Twistor geometry of a pair of second order ODEs

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.

Final version to appear in the Communications in Mathematical Physics. The procedure of recovering a system of torsion-fee ODEs from the heavenly equation has been clarified. The proof of Prop 7.1 has been expanded. Dedicated to Mike Eastwood on the occasion of his 60th birthday

References in corpus (1)

• Reduction of beta-integrable 2-Segre structures

Cited by in corpus (14)

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• Self-Dual Conformal Gravity
• Ordinary differential equations described by their Lie symmetry algebra
• Metrizable isotropic second-order differential equations and Hilbert's fourth problem
• Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• The Cayley cubic and differential equations
• Almost every path structure is not variational
• Webs and the Plebański equation
• Symmetry gaps for higher order ordinary differential equations
• Conformally related vacuum gravitational waves, and their symmetries