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arxiv: 2605.00403 · v2 · submitted 2026-05-01 · 🧮 math-ph · math.MP

Recognition: 2 theorem links

· Lean Theorem

Generalized Fourier Transforms for Momentum-Space Construction on Riemannian Manifolds

Seramika Ariwahjoedi , Muhammad Farchani Rosyid , Andika Kusuma Wijaya
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:54 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords generalized Fourier transformRiemannian manifoldLaplace-Beltrami operatorspectral decompositionParseval-Plancherel theoremmomentum spaceKilling vectorsMASA
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The pith

A generalized Fourier transform on Riemannian manifolds is defined via spectral decomposition of the Laplace-Beltrami operator and satisfies a generalized Parseval-Plancherel theorem under minimal conditions.

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

The paper constructs a Generalized Fourier Transform for any Riemannian manifold by using the eigenfunctions of the Laplace-Beltrami operator to decompose functions into a momentum-space representation. The transform must act as an isometric isomorphism whose kernel diagonalizes the operator. Spectral degeneracy is handled by introducing a local symmetry-adapted maximal Abelian commuting set of operators built from the manifold's geometry, such as those coming from Killing vectors. This produces momentum label spaces whose structure encodes the available symmetries. The construction distinguishes changes that come from actual isometries, which preserve the transform, from changes that merely alter the degeneracy resolution and can produce different labelings while remaining unitarily equivalent.

Core claim

Under the requirements that the GFT is an isometric isomorphism and its kernel diagonalizes the Laplace-Beltrami operator, the transform obeys a generalized Parseval-Plancherel theorem. Spectral degeneracy is resolved by a fiberwise maximal Abelian commuting set (MASA) built from geometric differential operators, particularly those from Killing data when available. The resulting momentum spaces encode symmetry constraints, and unitary equivalences are classified by whether they arise from isometries or from different separation schemes.

What carries the argument

The Generalized Fourier Transform (GFT) defined through spectral decomposition of the Laplace-Beltrami operator, with degeneracy resolved by a local symmetry-adapted fiberwise MASA constructed from geometric operators.

If this is right

  • The momentum label spaces reflect geometric symmetry constraints of the manifold.
  • Unitary changes induced by true isometries preserve the GFT structure, whereas changes in degeneracy resolution schemes can produce inequivalent k-space labelings.
  • Manifolds admit a dual classification by MASA completeness and Stackel separability together with the topology of the momentum space.
  • The construction supplies a basis for curved-space mode decompositions suitable for later dynamical applications.

Where Pith is reading between the lines

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

  • On manifolds with known symmetries such as spheres or hyperbolic spaces, the method should recover adapted bases like spherical harmonics or hyperbolic plane waves.
  • The same spectral construction could supply natural mode expansions for field theories on curved backgrounds once the dynamical equations are expressed in the GFT basis.
  • For non-compact manifolds the continuous parts of the spectrum would still require the MASA to produce well-defined continuous momentum labels.
  • Specializing to flat Euclidean space with Cartesian coordinates should reproduce the ordinary Fourier transform as a consistency check.

Load-bearing premise

The degenerate sectors of the Laplace-Beltrami spectrum can be resolved by a local, symmetry-adapted maximal Abelian commuting set of geometric differential operators.

What would settle it

A Riemannian manifold on which no such fiberwise MASA exists that fully resolves all degeneracies while preserving the isometric isomorphism property of the transform.

Figures

Figures reproduced from arXiv: 2605.00403 by Andika Kusuma Wijaya, Muhammad Farchani Rosyid, Seramika Ariwahjoedi.

Figure 1
Figure 1. Figure 1: Commutative diagram representing the local coord view at source ↗
Figure 2
Figure 2. Figure 2: The GFT classification, based on MASA completeness view at source ↗
Figure 3
Figure 3. Figure 3: Classification of GFT based on the topology of view at source ↗
Figure 4
Figure 4. Figure 4: Killing flows on T 2 with varying slopes. (i) Rational slope; Top, from left to right: tan ε = {1/5, 1/10} ; Bottom, from left to right tan ε = {5, 10} . (ii) Irrational slope; Top: tan ε = 1/ √ 2; from left to right: 1000 time steps, 4000 time steps. Bottom: tan ε = √ 2; from left to right: 1000 time steps, 4000 time steps. Notice that for the irrational slopes, the Killing integral curve covers the surfa… view at source ↗
read the original abstract

