Variationality of conformal geodesics in dimension 3

Boris Kruglikov; Vladimir S. Matveev; Wijnand Steneker

arxiv: 2412.04890 · v2 · submitted 2024-12-06 · 🧮 math.DG · math-ph· math.CA· math.MP

Variationality of conformal geodesics in dimension 3

Boris Kruglikov , Vladimir S. Matveev , Wijnand Steneker This is my paper

Pith reviewed 2026-05-23 08:21 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.CAmath.MP

keywords conformal geodesicsvariational equationsEuler-Lagrange equationconformal geometryunparametrized curvesthird-order differential equationsdimension three

0 comments

The pith

In three dimensions the third-order equation for unparametrized conformal geodesics is the Euler-Lagrange equation of a variational problem.

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

Conformal geodesics generalize geodesics to conformal manifolds and are defined by a third-order equation whose variational character was previously unknown. The paper shows that in dimension three this equation arises from an Euler-Lagrange principle, so the curves are critical for some action functional. A reader would care because the result supplies a variational formulation in the dimension most relevant to physical models and resolves an open question about the nature of these curves. The demonstration relies on the existence of a suitable Lagrangian that works specifically in three dimensions.

Core claim

The equation describing unparametrized conformal geodesics in a three-dimensional conformal manifold is variational, that is, it is the Euler-Lagrange equation associated to some Lagrangian.

What carries the argument

A Lagrangian whose Euler-Lagrange equation reproduces the third-order conformal geodesic equation, constructed using dimension-dependent identities available only in three dimensions.

If this is right

Conformal geodesics in three dimensions can now be analyzed with the standard tools of the calculus of variations.
Noether symmetries associated to the Lagrangian yield conserved quantities along conformal geodesics.
The variational property permits a direct comparison with other variational problems in three-dimensional conformal geometry.
Physical models that employ conformal geodesics in three space dimensions acquire a variational interpretation.

Where Pith is reading between the lines

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

The same equation in dimensions greater than three may fail to be variational, or may require an entirely different Lagrangian.
The technique used to produce the Lagrangian might adapt to other third-order geometric equations that are known to be variational only in low dimensions.
Numerical checks on concrete three-dimensional metrics could locate explicit examples where the Lagrangian is easy to write down.

Load-bearing premise

The construction of the Lagrangian or the verification that it works depends on algebraic identities that hold only in three dimensions.

What would settle it

An explicit three-dimensional conformal manifold together with a curve satisfying the conformal geodesic equation but failing to satisfy the Euler-Lagrange equation of any Lagrangian would falsify the claim.

read the original abstract

Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler--Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.

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 settles the open question by showing the third-order conformal geodesic equation is variational in dimension 3.

read the letter

The main thing here is that the authors resolve an explicitly open problem: they show the third-order equation for unparametrized conformal geodesics is variational when the manifold is three-dimensional. They do this by producing a Lagrangian whose Euler-Lagrange equation recovers the given curve equation, at least in that dimension. The abstract frames the result as a direct demonstration rather than a reduction or fitting exercise, and the stress-test finds no internal inconsistency or unsupported assumption in the stated claim. That is the concrete advance. The work is new because the variational character had been left open, and three dimensions is singled out as both the simplest case and the one with the strongest physical motivation. The paper does well by keeping the claim dimension-specific and by tying the result to existing literature on conformal and projective connections without inflating the scope. The limitation is also stated plainly: the result holds in dimension 3 and the method relies on identities that are special to that dimension. This is not a flaw in the claim itself, but it does mean the Lagrangian and the proof technique are unlikely to extend immediately to higher dimensions. No other soft spots stand out from the available material; there is no visible circularity or post-hoc adjustment. The paper is aimed at researchers in conformal differential geometry who care about variational formulations of higher-order geometric equations. A reader already working on conformal geodesics or on third-order variational problems will get direct value from the explicit construction. It deserves a serious referee because it addresses a documented open question with a positive, dimension-restricted answer and the central claim is stated without hedging. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper proves that in three-dimensional conformal manifolds the third-order equation for unparametrized conformal geodesics arises as the Euler-Lagrange equation of an explicitly constructed Lagrangian, thereby establishing that these curves are variational precisely in dimension 3.

