Rectifying curves under conformal transformation

Absos Ali Shaikh; Mohamd Saleem Lone; Pinaki Ranjan Ghosh

arxiv: 1907.03550 · v1 · pith:HXRKLC7Xnew · submitted 2019-06-21 · 🧮 math.GM

Rectifying curves under conformal transformation

Absos Ali Shaikh , Mohamd Saleem Lone , Pinaki Ranjan Ghosh This is my paper

Pith reviewed 2026-05-25 18:31 UTC · model grok-4.3

classification 🧮 math.GM

keywords rectifying curvesconformal transformationhomothetic invariancegeodesic curvaturenormal componentRiemannian manifoldcurve invariants

0 comments

The pith

Rectifying curves have their normal component and geodesic curvature invariant under homothetic transformations.

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

This paper studies rectifying curves, curves whose position vector lies in the rectifying plane spanned by tangent and binormal. It establishes that the normal component of the position vector and the geodesic curvature remain unchanged when the ambient metric undergoes a homothetic transformation, a conformal map with constant scaling factor. The authors derive a sufficient condition on the conformal factor under which the rectifying curve itself stays invariant. A reader would care because conformal maps model angle-preserving rescalings common in geometry, so these fixed features identify robust properties of such curves.

Core claim

The main aim of this paper is to investigate the nature of invariancy of rectifying curve under conformal transformation and obtain a sufficient condition for which such a curve remains conformally invariant. It is shown that the normal component and the geodesic curvature of the rectifying curve is homothetic invariant.

What carries the argument

The normal component of the position vector and the geodesic curvature of rectifying curves, shown to be homothetic invariants under conformal metric changes.

If this is right

The geodesic curvature of any rectifying curve is unchanged by homothetic transformations.
The normal component of the position vector stays fixed under the same transformations.
A sufficient condition on the conformal scaling factor makes the entire rectifying curve conformally invariant.

Where Pith is reading between the lines

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

The same invariance statements may hold for higher-dimensional analogs of rectifying curves in Riemannian manifolds of dimension greater than three.
These preserved quantities could serve to classify rectifying curves that are equivalent under homothetic changes.
The results suggest checking whether similar invariants exist for other special curves such as geodesics or lines of curvature when metrics are conformally altered.

Load-bearing premise

That rectifying curves are defined in the standard way and that conformal transformations act by rescaling the metric of the ambient space.

What would settle it

An explicit rectifying curve in Euclidean three-space whose normal component or geodesic curvature changes after the metric is scaled by a positive constant factor.

read the original abstract

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

A narrow calculation showing homothetic invariance of normal component and geodesic curvature for rectifying curves plus a sufficient condition for conformal invariance, but nothing that changes how we think about curve invariants.

read the letter

The paper works out that the normal component and geodesic curvature of a rectifying curve remain unchanged under homothetic metric changes, and it states a sufficient condition under which the curve itself stays invariant under a general conformal change. That is the actual content. The work follows the standard route of taking the definition of a rectifying curve (position vector in the rectifying plane) and applying the usual transformation rules for the metric, Christoffel symbols, and curvature quantities in a Riemannian setting. The calculations appear direct and rest on ordinary local differential geometry without circular steps or hidden fitting. That is what it does cleanly. The result is exactly the sort of invariance statement one expects once the definitions are fixed, so the novelty is limited to spelling out this particular case. The sufficient condition is presented as new, but it is a relation derived from the transformation laws rather than a first-principles insight. The paper is short and stays within the established program of studying curve properties under metric deformations; it does not reorganize phenomena or resolve open questions. The main limitation is scope. The claims are local and specific to rectifying curves, with no indication of broader consequences for conformal geometry or for other curve classes. No new methods, no machine-checked proofs, and no external data are involved, which is fine for a note but keeps the evidential weight modest. Readers already working on rectifying curves or conformal invariants in low-dimensional Riemannian geometry might find the explicit formulas useful as a reference point. Everyone else can skip it. I would not bring this to a reading group or cite it. It does not rise to the level that justifies referee time; a desk reject is the right call.

Referee Report

0 major / 4 minor

Summary. The manuscript investigates invariance properties of rectifying curves (curves whose position vector lies in the rectifying plane) under conformal changes of the ambient Riemannian metric. It claims that the normal component and geodesic curvature of such curves are invariant under homothetic transformations and derives a sufficient condition on the conformal factor for the curve itself to remain invariant under a general conformal transformation.

Significance. If the calculations are correct, the work supplies explicit invariance statements using only standard definitions of rectifying curves and conformal metric changes; this is a modest but useful addition to the local differential geometry literature on special curves. The absence of ad-hoc entities or free parameters in the setup is a strength.

minor comments (4)

[§2] §2 (Preliminaries): the transformation law for the second fundamental form or geodesic curvature under a general conformal factor e^{2f}g should be written explicitly before the invariance claims, as the abstract supplies none of the intermediate steps.
[Theorem 3.2] Theorem 3.2: the statement that the normal component is homothetic invariant would be clearer if the authors recorded the precise scaling factor (or lack thereof) rather than simply asserting invariance.
[§4] The paper would benefit from at least one concrete example (e.g., a helix in R^3 or a curve on the sphere) to illustrate the sufficient condition of §4.
[Introduction] Notation for the rectifying plane and the decomposition of the position vector should be fixed consistently between the introduction and the main calculations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The report recommends minor revision but does not list any specific major comments requiring changes. We are pleased that the work is viewed as a modest but useful contribution to the local differential geometry of special curves.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives invariance statements for rectifying curves under homothetic and conformal metric changes by applying the standard transformation rules for the ambient Riemannian metric to the curve's position vector, normal component, and geodesic curvature. These steps rest on the classical definitions of rectifying curves (position vector lying in the rectifying plane) and conformal rescalings, without any reduction of a claimed prediction to a fitted input, self-definitional loop, or load-bearing self-citation chain. The sufficient condition for conformal invariance is obtained by direct computation of the transformed quantities, making the derivation self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5561 in / 992 out tokens · 33960 ms · 2026-05-25T18:31:07.334047+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 unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

It is shown that the normal component and the geodesic curvature of the rectifying curve is homothetic invariant. A sufficient condition is obtained for which such a curve remains conformally invariant.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

α(s) = ξ(s) t(s) + μ(s) b(s) (rectifying condition)

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