• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.293-303
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• 2018

In this paper, we consider biharmonic slant curves in S-space forms. We obtain a main theorem, which gives us four different cases to find curvature conditions for these curves. We also give examples of slant curves in ${\mathbb{R}}^{2n+s}(-3s)$.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.417-424
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• 2017

In this paper, we study slant curves in a 3-dimensional almost f-Kenmotsu manifold with proper mean curvature vector field.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.310-324
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• 2022

In this paper, we define timelike curves in R⁴₂ and characterize such curves in terms of Frenet frame. Also, we examine the timelike helices of R⁴₂, taking into account their curvatures. In addition, we study timelike slant helices, timelike B₁-slant helices, timelike B₂-slant helices in four dimensional semi-Euclidean space, R⁴₂. And then we obtain an approximate solution for the timelike B₁ slant helix with Taylor matrix collocation method.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.967-977
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• 2020

In this article, we find the necessary and sufficient condition for a proper biharmonic Frenet curve in the Lorentzian Sasakian space forms 𝓜³₁(H) except the case constant curvature -1. Next, we find that for a slant curve in a 3-dimensional Sasakian Lorentzian manifold, its ratio of "geodesic curvature" and "geodesic torsion -1" is a constant. We show that a proper biharmonic Frenet curve is a slant pseudo-helix with 𝜅² - 𝜏² = -1 + 𝜀₁(H + 1)𝜂(B)² in the Lorentzian Sasakian space forms x1D4DC³₁(H) except the case constant curvature -1. As example, we classify proper biharmonic Frenet curves in 3-dimensional Lorentzian Heisenberg space, that is a slant pseudo-helix.

• Honam Mathematical Journal
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• v.45 no.1
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• pp.130-145
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• 2023

In this paper, we define some integral curves connected with a framed curve in Euclidean 3-space. These curves include framed generalized principal-direction curve, framed generalized binormal-direction curve, framed principal-donor curve and framed Darboux-direction curve. We obtain some relations between the framed curvatures of new defined framed curves and framed curvatures of given framed curve. By using the obtained relationships we give some characterizations for such curves. We also give methods for constructing framed helix and framed slant helix from planar curves.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.263-285
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• 2023

In this paper, we investigate the spherical indicatrices of a new relationship between Bertrand pair curves in Euclidean 3-space. We obtain necessary and sufficient conditions for this type of Bertrand pair curves to be slant helix, and provide an example.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1269-1281
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• 2020

In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric 3-manifold equipped with the canonical affine connection ${\hat{\nabla}}^t$ is a slant helix with pseudo-Hermitian curvature ${\hat{\kappa}}={\mid}q{\mid}\;sin\;{\theta}$ and pseudo-Hermitian torsion ${\hat{\tau}}=q\;cos\;{\theta}$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric 3-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve γ in a quasi-Sasakian 3-manifold M is a slant curve with curvature κ = |(t - α) cos θ + q| sin θ and torsion τ = α + {(t - α) cos θ + q} cos θ. These curves are not helices, in general. Note that if the ambient space M is an α-Sasakian 3-manifold, then γ is a slant helix.

• 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.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1309-1320
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• 2018

In this paper, we show that proper biharmonic spacelike curve ${\gamma}$ in Lorentzian Heisenberg space (${\mathbb{H}}_3$, g) is pseudo-helix with ${\kappa}^2-{\tau}^2=-1+4{\eta}(B)^2$. Moreover, ${\gamma}$ has the spacelike normal vector field and is a slant curve. Finally, we find the parametric equations of them.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.377-388
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• 2018

In this paper, we define new quaternionic associated curves called quaternionic principal-direction curves and quaternionic principal-donor curves. We give some properties and relationships between Frenet vectors and curvatures of these curves. For spatial quaternionic curves, we give characterizations for quaternionic helices and quaternionic slant helices by means of their associated curves.