Curvature equivalence for Legendre curves in the unit tangent bundle over Euclidean plane

Masatomo Takahashi; Minoru Yamamoto; Nozomi Nakatsuyama

arxiv: 2604.07898 · v1 · submitted 2026-04-09 · 🧮 math.DG

Curvature equivalence for Legendre curves in the unit tangent bundle over Euclidean plane

Nozomi Nakatsuyama , Masatomo Takahashi , Minoru Yamamoto This is my paper

Pith reviewed 2026-05-10 18:08 UTC · model grok-4.3

classification 🧮 math.DG

keywords Legendre curvecurvature equivalenceunit tangent bundleEuclidean planeclassificationdifferential geometry

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The pith

Legendre curves in the unit tangent bundle over the Euclidean plane are classified up to curvature equivalence both locally and globally.

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

The paper defines a curvature equivalence relation on Legendre curves, which are plane curves with an attached moving frame inside the unit tangent bundle over the Euclidean plane. It then studies the properties of this relation and proves both local and global classification results for the curves under the equivalence. A sympathetic reader would care because the relation reduces the problem of distinguishing curves to a comparison of their curvature functions, leveraging the already-known existence and uniqueness theorems for Legendre curvature. If the classifications hold, then curves that look different geometrically can be identified as the same when their curvatures match in the appropriate sense.

Core claim

We introduce an equivalence relation for Legendre curves called the curvature equivalence. We investigate properties of the curvature equivalence and give local and global classifications of Legendre curves under the curvature equivalence.

What carries the argument

The curvature equivalence relation, which identifies two Legendre curves when they share the same Legendre curvature (up to the natural action that preserves the bundle structure).

If this is right

Legendre curves are determined up to equivalence by their curvature functions alone.
Local classification reduces to matching the curvature as a function of arc length or parameter.
Global classification accounts for the topological or closed nature of the curves in the bundle.
Properties of the equivalence allow invariants preserved by the relation to be read off directly from the curvature.

Where Pith is reading between the lines

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

The same equivalence idea could be tested on Legendre curves in other contact manifolds or higher-dimensional Euclidean spaces.
Applications in path planning or robotics might use curvature equivalence to simplify comparisons of framed trajectories.
One could ask whether the equivalence classes admit natural moduli spaces or whether they interact with known invariants like total curvature.

Load-bearing premise

The Legendre curvature is well-defined on the unit tangent bundle and the proposed equivalence relation yields meaningful and non-trivial classifications without additional hidden assumptions on the curves.

What would settle it

Finding two Legendre curves whose Legendre curvatures agree pointwise but which cannot be related by any local or global transformation preserving the equivalence, or exhibiting a curve whose curvature fails to determine its equivalence class uniquely.

read the original abstract

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this paper, we introduce an equivalence relation for Legendre curves called the curvature equivalence. We investigate properties of the curvature equivalence and give local and global classifications of Legendre curves under the curvature equivalence.

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

They define a curvature equivalence on Legendre curves and classify them locally and globally using the standard existence-uniqueness theorem for the curvature.

read the letter

This paper introduces an equivalence relation on Legendre curves in the unit tangent bundle over the Euclidean plane, where two curves are equivalent if they share the same Legendre curvature function. It then works out basic properties of the relation and gives both local and global classifications of the curves under it. The approach sits directly on top of the known existence and uniqueness theorems for the curvature, so the classifications amount to grouping curves whose curvature functions match after the natural reparametrization. That keeps the arguments short and avoids hidden assumptions. The separation into local and global cases is handled cleanly, and nothing in the setup looks circular. The main limitation is the narrow scope. This is a refinement inside framed-curve theory rather than a result that reaches outside it or produces invariants with wider use. The classifications are useful for organizing examples in this setting, but they do not appear to open new directions or connect to other parts of differential geometry. Readers who already work on Legendre curves or plane contact geometry will see the value right away. For anyone else the paper is too specialized to repay the time. I would send it to a referee. The logic is direct, the claims are modest, and the details can be checked without much extra work.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces an equivalence relation called curvature equivalence for Legendre curves in the unit tangent bundle over the Euclidean plane. It investigates properties of this relation and provides local and global classifications of Legendre curves under the equivalence, building on the existence and uniqueness theorems for the Legendre curvature.

Significance. If the classifications are rigorously established, the work extends the standard fundamental theorem for Legendre curves by grouping them according to their curvature functions. This could offer a useful organizational tool for studying these curves in the tangent bundle setting, with potential applications to broader questions in curve geometry and contact structures.

minor comments (2)

