Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations
Painlevé metrics are a class of Riemannian metrics that generalize the well-known separable metrics of Stäckel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separa...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Zitieren: | Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations / T. Daudé, N. Kamran, F. Nicoleau // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 48 назв. — англ. |
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Daudé, T. Kamran, N. Nicoleau, F. 2025-12-04T13:02:39Z 2019 Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations / T. Daudé, N. Kamran, F. Nicoleau // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 48 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53B21; 70H20; 81Q80 arXiv: 1903.10573 https://nasplib.isofts.kiev.ua/handle/123456789/210226 https://doi.org/10.3842/SIGMA.2019.069 Painlevé metrics are a class of Riemannian metrics that generalize the well-known separable metrics of Stäckel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary differential equations, which characterizes the Stäckel case. Painlevé metrics in dimension n thus admit, in general, only r<n linearly independent Poisson-commuting quadratic first integrals of the geodesic flow, where r denotes the number of groups of variables. Our goal in this paper is to carry out for Painlevé metrics the generalization of the analysis, which has been extensively performed in the Stäckel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlevé metrics of the Robertson separability conditions for the Helmholtz equation, which are familiar from the Stäckel case, and a formulation thereof in terms of the vanishing of the off-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for Stäckel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist r<n linearly independent second-order differential operators that commute with the Laplace-Beltrami operator and that are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations that are compatible with the separation into blocks of variables of the Helmholtz equation for Painlevé metrics, leading to solutions that are R-separable in blocks. The paper concludes with a set of open questions and perspectives. We thank Willard Miller, Jr., Askold Perelomov, and Petar Topalov for having kindly answered our queries on Painlevé metrics. We are also grateful to Claudia Chanu for her helpful replies to our questions pertaining to R-separability and the Robertson conditions for Stäckel metrics. Finally, we thank the referees for their detailed comments and constructive suggestions. Thierry Daudé’s research is supported by the JCJC French National Research Projects Horizons, No. ANR-16-CE40-0012-01, Niky Kamran’s research is supported by NSERC grant RGPIN 105490-2018, and Francois Nicoleau’s research is supported by the French National Research Project GDR Dynqua. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations |
| spellingShingle |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations Daudé, T. Kamran, N. Nicoleau, F. |
| title_short |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations |
| title_full |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations |
| title_fullStr |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations |
| title_full_unstemmed |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations |
| title_sort |
separability and symmetry operators for painlevé metrics and their conformal deformations |
| author |
Daudé, T. Kamran, N. Nicoleau, F. |
| author_facet |
Daudé, T. Kamran, N. Nicoleau, F. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Painlevé metrics are a class of Riemannian metrics that generalize the well-known separable metrics of Stäckel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary differential equations, which characterizes the Stäckel case. Painlevé metrics in dimension n thus admit, in general, only r<n linearly independent Poisson-commuting quadratic first integrals of the geodesic flow, where r denotes the number of groups of variables. Our goal in this paper is to carry out for Painlevé metrics the generalization of the analysis, which has been extensively performed in the Stäckel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlevé metrics of the Robertson separability conditions for the Helmholtz equation, which are familiar from the Stäckel case, and a formulation thereof in terms of the vanishing of the off-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for Stäckel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist r<n linearly independent second-order differential operators that commute with the Laplace-Beltrami operator and that are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations that are compatible with the separation into blocks of variables of the Helmholtz equation for Painlevé metrics, leading to solutions that are R-separable in blocks. The paper concludes with a set of open questions and perspectives.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210226 |
| citation_txt |
Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations / T. Daudé, N. Kamran, F. Nicoleau // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 48 назв. — англ. |
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| first_indexed |
2025-12-07T21:24:48Z |
| last_indexed |
2025-12-07T21:24:48Z |
| _version_ |
1850886260225212416 |