Twistor geometry of a pair of second order ODEs

Details Twistor geometry of a pair of second order ODEs Twistor geometry of a pair of second order ODEs

2012-03-19

arxiv.org/abs/1203.4158v2

Mathematics

Differential Geometry

Includes Theorem 4.1. relating torsion free path geometries to solutions of the 2nd heavenly equation, as well as a new Sectio

Scientific paper

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. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.

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