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

Abstract

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 6${}^{\mathrm{th}}$ order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7${}^{\mathrm{th}}$ order nonlinear ODE described in Dunajski and Sokolov. We show that the 6${}^{\mathrm{th}}$ order ODE can be reduced to a 3${}^{\mathrm{rd}}$ order nonlinear ODE that is a generalised Chazy equation. The 7${}^{\mathrm{th}}$ 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 $F(q)=q^m$. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split $G_2$ as their group of symmetries.

References in corpus (2)

• New relations between $G_2$-geometries in dimensions 5 and 7
• The dancing metric, ${\mathrm{G}}_2$-symmetry and projective rolling

Cited by in corpus (2)

• The Chazy XII Equation and Schwarz Triangle Functions
• Homogeneous Real (2,3,5) Distributions with Isotropy