• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1265-1280
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• 2023

The aim of this paper is to investigate Betchov-Da Rios equation by using null Cartan, pseudo null and partially null curve in Minkowski spacetime. Time derivative formulas of frame of s parameter null Cartan, pseudo null and partially null curve are examined, respectively. By using the obtained derivative formulas, new results are given about the solution of Betchov-Da Rios equation. The differential geometric properties of these solutions are obtained with respect to Lorentzian causal character of s parameter curve. For a solution of Betchov-Da Rios equation, it is seen that null Cartan s parameter curves are space curves in three-dimensional Minkowski space. Then all points of the soliton surface are flat points of the surface for null Cartan and partially null curve. Thus, it is seen from the results obtained that there is no surface corresponding to the solution of Betchov-Da Rios equation by using the pseudo null s parameter curve.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.610-618
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• 2023

In this paper, we obtain (k, m)-type slant helices for a null Cartan curve with the Bishop frame in Minkowski space E⁴₁.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.561-576
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• 2018

In this paper, we define null Cartan and pseudo null Bertrand curves in Minkowski space ${\mathbb{E}}^3_1$ according to their Bishop frames. We obtain the necessary and sufficient conditions for pseudo null curves to be Bertand B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Bertrand B-curves, by considering the cases when their Bertrand B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Bertrand B-curve pairs.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.885-899
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• 2013

In this paper, we study null helices, null slant helices and Cartan slant helices in $\mathb{E}^3_1$. Using some associated curves, we characterize the null helices and the Cartan slant helices and construct them. Also, we study a space-like curve with the principal normal vector field which is a degenerate plane curve.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.635-645
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• 2012

In this study, we investigate the curvature functions of ruled surface with lightlike ruling for a null curve with Cartan frame in Minkowski 3-space. Also, we give relations between the curvature functions of this ruled surface and curvature functions of central normal surface. Finally, we use the curvature theory of the ruled surface for determine differential properties of a robot end-effector motion.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.467-477
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• 2016

In this paper, we study the timelike Bertrand curves in Minkowski 3-space. Since the principal normal vector of a timelike curve is spacelike, the Bertrand mate curve of this curve can be a timelike curve, a spacelike curve with spacelike principal normal or a Cartan null curve, respectively. Thus, by considering these three cases, we get the necessary and sufficient conditions for a timelike curve to be a Bertrand curve. Also we give the related examples.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.1003-1016
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• 2016

In this paper, we introduce a new kind of slant helix for null curves called null $W_n$-slant helix and we give a definition of new harmonic curvature functions of a null curve in terms of $W_n$ in (n + 2)-dimensional Lorentzian space $M^{n+2}_1$ (for n > 3). Also, we obtain a characterization such as: "The curve ${\alpha}$ s a null $W_n$-slant helix ${\Leftrightarrow}H^{\prime}_n-k_1H_{n-1}-k_2H_{n-3}=0$" where $H_n,H_{n-1}$ and $H_{n-3}$ are harmonic curvature functions and $k_1,k_2$ are the Cartan curvature functions of the null curve ${\alpha}$.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.81-92
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• 2008

In this paper, the trajectory scroll in 3-dimensional Minkowski space-time $E_1^3$ is given by a firmly connected oriented line moving with Cartan frame along curve. Some theorems and results between curvatures of base curve and distribution parameter of this surface are obtained. Moreover, some theorems and results related to being developable and minimal of this surface are given. And also, some relationships among geodesic curvature, geodesic torsion and the curvatures of base curve of trajectory scroll are found.