Some characterizations of Rectifying and osculating curves on a smooth immersed surface

Absos Ali Shaikh; Pinaki Ranjan Ghosh

arxiv: 1906.10520 · v1 · pith:I7I3MZW6new · submitted 2019-06-19 · 🧮 math.GM

Some characterizations of Rectifying and osculating curves on a smooth immersed surface

Absos Ali Shaikh , Pinaki Ranjan Ghosh This is my paper

Pith reviewed 2026-05-25 20:04 UTC · model grok-4.3

classification 🧮 math.GM

keywords rectifying curvesosculating curvesimmersed surfacesisometrynormal curvatureposition vector componentstangent vector

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The pith

Rectifying and osculating curve position vectors on a surface are invariant under isometry precisely when normal curvature is invariant or the vector aligns with the tangent.

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

The paper computes the components of the position vectors of rectifying and osculating curves on a smooth immersed surface using the frame formed by the tangent vector T, the surface normal N, and their cross product T times N. It then checks how these components transform when the surface is subjected to an isometry. The central result states that the components remain unchanged if and only if the normal curvature stays the same or the position vector lies along the tangent direction. This gives a direct link between the decomposition in the chosen frame and the preservation properties of the curves under surface mappings that preserve distances.

Core claim

The components of position vectors of rectifying and osculating curves along T, N, T times N are invariant under isometry of surfaces if and only if either the normal curvature of the curve is invariant or the position vector of the curve is in the direction of the tangent vector to the curve.

What carries the argument

The reference frame {T, N, T times N} attached to the curve, used to decompose the position vector into components whose invariance under isometry is then characterized.

If this is right

If normal curvature is preserved by the isometry then all three components of the position vector stay the same.
If the position vector lies exactly along T then the components are unchanged even when normal curvature varies.
The stated iff condition supplies a complete criterion for deciding when a rectifying or osculating curve keeps its frame-decomposed position data under any isometry of the ambient surface.

Where Pith is reading between the lines

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

The result could be used to classify families of curves that remain rectifying after an isometric deformation even when curvature is not preserved.
One might check the same decomposition on curves defined by other geometric conditions, such as lines of curvature, to see whether analogous invariance statements hold.
The characterization might extend to the study of how the Frenet frame itself transforms under the isometry when the curve is rectifying.

Load-bearing premise

The reference frame {T, N, T times N} is well-defined for the rectifying and osculating curves on the smooth immersed surface, permitting direct computation of the position vector components along each basis vector.

What would settle it

An explicit pair of isometric surfaces containing a rectifying curve where the normal curvature differs between them and the position vector is not aligned with the tangent, yet the three components of the position vector remain numerically identical after the isometry.

read the original abstract

The present paper deals with some characterizations of rectifying and osculating curves on a smooth surface with respect to the reference frame $\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\}$. We have computed the components of position vectors of rectifying and osculating curves along $\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}$ and then investigated their invariancy under isometry of surfaces, and it is shown that they are invariant iff either the normal curvature of the curve is invariant or the position vector of the curve is in the direction of the tangent vector to the curve.

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 invariance claim under surface isometries does not hold because position vectors are extrinsic and not preserved by intrinsic isometries.

read the letter

The paper's main result on invariance of position vector components under isometries of surfaces has a basic problem. An isometry preserves the first fundamental form but does not fix the ambient position vector r or its projections onto the frame {T, N, T×N}. Different isometric embeddings of the same abstract surface can place it anywhere in R^3, so the components cannot be invariant in general. The abstract states the iff condition without addressing this gap, and the stress-test concern lands directly on the claim as written.

Referee Report

1 major / 1 minor

Summary. The paper computes the components of the position vectors of rectifying and osculating curves on a smooth immersed surface in the frame {T, N, T×N} and claims to characterize their invariance under isometries of the surface, showing that the components are invariant if and only if the normal curvature is invariant or the position vector lies along T.

Significance. If the invariance result holds under a well-defined notion of isometry that preserves the relevant quantities, it would provide extrinsic characterizations of these curves in terms of invariance properties. The work builds on standard differential geometry of curves on surfaces but does not appear to include machine-checked proofs or reproducible code.

major comments (1)

[Abstract] Abstract (and central claim): the asserted invariance of position-vector components under 'isometry of surfaces' is not secured by the definitions. An isometry preserves the first fundamental form but does not canonically preserve the ambient position vector r or its inner products with the Darboux frame {T, N, T×N}, since distinct isometric embeddings into R^3 can translate or rotate the surface relative to the origin. The iff statement therefore requires an unstated assumption that isometries extend to rigid motions of the ambient space or that position vectors are identified across embeddings; without this, the claim reduces to a statement about a fixed embedding rather than intrinsic invariance.

