Local description of gl-regular Haantjes operators

Alexey V. Bolsinov; Andrey Yu. Konyaev; Vladimir S. Matveev

arxiv: 2606.04985 · v1 · pith:5L6E2K2Znew · submitted 2026-06-03 · 🧮 math.DG · math-ph· math.AP· math.MP· nlin.SI

Local description of gl-regular Haantjes operators

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

Pith reviewed 2026-06-28 04:14 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.APmath.MPnlin.SI

keywords Haantjes operatorsHaantjes torsiongl-regularNijenhuis torsionsplitting theoremcomplex eigenvaluesdifferential geometry

0 comments

The pith

Gl-regular Haantjes operators admit a complete local description.

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

The paper establishes a complete local description of gl-regular Haantjes operators, which are (1,1)-tensor fields with vanishing Haantjes torsion under the gl-regular condition. This description specifies their structure near each point using suitable coordinates. It includes a splitting theorem that applies to general Haantjes operators and, more broadly, to operators whose generalised Nijenhuis torsion vanishes at any level. The work also fully addresses the previously overlooked case of complex eigenvalues. A sympathetic reader would care because these tensors appear in the local analysis of integrable systems and geometric structures defined by tensor fields.

Core claim

Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex.

What carries the argument

The gl-regular condition on a (1,1)-tensor field, which enables the local normal form description when the Haantjes torsion vanishes.

If this is right

General Haantjes operators split into components that can be analysed separately.
The splitting extends to any level of vanishing generalised Nijenhuis torsion.
The local description holds without restriction when eigenvalues are complex.

Where Pith is reading between the lines

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

The splitting theorem suggests a way to reduce questions about higher-order torsion conditions to simpler cases.
The local forms may connect to the study of recursion operators in integrable systems.

Load-bearing premise

The (1,1)-tensor field is assumed to be gl-regular, a regularity condition on its eigenvalues and associated distributions.

What would settle it

A gl-regular Haantjes operator whose local expression in coordinates fails to match the described normal form would disprove the completeness of the local description.

read the original abstract

We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex; the latter case was ignored in many previous papers on this and related topics.

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.

Desk Editor's Note private letter to a colleague

The paper supplies the missing local normal form for gl-regular Haantjes operators and adds a splitting result that works without regularity.

read the letter

The punchline is that this work finishes the local classification of gl-regular Haantjes operators by treating the complex-eigenvalue case that earlier papers skipped. It also gives a splitting theorem for the non-regular situation and extends the idea to vanishing generalized Nijenhuis torsion at any level.

What is actually new is the explicit local coordinate description under the gl-regular hypothesis together with the separate handling of complex eigenvalues. The splitting result is a useful byproduct because it applies more broadly. The abstract makes clear that the authors are filling a documented omission rather than re-deriving known real-eigenvalue forms.

The paper does well by stating the standing assumptions up front and by separating the regular and non-regular cases. That separation keeps the main theorem clean.

The soft spot is the gl-regular condition itself. The result is conditional on it, and the abstract does not indicate how often the condition holds or how restrictive the coordinate form becomes once it is imposed. Without the explicit block expressions or the proof steps, it is hard to judge whether the description is as usable as claimed or whether hidden technicalities appear in the complex case. The citation pattern looks standard for the subfield, with no obvious self-reference loop.

This is for readers who already work with (1,1)-tensors and integrability conditions on manifolds. A specialist who needs the normal form or who wants to apply the splitting theorem will get direct value. It deserves a serious referee because the claim is precise, the gap it fills is real, and the additional results are stated cleanly.

Referee Report

0 major / 0 minor

Summary. The paper studies Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. The main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex.

