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In this article Hasimoto surfaces in Galilean space $$G_{3}$$ G 3 will be considered, Gauss curvature (K) and Mean curvature (H) of Hasimoto surfaces $$\chi =\chi (s,t)$$ χ = χ ( s , t ) will be investigated, some characterization of s-curves and t-curves of Hasimoto surfaces in Galilean space $$G_{3}$$ G 3 will be introduced. Example of Hasimoto surfaces will be illustrated.

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In this article, we present the equiform parameter and define the equiform-Bishop frame in Minkowski 3-space E 3 1 . Additionally, we investigate the equiform-Bishop formulas of the equiform spacelike case in Minkowski 3-space.Furthermore, some results of equiform spacelike normal curves according to the equiform-Bishop frame in E 3 1 are considered. KEYWORDS equiform-Bishop frame, equiform curvatures, Minkowski Space, normal curves PRELIMINARIESThe Minkowski space E 3 1 is the space R 3 , equipped with the metric g, where g is given bywhere (x 1 , x 2 , x 3 ) is a coordinate system of E 3 1 . Let v be any vector in E 3 1 , then the vector v is spacelike, timelike, or null (lightlike) if g(v, v) > 0 or v = 0, g(v, v) < 0 or g(v, v) = 0, and v ≠ 0. The causal character of a vector in Minkowski space is the property to be spacelike, timelike, or null (lightlike).

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In this paper, we construct helicoidal surfaces in the three dimensional Galilean space G 3. The First and the Second Fundamental Forms for such surfaces will be obtained. Also, mean and Gaussian curvature given by smooth functions will be derived. We consider the Galilean 3−space with a linear density e φ and construct a weighted helicoidal surfaces in G 3 by solving a second order non-linear differential equation. Moreover, we discuss the problem of finding explicit parameterization for the helicoidal surfaces in G 3 .

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In this article, we introduce the notion of a spatial quaternionic Bertrand curves in G 3 and give some characterizations of such curves. Furthermore, we introduce spatial quaternionic (1, 3)−Bertrand curves in G 4 .