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.

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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