Significance. If the local description holds, the classification advances the geometric understanding of these tensor fields by supplying an explicit local form under the gl-regular hypothesis. The splitting theorem for the general case and the treatment of complex eigenvalues are explicit strengths, as the latter fills a documented gap left by prior work on the topic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough reading and positive evaluation of the manuscript. We are pleased that the main contributions—the complete local description of gl-regular Haantjes operators, the splitting theorem, and the treatment of complex eigenvalues—were recognized as advancing the geometric understanding of these tensor fields. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a classification theorem establishing a local coordinate description for gl-regular Haantjes operators (vanishing Haantjes torsion under a regularity hypothesis). The gl-regular condition is a standing assumption invoked in the main theorem statement rather than derived from the torsion condition. No provided equations, definitions, or steps reduce the claimed local form to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The splitting theorem and complex-eigenvalue analysis are presented as independent extensions. The derivation chain is self-contained as a standard differential-geometric classification result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a classification theorem in differential geometry. It relies on the standard axioms of smooth manifold theory and the algebraic definition of Haantjes torsion; no free parameters, no invented entities, and no ad-hoc assumptions beyond the gl-regularity condition are visible in the abstract.

axioms (2)

standard math The underlying manifold is smooth and the (1,1)-tensor field is smooth.
Invoked implicitly by the statement that the operator is a (1,1)-tensor field on a manifold.
standard math The Haantjes torsion is a well-defined tensorial object whose vanishing is a coordinate-independent condition.
Central to the definition of a Haantjes operator.

pith-pipeline@v0.9.1-grok · 5636 in / 1301 out tokens · 28094 ms · 2026-06-28T04:14:52.398693+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 9 canonical work pages

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A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev.Duality of operator Frobenius algebras and solution of Eisenhart–St¨ ackel problem in the non-diagonal case. 2026.doi:10.48550/ arXiv.2604.03195. arXiv:2604.03195 [math.DG]

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A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev.Existence of Riemannian invariants for integrable systems of hydrodynamic type. 2026.doi:10.48550/arXiv.2602.18687. arXiv:2602.18687 [nlin.SI]

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Haantjes algebras and diagonalization

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2021
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Haantjes algebras of classical integrable systems

P. Tempesta and G. Tondo. “Haantjes algebras of classical integrable systems”. In:Ann. Mat. Pura Appl. (4)201.1 (2022), pp. 57–90

2022
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Higher Haantjes brackets and integrability

P. Tempesta and G. Tondo. “Higher Haantjes brackets and integrability”. In:Comm. Math. Phys.389.3 (2022), pp. 1647–1671

2022
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Haantjes structures for the Jacobi-Calogero model and the Benenti systems

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The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method

S. P. Tsar¨ ev. “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”. In:Izv. Akad. Nauk SSSR Ser. Mat.54.5 (1990), pp. 1048–1068.doi:10.1070/IM1991v037n02ABEH002069

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S. P. Tsar¨ ev. “Integrability of equations of hydrodynamic type from the end of the 19th to the end of the 20th century”. In:Integrability: the Seiberg-Witten and Whitham equations (Edinburgh, 1998). Gordon and Breach, Amsterdam, 2000, pp. 251–265. 32

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[1] [1]

General algebraic identities for the Nijenhuis and Haantjes tensors

O. I. Bogoyavlenski˘ ı. “General algebraic identities for the Nijenhuis and Haantjes tensors”. In:Izv. Ross. Akad. Nauk Ser. Mat.68.6 (2004), pp. 71–84.doi:10 . 1070 / IM2004v068n06ABEH000511

2004

[2] [2]

Nijenhuis geometry

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Nijenhuis geometry”. In:Adv. Math. 394 (2022), Paper No. 108001, 52.doi:10.1016/j.aim.2021.108001

work page doi:10.1016/j.aim.2021.108001 2022

[3] [3]

Nijenhuis geometry III: gl-regular Nijenhuis operators

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Nijenhuis geometry III: gl-regular Nijenhuis operators”. In:Rev. Mat. Iberoam.40.1 (2024), pp. 155–188

2024

[4] [4]

