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On separation of variables for symmetric spaces of rank 1
[Sur la séparation des variables pour les espaces symétriques de rang 1]
Alexey Bolsinov1 ; Holger R. Dullin2 ; Vladimir S. Matveev3 orcid ; Yuri Nikolayevsky4
1Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK and Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
2School of Mathematics and Statistics, University of Sydney, Australia
3Institut für Mathematik, Friedrich Schiller Universität Jena, 07737 Jena, Germany
4Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Victoria, 3086, Australia
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 267-278

We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu (2020) by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space $\mathbb{C} P^n$ and on complex hyperbolic space $\mathbb{C} H^n$, with $n\ge 2$, separating coordinates necessarily have precisely $n$ ignorable coordinates. In view of results of Boyer et al. (1983, 1985) and later results of Winternitz et al. (1994), this completes the description of separation of variables on $\mathbb{C} P^n$ for all $n$ and on $\mathbb{C} H^n$ for $n=2,3$.

Nous étudions l’existence et l’inexistence de coordonnées diagonales et séparantes pour les espaces symétriques riemanniens de rang $1$. Nous généralisons les résultats de Gauduchon et Moroianu (2020) en montrant qu’un espace symétrique de rang $1$ possède des coordonnées diagonales si et seulement s’il a une courbure sectionnelle constante. Cela implique que la séparation orthogonale des variables sur un espace symétrique de rang $1$ n’est possible que dans le cas d’une courbure sectionnelle constante. Nous montrons que, sur l’espace projectif complexe $\mathbb{C}P^{n}$ et sur l’espace hyperbolique complexe $\mathbb{C}H^{n}$, avec $n \ge 2$, les coordonnées séparées possèdent nécessairement exactement $n$ coordonnées négligeables (aussi appelées ignorables, celles qui correspondent à des champs de vecteurs de Killing). Compte tenu des résultats de Boyer et al. (1983, 1985) et des résultats ultérieurs de Winternitz et al. (1994), cela complète la description de la séparation des variables sur $\mathbb{C}P^{n}$ pour tout $n$ et sur $\mathbb{C}H^{n}$ pour $n=2,3$.

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DOI : 10.5802/crmath.823
Classification : 53C35, 32M15, 37J11, 37J35, 37J30, 53D20, 70H06, 70H15, 70H20

Alexey Bolsinov  1   ; Holger R. Dullin  2   ; Vladimir S. Matveev  3   ; Yuri Nikolayevsky  4

1 Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK and Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
2 School of Mathematics and Statistics, University of Sydney, Australia
3 Institut für Mathematik, Friedrich Schiller Universität Jena, 07737 Jena, Germany
4 Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Victoria, 3086, Australia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {On separation of variables for symmetric spaces of rank~1},
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Alexey Bolsinov; Holger R. Dullin; Vladimir S. Matveev; Yuri Nikolayevsky. On separation of variables for symmetric spaces of rank 1. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 267-278. doi: 10.5802/crmath.823

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