We extend Fourier analysis to curved spaces by defining a Generalized Fourier Transform (GFT) on any Riemannian manifold $\Sigma$ via spectral decomposition. Under minimal requirements that the transform is an isometric isomorphism and has a kernel diagonalizing the Laplace-Beltrami operator, we prove that the GFT satisfies a generalized Parseval-Plancherel theorem. To resolve the spectral degeneracy that obscures "momentum space" in such settings, we require the degenerate sector to be resolved by a local, symmetry-adapted maximal Abelian commuting set (a fiberwise MASA), constructed from geometric differential operators, most notably from Killing data when such symmetries are available. We provide a constructive algorithm for generating these commuting operators and show that the resulting momentum label spaces $\mathcal{F}$ (discrete, continuous, or mixed) reflect geometric symmetry constraints. We introduce a dual classification: (i) by MASA completeness and Stackel separability, and (ii) by the topology of $\mathcal{F}$. Finally, we distinguish unitary changes induced by true isometries (which preserve the GFT structure) from changes of coordinate-adapted degeneracy resolution/separation schemes, which may induce inequivalent $k$-space labelings (e.g. Cartesian vs spherical constructions in $\mathbb{R}^{3}$) while remaining unitarily equivalent on $\mathcal{L}^{2}\left[\Sigma\right]$. This symmetry-adapted harmonic analysis is intended as a foundation for curved-space mode decompositions; dynamical applications are developed in the subsequent work.

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

2 major / 3 minor

Summary. The paper introduces the Generalized Fourier Transform (GFT) on Riemannian manifolds via the spectral decomposition of the Laplace-Beltrami operator. Under the minimal assumptions that the GFT is an isometric isomorphism and its kernel diagonalizes the Laplace-Beltrami operator, it proves a generalized version of the Parseval-Plancherel theorem. The paper further develops a method to resolve spectral degeneracies using a fiberwise maximal Abelian commuting set (MASA) constructed from geometric differential operators, such as those derived from Killing vectors. It provides a constructive algorithm for these operators, classifies the resulting momentum label spaces F by MASA completeness, Stackel separability, and topology, and distinguishes between unitary transformations induced by isometries and those from different degeneracy resolution schemes.

Significance. This framework extends classical Fourier analysis to general curved spaces in a symmetry-aware manner. If the claims hold, it could serve as a foundation for mode decompositions in curved-space physics and geometry. The constructive algorithm and the careful distinction between different types of unitary equivalences are notable strengths that could facilitate practical applications on symmetric manifolds. The absence of free parameters and the grounding in spectral theory are positive aspects.

major comments (2)
  1. [Theorem on generalized Parseval-Plancherel] The generalized Parseval-Plancherel theorem is asserted to follow directly from the isometry and spectral theorem for the Laplace-Beltrami operator, but the manuscript must explicitly address how the Plancherel identity is formulated when the fiberwise MASA is used to label the degenerate eigenspaces, particularly in the mixed discrete-continuous case for F (see the theorem statement and its proof).
  2. [MASA construction and algorithm] The weakest assumption—that the degenerate sector can always be resolved by a local, symmetry-adapted fiberwise MASA constructed from geometric differential operators—requires a general existence argument or precise conditions on the manifold for the construction to be maximal Abelian; this is load-bearing for the claim that the GFT applies to any Riemannian manifold (see the section introducing the MASA and the constructive algorithm).
minor comments (3)
  1. [Abstract and notation section] Standardize notation: replace inconsistent forms such as L^{2}[Σ] and L^{2}(Σ) with a single convention throughout, and define the momentum space F at its first appearance rather than relying on context.
  2. [Definition of GFT] Add a brief reference to the classical spectral theorem for self-adjoint elliptic operators on compact or complete Riemannian manifolds to anchor the GFT definition.
  3. [Classification and examples] The classification of F by topology and MASA completeness would be strengthened by at least one explicit worked example (e.g., S^{2} or H^{2}) showing how different resolutions yield inequivalent labelings while remaining unitarily equivalent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions, which have helped us improve the clarity and precision of the manuscript. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: The generalized Parseval-Plancherel theorem is asserted to follow directly from the isometry and spectral theorem for the Laplace-Beltrami operator, but the manuscript must explicitly address how the Plancherel identity is formulated when the fiberwise MASA is used to label the degenerate eigenspaces, particularly in the mixed discrete-continuous case for F (see the theorem statement and its proof).

