Invariant Nijenhuis Tensors and Integrable Geodesic Flows
We study invariant Nijenhuis (1,1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bundle T*(G/K). Such a tensor induces an invariant Poi...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2019 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2019
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210239 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Invariant Nijenhuis Tensors and Integrable Geodesic Flows / K. Lompert, A. Panasyuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862602490047889408 |
|---|---|
| author | Lompert, K. Panasyuk, A. |
| author_facet | Lompert, K. Panasyuk, A. |
| citation_txt | Invariant Nijenhuis Tensors and Integrable Geodesic Flows / K. Lompert, A. Panasyuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study invariant Nijenhuis (1,1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bundle T*(G/K). Such a tensor induces an invariant Poisson tensor Π₁ on T*(G/K), which is Poisson compatible with the canonical Poisson tensor ΠT*(G/K). This Poisson pair can be reduced to the space of G-invariant functions on T*(G/K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions for the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G/K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G=G₁⋅G₂, where G/Gᵢ are symmetric spaces, K=G₁∩G₂.
|
| first_indexed | 2025-12-07T21:24:52Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210239 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:52Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Lompert, K. Panasyuk, A. 2025-12-04T13:08:20Z 2019 Invariant Nijenhuis Tensors and Integrable Geodesic Flows / K. Lompert, A. Panasyuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J15; 37J35; 53D25 arXiv: 1812.04511 https://nasplib.isofts.kiev.ua/handle/123456789/210239 https://doi.org/10.3842/SIGMA.2019.056 We study invariant Nijenhuis (1,1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bundle T*(G/K). Such a tensor induces an invariant Poisson tensor Π₁ on T*(G/K), which is Poisson compatible with the canonical Poisson tensor ΠT*(G/K). This Poisson pair can be reduced to the space of G-invariant functions on T*(G/K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions for the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G/K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G=G₁⋅G₂, where G/Gᵢ are symmetric spaces, K=G₁∩G₂. We are very grateful to anonymous referees for their useful remarks, which allowed us to essentially improve the quality of our paper in its final version. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Invariant Nijenhuis Tensors and Integrable Geodesic Flows Article published earlier |
| spellingShingle | Invariant Nijenhuis Tensors and Integrable Geodesic Flows Lompert, K. Panasyuk, A. |
| title | Invariant Nijenhuis Tensors and Integrable Geodesic Flows |
| title_full | Invariant Nijenhuis Tensors and Integrable Geodesic Flows |
| title_fullStr | Invariant Nijenhuis Tensors and Integrable Geodesic Flows |
| title_full_unstemmed | Invariant Nijenhuis Tensors and Integrable Geodesic Flows |
| title_short | Invariant Nijenhuis Tensors and Integrable Geodesic Flows |
| title_sort | invariant nijenhuis tensors and integrable geodesic flows |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210239 |
| work_keys_str_mv | AT lompertk invariantnijenhuistensorsandintegrablegeodesicflows AT panasyuka invariantnijenhuistensorsandintegrablegeodesicflows |