ON THE EQUIFORM DIFFERENTIAL GEOMETRY OF AW( k )-TYPE CURVES IN PSEUDO-GALILEAN 3-SPACE

. The aim of this paper is to study AW( k )-type (1 ≤ k ≤ 3) curves according to the equiform diﬀerential geometry of the pseudo-Galilean space G 13 . We give some geometric properties of AW( k ) and weak AW( k )-type curves. Moreover, we give some relations between the equiform curvatures of these curves. Finally, examples of some special curves are given and plotted to support our main results.


Introduction
The geometry of space is associated with mathematical group. The idea of invariance of geometry under transformation group may imply that, on some spacetimes of maximum symmetry there should be a principle of relativity which requires the invariance of physical laws without gravity under transformations among inertial systems [1]. The theory of curves and the curves of constant curvature in the equiform differential geometry of the isotropic spaces I 1 3 , I 2 3 and the Galilean space G 3 are described in [2] and [3], respectively. The pseudo-Galilean space is one of the real Cayley-Klein spaces. It has projective signature (0, 0, +, −) according to [2]. The absolute of the pseudo-Galilean space is an ordered triple {w, f, I} where w is the ideal plane, f a line in w and I is the fixed hyperbolic involution of the points of f . In [4], from the differential geometric point of view, K. Arslan and A. West defined the notion of AW(k)-type submanifolds. Since then, many works have been done related to AW(k)-type submanifolds (see, for example, [5][6][7][8][9][10]). In [9], Özgür and Gezgin studied a Bertrand curve of AW(k)-type and furthermore, they showed that there is no such Bertrand curve of AW(1) and AW(3)-types if and only if it is a right circular helix. In addition, they studied weak AW(2)-type and AW(3)-type conical geodesic curves in Euclidean 3-space E 3 . Besides, In 3-dimensional Galilean space and Lorentz space, the curves of AW(k)-type were investigated in [6,8]. In [7], the authors gave curvature conditions and characterizations related to AW(k)-type curves in E n and in [10], the authors investigated curves of AW(k)-type in the 3-dimensional null cone. This paper is organized as follows. In section 2, the basic notions and properties of a pseudo-Galilean geometry are reviewed. In section 3, properties of the equiform geometry of the pseudo-Galilean space G 1 3 are given. Section 4 contains a study of AW(k)-type equiform Frenet curves. Finally, some examples of special curves in G 1 3 are included in section 5.

Basic concepts
In this section, we recall some basic notions from pseudo-Galilean geometry [11,12]. In the inhomogeneous affine coordinates for points and vectors (point pairs) the similarity group H 8 of G 1 3 has the following form where a, b, c, d, e, f, r and θ are real numbers. Particularly, for b = r = 1, the group (2.1) becomes the group B 6 ⊂ H 8 of isometries (proper motions) of the pseudo-Galilean space G 1 3 . The motion group leaves invariant the absolute figure and defines the other invariants of this geometry. It has the following form According to the motion group in the pseudo-Galilean space, there are non-isotropic vectors A(A 1 , A 2 , A 3 ) (for which holds A 1 = 0) and four types of isotropic vectors: spacelike ( and two types of lightlike vectors (A 1 = 0, A 2 = ±A 3 ). The scalar product of two vectors u = (u 1 , u 2 , u 3 ) and v = (v 1 , v 2 , v 3 ) in G 1 3 is defined by We introduce a pseudo-Galilean cross product in the following way where j = (0, 1, 0) and k = (0, 0, 1) are unit spacelike and timelike vectors, respectively. Let us recall basic facts about curves in G 1 3 , that were introduced in [13][14][15]. A curve γ(s) = (x(s), y(s), z(s)) is called an admissible curve if it has no inflection points (γ ×γ = 0) and no isotropic tangents (ẋ = 0) or normals whose projections on the absolute plane would be lightlike vectors (ẏ = ±ż). An admissible curve in G 1 3 is an analogue of a regular curve in Euclidean space [12]. For an admissible curve γ(s) : I ⊆ R → G 1 3 , the curvature κ(s) and torsion τ (s) are defined by expressed in components. Hence, for an admissible curve γ : I ⊆ R → G 1 3 parameterized by the arc length s with differential form ds = dx is given by The formulas (2.3) have the following form The associated trihedron is given by where = +1 or = −1, chosen by criterion det(e 1 , e 2 , e 3 ) = 1, that means The curve γ given by (2.4) is timelike (resp. spacelike) if e 2 (s) is a spacelike (resp. timelike) vector. The principal normal vector or simply normal is spacelike if = +1 and timelike if = −1. For derivatives of the tangent e 1 , normal e 2 and binormal e 3 vector fields, the following Frenet formulas in G 1 3 hold: 3. Frenet equations according to the equiform geometry of G 1 3 This section contains some important facts about equiform geometry. The equiform differential geometry of curves in the pseudo-Galilean space G 1 3 has been described in [11]. In the equiform geometry a few specific terms will be introduced. So, let γ(s) : I → G 1 3 be an admissible curve in the pseudo-Galilean space G 1 3 , the equiform parameter of γ is defined by where ρ = 1 κ is the radius of curvature of the curve γ. Then, we have Let h be a homothety with center at origin and the coefficient µ. If we putγ = h(γ), then it follows wheres is the arc-length parameter ofγ andρ is the radius of curvature of this curve. Therefore, σ is an equiform invariant parameter of γ (see [11]).
Now we define the Frenet formulas of the curve γ with respect to its equiform invariant parameter σ in is called a tangent vector of the curve γ. From (2.6) and (3.1), we get Also, the principal normal and the binormal vectors are respectively, given by It is easy to show that {T, N, B} is an equiform invariant frame of γ. On the other hand, the derivatives of these vectors with respect to σ are given by The functions K : I → R defined by K =ρ is called the equiform curvature of the curve γ and T : I → R defined by T = ρτ = τ κ is called the equiform torsion of this curve. In the light of this, the formulas (3.4) analogous to the Frenet formulas in the equiform geometry of the pseudo-Galilean space G 1 3 can be written as  The equiform parameter σ = κ(s)ds for closed curves is called the total curvature, and it plays an important role in global differential geometry of Euclidean space. Also, the function τ κ has been already known as a conical curvature and it also has interesting geometric interpretation.
3 be a Frenet curve in the equiform geometry of G 1 3 , the following statements are true ( for more details, see [11,13] ): (1) If γ(s) is an isotropic logarithmic spiral in G    (1) equiform AW(1) if they satisfy Q 3 = 0, weak equiform AW(2) if they satisfy    Proof. Since γ is of type equiform AW(1), then from (4.3), we obtain As we know, the vectors N and B are linearly independent, so we can write The converse statement is straightforward and therefore, the proof is completed.

Computational examples
We consider some examples (timelike and spacelike curves [11,12]) which characterize equiform gen-  Also, the associated trihedron is given by (a cosh (bs) + b sinh (bs)) , For this curve, the equiform vector fields are obtained as follows respectively.
It follows that Here, the equiform differential vectors respectively, are as follows Equiform curvature and equiform torsion are calculated as follows Example 5.5. Let r : I −→ G 1 3 , I ⊆ R be a equiform timelike isotropic logarithmic spiral which parameterized by the arc length s with differential form ds = dx, and is given by

Conclusion
In this paper, we have considered some special curves of equiform AW(k)-type of the pseudo-Galilean 3space. Also, using the equiform curvature conditions of these curves, the necessary and sufficient conditions for them to be equiform AW(k) and weak equiform AW(k)-types are obtained. Furthermore, some examples to support our main results are given and plotted.