Hasimoto surfaces in Galilean space G3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{3}$$\end{document}

In this article Hasimoto surfaces in Galilean space G3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{3}$$\end{document} will be considered, Gauss curvature (K) and Mean curvature (H) of Hasimoto surfaces χ=χ(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi =\chi (s,t)$$\end{document} will be investigated, some characterization of s-curves and t-curves of Hasimoto surfaces in Galilean space G3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{3}$$\end{document} will be introduced. Example of Hasimoto surfaces will be illustrated.

1. Lines, which do not cross the absolute line f is called proper non-isotropic lines. 2. The lines, which not belong to the ideal plane ω but intersect the absolute line f is called the proper isotropic lines. 3. All lines of the ideal plane ω except f are called proper non-isotropic lines. 4. The absolute line f.
The norm of the vector − → u = (u 1 , u 2 , u 3 ) can be written as The vector product of − → u = (u 1 , u 2 , u 3 ) and − → v = (v 1 , v 2 , v 3 ) in Galilean space G 3 is defined by The curve r(s) = (s, y(s), z(s)) is called the admissible curve. The associated invariant trihedron (Frenet invariant) T, N , and B for r(s) is given by the following equations. where T is the Tangent vector to r(s), N is the Normal vector to r(s), and B is the Binormal of r(s). Also k(s) is called the curvature function of the admissible curve r(s), and is denoted by the relation and τ (s) is the torsion function of the admissible curve r(s) and is given by the following equation The Frenet equations in Galilean space G 3 for the a admissible curve r(s) can be written as A C n -surface M, n ≥ 1 , immersed in Galilean space r : U → M, U belongs to R 2 , is denoted by χ(s, t) = (x(s, t), y(s, t), z(s, t)).
First fundamental form for the surface χ(s, t) is denoted by I and is given by the following equation.
where the symbols g i = x i is the derivatives of the first coordinates function x(s, t) with respect to s and t, and h ij =r i .r j the Euclidean inner product of the projection r k onto yz-plane . Furthermore, Gauss curvature K is denoted by (1) The vector N = 1 W (0, x t z s − x s z t , x s y t − x t y s ) is the normal vector to the surface χ(s, t).
is called a side tangential vector which is tangent plane to surface M.

Main results
In this section we will introduce Frenet equations of curves in both directions s, and t parameters. For Hasimoto surface χ(s, t) , we will obtain Gauss Curvature (K), Mean Curvature (H), and we will prove that Hasimoto surfaces are Weingarten surfaces. Also we obtain the necessary and sufficient conditions for the t-curves of Hasimoto surface χ(s, t) to be geodesic curves, or to be asymtotic curves. Also, we give conditions of the parameter curves to be lines of curvature. Finally, we give characterization for the s-parameter curves to be principal direction for Hasimoto surface χ(s, t) . At the end of this section example of Hasimoto surface in Galilean space G 3 is introduced.
respect to the parameter s is given by the following equations The Frenet Equations T · , N · and B · with respect to the parameter t, is obtained by the following equations where k = 0 is the curvature and τ is the torsion for the curve χ = χ(s, t) ∀t.

Proof
Frenet equations T ′ , N ′ and B ′ with respect to s is given directly from Frenet equation in Galilean space G 3 (1). Suppose that we have the differentiable functions α, β, γ and η where Our aim is to find α, β, γ and η functions interms of the curvature and torsion functions for the s-curve χ = χ(s, t) for all t. Using the conditions T ts = T st and N ts = N st we obtain i.e.
From the condition χ st = χ ts we get the the following equations substituting from Eqs. (8,9) we give the system in (6).
In the following theorem we will prove that Gaussian curvature K for Hasimoto surface equal to zero and the mean curvature H is equal to .

Proof
Suppose that χ(s, t) = (x(s, t), y(s, t), z(s, t)) is a parametrization of the surface χ(s, t) where the parameters s, t ∈ R , and x(s, t), y(s, t), z(s, t) ∈ C 3 . The normal of the surface is given by N = −N since χ s = T we obtain χ st = −kτ N from the property of Hasimoto surfaces r t = kB, we have r ts = k s B − kτ N therefore k s = 0 . By using the statement (4) of the second fundamental form we give hence, Gauss curvature K of Hasimoto surfaces χ(s, t) identically zero.
Mean curvature H of Hasimoto surface is given by Since Gauss Curvature of Hasimoto surfaces in Galilean space G 3 equal zero the following corollary is true. surface χ(s, t) is a Weingarten surface in G 3 .

Proof
The identically Jacobi equation Therefore, Hasimoto surface χ(s, t) is Weingarten surface.
The curve r(s) is a geodesic curve if and only if it has geodesic curvature equal to zero (k g = 0) , the curve is called asymptotic is its normal curvature k n = 0 In the following theorems we give some properties for the s-curves and t-curves of Hasimoto surface χ(s, t) to be geodesic curves and asymptotic curves in G 3 .

Theorem 3
Let χ(s, t) be a Hasimoto surface in G 3 . Then the following statements are satisfied 1. The s-curves of χ(s, t) are geodesic curves. 2. The t-curves of χ(s, t) are geodesic curves, ⇐⇒ the curvature of the t-curves of χ(s, t) equal to zero for all s (k t = 0).

Proof
1. For the s-curves of the Hasimoto χ(s, t) for all t, the geodesic curvature is obtain from the following relation k g = S · χ ss = (N × T) · (kN) = 0 , which proof the statement 1.
2. The geodesic curvature for the t-curves of the Hasimoto surfaceχ(s, t) for all s is k g = S · χ tt = (−n × T) · (k t B + kτ 2 N) =k t .
2. For t-curves we have χ tt = k t B+kτ 2 N, k n = −N · (k t B+kτ 2 N) = −kτ 2 i.e. t-curves are asymptotic curves of Hasimoto surface ⇐⇒ kτ 2 = 0 but k = 0 therefore τ 2 = 0 this means that τ must equal zero. Principal direction are tangent directions of a curve r(s) on a surface if the normal field of the surface satisfy det(α · , N , N · ) = 0 this condition essential for principal directions in Euclidean space [15].