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

RANDALL, Matthew James. Flat (2,3,5)-Distributions and Chazy's Equations. Symmetry, Integrability and Geometry: Methods and Applications. Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine, 2016, vol. 12, No 29, p. "nestrankovano", 28 pp. ISSN 1815-0659. Available from: https://dx.doi.org/10.3842/SIGMA.2016.029.

Basic information

• Original name: Flat (2,3,5)-Distributions and Chazy's Equations
• Authors: RANDALL, Matthew James (554 New Zealand, guarantor, belonging to the institution).
• Edition: Symmetry, Integrability and Geometry: Methods and Applications, Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine, 2016, 1815-0659.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10101 Pure mathematics
• Country of publisher: Ukraine
• Confidentiality degree: is not subject to a state or trade secret
• WWW: URL
• Impact factor: Impact factor: 0.765
• RIV identification code: RIV/00216224:14310/16:00088799
• Organization unit: Faculty of Science
• Doi: http://dx.doi.org/10.3842/SIGMA.2016.029
• UT WoS: 000374454900001
• Keywords in English: generic rank two distribution in dimension five; conformal geometry; Chazy's equations.
• Tags: AKR, rivok
• Tags: International impact, Reviewed

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 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.

Links

• GBP201/12/G028, research and development project: Name: Ústav Eduarda Čecha pro algebru, geometrii a matematickou fyziku

Investor: Czech Science Foundation