On Separation of Variables for Symmetric Spaces of Rank1

2025/04/07 by Bolsinov, Alexey, Dullin, Holger R., Matveev, Vladimir. S. +1
#32M15 #37J11 #37J30 #37J35 #53C35 #53D20 #70H06 #70H15 #70H20 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2504.04999

Abstract

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 ℂPⁿ and on complex hyperbolic space ℂHⁿ, with n≥ 2, separating coordinates necessarily have precisely n ignorable coordinates. In view of results of Boyer et al, 1983 and 1985, and later results of Winternitz et al, 1994, this completes the description of separation of variables on ℂPⁿ for all n and on ℂHⁿ for n=2,3.

Related

• Orthogonal separation of variables for spaces of constant curvature
• Existence of non-isotropic conjugate points on rank one normal homogeneous spaces
• Harmonic Riemannian submersions between Riemannian symmetric spaces of noncompact type
• Rigidity of compact rank one symmetric spaces
• Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one\n symmetric spaces