Abstract
In this paper, by using the Darboux frame of null curves, we define null Bertrand partner -curves and present the relations between curvatures of these curves in Minkowski 3-space . In addition, we obtain some special results. Finally, by considering surface construction methods, we provide examples for null Bertrand partner -curves in .
1 Introduction
The associated curve of a given curve is a fascinating subject of differential geometry. So, finding such a curve is an interesting problem. Many geometers have investigated this problem in different spaces. The well-known examples of associated curves are Bertrand and Mannheim curves in the Euclidean 3-space. A Bertrand curve is a curve that shares its principal normal vectors with another curve and is characterized by the property that , where are constants []. Similarly, Mannheim curves are special curves for which the principal normal of one of the curves is linearly dependent on the binormal vector of the other curve.
Considering the curves on surfaces is more interesting and provides an idea for defining new types of associated curves on surfaces. We note that a new type of Bertrand curve has been defined on surfaces and is called the Bertrand partner -curves [, ]. In this definition, the authors have considered the Darboux frames of surface curves and obtained some characterizations of those curves.
Moreover, studying a concept of Euclidean space within Minkowski space is particularly interesting since the curves of this space are related to physics and the theory of relativity. A timelike curve corresponds to the path of an observer moving slower than the speed of light, a null curve corresponds to the observer moving at the speed of light, and a spacelike curve corresponds to an observer moving faster than light []. Particularly, null curves have extra importance since the classical relativistic string is a surface or world-sheet in Minkowski space, which satisfies the Lorentzian analog of the minimal surface equation []. Moreover, string equations are useful tools for simplifying the wave equation and a few additional simple equations. For instance, the solution of a two-dimensional (2D) wave equation shows that strings are related to null curve pairs, and if the string is open, it is related to a single null curve [, ].
In this paper, we define null Bertrand partner -curves lying on spacelike surfaces and present characterizations for these associated null curves. We obtain relations between curvatures of null Bertrand partner -curves. Finally, we provide some examples for null Bertrand partner -curves in Minkowski 3-space .
2 Preliminaries
The Minkowski 3-space is the real vector space provided with the standard flat metric given by , where is a rectangular coordinate system of . An arbitrary vector in can have one of three Lorentzian causal characters; it can be spacelike if or , timelike if , and null (light-like) if and . Similarly, an arbitrary curve can locally be spacelike, timelike, or null (light-like) if all of its velocity vectors are spacelike, timelike, or null (light-like), respectively. For any vectors and in , the Lorentz vector product of and is defined as follows:where and [, ].
is used to denote the moving frame along the null curve in . For an arbitrary null curve , the following Frenet formulas are givenwhere and “” denotes the derivative with respect to the arc length parameter [].
Similar to the curves, a surface in can be timelike or spacelike. A surface is called a timelike (spacelike) surface if the induced metric on the surface is a Lorentz metric (positive definite Riemannian metric); i.e., the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector, where is an open set in [].
Let be a spacelike surface in defined on an open set , and let us consider a null curve on with Frenet frame . Since lies on , there exists another frame along , which is called the Darboux frame of and is denoted by . In this frame, is the unit tangent of the curve, is the unit normal of the surface along , and is the unique vector obtained bywhere
Therefore, the Darboux formula of the moving frame is
In these formulas, , and are called the geodesic curvature, the normal curvature, and the geodesic torsion, respectively. Henceforth, we use “quote” to denote the derivative with respect to the arc length parameter of [, ].
3 Null Bertrand partner -curves on spacelike surfaces in
In this section, by considering the Darboux frame of null curves, we define null Bertrand partner -curves and provide the characterizations of these curves in .
Definition 1Letandbe oriented spacelike surfaces in, and let us consider the unit-speed null curvesandlying fully onand, respectively. The Darboux frames of null curvesandare denoted byand, respectively. If there exists a corresponding relationship between the curvesandsuch that at the corresponding points of the curves, the Darboux frame elementofcoincides with the Darboux frame elementof, thenis called a null Bertrand-curve andis called a null Bertrand partner-curve of. Then, the pairis said to be a null Bertrand-pair.
Letandbe oriented spacelike surfaces in, and let null curvesandwith non-zero normal curvaturesandlie onand, respectively. Then,andare null Bertrand partner-curves if and only if the following equality holds:
Proof. Suppose that the pair is a null Bertrand -pair. The Darboux frames of and are denoted by and , respectively. Then, by the definition, we can assume thatfor some smooth function . By taking the derivative of Equation 4 with respect to and applying the Darboux Formula 2, we obtain
Since the direction of coincides with the direction of , the inner product of Equation 5 with yields
Thus, is a non-zero constant. Now, equality Equation 5 can be written as
Taking the inner product of Equation 7 with itself, we obtain
From Equation 8, we obtain
Therefore, Equation 7 can be written as follows:
By taking the derivative of Equation 10, we obtainand taking the inner product of Equation 11 with itself, we obtainwhich yields Equation 3.
Conversely, we assume that Equation 3 holds. For a non-zero constant , we define a curve as
We will prove that is a null Bertrand -curve and that is the null Bertrand partner -curve of . By taking the derivative of Equation 13 with respect to twice, we obtainandrespectively. Taking the cross-product of Equation 15 and Equation 14, we obtain
Without loss of generality, taking the inner product of Equation 14 with itself yields . Thus, Equation 14 can be written as
Finally, the cross-product of Equation 16 and Equation 17 shows that the Darboux frame element of coincides with the Darboux frame element of at the corresponding points of the curves; i.e., the curves and are null Bertrand -pair curves.
