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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 .

In this paper, depending on the Darboux instantaneous rotation vector of a solid perpendicular trihedron in the Minkowski 3-space R j 3 =[ R 3 , (+,+,-)] the Frenet instantaneous rotation vector was stated for a space-like curve (c) with the principal normal n being a time-like vector. The Darboux instantaneous rotation vector for the Darboux trihedron was found when the curve (c) is on a time-like surface. Some theorems and results giving the relations between two frames were stated and proved.

Abstract. In this study, by using Laplace and normal Laplace operators, we give some characterizations for the Darboux instantaneous rotation vector field of the curves in the Euclidean 3-space E 3 . Further, we give necessary and sufficient conditions for unit speed space curves to have 1-type Darboux vectors. Moreover, we obtain some characterizations of helices according to Darboux vector.

In this paper, we obtained some characterizations of timelike curves according to Bihop frame in Minkowski 3-space 3 1 E by using Laplacian operator and Levi-Civita connection. Furthermore we gave the general differential equations which characterize the timelike curves according to the Bishop Darboux vector and the normal Bishop Darboux vector.