    Authors: The referee is correct that the role of the MASA in the Plancherel identity deserves explicit treatment, especially for mixed spectra. The identity itself derives solely from the isometry of the GFT, independent of the specific choice of labels within degenerate subspaces. However, the measure on the momentum space F is defined using the MASA to parametrize the degeneracy. In the mixed discrete-continuous case, the continuous part of F is equipped with a measure that incorporates the volume factors from the MASA coordinates on the fibers. We will revise the proof of the theorem to include a clarifying subsection that details how the MASA enters the formulation of the Plancherel identity in these cases, ensuring the statement is unambiguous. revision: yes

  2. Referee: The weakest assumption—that the degenerate sector can always be resolved by a local, symmetry-adapted fiberwise MASA constructed from geometric differential operators—requires a general existence argument or precise conditions on the manifold for the construction to be maximal Abelian; this is load-bearing for the claim that the GFT applies to any Riemannian manifold (see the section introducing the MASA and the constructive algorithm).

    Authors: We appreciate this observation regarding the scope. The core GFT is defined for any Riemannian manifold satisfying the minimal assumptions of being an isometric isomorphism with a kernel that diagonalizes the Laplace-Beltrami operator; this does not depend on the MASA. The MASA construction is presented as a method to resolve degeneracies when geometric symmetries (such as Killing vectors) permit it, allowing for a unique momentum-space labeling. The manuscript states that we 'require' the degenerate sector to be resolved in this manner for the full construction, rather than asserting universal existence. For manifolds without adequate symmetries, full resolution may not be achievable via geometric operators, and the resulting F reflects the unresolved degeneracy. The algorithm is constructive in the presence of such operators. To address the referee's concern, we will add explicit statements in the introduction and the MASA section specifying the conditions (e.g., sufficient isometries leading to a complete MASA) under which the construction yields a maximal Abelian set, and clarify that the framework applies generally but the momentum labels are fully resolved only when these conditions hold. revision: yes

Circularity Check

1 steps flagged

Main theorem is tautological given the isometric-isomorphism requirement in its own definition

specific steps
  1. self definitional [Abstract]
    "Under minimal requirements that the transform is an isometric isomorphism and has a kernel diagonalizing the Laplace-Beltrami operator, we prove that the GFT satisfies a generalized Parseval-Plancherel theorem."

    The generalized Parseval-Plancherel theorem asserts that the transform preserves the L2 inner product (or norm). Requiring the GFT to be an isometric isomorphism therefore makes the theorem hold by the definition of isometry; the 'proof' adds no independent content beyond restating the assumption.

full rationale

The paper defines the GFT via spectral decomposition of the Laplace-Beltrami operator and then states that, under the explicit requirement that this transform be an isometric isomorphism whose kernel diagonalizes the operator, it satisfies a generalized Parseval-Plancherel theorem. Because the Parseval-Plancherel identity is precisely the L2-norm preservation property of an isometric isomorphism, the claimed theorem reduces directly to the assumption by definition with no additional derivation. The MASA construction for degeneracy resolution is independent of this step and does not affect the circularity of the central claim. No other load-bearing reductions to self-citation or fitted inputs appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the spectral theorem for the Laplace-Beltrami operator on Riemannian manifolds and the existence of suitable geometric differential operators (including Killing fields when available) that can be assembled into a fiberwise MASA. No free parameters are introduced; the construction is presented as geometric and constructive.

axioms (2)
  • standard math The Laplace-Beltrami operator on a Riemannian manifold admits a spectral decomposition that allows an isometric isomorphism between L2 functions and a suitable label space.
    Invoked as the basis for defining the GFT kernel that diagonalizes the operator.
  • domain assumption When symmetries are present, Killing vector fields generate differential operators that can be used to construct a maximal Abelian commuting set resolving spectral degeneracy.
    Stated as the source for the fiberwise MASA in the degeneracy-resolution step.
invented entities (2)
  • Generalized Fourier Transform (GFT) no independent evidence
    purpose: To extend Fourier analysis to arbitrary Riemannian manifolds via spectral decomposition while satisfying an isometric isomorphism property.
    Newly defined transform whose kernel diagonalizes the Laplace-Beltrami operator.
  • fiberwise MASA no independent evidence
    purpose: To resolve spectral degeneracy and produce unique momentum labels reflecting geometric symmetries.
    Constructed locally from geometric differential operators; the paper presents it as a new application in this context.

