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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148091 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862575211817205760 |
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| author | Bucataru, I. Muzsnay, Z. |
| author_facet | Bucataru, I. Muzsnay, Z. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-11-26T11:50:12Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148091 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T11:50:12Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
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| spelling | Bucataru, I. Muzsnay, Z. 2019-02-16T20:53:19Z 2019-02-16T20:53:19Z 2011 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 https://nasplib.isofts.kiev.ua/handle/123456789/148091 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. The work of IB was supported by the Romanian National Authority for Scientific Research,
 CNCS UEFISCDI, project number PN-II-RU-TE-2011-3-0017. The work of Z.M. has been
 supported by the Hungarian Scientific Research Fund (OTKA) Grant K67617. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Projective Metrizability and Formal Integrability Article published earlier |
| spellingShingle | Projective Metrizability and Formal Integrability Bucataru, I. Muzsnay, Z. |
| title | 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_short | Projective Metrizability and Formal Integrability |
| title_sort | projective metrizability and formal integrability |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148091 |
| work_keys_str_mv | AT bucatarui projectivemetrizabilityandformalintegrability AT muzsnayz projectivemetrizabilityandformalintegrability |