Theorem 2 has the following corollaries.
Corollary 1The distance between the corresponding points of null Bertrand curves is constant and is given by.
Corollary 2Let the pairbe a null Bertrand-pair. Then, the geodesic torsion ofis a non-zero constant and is given by.
Corollary 3There is no null Bertrand-curvethat is a principal line; i.e.,.
Letandbe null Bertrand partner-curves with non-zero normal curvaturesand, respectively. Then,
Proof. Based on the definition, we can assume thatfor a non-zero constant . By taking the derivative of Equation 19 with respect to twice and applying the Darboux formulas, we obtainandrespectively. By substituting Equation 20 into Equation 21, we obtain
Since and are null Bertrand partner curves, we obtain the desired equation
Letandbe null Bertrand partnercurves lying on surfacesand, respectively. Then, the geodesic torsion ofis constant and is given by.
Proof. From (Equation 3), we obtainand by substituting Equation 24 into Equation 23, we obtain
From corollary 4 and theorem 7, we have the following corollary.
Corollary 4The relationship between the geodesic torsions of null Bertrand partnercurvesandis given by
Corollary 5Letandbe null Bertrand partnercurves. Then, the curvatures ofandhold
Proof. It is proven based on Equation 22.
Corollary 6Letandbe null Bertrand partnercurves. Then,andare geodesic null Bertrand partnercurves onandif and only if
Proof. The proof is clear from Equation 26.
4 Examples
In this section, we provide some examples of null Bertrand partner curves. For this purpose, we use a method related to the construction of spacelike surfaces [].
Example 1Let us consider the null curve. Then, by using the method proposed by [], the spacelike surfacecontainingas a geodesic is obtained as, wherewhere (Figure 1). Then, by using the Darboux frame components, the Bertrand partner curve of is obtained asThen, we can construct a spacelike surface with null geodesic as , whereand (Figure 2).
FIGURE 1
FIGURE 2
Example 2Letbe a null curve. Similarly, by using the method described by [], the surfacecontainingas a geodesic is constructed as, whereand and (Figure 3). Then, by using the Darboux frame, the Bertrand partner curve of is obtained asNow, the surface containing as a geodesic is constructed as , whereand and (Figure 4).
FIGURE 3
FIGURE 4
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.
Author contributions
TK: Writing – original draft.
Funding
The author declares that no financial support was received for the research and/or publication of this article.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author declares that no Generative AI was used in the creation of this manuscript.
Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.
MatsudaHYorozuS. Notes on bertrand curves. Yokohama Math J (2003) 50:41–58.
2.
KazazMUğurluHHÖnderMOralS. Bertrand partner D-curves in the euclidean 3-space E3. Afyon Kocatepe Univ J Sci Eng (2016) 16:76–83. 10.5578/fmbd.25270
3.
KazazMUğurluHHÖnderMOralS. Bertrand partner D-curves in the minkowski 3-space. Math Sci Appl E-Notes (2014) 2(1):68–82.
4.
El NaschieMS. Einstein’s dream and fractal geometry. Chaos Solitons Fractals (2005) 24:1–5. 10.1016/j.chaos.2004.09.001
5.
HughstonLPShawWT. Constraint-free analysis of relativistic strings. Classical Quan Gravity (1988) 5:69–72. 10.1088/0264-9381/5/3/001
6.
HughstonLPShawWT. Real classical string. Proc R Soc Lond Ser A (1987) 414:415–422.
7.
O‘NeillB. Semi-riemannian geometry with applications to relativity. New York: Academic Press (1983).
8.
WalraveJLeuvenKU. Curves and surfaces in minkowski space. Leuven: Fac of Science (1995). PhD thesis.
9.
DuggalKLBejancuA. Lightlike submanifolds of Semi-Riemannian manifolds and applications. Dordrecht: Kluwer Academic Publishers (1996). p. 54–75.
10.
BeemJKEhrlichPE. Global lorentzian geometry. New York: Marcel Dekker (1981).
11.
ÇökenACÇiftçiÜ. On null curves on surfaces and null vectors in lorentz space. Süleyman Demirel Univ J Sci (2007) 2(1):111–116.
12.
ŞaffakGKasapE. Family of surface with a common null geodesic. Int J Phys Sci (2009) 4(8):428–433.
Summary
Keywords
null curve, Bertrand, string, spacelike surfaces, partner curves
Citation
Kahraman T (2025) Null Bertrand partner -curves on spacelike surfaces. Front. Phys. 13:1611559. doi: 10.3389/fphy.2025.1611559
Received
15 April 2025
Revised
20 September 2025
Accepted
29 October 2025
Published
11 December 2025
Volume
13 - 2025
Edited by
Clemente Cesarano, Università Telematica Internazionale Uninettuno, Italy
Reviewed by
Yanlin Li, Hangzhou Normal University, China
Janos Polonyi, Université de Strasbourg, France
Updates
Copyright
© 2025 Kahraman.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Tanju Kahraman, tanju.kahraman@cbu.edu.tr
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.