Abstract. In this paper, we give some characterization for a osculating curve in 3-dimensional Euclidean space and we define a osculating curve in the Euclidean 4-space as a curve whose position vector always lies in orthogonal complement Bi of its first binormal vector field Si. In particular, we study the osculating curves in E 4 and characterize such curves in terms of their curvature functions.

We define normal curves in Minkowski space-time E41. In particular, we characterize the spacelike normal curves in E41 whose Frenet frame contains only non-null vector fields, as well as the timelike normal curves in E41 , in terms of their curvature functions. Moreover, we obtain an explicit equation of such normal curves with constant curvatures.

In this paper, we firstly give the necessary and sufficient conditions for null, pseudo null and partially null curves in Minkowski space-time to be normal curves. We prove that the null, pseudo null and partially null normal curves have a common property that their orthogonal projection onto non-degenerate hyperplane of E 4 1 or onto lightlike 2-plane of E 4 1 is the corresponding rectifying curve. Finally, we give some examples of such curves in E 4 1 .

In this paper, we define the Bishop frame of a null Cartan curve in Minkowski space-time [Formula: see text]. We obtain the Bishop’s frame equations of a null Cartan curve which lies in the timelike hyperplane of [Formula: see text]. We show that a null Cartan cubic lying in the timelike hyperplane of [Formula: see text] has two Bishop frames, one of which coincides with its Cartan frame. We also derive the Bishop’s frame equation of the null Cartan curve which has the third Cartan curvature [Formula: see text]. As an application, we find a solution of the null Betchov-Da Rios vortex filament equation in terms of a null Cartan curve and its Bishop frame, which generates a timelike Hasimoto surface.

In this paper, we define k-type spacelike slant helices lying on a lightlike surface in Minkowski space E 3 1 according to their Darboux frame for k ∈ {0, 1, 2}. We obtain the necessary and the sufficient conditions for spacelike curves with non-null and null principal normal lying on lightlike surface to be the k-type spacelike slant helices in terms of their geodesic curvature, normal curvature and geodesic torsion. Additionally, we determine their axes and show that the Darboux frame of a spacelike curve lying on a lightlike surface coincides with its Bishop frame if and only if it has zero geodesic torsion. Finally, we give some examples.

In this paper, we introduce the notion of the second kind twisted surfaces in Minkowski 3-space. We classify all non-degenerate second kind twisted surfaces in terms of flat, minimal, constant Gaussian and constant mean curvature surfaces, with respect to a chosen lightlike transversal bundle. We also prove that a lightlike second kind twisted surfaces, with respect to a chosen lightlike transversal vector bundle, are the lightcones, the lightlike binormal surfaces over pseudo null base curve and the lightlike ruled surfaces with null rulings whose base curve lies on lightcone.

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In this paper, we give some characterization for a osculating curve in 3-dimensional Euclidean space and we define a osculating curve in the Euclidean 4-space as a curve whose position vector always lies in orthogonal complement