minor comments (1)

[Abstract] The abstract refers to 'computations' and an 'invariance theorem' but provides no explicit equations or steps; the full manuscript should include the component derivations and the precise definition of the isometry action on the frame and position vector.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the central claim. We address the concern regarding the notion of isometry and invariance below, and we will revise the manuscript to clarify the assumptions.

read point-by-point responses

Referee: Abstract (and central claim): the asserted invariance of position-vector components under 'isometry of surfaces' is not secured by the definitions. An isometry preserves the first fundamental form but does not canonically preserve the ambient position vector r or its inner products with the Darboux frame {T, N, T×N}, since distinct isometric embeddings into R^3 can translate or rotate the surface relative to the origin. The iff statement therefore requires an unstated assumption that isometries extend to rigid motions of the ambient space or that position vectors are identified across embeddings; without this, the claim reduces to a statement about a fixed embedding rather than intrinsic invariance.

Authors: We acknowledge that the manuscript does not explicitly articulate the assumption that the isometries of the immersed surface are realized by (or extend to) rigid motions of the ambient R^3. In the context of the work, the invariance is understood with respect to congruences, i.e., isometries of R^3 that map one immersed surface to another while transporting the Darboux frame and the position vector accordingly. Under this interpretation the stated iff condition holds. Nevertheless, the referee is correct that this must be stated clearly to avoid ambiguity between intrinsic surface isometries and ambient congruences. We will revise the abstract, introduction, and the statement of the main theorem to specify that the isometries are induced by isometries of R^3 (hence preserving the relevant inner products after transport). This revision will be made in the next version. revision: yes

Circularity Check

0 steps flagged

No circularity: direct component calculations yield independent characterization

full rationale

The paper states it computes the components of position vectors of rectifying and osculating curves along the Darboux-type frame {T, N, T×N} and then derives the invariance condition under surface isometries. This is a standard differential-geometric computation from the definitions of the curves, the frame, and the first fundamental form; the iff statement follows from equating the transformed components to the original ones. No parameter is fitted and then relabeled as a prediction, no self-citation is invoked as a uniqueness theorem, and no ansatz is smuggled in. The derivation chain is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions from differential geometry without introducing free parameters or new entities.

axioms (1)

domain assumption The surface is smooth and immersed so that the tangent vector T and normal vector N are well-defined everywhere along the curve.
This is required to construct the reference frame {T, N, T×N} and to speak of normal curvature.

pith-pipeline@v0.9.0 · 5627 in / 1197 out tokens · 27892 ms · 2026-05-25T20:04:06.945784+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

?

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

We have computed the components of position vectors of rectifying and osculating curves along T, N, T×N and then investigated their invariancy under isometry of surfaces, and it is shown that they are invariant iff either the normal curvature of the curve is invariant or the position vector of the curve is in the direction of the tangent vector to the curve.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

κn(s) = Lu′² + 2Mu′v′ + Nv′² (normal curvature from second fundamental form)

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

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Pressley, A., Elementary diﬀerential geometry , Springer-Verlag, 2001

work page 2001
[2]

P., Diﬀerential geometry of curves and surfaces , Prentice-Hall, Inc, New Jersey, 1976

do Carmo, M. P., Diﬀerential geometry of curves and surfaces , Prentice-Hall, Inc, New Jersey, 1976

work page 1976
[3]

Chen, B.-Y., What does the position vector of a space curve always lie in it s rectifying plane ?, Amer. Math. Monthly, 110 (2003), 147-152

work page 2003
[4]

and Dillen, F., Rectfying curve as centrode and extremal curve

Chen, B.-Y. and Dillen, F., Rectfying curve as centrode and extremal curve. , Bull. Inst. Math. Acad. Sinica, 33, no. 2, (2005), 77-90

work page 2005
[5]

and Nesovic, E., Some characterizations of null, pseudo null and partially n ull rectifying curves in Minkowski space-time , Taiwanese J

Ilarslan, K. and Nesovic, E., Some characterizations of null, pseudo null and partially n ull rectifying curves in Minkowski space-time , Taiwanese J. Math., 12, no. 5, (2008), 1035-1044

work page 2008
[6]

and Alshammari, S

Deshmukh, S., Chen, B.-Y. and Alshammari, S. H., On rectifying curves in Euclidean 3-space , Turk. J. Math., 42 (2018), 609-620

work page 2018
[7]

and Ilarslan, K., Characterizations of the position vector of a surface curve in Euclidean 3-space, An

