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Twistor Geometry of a Pair of Second Order ODEs

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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.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Stephen Casey & Maciej Dunajski

  2. The Mathematical Institute, Oxford University, 24-29 St Giles, Oxford, OX1 3LB, UK

    Paul Tod

Authors
  1. Stephen Casey
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  2. Maciej Dunajski
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  3. Paul Tod
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Correspondence to Maciej Dunajski.

Additional information

Communicated by P. T. Chruściel

Dedicated to Mike Eastwood on the occasion of his 60th birthday

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Casey, S., Dunajski, M. & Tod, P. Twistor Geometry of a Pair of Second Order ODEs. Commun. Math. Phys. 321, 681–701 (2013). https://doi.org/10.1007/s00220-013-1729-7

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  • DOI: https://doi.org/10.1007/s00220-013-1729-7

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