Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields
Every Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in velocities integral of the geodesic flow is a polynomial in the linear integrals)....
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.624/ |
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| author | Matveev, Vladimir S. Nikolayevsky, Yuri |
| author_facet | Matveev, Vladimir S. Nikolayevsky, Yuri |
| author_sort | Matveev, Vladimir S. |
| collection | DOAJ |
| description | Every Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in velocities integral of the geodesic flow is a polynomial in the linear integrals). This fact led to the natural question on whether this property is shared by Killing tensor fields on all Riemannian symmetric spaces. We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces $\mathbb{H} P^n, \, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$. |
| format | Article |
| id | doaj-art-c97608e0aeb548b2a277ed1804a76593 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-c97608e0aeb548b2a277ed1804a765932025-02-07T11:23:08ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G91043104910.5802/crmath.62410.5802/crmath.624Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fieldsMatveev, Vladimir S.0Nikolayevsky, Yuri1Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, 07737 Jena, GermanyDepartment of Mathematical and Physical Sciences, La Trobe University, VIC 3086, AustraliaEvery Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in velocities integral of the geodesic flow is a polynomial in the linear integrals). This fact led to the natural question on whether this property is shared by Killing tensor fields on all Riemannian symmetric spaces. We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces $\mathbb{H} P^n, \, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.624/quadratic Killing tensorsymmetric spaceCayley projective planeQuaternionic projective space |
| spellingShingle | Matveev, Vladimir S. Nikolayevsky, Yuri Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields Comptes Rendus. Mathématique quadratic Killing tensor symmetric space Cayley projective plane Quaternionic projective space |
| title | Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields |
| title_full | Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields |
| title_fullStr | Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields |
| title_full_unstemmed | Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields |
| title_short | Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields |
| title_sort | quadratic killing tensors on symmetric spaces which are not generated by killing vector fields |
| topic | quadratic Killing tensor symmetric space Cayley projective plane Quaternionic projective space |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.624/ |
| work_keys_str_mv | AT matveevvladimirs quadratickillingtensorsonsymmetricspaceswhicharenotgeneratedbykillingvectorfields AT nikolayevskyyuri quadratickillingtensorsonsymmetricspaceswhicharenotgeneratedbykillingvectorfields |