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Abstract. In this paper, we give the differential equations characterizing the timelike and spacelike curves of constant breadth in Minkowski 3-space E 3 1 . Furthermore, we give a criterion for a timelike or spacelike curve to be the curve of constant breadth in E 3 1 . As an example, the obtained results are applied to the case ρ = const. and k 2 = const., and are discussed.

We study 1-type curves by using the mean curvature vector field of the curve. We also study biharmonic curves whose mean curvature vector field is in the kernel of Laplacian. We give some theorems for them in the Euclidean 3-space. Moreover we give some characterizations and results for a Frenet curve in the same space.

In this paper, we give the definitions of quaternionic Smarandache curves in 3-dimensional Euclidean space 3 E and we investigate some differential geometric properties of these curves.

In this paper, we give some characterizations for spacelike helices in Minkowski space-time E 4 1 . We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space-time E 4 1 .