Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space

We introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space . Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. And we give an example to illustrate these curves. 1. Introduction In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the characterization of a regular curve. In the solution of the problem, the curvature functions and of a regular curve have an effective role. It is known that the shape and size of a regular curve can be determined by using its curvatures and . Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. The axiom is said to be Smarandachely denied if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. By definition, if the position vector of a curve is composed by the Frenet frame’s vectors of another curve , then the curve is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–8]. For instance, the special Smarandache curves according to Darboux frame in are characterized in [9]. In this paper, we define Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski -space . Inspired by the previous papers we investigate the geodesic curvature and the Sabban frame’s vectors of Smarandache curves. In Section 2, we explain the basic concepts of Minkowski -space and give Lorentzian Darboux frame that will be used throughout the paper. Section 3 is devoted to the study of four Smarandache curves, -Smarandache curve, -Smarandache curve, -Smarandache curve, and -Smarandache curve by considering the relationship with invariants , , and of curve on spacelike surface in Minkowski -space . Also, we give some characterizations on the curves when the curve is an asymptotic curve or a principal curve. Finally, we illustrate these curves with an example. 2. Basic Concepts The Minkowski 3-space is the Euclidean 3-space provided with the standard flat