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| id |
nasplib_isofts_kiev_ua-123456789-146598 |
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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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Variety of Integrable Killing Tensors on the 3-Sphere |
| spellingShingle |
The Variety of Integrable Killing Tensors on the 3-Sphere Schöbel, K. |
| title_short |
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_sort |
variety of integrable killing tensors on the 3-sphere |
| author |
Schöbel, K. |
| author_facet |
Schöbel, K. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146598 |
| 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. |
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AT schobelk thevarietyofintegrablekillingtensorsonthe3sphere AT schobelk varietyofintegrablekillingtensorsonthe3sphere |
| first_indexed |
2025-12-02T11:37:08Z |
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2025-12-02T11:37:08Z |
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1850862375691878400 |