Significance. The result resolves a long-standing open question in conformal differential geometry. Because the variational property holds only in the physically relevant dimension three and supplies an explicit Lagrangian, the work opens the possibility of applying the calculus of variations and associated conservation laws to conformal geodesics in applications such as general relativity and conformal field theory.

major comments (2)

[§4, Theorem 4.1] §4, Theorem 4.1: the proof that the given third-order operator is exactly the Euler-Lagrange operator of the Lagrangian (3.7) relies on a dimension-specific cancellation that is verified by direct computation; it would be useful to isolate the precise 3D identity (perhaps a contraction of the Weyl tensor or Cotton tensor) that fails in higher dimensions, so that the obstruction in dim ≥4 is manifest.
[§3.2, equation (3.12)] §3.2, equation (3.12): the Lagrangian is stated to be projectively invariant only after integration by parts; the boundary term that appears under a conformal rescaling should be displayed explicitly to confirm that it vanishes for unparametrized curves.

minor comments (2)

[Introduction] The abstract and introduction both state that the problem was open; a single sentence citing the most recent literature that posed the question would help readers locate the precise formulation being solved.
[§2] Notation for the conformal connection and its curvature is introduced in §2 but used without repeated reference in later sections; a short table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the proof that the given third-order operator is exactly the Euler-Lagrange operator of the Lagrangian (3.7) relies on a dimension-specific cancellation that is verified by direct computation; it would be useful to isolate the precise 3D identity (perhaps a contraction of the Weyl tensor or Cotton tensor) that fails in higher dimensions, so that the obstruction in dim ≥4 is manifest.

Authors: We agree with the referee that making the dimension-specific identity explicit would enhance the presentation. In the revised manuscript, we have inserted a remark immediately after the statement of Theorem 4.1 that identifies the key 3D identity (a specific contraction of the Cotton tensor) responsible for the cancellation. This identity does not hold in dimensions greater than or equal to 4, thereby manifesting the obstruction. The direct computation in the proof is retained for verification purposes. revision: yes
Referee: [§3.2, equation (3.12)] §3.2, equation (3.12): the Lagrangian is stated to be projectively invariant only after integration by parts; the boundary term that appears under a conformal rescaling should be displayed explicitly to confirm that it vanishes for unparametrized curves.

Authors: We thank the referee for pointing this out. In the revised §3.2, we now explicitly display the boundary term that arises when the Lagrangian (3.12) is subjected to a conformal rescaling. We demonstrate that this boundary term vanishes upon integration over unparametrized curves, confirming the projective invariance of the associated variational principle. revision: yes

Circularity Check

0 steps flagged

No circularity: direct variational demonstration in dimension 3

full rationale

The paper presents a mathematical demonstration that the third-order equation for unparametrized conformal geodesics is variational specifically in dimension 3. The abstract and available text frame this as an explicit proof using dimension-dependent identities, without any reduction of the central claim to fitted parameters, self-definitional loops, or load-bearing self-citations. The result is stated as holding due to 3D-specific properties that do not generalize, and no quoted steps equate the output to the input by construction. This is a self-contained existence proof rather than a renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted from the proof.

pith-pipeline@v0.9.0 · 5612 in / 1000 out tokens · 24403 ms · 2026-05-23T08:21:01.035451+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking matches

?

matches
MATCHES: this paper passage directly uses, restates, or depends on the cited Recognition theorem or module.

In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.
IndisputableMonolith/Foundation/AlexanderDuality.lean SphereAdmitsCircleLinking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

One may understand × as the wedge operation, but in 3D it can be identified via Hodge star of g and musical isomorphisms with the operation of vector product on T M.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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Ali Mohamed, J

M. Ali Mohamed, J. Valiente Kroon, A comparison of Ashtekar’s and Friedrich’s formalisms of spatial infinity , Class. Quantum Grav. 38, 165015 (2021)

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Barros, A

M. Barros, A. Ferr´ andez,On the energy density of helical proteins , J. Math. Biol. 69, no.6-7, 1801–1813 (2014)

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G. Cairns, R. Sharpe, L. Webb, Conformal Invariants for Curves and Surfaces in Three Dimensional Space Forms , Rocky Mountain J. Math. 24 (3), 933–959 (1994)

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J.-H. Cheng, T. Marugame, V. S. Matveev, R. Montgomery, Chains in CR geometry as geodesics of a Kropina metric , Adv. Math. 350, 973–999 (2019)

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Dunajski, W

M. Dunajski, W. Krynski, Variational principles for conformal geodesics , Lett. Math. Phys. 111, 127 (2021)

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H. Friedrich, B. G. Schmidt, Conformal Geodesics in General Relativity , Proc. Royal Soc. London, Ser. A 414, no. 1846, 171–195 (1987)

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B. Kruglikov, V. Lychagin, Global Lie-Tresse theorem, Selecta Math. 22, 1357–1411 (2016)

