On separation of variables for symmetric spaces of rank 1

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 ℂ P n and on complex hyperbolic space ℂ H n , with n ≥ 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 ℂ P n for all n and on ℂ H n for n = 2 , 3 .