[Abstract] The abstract asserts the validity of existence and uniqueness theorems for the Legendre curvature without indicating whether these are recalled from prior literature or proven anew in the manuscript; a brief reference to the relevant section would clarify this.
The manuscript would benefit from at least one explicit example of a Legendre curve, its curvature function, and the resulting equivalence class to illustrate the local and global classification statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recommending minor revision. No specific major comments appear in the report, so we interpret the recommendation as a request for minor clarifications or improvements to presentation. We will incorporate any such changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the curvature equivalence relation directly from the Legendre curvature function of the curve in the unit tangent bundle. It invokes standard existence and uniqueness theorems for such curvatures (valid by the fundamental theorem for framed curves in Euclidean geometry) to obtain local and global classifications under the equivalence. No step reduces a claimed prediction or classification result to a fitted parameter, self-referential definition, or load-bearing self-citation whose content is unverified within the paper. The construction extends classical curve theory without internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the work rests on standard differential-geometric background rather than new axioms or fitted parameters.

axioms (1)

standard math Standard axioms of smooth manifold theory, Euclidean geometry, and the existence of the unit tangent bundle with its canonical contact structure.
Invoked implicitly when defining Legendre curves and their curvature.

invented entities (1)

Curvature equivalence relation no independent evidence
purpose: To partition Legendre curves into equivalence classes for classification
Newly defined in the paper from the Legendre curvature function.

pith-pipeline@v0.9.0 · 5361 in / 1169 out tokens · 48226 ms · 2026-05-10T18:08:50.428631+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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T. Fukunaga, M. Takahashi, On convexity of simple closed frontals. Kodai Math. J. 39, 2016, 389–398

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Salarinoghabi and F

M. Salarinoghabi and F. Tari, Flat and round singularit y theory of plane curves. Q. J. Math. , 68, 2017, 1289–1312. Nozomi Nakatsuyama, Muroran Institute of Technology, Muroran 050-8585, Japan, E-mail address: 25096009b@muroran-it.ac.jp Masatomo Takahashi, Muroran Institute of Technology, Muroran 050-8585, Japan, E-mail address: masatomo@muroran-it.ac.jp...

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V. I. Arnol’d, Singularities of Caustics and Wave Fronts . Mathematics and Its Applications 62 Kluwer Academic Publishers, 1990

work page 1990

[2] [2]

V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Diﬀerentiable Maps vol. I. Birkh¨ auser, 1986

work page 1986

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J. W. Bruce and T. J. Gaﬀney, Simple singularities of mappi ngs C,0 → C2,0. J. London Math. Soc. (2). 26, 1982, 465–474

work page 1982

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J. W. Bruce and P. J. Giblin, Curves and singularities. A geometrical introduction to sin gularity theory. Second edition . Cambridge University Press, Cambridge, 1992

work page 1992

[5] [5]

M. P. do Carmo, Diﬀerential geometry of curves and surfaces. Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliﬀs, N.J., 1976

work page 1976

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F. S. Dias and F. Tari, On vertices and inﬂections of plane curves. J. Singul. 17, 2018, 70–80

work page 2018

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Fukunaga and M

T. Fukunaga and M. Takahashi, Existence and uniqueness f or Legendre curves. J. Geom. 104, 2013, 297–307

work page 2013

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Fukunaga, M

T. Fukunaga, M. Takahashi, On convexity of simple closed frontals. Kodai Math. J. 39, 2016, 389–398

work page 2016

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C. G. Gibson, Elementary geometry of diﬀerentiable curves. An undergrad uate introduction . Cambridge University Press, Cambridge, 2001

work page 2001

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A. Gray, E. Abbena and S. Salamon, Modern diﬀerential geometry of curves and surfaces with Mathematica. Third edition. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boc a Raton, FL, 2006

work page 2006

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Ishikawa, Singularities of Curves and Surfaces in Various Geometric Pr oblems

G. Ishikawa, Singularities of Curves and Surfaces in Various Geometric Pr oblems. CAS Lecture Notes 10, Exact Sciences, 2015. 11

work page 2015

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Ishikawa, Singularities of frontals

G. Ishikawa, Singularities of frontals. Singularities in generic geometry , 55–106, Adv. Stud. Pure Math., 78, Math. Soc. Japan, Tokyo, 2018

work page 2018

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Ishikawa, Recognition problem of frontal singulari ties

G. Ishikawa, Recognition problem of frontal singulari ties. J. Singul. 21, 2020, 149–166

work page 2020

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Izumiya, M

S. Izumiya, M. C. Romero-Fuster, M. A. S. Ruas and F. Tari , Diﬀerential Geometry from a Singularity Theory Viewpoint . World Scientiﬁc Pub. Co Inc. 2015

work page 2015

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Nakatsuyama and M

N. Nakatsuyama and M. Takahashi, On vertices of frontal s in the Euclidean plane. Bull. Braz. Math. Soc. (N.S.) 55, 2024, Paper No. 35, 21 pp

work page 2024

[16] [16]

Salarinoghabi and F

M. Salarinoghabi and F. Tari, Flat and round singularit y theory of plane curves. Q. J. Math. , 68, 2017, 1289–1312. Nozomi Nakatsuyama, Muroran Institute of Technology, Muroran 050-8585, Japan, E-mail address: 25096009b@muroran-it.ac.jp Masatomo Takahashi, Muroran Institute of Technology, Muroran 050-8585, Japan, E-mail address: masatomo@muroran-it.ac.jp...

work page 2017