Bertrand Curves of AW(k)-Type in the Equiform Geometry of the Galilean Space

and Applied Analysis 3 Thus, the formula analogous to the Frenet formula in the equiform geometry of the Galilean space has the following form:


Introduction
A Galilean space may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. This limit transition corresponds to the limit transition from the special theory of relativity to classical mechanics. On the other hand, Galilean space-time plays an important role in nonrelativistic physics. The fact that the fundamental concepts such as velocity, momentum, kinetic energy, and principles; laws of motion and conservation laws of classical physics are expressed in terms of Galilean space [1]. As it is well known, 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. Besides, the theory of curves and the curves of constant curvature in the equiform differential geometry of the isotropic spaces 1 3 and 2 3 and the Galilean space 3 are described in [2,3], respectively. Although the equiform geometry has minor importance related to the usual one, the curves that appear here in the equiform geometry can be seen as generalizations of well-known curves from the above mentioned geometries and therefore could have been of research interest. Many interesting results on curves of AW( )-type have been obtained by many mathematicians (see [4][5][6][7]). For example, in [4],Özgür and Gezgin studied a Bertrand curve of AW( )-type, and furthermore they showed that there was no such Bertrand curve of AW(1)-type and it was of AW(3)-type if and only if it was a right circular helix. In addition they studied weak AW(2)-type and AW(3)type conical geodesic curves in 3 . Besides, in 3-dimensional Galilean space and Lorentz space, the curves of AW( )-type were investigated by Külahcı et al. [8] and Külahcı and Ergüt [6], respectively. Kızıltug and Yaylı investigated quaternionic AW( )-type curves [9]. Also, Kızıltug and Yaylı [7] studied curves of AW( )-type in three Lie groups and gave some interesting results.
The purpose of the present paper is to provide AW( )type curves in the equiform geometry of the Galilean space 2 Abstract and Applied Analysis (absolute line) in , and is the fixed elliptic involution of points of .
In the nonhomogeneous coordinates the similarity group 8 has the form = 11 + 12 , = 21 + 22 + 23 cos + 23 sin , where and are real numbers [10]. In what follows the real numbers 12 and 23 will play the special role. In particular, for 12 = 23 = 1, (1) defines the group 6 ⊂ 8 of isometries of the Galilean space 3 . The Galilean scalar product can be written as where = ( 1 , 1 , 1 ) and = ( 2 , 2 , 2 ). It leaves invariant the Galilean norm of the vector defined by A curve : ⊂ R → 3 of the class ∞ in the Galilean space 3 is defined by the parameterization where is a Galilean invariant arc-length of . Then the curvature ( ) and the torsion ( ) are given by, respectively, On the other hand, the Frenet vectors of ( ) in 3 are defined by ( ) =̇( ) = (1,̇( ) ,̇( )) , The vectors , , are called the vectors of tangent, principal normal, and binormal of , respectively. For their derivatives, the following Frenet formula satisfies [10]:

Frenet Formulas in Equiform Geometry in 3
Let : → 3 be a curve in the Galilean space 3 . We define the equiform parameter of by where = (1/ ) is the radius of curvature of the curve . Then, we have = .
Let ℎ be a homothety with the center in the origin and the coefficient . If we put̃= ℎ( ), then it follows wherẽis the arc-length parameter of̃and̃the radius of curvature of this curve. Therefore, is an equiform invariant parameter of [10]. From now on, we define the Frenet formula of the curve with respect to the equiform invariant parameter in 3 . The vector is called a tangent vector of the curve . From (7) and (9) we get We define the principal normal vector and the binormal vector by Then, we easily show that { , , } are an equiform invariant orthonormal frame of the curve . On the other hand, the derivations of these vectors with respect to are given by is called the equiform curvature of the curve .
Definition 2. The function T : → R defined by is called the equiform torsion of the curve .
Abstract and Applied Analysis 3 Thus, the formula analogous to the Frenet formula in the equiform geometry of the Galilean space has the following form: = K ⋅ + , The equiform parameter = ∫ ( ) for closed curves is called the total curvature, and it plays an important role in global differential geometry of Euclidean space. Also, the function ( / ) has been already known as a conical curvature, and it also has interesting geometric interpretation.
Remark 3. Let : → 3 be a curve in the equiform geometry of the Galilean space 3 . So the following statements are true (see for details [2,10]).
Notation. Let us write As the definition of AW( )-type curves in [5], we have the following definition. Definition 6. Curves (of osculating order 3) in the equiform geometry of the Galilean space are given as (i) of type weak AW(2) if they satisfy (ii) of type weak AW(3) if they satisfy where * (i) If is an isotropic logarithmic spiral in 3 , then is type weak AW(2) curve.
(ii) If is a circular helix in 3 , then is type weak AW (2) curve.
(iii) If is an isotropic circle in 3 , then is type weak AW(2) curve.
Proof. By using Remark 3 and Proposition 7, we have the results. Proof. It is obvious from Remark 3 and Proposition 9. Proof. If is an isotropic logarithmic spiral or circular helix, then from Remark 3 we have, respectively, Substituting (27) and (28) in (26), we get, respectively, Since K( ) is nonzero constant and T( ) is nonzero constant, this is impossible, so is not isotropic logarithmic spiral or circular helix of type weak AW(3).
The converse statement is trivial. Hence our theorem is proved.

