Flat (2,3,5)-Distributions and Chazy's Equations
n 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...
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Інститут математики НАН України
2016
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| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Zitieren: | Flat (2,3,5)-Distributions and Chazy's Equations / M. Randall // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1477242019-02-16T01:25:34Z Flat (2,3,5)-Distributions and Chazy's Equations Randall, M. n 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. 2016 Article Flat (2,3,5)-Distributions and Chazy's Equations / M. Randall // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58A30; 53A30; 34A05; 34A34 DOI:10.3842/SIGMA.2016.029 http://dspace.nbuv.gov.ua/handle/123456789/147724 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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n 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. |
| format |
Article |
| author |
Randall, M. |
| spellingShingle |
Randall, M. Flat (2,3,5)-Distributions and Chazy's Equations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Randall, M. |
| author_sort |
Randall, M. |
| title |
Flat (2,3,5)-Distributions and Chazy's Equations |
| title_short |
Flat (2,3,5)-Distributions and Chazy's Equations |
| title_full |
Flat (2,3,5)-Distributions and Chazy's Equations |
| title_fullStr |
Flat (2,3,5)-Distributions and Chazy's Equations |
| title_full_unstemmed |
Flat (2,3,5)-Distributions and Chazy's Equations |
| title_sort |
flat (2,3,5)-distributions and chazy's equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147724 |
| citation_txt |
Flat (2,3,5)-Distributions and Chazy's Equations / M. Randall // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT randallm flat235distributionsandchazysequations |
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2025-07-11T02:43:10Z |
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