pith-pipeline@v0.9.0 · 5581 in / 1745 out tokens · 71343 ms · 2026-05-12T04:54:06.343713+00:00 · methodology

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Reference graph

Works this paper leans on

105 extracted references · 105 canonical work pages

  1. [1]

    The definition of the momentum domain ( 6) assumes our MASA includes the Laplace-Beltrami operator △

    Formalization: Invariance vs Representation-Dependen ce of k-Space. The definition of the momentum domain ( 6) assumes our MASA includes the Laplace-Beltrami operator △ . In many practical cases, requiring △ to be explicitly in the generator set would cause conflicts: either the subalgebra is redundant, not maximal, or the sets containing non-local operato ...

    work page

  2. [2]

    To accommodate cases where △ is not a generator but a function of the generators, we generalize the definition of k-space as follows: Definition

    is valid for the case where our MASA includes the Laplace-Beltrami operator △ . To accommodate cases where △ is not a generator but a function of the generators, we generalize the definition of k-space as follows: Definition. (k-space as joint spectrum/label space). Let D = { ˆO1,..., ˆOm } , m ≤ n, be a commuting family of (essentially) self-adjoint operat...

    work page

  3. [3]

    Then: (1) ˆUφ =φ ∗ is unitary on L2 (Σ,dµ Σ ), (2) ˆUφ =φ ∗ commute with the Laplace-Beltrami operator: φ ∗ △ = △ φ ∗

    and let ˆUφ =φ ∗ be the pullback operator on H defined by ( 37). Then: (1) ˆUφ =φ ∗ is unitary on L2 (Σ,dµ Σ ), (2) ˆUφ =φ ∗ commute with the Laplace-Beltrami operator: φ ∗ △ = △ φ ∗ . (3) If D = { ˆOi }m i=1 is a MASA for the commutant △ with joint spectrum F , then the transformed family D′ = { ˆUφ ˆOi ˆU † φ }m i=1 also commute with △ , and their joint ...

    work page

  4. [4]

    St ¨ackel Manifold and St ¨ackel Coordinate A Riemannian manifold (Σ , q) is a St¨ ackel manifold / St¨ ackel-separable if the Hamilton-Jacobi equation for the geodesic flow admits a complete additive separation for each of its va riables [ 31, 53, 54]. Equivalently, there exists a local coordinate chart (the St¨ ackel coordinate) in Σ that allows t he sep...

    work page

  5. [5]

    Building on the MASA algorithm, we outline the constructive procedure to find St¨ ackel coordinates

    A Constructive Algorithm for St ¨ackel Coordinate Let us return to the algorithm to find the MASA in Section IV B. Building on the MASA algorithm, we outline the constructive procedure to find St¨ ackel coordinates. Afte r Step 2, we could obtain the maximal commuting sets of Killing vectors {σa}, a = 1,...,m where m<n . The next steps are the following: St...

    work page

  6. [6]

    Solve the rank-2 Killing-tensor equation ( 30) to obtain the symmetric tensors K =β pˆκ p, β p ∈ R, and ˆκ p is the basis/generator that construct the space of solution K

    work page

  7. [7]

    This could be done by solving LKK = 0, K is the Killing vector, or simply by checking Lσ a ˆκ p = 0

    Construct a subspace inside K0 ⊂ K where all its elements commute with the Killing vectors {ˆσa}. This could be done by solving LKK = 0, K is the Killing vector, or simply by checking Lσ a ˆκ p = 0

    work page

  8. [8]

    If the rank of this space is lower than ( n − 1 − r), the manifold does not possess a St¨ ackel coordinate and hence a non-St¨ ackel manifold

    Find a maximal Abelian set in K0, i.e., {ˆκ a} where every element inside are in involution [ˆ κ a, ˆκ b]NS = 0. If the rank of this space is lower than ( n − 1 − r), the manifold does not possess a St¨ ackel coordinate and hence a non-St¨ ackel manifold. Step 5: Obtain the remaining non-cyclic coordinate from the second order Killing tensors. Choose (n −...

    work page

  9. [9]

    As written, they would not give a stan dard eigenproblem unless ˆκ p =κ ij p∂i ⊗ ∂j is an endomorphism

