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Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2016, том 12
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Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
Інший ID:
2010 Mathematics Subject Classification: 53D10; 35A30; 58A30; 14M15
DOI:10.3842/SIGMA.2016.032
DOI:10.3842/SIGMA.2016.032
Посилання:
Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics / G. Manno, G. Moreno // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue on Analytical Mechanics and Dif ferential Geometry in honour
of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html.
The authors wish to express their gratitude towards the anonymous referees whose comments
contributed to shape the paper into its final form. The authors thank C. Ciliberto, E. Ferapontov
and F. Russo for stimulating discussions. The research of the first author has been partially
supported by the project “Finanziamento giovani studiosi – Metriche proiettivamente equivalenti,
equazioni di Monge–Amp`ere e sistemi integrabili”, University of Padova 2013–2015, by
the project “FIR (Futuro in Ricerca) 2013 – Geometria delle equazioni dif ferenziali”. The
research of the second author has been partially supported by the Marie Sk lodowska–Curie
Action No 654721 “GEOGRAL”, by the University of Salerno, and by the project P201/12/G028
of the Czech Republic Grant Agency (GA CR). Both the authors are members of G.N.S.A.G.A. ˇ
of I.N.d.A.M.
Дата:
2016
Переглядів:
529
Завантажень:
242
Анотація:
This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called ''meta-symplectic structure'' associated with the 8D prolongation M⁽¹⁾ of a 5D contact manifold M. We write down a geometric definition of a third-order Monge-Ampère equation in terms of a (class of) differential two-form on M⁽¹⁾. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge-Ampère equations, herewith called of Goursat type.