Camci, C., Kula, L. and Ilarslan, K., Characterizations of the position vector of a surface curve in Euclidean 3-space, An. St. Univ. Ovidius Constanta, 19, no. 3, (2011), 59-70

work page 2011
[8]

Ilarslan, M

K. Ilarslan, M. Sakaki and A. U¸ cum, On osculating, normal and rectifying bi-null curves in R5 2, Novi Sad J. Math. , 48 (2018), 9-20

work page 2018
[9]

Shaikh, A. A. and Ghosh, P. R., Rectifying curves on a smooth surface immersed in the Euclid ean space, to appear in Indian J. Pure Appl. Math., 2018

work page 2018
[10]

Shaikh, A. A. and Ghosh, P. R., Rectifying and osculating curves on a smooth surface , to appear in Indian J. Pure Appl. Math., 2018

work page 2018
[11]

Shaikh, A. A. and Ghosh, P. R., Curves on a smooth surface with position vectors lie in the ta ngent plane , to appear in Indian J. Pure Appl. Math., 2019

work page 2019
[12]

Shaikh, A. A. Lone, M. S., and Ghosh, P. R., Normal curves on a smooth immersed surface , arXiv:1906.04738. 10 A. A. SHAIKH AND P. R. GHOSH Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India E-mail address : aask2003@yahoo.co.in, aashaikh@math.buruniv.ac.in Department of Mathematics, University of Burdwan, Golap...

work page internal anchor Pith review Pith/arXiv arXiv 1906

[1] [1]

Pressley, A., Elementary diﬀerential geometry , Springer-Verlag, 2001

work page 2001

[2] [2]

P., Diﬀerential geometry of curves and surfaces , Prentice-Hall, Inc, New Jersey, 1976

do Carmo, M. P., Diﬀerential geometry of curves and surfaces , Prentice-Hall, Inc, New Jersey, 1976

work page 1976

[3] [3]

Chen, B.-Y., What does the position vector of a space curve always lie in it s rectifying plane ?, Amer. Math. Monthly, 110 (2003), 147-152

work page 2003

[4] [4]

and Dillen, F., Rectfying curve as centrode and extremal curve

Chen, B.-Y. and Dillen, F., Rectfying curve as centrode and extremal curve. , Bull. Inst. Math. Acad. Sinica, 33, no. 2, (2005), 77-90

work page 2005

[5] [5]

and Nesovic, E., Some characterizations of null, pseudo null and partially n ull rectifying curves in Minkowski space-time , Taiwanese J

Ilarslan, K. and Nesovic, E., Some characterizations of null, pseudo null and partially n ull rectifying curves in Minkowski space-time , Taiwanese J. Math., 12, no. 5, (2008), 1035-1044

work page 2008

[6] [6]

and Alshammari, S

Deshmukh, S., Chen, B.-Y. and Alshammari, S. H., On rectifying curves in Euclidean 3-space , Turk. J. Math., 42 (2018), 609-620

work page 2018

[7] [7]

and Ilarslan, K., Characterizations of the position vector of a surface curve in Euclidean 3-space, An

Camci, C., Kula, L. and Ilarslan, K., Characterizations of the position vector of a surface curve in Euclidean 3-space, An. St. Univ. Ovidius Constanta, 19, no. 3, (2011), 59-70

work page 2011

[8] [8]

Ilarslan, M

K. Ilarslan, M. Sakaki and A. U¸ cum, On osculating, normal and rectifying bi-null curves in R5 2, Novi Sad J. Math. , 48 (2018), 9-20

work page 2018

[9] [9]

Shaikh, A. A. and Ghosh, P. R., Rectifying curves on a smooth surface immersed in the Euclid ean space, to appear in Indian J. Pure Appl. Math., 2018

work page 2018

[10] [10]

Shaikh, A. A. and Ghosh, P. R., Rectifying and osculating curves on a smooth surface , to appear in Indian J. Pure Appl. Math., 2018

work page 2018

[11] [11]

Shaikh, A. A. and Ghosh, P. R., Curves on a smooth surface with position vectors lie in the ta ngent plane , to appear in Indian J. Pure Appl. Math., 2019

work page 2019

[12] [12]

Shaikh, A. A. Lone, M. S., and Ghosh, P. R., Normal curves on a smooth immersed surface , arXiv:1906.04738. 10 A. A. SHAIKH AND P. R. GHOSH Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India E-mail address : aask2003@yahoo.co.in, aashaikh@math.buruniv.ac.in Department of Mathematics, University of Burdwan, Golap...

work page internal anchor Pith review Pith/arXiv arXiv 1906