Projective Metrizability and Formal Integrability
The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order part...
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| Видавець: | Інститут математики НАН України |
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| Дата: | 2011 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2011
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148091 |
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| Цитувати: | Projective Metrizability and Formal Integrability / I. Bucataru, Z. Muzsnay // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. |
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irk-123456789-1480912019-02-17T01:25:46Z Projective Metrizability and Formal Integrability Bucataru, I. Muzsnay, Z. The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P₁ and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P₁ using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P₁ is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P₁, and this obstruction is due to the curvature tensor of the induced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable. 2011 Article Projective Metrizability and Formal Integrability / I. Bucataru, Z. Muzsnay // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 49N45; 58E30; 53C60; 58B20; 53C22 DOI: http://dx.doi.org/10.3842/SIGMA.2011.114 http://dspace.nbuv.gov.ua/handle/123456789/148091 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P₁ and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P₁ using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P₁ is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P₁, and this obstruction is due to the curvature tensor of the induced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable. |
| format |
Article |
| author |
Bucataru, I. Muzsnay, Z. |
| spellingShingle |
Bucataru, I. Muzsnay, Z. Projective Metrizability and Formal Integrability Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Bucataru, I. Muzsnay, Z. |
| author_sort |
Bucataru, I. |
| title |
Projective Metrizability and Formal Integrability |
| title_short |
Projective Metrizability and Formal Integrability |
| title_full |
Projective Metrizability and Formal Integrability |
| title_fullStr |
Projective Metrizability and Formal Integrability |
| title_full_unstemmed |
Projective Metrizability and Formal Integrability |
| title_sort |
projective metrizability and formal integrability |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148091 |
| citation_txt |
Projective Metrizability and Formal Integrability / I. Bucataru, Z. Muzsnay // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT bucatarui projectivemetrizabilityandformalintegrability AT muzsnayz projectivemetrizabilityandformalintegrability |
| first_indexed |
2023-05-20T17:29:11Z |
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2023-05-20T17:29:11Z |
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1796153406352523264 |