    Notice that {ˆκ p} is a set of rank (2,0) tensors. As written, they would not give a stan dard eigenproblem unless ˆκ p =κ ij p∂i ⊗ ∂j is an endomorphism. So let us define define an endomorphism ˆ κ ∗ p = (κ p)i j∂i ⊗ dxj, where its components are obtained by metric contraction as follows : (κ p)i j :=κ ik p qjk

    work page

  10. [10]

    Diagonalize each ˆκ ∗ p to obtain the eigenfunction 1-form (eigenline) e(p) λ , i.e.: ˆκ ∗ pe(p) λ =λ (p)e(p) λ , (41) with λ (p) is its corresponding eigenvalue

    work page

  11. [11]

    Check the Frobenius integrability condition of the eigenfunction e(p) λ , namely [ 61, 62]: e(p) λ ∧ de(p) λ = 0. (42) If this is satisfied, e(p) λ is integrable by the Frobenius theorem and could be written as e(p) λ = g[x]dq(p) λ [x]; otherwise, find a linear combination of all possible eigenfunction e(p) λ inside the degenerate eigenspace Hx|λ ⊂ H x. If ...

    work page

  12. [12]

    The last variable χ is obtained by integrating any scalar function whose gradient annihilates all symmetry directions, namely: α (a)i∂iχ = 0, β (p)ij∂jχ = 0

    q(p) λ [x] : vp is the remaining ( n − 1 − r) coordinate, which, together with {ua} gives (n − 1) coordinate variables. The last variable χ is obtained by integrating any scalar function whose gradient annihilates all symmetry directions, namely: α (a)i∂iχ = 0, β (p)ij∂jχ = 0. Locally, a solution exists provided the annihilator distribution is integr able...

    work page

  13. [13]

    The St¨ ackel coordinate is {ua,v p,χ }, with a = 1,...,r, p =r + 1,...,n − 1. One could verify that ¯x = {ua := ¯xa,v p := ¯xp,χ := ¯xn} is St¨ ackel by showing that the Laplace-Beltrami equation and the Hamilton-Jacobi equation are separable in this coordinate. 24

    work page

  14. [14]

    The R-Separation Trick In St¨ ackel coordinate, the metricq of a St¨ ackel manifold Σ could always be written in a diagonal form (wit h diagonal components qi[¯xi]): ds2 = n∑ i=1 qi[¯xi] ( d¯xi) 2 = r∑ a=1 qa (d¯xa)2 + n∑ b=r+1 qb[¯xb] ( d¯xb) 2 , (43) where qa = ca is a constant (because ¯ xa for a = 1,...,r are cyclic) and its qb[¯xb] is only a function...

    work page

  15. [15]

    Semi-Algorithm and Completeness at Rank m The operator construction procedure in Section IV B is a semi-algor ithm in the sense that it can certify completeness of the MASA, but not its incompleteness. We say that the MASA is com plete at Killing-tensor rank m if by rankm one succesfully identifies a maximal set of ( n − 1) mutually commuting operators in ...

    work page

  16. [16]

    algebraically incomplete

    Why Rank 2? The St ¨ackel Certificate The choice m0 = 2 is not arbitrary. Rank-2 Killing tensors admit several distinguishe d physical and mathematical features that make them the natural cutoff for a standard GFT c lassification: (1): Natural companions to the Laplace-Beltrami operator : △ is second order and canonically built from the metric; the next simp...

    work page

  17. [17]

    incomplete

    Completeness Classification at Rank m = 2 Here, we classify the GFT systems based on their MASA completenes s/ incompleteness at rank 2 and the existence of St¨ ackel coordinate. It must be emphasized that this classification is restricted to the cutoff m = 2. The completeness certificate is absolute: once the ( n − 1)-quota is achieved at a finite order m, hi...

    work page

  18. [18]

    This arises in non- compact, unbounded manifolds (e.g., Euclidean plane R2 and hyperbolic plane H2 with Cartesian MASA)

    Topological Classification We classify the GFT into three categories based on the structure o f the joint spectrum σ ( ˆO1,..., ˆOm ) (the topology of the k-space F ): • Continuous (C) : Occurs when the spectrum is purely continuous. This arises in non- compact, unbounded manifolds (e.g., Euclidean plane R2 and hyperbolic plane H2 with Cartesian MASA). The...

    work page

  19. [19]

    standard

    The 3 × 3 Classification Chart Together with the topological classification, we could construct a u nified taxonomy of GFTs on Riemannian mani- folds, based on their algebraic completeness (MASA completeness in m = 2 and Helmholtz separability): Type I-III from Section VI A, and spectral (Fourier dual-space) topology: C lass D, C, SD from Section VI B. The n...

