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 obtain the Frenet equations of a pseudo null and a partially null curves, lying fully in the semi-Euclidean space R 4 2 , and classify all such curves with constant curvatures. (2000): 53C50, 53C40.
Mathematics Subject Classification
In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.
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 .
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