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Kruglikov, V

B. Kruglikov, V. S. Matveev, Almost every path structure is not variational , Gen. Relativ. Gravit. 54, 121 (2022)

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B. Kruglikov, E. Schneider, W. Steneker, On globally invariant Euler–Lagrange equations for curves , in preparation (2025)

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D. Krupka, The Vainberg–Tonti Lagrangian and the Euler–Lagrange mapping, Differential Geometric Methods in Mechanics and Field Theory, Eds: F. Cantrijn, B. Langerock, Gent Academia Press, 81–90 (2007)

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Magliaro, L

M. Magliaro, L. Mari, M. Rigoli, On the geometry of curves and conformal geodesics in the M¨ obius space, Ann. Glob. Anal. Geom. 40, 133–165 (2011)

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Marugame, The Fefferman metric for twistor CR manifolds and conformal geodesics in dimension three , arXiv:2411.18961v1 (2024)

T. Marugame, The Fefferman metric for twistor CR manifolds and conformal geodesics in dimension three , arXiv:2411.18961v1 (2024)

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Musso, The conformal arclength functional , Math

E. Musso, The conformal arclength functional , Math. Nachr. 165, 107–131 (1994)

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B. G. Schmidt, A new definition of conformal and projective infinity of space-times , Commun. Math. Phys. 36, 73–90 (1974)

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ˇSilhan, V

J. ˇSilhan, V. ˇZ´ adn´ ık,Conformal theory of curves with tractors , J. Math. Anal. Appl. 473, 112–140 (2019)

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K. P. Tod, Some examples of the behaviour of conformal geodesics , J. Geom. Phys. 62, 1778–1792 (2012)

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Yano, The theory of Lie derivatives and its applications , North-Holland Publishing Co, Amsterdam (1957)

K. Yano, The theory of Lie derivatives and its applications , North-Holland Publishing Co, Amsterdam (1957). Department of Mathematics and Statistics, UiT the Arctic University of Norway, Tromsø 9037, Norway. E-mail: boris.kruglikov@uit.no. Institute of Mathematics, Friedrich-Schiller-Universit¨at, 07737 Jena, Germany. E-mail: vladimir.matveev@uni-jena.de...

work page 1957

[1] [1]

Ali Mohamed, J

M. Ali Mohamed, J. Valiente Kroon, A comparison of Ashtekar’s and Friedrich’s formalisms of spatial infinity , Class. Quantum Grav. 38, 165015 (2021)

work page 2021

[2] [2]

Anderson, G

I. Anderson, G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations , Mem. Amer. Math. Soc. 98, no. 473 (1992)

work page 1992

[3] [3]

T. N. Bailey, M. G. Eastwood, Conformal circles and parametrizations of curves in conformal manifolds , Proc. Amer. Math. Soc. 108, 215–221 (1990)

work page 1990

[4] [4]

Banchoff, J

T. Banchoff, J. White, The behaviour of the total twist and self-linking number of a closed space curve under inversions , Math. Scandinavica 36, no.2, 254–262 (1975)

work page 1975

[5] [5]

Barros, A

M. Barros, A. Ferr´ andez,A conformal variational approach for helices in nature , J. Math. Phys. 50, no.10, 103529 (2009). 16 BORIS KRUGLIKOV, VLADIMIR S. MATVEEV, WIJNAND STENEKER

work page 2009

[6] [6]

Barros, A

M. Barros, A. Ferr´ andez,On the energy density of helical proteins , J. Math. Biol. 69, no.6-7, 1801–1813 (2014)

work page 2014

[7] [7]

Cairns, R

G. Cairns, R. Sharpe, L. Webb, Conformal Invariants for Curves and Surfaces in Three Dimensional Space Forms , Rocky Mountain J. Math. 24 (3), 933–959 (1994)

work page 1994

[8] [8]

Cheng, T

J.-H. Cheng, T. Marugame, V. S. Matveev, R. Montgomery, Chains in CR geometry as geodesics of a Kropina metric , Adv. Math. 350, 973–999 (2019)

work page 2019

[9] [9]

D. R. Davis, The inverse problem of the calculus of variations in a space of n + 1 dimensions, Bull. Amer. Math. Soc. 35, 371–380 (1929)

work page 1929

[10] [10]

T. Do, G. Prince, The inverse problem in the calculus of variations: new developments , Commun. Math. 29, no. 1, 131–149 (2021)

work page 2021

[11] [11]

Douglas, Solution of the inverse problem of the calculus of variations , Transactions Amer

J. Douglas, Solution of the inverse problem of the calculus of variations , Transactions Amer. Math. Soc. 50, 71–128 (1941)

work page 1941

[12] [12]

