Publication details

Flat (2,3,5)-Distributions and Chazy's Equations

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RANDALL Matthew James

Year of publication 2016
Type Article in Periodical
Magazine / Source Symmetry, Integrability and Geometry: Methods and Applications
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords generic rank two distribution in dimension five; conformal geometry; Chazy's equations.
Description In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2,3,5)-distributions determined by a single function of the form F(q), the vanishing condition for the curvature invariant is given by a 6th order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7th order nonlinear ODE described in Dunajski and Sokolov. We show that the 6th order ODE can be reduced to a 3rd order nonlinear ODE that is a generalised Chazy equation. The 7th order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat (2,3,5)-distributions not of the form F(q)=qm. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split G2 as their group of symmetries.
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