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arxiv: 2212.01605 · v2 · submitted 2022-12-03 · 🧮 math.DG · math-ph· math.MP· nlin.SI

Orthogonal separation of variables for spaces of constant curvature

Alexey V. Bolsinov , Andrey Yu. Konyaev , Vladimir S. Matveev This is my paper

Pith reviewed 2026-05-24 09:51 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPnlin.SI MSC 53C2037J35
keywords orthogonal separation of variablesconstant curvature spacesKilling tensorsStäckel matricesintegrable systemsHamilton-Jacobi equationcoordinate transformations
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The pith

All orthogonal separating coordinates on constant curvature spaces of any dimension and signature can be explicitly built from flat ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to construct every orthogonal coordinate system that separates variables on spaces of constant curvature, no matter the dimension or the signature of the metric. It supplies explicit maps from these coordinates to flat or generalised flat ones and gives formulas for the Killing tensors and Stäckel matrices that realise the separation. A reader would care because this converts an abstract classification problem into a practical recipe for writing down conserved quantities and solving the Hamilton-Jacobi equation on these spaces. The construction is uniform and covers all cases without extra restrictions.

Core claim

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and the Stäckel matrices.

What carries the argument

The explicit transformation between orthogonal separating coordinates and flat or generalised flat coordinates, which yields the associated Killing tensors and Stäckel matrices.

If this is right

  • A complete explicit list of all orthogonal separating systems exists in every dimension and signature.
  • The integrals of motion for geodesic flow or other Hamiltonians can be written directly from the Stäckel matrix.
  • Solutions of the Hamilton-Jacobi equation become available by transforming to the flat coordinates.
  • The same formulas work uniformly for both positive-definite and indefinite metrics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction to flat coordinates may classify separating systems on spaces whose curvature is close to constant.
  • In Lorentzian signature the listed coordinates could simplify geodesic equations on spaces like de Sitter space.
  • Direct verification in two or three dimensions should recover all classical examples such as spherical or parabolic coordinates.

Load-bearing premise

The geometric properties of constant curvature together with the definitions of orthogonality and separation are enough to produce a complete explicit list of all such coordinate systems.

What would settle it

An orthogonal separating coordinate system on a constant curvature space whose Killing tensors or separation constants cannot be recovered from the transformation formulas to flat coordinates.

Figures

Figures reproduced from arXiv: 2212.01605 by Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev.

Figure 1
Figure 1. Figure 1: An example of an in-tree, structure of its labels and the form of the corresponding metric (5). [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and the St\"ackel matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper constructs all orthogonal separating coordinates on constant-curvature spaces of arbitrary dimension and signature. It supplies explicit coordinate transformations to flat or generalised-flat coordinates, together with closed-form expressions for the associated Killing tensors and Stäckel matrices.

Significance. If the claimed constructions are complete and explicit, the result supplies a definitive classification of orthogonal separable systems in spaces of constant curvature. Such a classification is directly useful for integrable systems, Hamilton-Jacobi theory, and the study of Killing tensors in pseudo-Riemannian geometry of any signature.

minor comments (3)
  1. [Abstract] Notation for the signature (p,q) and the dimension n is introduced only in the body; repeating the conventions in the abstract would improve immediate readability.
  2. [§3] Several displayed equations in §3 contain indices that are summed without explicit summation signs; adding the summation convention statement once at the beginning of the section would remove ambiguity.
  3. [References] The reference list omits two standard works on Stäckel systems in constant curvature (e.g., the 1990s papers by Kalnins & Miller) that are cited in the text; these should be added for completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript, including the recommendation to accept. No major comments or criticisms were raised that require a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; explicit construction from geometric definitions

full rationale

The paper claims an explicit, complete construction of all orthogonal separating coordinates (plus associated objects) on constant-curvature manifolds of arbitrary dimension and signature. This is positioned as a direct derivation from the definitions of constant curvature, orthogonality, and additive separation of variables, without any reduction of predictions to fitted parameters, self-definitional loops, or load-bearing self-citations. No equations or steps are shown that equate outputs to inputs by construction. The result is therefore self-contained as a classification theorem rather than a circular renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; the work is described as a construction within standard differential geometry of constant-curvature spaces.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Integrable geodesic flows with simultaneously diagonalisable quadratic integrals

    math.DG 2024-03 unverdicted novelty 7.0

    n functionally independent commutative quadratic integrals for a geodesic flow that are simultaneously diagonalisable imply the metric comes from the Stäckel construction and admits orthogonal separation of variables.

Reference graph

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