Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Vladimir S. Matveev; Yuri Nikolayevsky

doi:10.5802/crmath.624

Article de recherche - Géométrie et Topologie

Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields
[Tenseurs de Killing quadratiques sur des espaces symétriques qui ne sont pas générés par des champs de Killing vectoriels]

Vladimir S. Matveev ¹ ; Yuri Nikolayevsky ²

¹ Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, 07737 Jena, Germany
² Department of Mathematical and Physical Sciences, La Trobe University, VIC 3086, Australia

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1043-1049.

Résumé
Abstract

Chaque champ tensoriel de Killing sur l’espace de courbure constante et sur l’espace projectif complexe peut être décomposé en la somme des produits tensoriels symétriques des champs vectoriels de Killing (de manière équivalente, chaque polynôme des intégrales de vitesses du flux géodésique est un polynôme dans les intégrales linéaires). Ce fait a conduit à la question naturelle de savoir si cette propriété est partagée par les champs tensoriels de Killing sur tous les espaces symétriques riemanniens. Nous répondons à cette question par la négative en construisant des exemples explicites de champs tensoriels de Killing quadratiques qui ne sont pas des formes quadratiques dans les champs vectoriels de Killing sur les espaces projectifs quaternioniques $ℍ P^{n}, n \geq 3$ , et sur le Cayley plan projectif $𝕆 P^{2}$ .

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 $ℍ P^{n}, n \geq 3$ , and on the Cayley projective plane $𝕆 P^{2}$ .

Reçu le : 2023-12-30
Accepté le : 2024-08-27
Publié le : 2024-11-04

DOI : 10.5802/crmath.624

Classification : 53C35, 53B20, 37J30, 37J35, 70H06, 22E46
Keywords: quadratic Killing tensor, symmetric space, Cayley projective plane, Quaternionic projective space
Mot clés : Tenseur de Killing quadratique, espace symétrique, plan projectif de Cayley, espace projectif quaternionique

Affiliations des auteurs :

Vladimir S. Matveev ¹ ; Yuri Nikolayevsky ²

¹ Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, 07737 Jena, Germany
² Department of Mathematical and Physical Sciences, La Trobe University, VIC 3086, Australia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Vladimir S. Matveev and Yuri Nikolayevsky},
     title = {Quadratic {Killing} tensors on symmetric spaces which are not generated by {Killing} vector fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1043--1049},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.624},
     language = {en},
}

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Vladimir S. Matveev; Yuri Nikolayevsky. Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1043-1049. doi : 10.5802/crmath.624. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.624/

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