Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Vladimir S. Matveev; Yuri Nikolayevsky

arxiv: 2312.16518 · v1 · submitted 2023-12-27 · 🧮 math.DG

Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Vladimir S. Matveev , Yuri Nikolayevsky This is my paper

Pith reviewed 2026-05-24 05:22 UTC · model grok-4.3

classification 🧮 math.DG

keywords Killing tensorssymmetric spacesquaternionic projective spacesCayley projective planegeodesic integrals

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The pith

Quadratic Killing tensors on quaternionic projective spaces and the Cayley plane are not always generated by Killing vector fields.

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

The paper examines whether every Killing tensor field on Riemannian symmetric spaces can be decomposed into symmetric tensor products of Killing vector fields, a property known to hold for spaces of constant curvature and complex projective spaces. By constructing explicit examples, it shows this is not the case for quaternionic projective spaces HP^n with n at least 3 and the Cayley projective plane OP^2. These quadratic Killing tensors satisfy the Killing equation but are not quadratic forms in the Killing vector fields. This provides a negative answer to the question for all symmetric spaces.

Core claim

We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces HP^n, n ≥ 3, and on the Cayley projective plane OP².

What carries the argument

Explicit constructions of quadratic Killing tensor fields on HP^n (n≥3) and OP² that satisfy the Killing equation but cannot be expressed as symmetric products of Killing vector fields.

If this is right

The decomposition property does not hold for all Riemannian symmetric spaces.
Quadratic integrals of the geodesic flow on these spaces are not necessarily polynomials in the linear integrals.
The question has a negative answer specifically for quaternionic and octonionic projective spaces of sufficient dimension.

Where Pith is reading between the lines

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

These counterexamples indicate that the structure of integrals for geodesic flows may be more complex on quaternionic symmetric spaces than on complex ones.
Similar non-standard Killing tensors might exist on other exceptional symmetric spaces.

Load-bearing premise

The explicit constructions provide tensor fields that satisfy the Killing equation without being reducible to products of Killing vector fields.

What would settle it

Verifying whether the constructed tensors on HP^n or OP² can be written as quadratic forms in Killing vector fields would confirm or refute the examples.

read the original abstract

Every Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in the velocities integral of the geodesic flow is a polynomial in the linear integrals). This fact led to the natural question on whether this property is shared by Killing tensor fields on all Riemannian symmetric spaces. We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces $\mathbb{H} P^n, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$.

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

Matveev and Nikolayevsky supply explicit quadratic Killing tensors on HP^n (n≥3) and OP² that lie outside the span of products of Killing vectors, giving a negative answer to the decomposition question for these spaces.

read the letter

The main point is straightforward: the paper produces concrete quadratic Killing tensors on the quaternionic projective spaces and the Cayley plane that cannot be written as symmetric products of Killing vector fields. This settles the open question left by the positive results on constant-curvature spaces and CP^n in the negative for these rank-one symmetric spaces. The authors state they give explicit constructions and verify the Killing equation while showing the tensors are irreducible in the required sense. That is the new content. Earlier literature had the decomposition property in the cases they checked; here the isometry groups of HP^n and OP² permit tensors that escape that pattern. The work is useful because it supplies verifiable examples rather than an abstract existence argument. The calculations rest on the standard symmetric-space structure and the specific division-algebra geometry of these manifolds, which is the right place to look for counterexamples. No obvious circularity or hidden fitting appears in the claim. A minor soft spot is that the paper stays tightly focused on these two families of spaces; it does not attempt a broader classification or explain why the property fails exactly here and not elsewhere, but that is a question for later work rather than a flaw in the present argument. The central negative result stands on the explicit tensors. Readers working on integrals of the geodesic flow on homogeneous spaces or on the algebraic structure of Killing tensors will find the examples directly relevant. Specialists in symmetric geometry who want to know the precise limits of the decomposition property should see this. The paper is worth sending to a serious referee because the claim is specific, the evidence is the constructions themselves, and the question it answers was already on the table.

Referee Report

0 major / 2 minor

Summary. The paper constructs explicit examples of quadratic Killing tensor fields on the quaternionic projective spaces HP^n (n ≥ 3) and the Cayley projective plane OP² that cannot be expressed as quadratic forms in the Killing vector fields. This provides a negative answer to the question of whether every Killing tensor on a Riemannian symmetric space arises as a symmetric product of Killing vectors, extending the known affirmative results for spaces of constant curvature and for complex projective space CP^n.

Significance. If the explicit constructions and verifications hold, the result is significant: it identifies the first known symmetric spaces where the decomposition property for quadratic Killing tensors fails, thereby sharpening the boundary between spaces where all geodesic integrals are generated by linear ones and those where genuinely new quadratic integrals exist. The constructions on rank-1 symmetric spaces of quaternionic and octonionic type supply concrete, falsifiable objects for further study of the geodesic flow.

minor comments (2)

The abstract states that the examples are 'explicit' but does not preview the coordinate system or the algebraic form used for the tensors; a single sentence indicating the ambient coordinates (e.g., homogeneous coordinates on HP^n) would improve immediate readability.
Notation for the symmetric product of vector fields is introduced without an explicit equation number; adding a displayed definition (e.g., K = X ⊙ Y) in §2 would eliminate any ambiguity when the same symbol appears later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending acceptance. The summary accurately captures the main contribution: explicit quadratic Killing tensors on HP^n (n≥3) and OP² that are not generated by Killing vector fields, providing a negative answer to the decomposition question for these symmetric spaces.