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev.Duality of operator Frobenius algebras and solution of Eisenhart–St¨ ackel problem in the non-diagonal case. 2026.doi:10.48550/ arXiv.2604.03195. arXiv:2604.03195 [math.DG]

Pith/arXiv arXiv 2026

[5] [5]

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev.Existence of Riemannian invariants for integrable systems of hydrodynamic type. 2026.doi:10.48550/arXiv.2602.18687. arXiv:2602.18687 [nlin.SI]

work page doi:10.48550/arxiv.2602.18687 2026

[6] [6]

B¨ ottcher and S

A. B¨ ottcher and S. M. Grudsky.Toeplitz matrices, asymptotic linear algebra, and functional analysis. Birkh¨ auser Verlag, Basel, 2000, pp. x+116.doi:10 . 1007 / 978 - 3-0348-8395-5

2000

[7] [7]

On operator fields in an upper triangular Toeplitz form

M. M. Chernin and A. Yu. Konyaev. “On operator fields in an upper triangular Toeplitz form”. In:Russ. J. Math. Phys.33.1 (2026), pp. 42–55.doi:10 . 1134 / S1061920825601697

2026

[8] [8]

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

E. V. Ferapontov and D. G. Marshall. “Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor”. In:Math. Ann.339.1 (2007), pp. 61–99

2007

[9] [9]

OnX m-forming sets of eigenvectors

J. Haantjes. “OnX m-forming sets of eigenvectors”. In:Indag. Math.17 (1955). Nederl. Akad. Wetensch. Proc. Ser. A58, pp. 158–162

1955

[10] [10]

Killing tensors and variable separation for Hamilton- Jacobi and Helmholtz equations

E. G. Kalnins and W. Miller Jr. “Killing tensors and variable separation for Hamilton- Jacobi and Helmholtz equations”. In:SIAM J. Math. Anal.11.6 (1980), pp. 1011–1026

1980

[11] [11]

Kol´ aˇ r, P

I. Kol´ aˇ r, P. W. Michor, and J. Slovak.Natural Operations in Differential Geometry. Springer-Verlag, 1993, pp. vi+443

1993

[12] [12]

When a (1,1)-tensor generates separation of variables of a certain metric

A. Yu. Konyaev, J. M. Kress, and V. S. Matveev. “When a (1,1)-tensor generates separation of variables of a certain metric”. In:J. Geom. Phys.195 (2024), Paper No. 105031, 9.doi:10.1016/j.geomphys.2023.105031

work page doi:10.1016/j.geomphys.2023.105031 2024

[13] [13]

Beyond recursion operators

Y. Kosmann-Schwarzbach. “Beyond recursion operators”. In:Geometric methods in physics XXXVI. Trends Math. Birkh¨ auser/Springer, Cham, 2019, pp. 167–180.doi:10. 1007/978-3-030-01156-7\_18

2019

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Integrable hierarchies and F-manifolds with compatible connection

P. Lorenzoni, S. Perletti, and K. van Gemst. “Integrable hierarchies and F-manifolds with compatible connection”. In:Comm. Math. Phys.406.4 (2025), Paper No. 91, 64

2025

[15] [15]

Semi-Hamiltonian properties of a class of non-diagonalisable systems of hydrodynamic type

P. Lorenzoni, S. Perletti, and K. van Gemst. “Semi-Hamiltonian properties of a class of non-diagonalisable systems of hydrodynamic type”. In:Comm. Math. Phys.407.3 (2026), Paper No. 46, 48

2026

[16] [16]

Haantjes manifolds with symmetry

F. Magri. “Haantjes manifolds with symmetry”. In:Teoret. Mat. Fiz.196.2 (2018), pp. 313–327.doi:10.4213/tmf9499

work page doi:10.4213/tmf9499 2018

[17] [17]

