The Variety of Integrable Killing Tensors on the 3-Sphere
Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S³ and give a set of isometry invariants for...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2014 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2014
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146598 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Variety of Integrable Killing Tensors on the 3-Sphere / K. Schöbel // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 46 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862660855829626880 |
|---|---|
| author | Schöbel, K. |
| author_facet | Schöbel, K. |
| citation_txt | The Variety of Integrable Killing Tensors on the 3-Sphere / K. Schöbel // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 46 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S³ and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on S³ in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron K₄.
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| first_indexed | 2025-12-02T11:37:08Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146598 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-02T11:37:08Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Schöbel, K. 2019-02-10T09:46:06Z 2019-02-10T09:46:06Z 2014 The Variety of Integrable Killing Tensors on the 3-Sphere / K. Schöbel // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A60; 14H10; 14M12 DOI:10.3842/SIGMA.2014.080 https://nasplib.isofts.kiev.ua/handle/123456789/146598 Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S³ and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on S³ in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron K₄. I would like to express my gratitude to Robert Milson for his motivation and the inspiring
 discussions about my findings. I would also like to thank Alexander P. Veselov for pointing out
 the link between my solution and moduli spaces of stable curves. Finally, I would like to thank
 the anonymous referees, who helped to improve the paper considerably with their comments
 and additional references. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Variety of Integrable Killing Tensors on the 3-Sphere Article published earlier |
| spellingShingle | The Variety of Integrable Killing Tensors on the 3-Sphere Schöbel, K. |
| title | The Variety of Integrable Killing Tensors on the 3-Sphere |
| title_full | The Variety of Integrable Killing Tensors on the 3-Sphere |
| title_fullStr | The Variety of Integrable Killing Tensors on the 3-Sphere |
| title_full_unstemmed | The Variety of Integrable Killing Tensors on the 3-Sphere |
| title_short | The Variety of Integrable Killing Tensors on the 3-Sphere |
| title_sort | variety of integrable killing tensors on the 3-sphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146598 |
| work_keys_str_mv | AT schobelk thevarietyofintegrablekillingtensorsonthe3sphere AT schobelk varietyofintegrablekillingtensorsonthe3sphere |