    work page

  20. [20]

    coordinate depe ndence

    Coordinate Transformations, Isometries, and Gauge Free dom Choosing a basis within each degenerate eigenspace of the Laplace- Beltrami operator is equivalent to choosing ad- ditional commuting self-adjoint operators (preferably local) whos e joint eigenfunctions resolve the degeneracy, i.e. a maximal commuting choice within each degenerate spectral fiber (...

    work page

  21. [21]

    momentum

    Toward a Unified Definition of Momentum At present the word “momentum” in physics terminology has severa l context-dependent meanings:

    work page

  22. [22]

    Canonical/Noether perspective (local): Momentum is a cotangen t vector, the conjugate variable to position that generates spatial translations; that is, the quantity p = ∂L ∂ ˙x ∈ T ∗ x Σ with ˙x = dx dλ (λ a real parameter) that is well-defined if we provide a scalar functional L[x, ˙x,λ ] on Σ. The Special and General Relativity perspective of momentum a...

    work page

  23. [23]

    momentum space

    The GFT/spectral perspective (global): Momentum is a spectra l label belonging to the eigenvalue set of a self-adjoint operator (MASA) on Σ . These joint eigen-labels form the familiar k-space. In our framework, the momentum space is the topological space of real spectral parame ters and its degeneracy F, that label the irreducible representations of the ...

    work page

  24. [24]

    geometrize

    The Remaining Freedom: Geometrizing the k-Space We noted in Section II that the GFT is not unique. Our framework rev eals three inherent levels of freedom in defining the spectral domain: • Basis rotation inside degenerate fibers. This is related to MASA (orthonormal basis) freedom : The choice of MASA will affect the orthonormal basis, consequently, fixes th...

    work page

  25. [25]

    M. Reed, B. Simon. Functional Analysis (Methods of Modern Mathematical Physics Volume I). Academic Press. 1980

    work page 1980

  26. [26]

    G. B. Folland. Fourier Analysis and its Applications . Wadsworth and Brooks. California. 1992

    work page 1992

  27. [27]

    Grigor’yan

    A. Grigor’yan. Heat Kernel and Analysis on Manifolds . American Mathematical Soc. 2009

    work page 2009

  28. [28]

    P. Berard. Spectral Geometry: Direct and Inverse Problems . Lecture Notes in Mathematics 1207. 1986

    work page 1986

  29. [29]

    S. A. Morris. Pontryagin Duality and the Structure of Locally Compact Abe lian Groups. London Math. Soc. Lecture Notes

    work page

  30. [30]

    Cambridge U. Press. 1977

    work page 1977

  31. [31]

    Reiter, J

    H. Reiter, J. D. Stegeman. Classical Harmonic Analysis and Locally Compact Groups . 2nd ed. Clarendon Press. Oxford. 2000

    work page 2000

  32. [32]

    W. Rudin. Fourier Analysis on Groups . D. van Nostrand Co. 1962

    work page 1962

  33. [33]

    Helgason

    S. Helgason. Geometric Analysis on Symmetric Spaces . Am. Math. Soc. Providence .1994

    work page 1994

  34. [34]

    Helgason

    S. Helgason. Groups and Geometric Analysis : 425-444. Academic Press. New York. 1984

    work page 1984

  35. [35]

    Helgason

    S. Helgason. The Fourier transform on symmetric spaces . ´Elie Cartan et les math´ ematiques d’aujourd’hui no.131: 151-164. 1984

    work page 1984

  36. [36]

    Mohanty, S

    P. Mohanty, S. K. Ray, R. P. Sarkar, A. Sitaram. The Helgason–Fourier Transform for Symmetric Spaces II . Journal of Lie Theory Volume 14: 227–242. Heldermann-Verlag. 2004

    work page 2004

  37. [37]

    Boujeddaine, M

    M. Boujeddaine, M. E. Kassimi, S. Fahlaoui. Helgason–Gabor–Fourier transform and uncertainty princi ples. International Journal of Wavelets, Multiresolution, and Information Pro cessing 19 (01): 2050056. 2021

    work page 2021

  38. [38]

    Helgasson

    S. Helgasson. Wave equations on homogeneous spaces. Lie Group Representations III. Lecture Notes in Math. 1077: 254-287. Springer-Verlag, New York. 1984

    work page 1984

  39. [39]

