The Smarandache Curves on S21 and Its Duality on H20

Curves as a subject of differential geometry have been intriguing for researchers throughout mathematical history and so they have been one of the interesting research fields. Regular curves play a central role in the theory of curves in differential geometry. In the theory of curves, there are some special curves such as Bertrand curves, Mannheim curves, involute and evolute curves, and pedal curves in which differential geometers are interested. A common approach to characterization of curves is to consider the relationship between the corresponding Frenet vectors of two curves. Bertrand and Mannheim curves are excellent examples for such cases. In the study of fundamental theory and the characterizations of space curves, the corresponding relations between the curves are a very fascinating problem. Recently, a new special curve is named according to the Sabban frame in the Euclidean unit sphere; Smarandache curve has been defined by Turgut and Yilmaz in Minkowski space-time [1]. Ali studied Smarandache curves with respect to the Sabban frame in Euclidean 3-space [2]. Then Taşköprü and Tosun studied Smarandache curves on S [3]. Smarandache curves have also been studied by many researchers [1, 4– 7]. Smarandache curves are one of the most important tools in Smarandache geometry. Smarandache geometry has an important role in the theory of relativity and parallel universes.There are many results related to Smarandache curves in Euclidean andMinkowski spaces, but Smarandache curves are getting more tedious and complicated when de Sitter space is concerned.A regular curve inMinkowski space-time, whose position vector is associated with Frenet frame vectors on another regular curve, is called a Smarandache curve [1]. In this paper, we define Smarandache curves on de Sitter surface according to the Sabban frame {α, t, η} in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame’s vectors of special Smarandache curves on de Sitter surface. Furthermore, we give some examples of special de Sitter and hyperbolic Smarandache curves in Minkowski 3-space.