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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2016 |
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Інститут математики НАН України
2016
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147724 |
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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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862635036972417024 |
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| author | Randall, M. |
| author_facet | Randall, M. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-11-30T17:03:05Z |
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| id | nasplib_isofts_kiev_ua-123456789-147724 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T17:03:05Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
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| spelling | Randall, M. 2019-02-15T18:43:43Z 2019-02-15T18:43:43Z 2016 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 https://nasplib.isofts.kiev.ua/handle/123456789/147724 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. This work is inspired by the paper of [6]. The author would like to thank Daniel An, Pawe l
 Nurowski, Travis Willse and the anonymous referees for comments. Part of this work is supported
 by the Grant agency of the Czech Republic P201/12/G028. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Flat (2,3,5)-Distributions and Chazy's Equations Article published earlier |
| spellingShingle | Flat (2,3,5)-Distributions and Chazy's Equations Randall, M. |
| title | 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_short | Flat (2,3,5)-Distributions and Chazy's Equations |
| title_sort | flat (2,3,5)-distributions and chazy's equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147724 |
| work_keys_str_mv | AT randallm flat235distributionsandchazysequations |