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...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
National Academy of Science of Ukraine
2011-12-01
|
| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2011.114 |
| _version_ | 1818253469144842240 |
|---|---|
| author | Ioan Bucataru Zoltán Muzsnay |
| author_facet | Ioan Bucataru Zoltán Muzsnay |
| author_sort | Ioan Bucataru |
| collection | DOAJ |
| 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 P_1 and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P_1 using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P_1 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 P_1, 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. |
| first_indexed | 2024-12-12T16:40:34Z |
| format | Article |
| id | doaj.art-4b498d5c1e9e4c8fbee42aad31721546 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-12T16:40:34Z |
| publishDate | 2011-12-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-4b498d5c1e9e4c8fbee42aad317215462022-12-22T00:18:35ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592011-12-017114Projective Metrizability and Formal IntegrabilityIoan BucataruZoltán MuzsnayThe 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 P_1 and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P_1 using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P_1 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 P_1, 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.http://dx.doi.org/10.3842/SIGMA.2011.114spraysprojective metrizabilitysemi-basic formspartial differential operatorsformal integrability |
| spellingShingle | Ioan Bucataru Zoltán Muzsnay Projective Metrizability and Formal Integrability Symmetry, Integrability and Geometry: Methods and Applications sprays projective metrizability semi-basic forms partial differential operators formal integrability |
| 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 |
| topic | sprays projective metrizability semi-basic forms partial differential operators formal integrability |
| url | http://dx.doi.org/10.3842/SIGMA.2011.114 |
| work_keys_str_mv | AT ioanbucataru projectivemetrizabilityandformalintegrability AT zoltanmuzsnay projectivemetrizabilityandformalintegrability |