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On Separation of Variables for Symmetric Spaces of Rank1

Alexey Bolsinov, Holger R. Dullin, Vladimir. S. Matveev, Yury Nikolayevsky

TL;DR

This work analyzes separation of variables on rank-1 Riemannian symmetric spaces, proving that a local diagonal coordinate system exists if and only if the space has constant sectional curvature, so orthogonal separation is possible only in that case. It shows nonexistence of both diagonal and non-orthogonal separating coordinates on $\mathbb{H}P^n$ ($n\ge2$), $\mathbb{O}P^2$, and $\mathbb{O}H^2$, while fully classifying separating coordinates on $\mathbb{C}P^n$ (all such coordinates are those known from prior work) and proving that any separating coordinates on $\mathbb{C}H^n$ have exactly $n$ ignorable coordinates; for $\mathbb{C}H^n$ this yields complete results for $n=2,3$. The analysis combines Killing tensor theory, Stäckel systems, and polar-action geometry to reduce the problem to abelian subalgebras of isometry algebras and totally geodesic submanifolds, offering an algorithm to classify separations on $\mathbb{C}H^n$ for arbitrary $n$ and outlining avenues toward higher-rank generalizations and connections to superintegrability.

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 $n=2,3$.

On Separation of Variables for Symmetric Spaces of Rank1

TL;DR

This work analyzes separation of variables on rank-1 Riemannian symmetric spaces, proving that a local diagonal coordinate system exists if and only if the space has constant sectional curvature, so orthogonal separation is possible only in that case. It shows nonexistence of both diagonal and non-orthogonal separating coordinates on (), , and , while fully classifying separating coordinates on (all such coordinates are those known from prior work) and proving that any separating coordinates on have exactly ignorable coordinates; for this yields complete results for . The analysis combines Killing tensor theory, Stäckel systems, and polar-action geometry to reduce the problem to abelian subalgebras of isometry algebras and totally geodesic submanifolds, offering an algorithm to classify separations on for arbitrary and outlining avenues toward higher-rank generalizations and connections to superintegrability.

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 and on complex hyperbolic space , with , separating coordinates necessarily have precisely 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 for all and on for .

Paper Structure

This paper contains 5 sections, 5 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Separation of variables
  3. Nonexistence of separating coordinates on HP"006E, HP"006E, OP"00B2 and OH"00B2
  4. Number of ignorable coordinates in separating coordinates on CPn and CHn
  5. Conclusion

Key Result

Theorem 1

Let $M^N$ be a Riemannian manifold equipped with a set of $r$ Killing vector fields $V_{N-r+1},\dots, V_N$ and $N-r$ quadratic Killing tensor fields $\overset{1}{K},\dots,\overset{N-r}{K}$. There locally exists a separating coordinate system $(x_1,\dots,x_{N-r}, t_{N-r+1}, \dots, t_N)$ on $M^N$ such Moreover, in the separating coordinate system, the Killing vector fields $V_j$ are constant linear

Theorems & Definitions (12)

  • Theorem 1: follows from Theorem 2.7 of BF1980 or § 7 in Benenti1990, Theorem 4 in KalninsMiller1981 and agafonov, see also BCR2001BCR2002
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • ...and 2 more