|
This article is cited in 5 scientific papers (total in 5 papers)
Flat $(2,3,5)$-Distributions and Chazy's Equations
Matthew Randall Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
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$^{\rm th}$ order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We show that the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ order nonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ 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.
Keywords:
generic rank two distribution in dimension five; conformal geometry; Chazy's equations.
Received: September 23, 2015; in final form March 14, 2016; Published online March 18, 2016
Citation:
Matthew Randall, “Flat $(2,3,5)$-Distributions and Chazy's Equations”, SIGMA, 12 (2016), 029, 28 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1111 https://www.mathnet.ru/eng/sigma/v12/p29
|
Statistics & downloads: |
Abstract page: | 249 | Full-text PDF : | 28 | References: | 38 |
|