmetry algebra of the same dimension. Some of these examples are $(2, 2)$ analogues of plane wave space--times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.