    A. Terras. Harmonic Analysis on Symmetric Spaces and Applications I . 1985. Springer-Verlag

    work page 1985

  40. [40]

    I. M. Gelfand, M. A. Naimark. On the imbedding of normed rings into the ring of operators on a Hilbert space . Mat. Sbornik. 12 (2): 197–217. 1943

    work page 1943

  41. [41]

    Hewitt, K

    E. Hewitt, K. A. Ross. Abstract Harmonic Analysis. Volume I: Structure of Topolog ical Groups Integration Theory Group Representations. Springer-Verlag. 1979

    work page 1979

  42. [42]

    M. E. Taylor. Noncommutative Harmonic Analysis . American Mathematical Society. 1986

    work page 1986

  43. [43]

    Carmona, P

    J. Carmona, P. Delorme, M. Vergne. Noncommutative harmonic analysis: in honor of Jacques Carm ona. Springer. 2004

    work page 2004

  44. [44]

    K. I. Gross. On the evolution of noncommutative harmonic analysis . Amer. Math. Monthly. 85 (7): 525–548. 1978

    work page 1978

  45. [45]

    H¨ ormander.Fourier integral operators I

    L. H¨ ormander.Fourier integral operators I . Acta Mathematica 127: 79–183. Springer Netherlands. 1970

    work page 1970

  46. [46]

    J. J. Duistermaat. Fourier Integral Operators. Progress in Mathematics. Birkh ¨auser. 1995

    work page 1995

  47. [48]

    S. A. Franchino-Vi˜ nas, S. Mignemi, J. J. Relancio.The beauty of curved momentum space. Proceedings of the Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity. 2022. https://arxiv.org/abs/2303.08220v1

    work page arXiv 2022

  48. [49]

    N. Jafari. Evolution of the concept of the curvature in the momentum spa ce. https://arxiv.org/abs/2404.08553

    work page arXiv

  49. [50]

    H. Hopf. ¨U ber die Abbildungen der dreidimensionalen Sph ¨are auf die Kugelfl ¨ache. Mathematische Annalen 104: 637 - 665. 1931

    work page 1931

  50. [51]

    C. N. Yang. Generalization of Dirac’s monopole to SU2 gauge fields . J. Math. Phys. 19: 320–328. 1978

    work page 1978

  51. [52]

    W. G. Unruh. Notes on black-hole evaporation. Phys. Rev. D 14 (4): 870–892. 1976

    work page 1976

  52. [53]

    S. A. Fulling. Nonuniqueness of Canonical Field Quantization in Riemanni an Space-Time. Phys. Rev. D 7 (10): 2850–2862. 1973

    work page 1973

  53. [54]

    P. C. W. Davies. Scalar production in Schwarzschild and Rindler metrics . J. Phys. A 8 (4): 609–616. 1975

    work page 1975

  54. [55]

    Aharonov, D

    Y. Aharonov, D. Bohm. Significance of Electromagnetic Potentials in the Quantum T heory. Phys. Rev. 115: 485. 1959

    work page 1959

  55. [56]

    L. P. Eisenhart. Separable systems of St ¨ackel. Ann. of Math. 35: 284–305. 1934

    work page 1934

  56. [57]

    E. G. Kalnins. On the separation of variables for the Laplace equation ∆Ψ + K 2Ψ = 0 in two and three-dimensional Minkowski space. SIAM J. Math. Anal. 6: 340–374. 1975

    work page 1975

  57. [58]

    S. Benenti. Intrinsic characterization of the variable separation in t he Hamilton-Jacobi equation . J. Math. Phys. 38: 6578–

    work page

  58. [59]

    S. Benenti. Separability in Riemannian manifolds . SIGMA 12: 013. 2016. https://arxiv.org/abs/1512.07833

    work page arXiv 2016

  59. [60]

    E. G. Kalnins, J. M. Kress, W. Miller, Jr. Separation of Variables and Superintegrability: The symme try of solvable systems . IOP Publishing Ltd. 2018

    work page 2018

  60. [61]

    Rajaratnam, R

    K. Rajaratnam, R. G. McLenaghan, C. Valero. Orthogonal Separation of the Hamilton–Jacobi Equation on S paces of Constant Curvature. SIGMA 12: 117. 2016. https://arxiv.org/pdf/1607.00712

    work page arXiv 2016

  61. [62]

