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

J 2016

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

RANDALL, Matthew James

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

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Ukraine

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

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.

Abstract

V originále

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

Citovat

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.

@article{1376635,
   author = {Randall, Matthew James},
   article_number = {29},
   doi = {http://dx.doi.org/10.3842/SIGMA.2016.029},
   keywords = {generic rank two distribution in dimension five; conformal geometry; Chazy's equations.},
   language = {eng},
   issn = {1815-0659},
   journal = {Symmetry, Integrability and Geometry: Methods and Applications},
   title = {Flat (2,3,5)-Distributions and Chazy's Equations},
   url = {http://www.emis.de/journals/SIGMA/2016/029/},
   volume = {12},
   year = {2016}
}

TY  - JOUR
ID  - 1376635
AU  - Randall, Matthew James
PY  - 2016
TI  - Flat (2,3,5)-Distributions and Chazy's Equations
JF  - Symmetry, Integrability and Geometry: Methods and Applications
VL  - 12
IS  - 29
SP  - "nestrankovano"
EP  - "nestrankovano"
PB  - Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
SN  - 18150659
KW  - generic rank two distribution in dimension five
KW  - conformal geometry
KW  - Chazy's equations.
UR  - http://www.emis.de/journals/SIGMA/2016/029/
N2  - 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.
ER  -

RANDALL, Matthew James. Flat (2,3,5)-Distributions and Chazy's Equations. \textit{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.

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