X n−1-forming sets of eigenvectors

A. Nijenhuis. “X n−1-forming sets of eigenvectors”. In:Indag. Math.13 (1951). Nederl. Akad. Wetensch. Proc. Ser. A54, pp. 200–212. 31

1951

[18] [18]

An invariant criterion for hydrodynamic integrability

M. V. Pavlov, S. I. Svinolupov, and R. A. Sharipov. “An invariant criterion for hydrodynamic integrability”. In:Funktsional. Anal. i Prilozhen.30.1 (1996), pp. 18–29, 96.doi:10.1007/BF02509552

work page doi:10.1007/bf02509552 1996

[19] [19]

V. V. Prasolov.Polynomials. second. Vol. 11. Algorithms and Computation in Mathe- matics. Springer-Verlag, Berlin, 2004, pp. xiv+301.doi:10.1007/978- 3- 642- 03980- 5

work page doi:10.1007/978- 2004

[20] [20]

Natural operators on tensor fields

E. G. Puninskii. “Natural operators on tensor fields”. In:Moscow Univ. Math. Bull.69.5 (2014). Translation of Vestnik Moskov. Univ. Ser. I Mat. Mekh.2014, no. 5, 58–62, pp. 225–228.doi:10.3103/S002713221405009X

work page doi:10.3103/s002713221405009x 2014

[21] [21]

Generalized Nijenhuis torsions and block- diagonalization of operator fields

D. Reyes, P. Tempesta, and G. Tondo. “Generalized Nijenhuis torsions and block- diagonalization of operator fields”. In:J. Nonlinear Sci.33.2 (2023), Paper No. 35, 24. doi:10.1007/s00332-022-09877-8

work page doi:10.1007/s00332-022-09877-8 2023

[22] [22]

Sur les tenseurs deV n aux directions principalesV n−1-normales

J. A. Schouten. “Sur les tenseurs deV n aux directions principalesV n−1-normales”. In: Colloque de G´ eom´ etrie Diff´ erentielle, Louvain, 1951. Georges Thone, Li` ege, 1951, pp. 67– 70

1951

[23] [23]

Serre.Systems of conservation laws

D. Serre.Systems of conservation laws. 2. Geometric structures, oscillations, and initial- boundary value problems, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000, pp. xii+269

1996

[24] [24]

Haantjes algebras and diagonalization

P. Tempesta and G. Tondo. “Haantjes algebras and diagonalization”. In:J. Geom. Phys. 160 (2021), Paper No. 103968, 21

2021

[25] [25]

Haantjes algebras of classical integrable systems

P. Tempesta and G. Tondo. “Haantjes algebras of classical integrable systems”. In:Ann. Mat. Pura Appl. (4)201.1 (2022), pp. 57–90

2022

[26] [26]

Higher Haantjes brackets and integrability

P. Tempesta and G. Tondo. “Higher Haantjes brackets and integrability”. In:Comm. Math. Phys.389.3 (2022), pp. 1647–1671

2022

[27] [27]

Haantjes structures for the Jacobi-Calogero model and the Benenti systems

G. Tondo and P. Tempesta. “Haantjes structures for the Jacobi-Calogero model and the Benenti systems”. In:SIGMA Symmetry Integrability Geom. Methods Appl.12 (2016), Paper No. 023, 18

2016

[28] [28]

The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method

S. P. Tsar¨ ev. “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”. In:Izv. Akad. Nauk SSSR Ser. Mat.54.5 (1990), pp. 1048–1068.doi:10.1070/IM1991v037n02ABEH002069

work page doi:10.1070/im1991v037n02abeh002069 1990

[29] [29]

Integrability of equations of hydrodynamic type from the end of the 19th to the end of the 20th century

S. P. Tsar¨ ev. “Integrability of equations of hydrodynamic type from the end of the 19th to the end of the 20th century”. In:Integrability: the Seiberg-Witten and Whitham equations (Edinburgh, 1998). Gordon and Breach, Amsterdam, 2000, pp. 251–265. 32

1998