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Monge-Ampère Systems with Lagrangian Pairs
The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based...
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Інститут математики НАН України
2015
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/147154 |
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irk-123456789-1471542019-02-14T01:24:48Z Monge-Ampère Systems with Lagrangian Pairs Ishikawa, G. Machida, Y. The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables ≥3. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Ampère systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3). 2015 Article Monge-Ampère Systems with Lagrangian Pairs / G. Ishikawa, Y. Machida // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58K20; 53A15; 53C42 DOI:10.3842/SIGMA.2015.081 http://dspace.nbuv.gov.ua/handle/123456789/147154 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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| language |
English |
| description |
The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables ≥3. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Ampère systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3). |
| format |
Article |
| author |
Ishikawa, G. Machida, Y. |
| spellingShingle |
Ishikawa, G. Machida, Y. Monge-Ampère Systems with Lagrangian Pairs Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Ishikawa, G. Machida, Y. |
| author_sort |
Ishikawa, G. |
| title |
Monge-Ampère Systems with Lagrangian Pairs |
| title_short |
Monge-Ampère Systems with Lagrangian Pairs |
| title_full |
Monge-Ampère Systems with Lagrangian Pairs |
| title_fullStr |
Monge-Ampère Systems with Lagrangian Pairs |
| title_full_unstemmed |
Monge-Ampère Systems with Lagrangian Pairs |
| title_sort |
monge-ampère systems with lagrangian pairs |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147154 |
| citation_txt |
Monge-Ampère Systems with Lagrangian Pairs / G. Ishikawa, Y. Machida // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT ishikawag mongeamperesystemswithlagrangianpairs AT machiday mongeamperesystemswithlagrangianpairs |
| first_indexed |
2023-05-20T17:26:43Z |
| last_indexed |
2023-05-20T17:26:43Z |
| _version_ |
1796153311293865984 |