Circularity Check

0 steps flagged

No circularity; claim rests on explicit constructions

full rationale

The paper's negative answer to the question is established by explicit constructions of quadratic Killing tensors on HP^n (n≥3) and OP². These are stated to satisfy the Killing equation while lying outside the span of symmetric products of Killing vector fields. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the abstract or described derivation chain. The constructions are presented as direct, verifiable examples on rank-1 symmetric spaces, making the central claim independent of any circular reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract mentions no free parameters, background axioms, or new postulated entities; the claim rests on explicit constructions whose details are not supplied here.

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discussion (0)

Forward citations

Cited by 1 Pith paper

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

Killing tensors on reducible spaces
math.DG 2024-01 unverdicted novelty 6.0

Killing tensors on products of Riemannian manifolds with one compact factor are reducible sums of factor tensors.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

Baez, The Octonions, Bull

J. Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205

work page 2002
[2]

Bolsinov, E

A. Bolsinov, E. Miranda, V. Matveev, S. Tabachnikov, Open problems, questions, and challenges in finite-dimensional integrable systems, Philos. Trans. Roy. Soc. A 376 (2018), 20170430

work page 2018
[3]

Eastwood, Killing tensors on complex projective space, arXiv: https://arxiv.org/abs/2309.00589

M. Eastwood, Killing tensors on complex projective space, arXiv: https://arxiv.org/abs/2309.00589

work page arXiv
[4]

F. R. Harvey, Spinors and Calibrations. Academic Press, Boston, 1990

work page 1990
[5]

Helgason, Differential geometry, Lie groups, and symmetric spaces

S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. New York, London, 1978

work page 1978
[6]

Freudenthal, Zur ebenen Oktavengeometrie, Indag

H. Freudenthal, Zur ebenen Oktavengeometrie, Indag. Math. 15 (1953), 195-200

work page 1953
[7]

McLenaghan, R

R. McLenaghan, R. Milson, R. Smirnov, Killing tensors as irreducible representations of the general linear group, C. R. Acad. Sci. Paris, Ser. I 339 (2004), 621-624

work page 2004
[8]

Samelson, Notes on Lie algebras

H. Samelson, Notes on Lie algebras. Universitext, Springer-Verlag, New York, 1990

work page 1990
[9]

Sumitomo, K

T. Sumitomo, K. Tandai, Killing tensor fields on the standard sphere and spectra of (n+1)/( (n-1) (2)) and O(n+1)/(O(n-1) O(2)) , Osaka Math. J. 20 (1983), 51-78

work page 1983
[10]

Sumitomo, K

T. Sumitomo, K. Tandai, On the centralizer of the Laplacian of P_n( ) and the spectrum of complex Grassmann manifold G_ 2,n-1 ( ) , Osaka J. Math. 22 (1985), 123-155

work page 1985
[11]

Thompson, Killing tensors in spaces of constant curvature, J

G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986), 2693-2699

work page 1986
[12]

\` E . B. Vinberg, A. L. Onishchik, A seminar on Lie groups and algebraic groups, Nauka, Moscow, 1988

work page 1988

[1] [1]

Baez, The Octonions, Bull

J. Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205

work page 2002

[2] [2]

Bolsinov, E

A. Bolsinov, E. Miranda, V. Matveev, S. Tabachnikov, Open problems, questions, and challenges in finite-dimensional integrable systems, Philos. Trans. Roy. Soc. A 376 (2018), 20170430

work page 2018

[3] [3]

Eastwood, Killing tensors on complex projective space, arXiv: https://arxiv.org/abs/2309.00589

M. Eastwood, Killing tensors on complex projective space, arXiv: https://arxiv.org/abs/2309.00589

work page arXiv

[4] [4]

F. R. Harvey, Spinors and Calibrations. Academic Press, Boston, 1990

work page 1990

[5] [5]

Helgason, Differential geometry, Lie groups, and symmetric spaces

S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. New York, London, 1978

work page 1978

[6] [6]

Freudenthal, Zur ebenen Oktavengeometrie, Indag

H. Freudenthal, Zur ebenen Oktavengeometrie, Indag. Math. 15 (1953), 195-200

work page 1953

[7] [7]

McLenaghan, R

R. McLenaghan, R. Milson, R. Smirnov, Killing tensors as irreducible representations of the general linear group, C. R. Acad. Sci. Paris, Ser. I 339 (2004), 621-624

work page 2004

[8] [8]

Samelson, Notes on Lie algebras

H. Samelson, Notes on Lie algebras. Universitext, Springer-Verlag, New York, 1990

work page 1990

[9] [9]

Sumitomo, K

T. Sumitomo, K. Tandai, Killing tensor fields on the standard sphere and spectra of (n+1)/( (n-1) (2)) and O(n+1)/(O(n-1) O(2)) , Osaka Math. J. 20 (1983), 51-78

work page 1983

[10] [10]

Sumitomo, K

T. Sumitomo, K. Tandai, On the centralizer of the Laplacian of P_n( ) and the spectrum of complex Grassmann manifold G_ 2,n-1 ( ) , Osaka J. Math. 22 (1985), 123-155

work page 1985

[11] [11]

Thompson, Killing tensors in spaces of constant curvature, J

G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986), 2693-2699

work page 1986

[12] [12]

\` E . B. Vinberg, A. L. Onishchik, A seminar on Lie groups and algebraic groups, Nauka, Moscow, 1988

work page 1988