On Separation of Variables for Symmetric Spaces of Rank1

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 $\mathbb{C}P^n$ 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 and 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 $n=2,3$.