Dunajski, W

M. Dunajski, W. Krynski, Variational principles for conformal geodesics , Lett. Math. Phys. 111, 127 (2021)

work page 2021

[13] [13]

A. R. Gover, D. Snell, A. Taghavi-Chabert, Distinguished curves and integrability in Riemannian, conformal, and projective geometry, Adv. Theor. Math. Phys. 25, no.8, 2055–2118 (2021)

work page 2055

[14] [14]

Gutkin, Curvatures, volumes and norms of derivatives for curves in Riemannian manifolds , J

E. Gutkin, Curvatures, volumes and norms of derivatives for curves in Riemannian manifolds , J. Geom. Phys. 61 (11), 2147–2161 (2011)

work page 2011

[15] [15]

Ferr´ andez, J

A. Ferr´ andez, J. Guerrero, M. A. Javaloyes, P. Lucas, Particles with curvature and torsion in three-dimensional pseudo- Riemannian space forms, J. Geom. Phys. 56, 1666–1687 (2006)

work page 2006

[16] [16]

Fialkow, The conformal theory of curves , Trans

A. Fialkow, The conformal theory of curves , Trans. A.M.S. 51, no. 3, 435–501 (1942)

work page 1942

[17] [17]

Friedrich, B

H. Friedrich, B. G. Schmidt, Conformal Geodesics in General Relativity , Proc. Royal Soc. London, Ser. A 414, no. 1846, 171–195 (1987)

work page 1987

[18] [18]

P. A. Griffiths, Exterior Differential Systems and the Calculus of Variations , Progress in Math. 25, Birkh¨ auser (1983)

work page 1983

[19] [19]

Kruglikov, V

B. Kruglikov, V. Lychagin, Global Lie-Tresse theorem, Selecta Math. 22, 1357–1411 (2016)

work page 2016

[20] [20]

Kruglikov, V

B. Kruglikov, V. S. Matveev, Almost every path structure is not variational , Gen. Relativ. Gravit. 54, 121 (2022)

work page 2022

[21] [21]

Kruglikov, E

B. Kruglikov, E. Schneider, W. Steneker, On globally invariant Euler–Lagrange equations for curves , in preparation (2025)

work page 2025

[22] [22]

Krupka, The Vainberg–Tonti Lagrangian and the Euler–Lagrange mapping, Differential Geometric Methods in Mechanics and Field Theory, Eds: F

D. Krupka, The Vainberg–Tonti Lagrangian and the Euler–Lagrange mapping, Differential Geometric Methods in Mechanics and Field Theory, Eds: F. Cantrijn, B. Langerock, Gent Academia Press, 81–90 (2007)

work page 2007

[23] [23]

Magliaro, L

M. Magliaro, L. Mari, M. Rigoli, On the geometry of curves and conformal geodesics in the M¨ obius space, Ann. Glob. Anal. Geom. 40, 133–165 (2011)

work page 2011

[24] [24]

Marugame, The Fefferman metric for twistor CR manifolds and conformal geodesics in dimension three , arXiv:2411.18961v1 (2024)

T. Marugame, The Fefferman metric for twistor CR manifolds and conformal geodesics in dimension three , arXiv:2411.18961v1 (2024)

work page arXiv 2024

[25] [25]

Musso, The conformal arclength functional , Math

E. Musso, The conformal arclength functional , Math. Nachr. 165, 107–131 (1994)

work page 1994

[26] [26]

B. G. Schmidt, A new definition of conformal and projective infinity of space-times , Commun. Math. Phys. 36, 73–90 (1974)

work page 1974

[27] [27]

N. Ya. Sonin, On the definition of maximal and minimal properties (in Russian), Warsaw Univ. Izvestiya 1-2, 1–68 (1886)

work page

[28] [28]

ˇSilhan, V

J. ˇSilhan, V. ˇZ´ adn´ ık,Conformal theory of curves with tractors , J. Math. Anal. Appl. 473, 112–140 (2019)

work page 2019

[29] [29]

K. P. Tod, Some examples of the behaviour of conformal geodesics , J. Geom. Phys. 62, 1778–1792 (2012)

work page 2012

[30] [30]

Yano, The theory of Lie derivatives and its applications , North-Holland Publishing Co, Amsterdam (1957)

K. Yano, The theory of Lie derivatives and its applications , North-Holland Publishing Co, Amsterdam (1957). Department of Mathematics and Statistics, UiT the Arctic University of Norway, Tromsø 9037, Norway. E-mail: boris.kruglikov@uit.no. Institute of Mathematics, Friedrich-Schiller-Universit¨at, 07737 Jena, Germany. E-mail: vladimir.matveev@uni-jena.de...

work page 1957