Narrow Results

In this paper, we show that there is no timelike translation surface with non-minimal constant mean curvature $H$, constructed by the sum of a timelike plane curve and a timelike space curve in Minkowski 3-space $E_1^3$.

In this paper, we introduce three new transformations and establish connections between moving non-null curves and soliton equations according to Bishop frame in Minkowski 3-space. Later we find formulas for differentials of these three new transformations associated with the nonlinear heat system and repulsive type nonlinear Schrödinger equation.

In this paper, we present three classes of non-lightlike curve evolution according to the Darboux frame and we give three geometric phases according to three classes in Minkowski 3-space.

We study the non-lightlike ruled surfaces in Minkowski 3-space with non-lightlike base curve $c(s)=\int (\alpha t+\beta n+\gamma b)ds$, where $t$, $n$, $b$ are the tangent, principal normal and binormal vectors of an arbitrary timelike curve $\mathrm{\Gamma }(s)$. Some important results of flat, minimal, II-minimal and II-flat non-lightlike ruled surfaces are studied. Finally, the following interesting theorem is proved: the only non-zero constant mean curvature (CMC) non-lightlike ruled surface is developable timelike ruled surface generated by binormal vector.

In this paper, we examine curves drawn by a timelike or spacelike curve$\alpha$ on the pseudo-hyperbolic sphere $H_0^2$ and the unit Lorentzian sphere $S_1^2$in three-dimensional Minkowski space ${ℝ}_1^3.$ We present Lorentz forces and magnetic curves produced by the Frenet frame $\{T_T,N_T,B_T\}$ on spherical indicatricies in a magnetic field of the curve $\alpha$. We calculate magnetic vector fields of magnetic curves for $T_T,N_T,B_T$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatricies. We obtain developability conditions for these surfaces. Finally, we illustrate magnetic flux surfaces with some interesting examples.

In this paper, we focus on geometry of evolutoids and pedaloids for timelike curves and spacelike curves on timelike surfaces. First, by studying an example of a pseudo-circle, we get some properties of timelike curve on timelike surface. Then, we introduce a moving frame along curves and describe the corresponding evolutoids of curves as ruled surfaces. After that, we investigate relationships between pedal surfaces of evolutoids and pedaloids of curves. Moreover, we investigate the singularities of the evolutoids and the pedaloids.

In the present paper, we consider general helix with spacelike slope axis for all possible types of curves in Minkowski $3$-space. We give the conditions under which the curves in Minkowski $3$-space have spacelike slope axis. In addition, we find the parametric equations of the curves. Also, we give the related examples and their graphics.

If the principal normal vector field of a curve makes a constant angle with constant direction, this curve is called as slant helix. In this paper, a slant helix is defined according to N-Bishop frame of the spacelike curve with a spacelike principal normal. Some characterizations of the slant helices are obtained according to spacelike curve N-Bishop frame with a spacelike principal normal, benefiting from the definition of the slant helices.