Corollary 14. If ( ) is an isotropic circle in 3 , then is of type AW(1) curve.
Proof. The proof is obvious from Remark 3 and Theorem 13.
where , , , and are differentiable functions. Since 2 ( ) and 3 ( ) are linearly dependent, coefficients determinant is equal to zero and hence one can write Here, Substituting these into (37), we obtain (35). Conversely if (35) holds, it is easy to show that is of type AW(2). This completes the proof.
Corollary 16. Let : → 3 be a curve (of osculating order 3) in the equiform geometry of the Galilean space 3 .

(i) If is an isotropic logarithmic spiral in 3 , then is of type AW(2) curve.
(ii) If is a circular helix in 3 , then there is not circular helix of type AW (2).

Bertrand Curves of AW( )-Type in the Equiform Geometry of 3
This section characterizes the curvatures of AW( )-type Bertrand curves in the equiform geometry of the Galilean space 3 . We provided some theorems and conclusion to show that there are Bertrand curves of weak AW(2)-type and AW(3)-type in the equiform geometry of the Galilean space 3 .
Definition 18. A curve : → 3 with ( ) ̸ = 0 is called a Bertrand curve if there exists a curvẽ: → 3 such that the principal normal lines of and̃at ∈ are equal. In this casẽis called a Bertrand mate of [11].
Abstract and Applied Analysis 5 The curvẽis called a Bertrand mate of and vice versa. A Frenet framed curve is said to be a Bertrand curve if it admits a Bertrand mate.
Hence, the proof is completed.
Theorem 21. Let : → 3 be a curve in the equiform geometry of the Galilean space 3
Abstract and Applied Analysis Taking = cot , we get This means that T is constant. The converse statement is trivial. Hence, theorem is proved.
Corollary 22. Let : → 3 be Bertrand curve in the equiform geometry of the Galilean space 3 . Then is a circular helix in 3 .
Proof. Since is Bertrand curve in the equiform geometry of the Galilean space 3 , from Theorems 19 and 21 we have Thus, is a circular helix in 3 . Hence, theorem is proved.
Theorem 23. Let : → 3 be Bertrand curve in the equiform geometry of the Galilean space 3 . Then is a weak AW(2)-type or AW(3)-type curve.
Proof. Now suppose that : → 3 is Bertrand curve in the equiform geometry of the Galilean space 3 . Then, from Theorems 19 and 21 we have if (60) is substituted into (25) and (39), which completes the proof of the theorem.
Theorem 24. Let : → 3 be Bertrand curve in the equiform geometry of the Galilean space 3 .Then is not a weak AW(3)-type or AW(2)-type curve.

Proof. Since
: → 3 is Bertrand curve according to the equiform geometry of the Galilean space 3 , then, (60) holds on . If (60) is substituted in (26) and (35), we get, respectively, Since T is nonzero constant, this is impossible. So, is not a weak AW(3)-type or type AW(2) curve. Hence, the theorem is proved.