    C. M. Chanu, G. Rastelli. Block-Separation of Variables: a Form of Partial Separatio n for Natural Hamiltonians. SIGMA 15: 013. 2019. https://arxiv.org/abs/1808.01889

    work page arXiv 2019

  62. [63]

    J. Peetre. Une caract´ erisation abstraite des op´ erateurs diff´ erentiels. Mathematica Scandinavica 7: 211–218. (1959)

    work page 1959

  63. [64]

    A. R. Gover, T. Leistner. Invariant prolongation of the Killing tensor equation. Annali di Matematica Pura ed Applicata (1923 -) 198: 307–334. 2019

    work page 1923

  64. [65]

    Eastwood

    M. Eastwood. Higher symmetries of the Laplacian . Annals of Mathematics, 161: 1645–1665. 2005

    work page 2005

  65. [66]

    Eastwood, T

    M. Eastwood, T. Leistner. Higher Symmetries of the Square of the Laplacian. IMA Vol. Math. Appl. 144: 319-338. Springer, New York. 2008. https://arxiv.org/abs/math/0610610

    work page arXiv 2008

  66. [67]

    O. P. Santillan. Killing-Yano tensors and some applications . J. Math. Phys. 53: 043509. 2012. https://arxiv.org/pdf/1108.0149

    work page arXiv 2012

  67. [68]

    Clemente

    G. Clemente. A curvature obstruction to integrability . Math. Commun. 28: 29–48. 2023. https://arxiv.org/abs/2108.03376

    work page arXiv 2023

  68. [69]

    Uhlenbeck

    K. Uhlenbeck. Generic properties of eigenfunctions . Amer. J. Math. 98 (4): 1059-1078. 1976

    work page 1976

  69. [70]

    B. Carter. Hamilton-Jacobi and Schr ¨odinger Separable Solutions of Einstein ’s Equations . Commun. Math. Phys. 10: 280–

    work page

  70. [71]

    E. G. Kalnins, W. Miller, Jr. Killing Tensors and Variable Separation for Hamilton-Jaco bi and Helmholtz Equations . SIAM J. Math. Anal. 11: 6. 1980

    work page 1980

  71. [72]

    S. Benenti. Orthogonal Separable Dynamical Systems. Differential Geometry and Its Applications Proc. Conf. Opav a (Czechoslovakia): 163-184. 1993

    work page 1993

  72. [73]

    P. Hartman. Ordinary Differential Equations : Second Edition. Vol. 38 of Classics in Applied Mathematics. SIAM e-books. 1982

    work page 1982

  73. [74]

    Nijenhuis

    A. Nijenhuis. Jacobi-type identities for bilinear differential concomita nts of certain tensor fields. II. Indagationes Mathe- maticae (Proceedings) 58: 398-403. 1955

    work page 1955

  74. [75]

    J. A. Schouten. Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications. Grundlehren der mathematischen Wissenschaften, 2nd ed. Springer. 1954

    work page 1954

  75. [76]

    Kol´ aˇ r, P

    I. Kol´ aˇ r, P. W. Michor, J. Slovak.Natural operations in differential geometry. Springer. 1993

    work page 1993

  76. [77]

    Busch, P

    P. Busch, P. J. Lahti, P. Mittelstaedt. The Quantum Theory of Measurement. Lecture Notes in Physics Monographs. 2nd ed. Springer. 2013

    work page 2013

  77. [78]

    G. F. Torres del Castillo. The St ¨ackel theorem in the Lagrangian formalism and the use of local times. Revista Mexicana de Fisica 67 (3): 44-451. 2021

    work page 2021

  78. [79]

    St¨ ackel.Uber die Integration der Hamilton-Jacobischen Differential Gleichung mittelst Separation der Variabeln

    P. St¨ ackel.Uber die Integration der Hamilton-Jacobischen Differential Gleichung mittelst Separation der Variabeln . Habil- itationsschrift. Halle, 1891

    work page

  79. [80]

    St¨ ackel.Uber Die Bewegung Eines Punktes In Einer N-Fachen Mannigfal tigkeit

    P. St¨ ackel.Uber Die Bewegung Eines Punktes In Einer N-Fachen Mannigfal tigkeit. Math. Ann. 42: 537–563. 1893

    work page

  80. [81]

    A. V. Tsiganov. The St ¨ackel systems and algebraic curves . J. Math. Phys. 40: 279–298. 1999

    work page 1999

Showing first 80 references.