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

Abstract

We consider curves of AW()-type () in the equiform geometry of the Galilean space . We give curvature conditions of curves of AW()-type. Furthermore, we investigate Bertrand curves in the equiform geometry of . We have shown that Bertrand curve in the equiform geometry of is a circular helix. Besides, considering AW()-type curves, we show that there are Bertrand curves of weak AW(2)-type and AW(3)-type. But, there are no such Bertrand curves of weak AW(3)-type and AW(2)-type.

1. 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 and and the Galilean space 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 [4–7]). 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 . 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. Kızıltuğ and Yaylı investigated quaternionic AW-type curves [9]. Also, Kızıltuğ 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 and provide the properties of Bertrand curve of AW-type in the equiform geometry of the Galilean space .

2. Preliminaries

The Galilean space is a Cayley-Klein space equipped with the projective metric of signature . The absolute figure of the Galilean space consists of an ordered triple , where is the ideal (absolute) plane, is the line (absolute line) in , and is the fixed elliptic involution of points of .

In the nonhomogeneous coordinates the similarity group has the form where and are real numbers [10]. In what follows the real numbers and will play the special role. In particular, for , (1) defines the group of isometries of the Galilean space . The Galilean scalar product can be written as where and . It leaves invariant the Galilean norm of the vector defined by A curve of the class in the Galilean space 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 are defined by The vectors , , are called the vectors of tangent, principal normal, and binormal of , respectively. For their derivatives, the following Frenet formula satisfies [10]:

3. Frenet Formulas in Equiform Geometry in

Let be a curve in the Galilean space . We define the equiform parameter of by where is the radius of curvature of the curve . Then, we have Let be a homothety with the center in the origin and the coefficient . If we put , then it follows where is the arc-length parameter of and the 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 . 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

Definition 1. The function defined by is called the equiform curvature of the curve .

Definition 2. The function defined by is called the equiform torsion of the curve .

Thus, the formula analogous to the Frenet formula in the equiform geometry of the Galilean space has the following form: 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 be a curve in the equiform geometry of the Galilean space . So the following statements are true (see for details [2, 10]).(i)If is an isotropic logarithmic spiral in , then and .(ii)If is an circular helix in , then and .(iii)If is an isotropic circle in , then and .

4. AW-Type Curves in Equiform Geometry in

Let be a curve in the Galilean space . The curve is called a Frenet curve of osculating order if its derivatives , , , are linearly dependent, and , , , are no longer linearly independent for all .

Proposition 4. Let be a curve in the equiform geometry of the Galilean space ; one has

Notation. Let us write

Remark 5. , , , are linearly dependent if and only if , , are linearly dependent.

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

Proposition 7. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space . Then is of type weak AW(2) if and only if

Corollary 8. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space .(i)If is an isotropic logarithmic spiral in , then is type weak AW(2) curve.(ii)If is a circular helix in , then is type weak AW(2) curve.(iii)If is an isotropic circle in , then is type weak AW(2) curve.

Proof. By using Remark 3 and Proposition 7, we have the results.

Proposition 9. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space . If is of type weak AW(3), then

Corollary 10. If is an isotropic circle in . Then is of type weak AW(3) curve.

Proof. It is obvious from Remark 3 and Proposition 9.

Corollary 11. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space . Then there is no isotropic logarithmic spiral or circular helix of type weak AW(3).

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 is nonzero constant and is nonzero constant, this is impossible, so is not isotropic logarithmic spiral or circular helix of type weak AW.

Definition 12. Curves (of osculating order 3) in the equiform geometry of the Galilean space are given as(i)of type AW(1) if they satisfy ,(ii)of type AW(2) if they satisfy (iii)of type AW(3) if they satisfy

Theorem 13. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space . Then is of type AW(1) if and only if

Proof. Since is a curve of type AW(1), we have . Then from (21), we have Furthermore, since and are linearly independent, we get The converse statement is trivial. Hence our theorem is proved.

Corollary 14. If is an isotropic circle in , then is of type AW(1) curve.

Proof. The proof is obvious from Remark 3 and Theorem 13.

Theorem 15. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space . Then is of type AW(2) if and only if

Proof. Suppose that is a Frenet curve of order 3; then from (20) and (21), we can write where , , , and are differentiable functions. Since and 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 be a curve (of osculating order 3) in the equiform geometry of the Galilean space .(i)If is an isotropic logarithmic spiral in , then is of type AW(2) curve.(ii)If is a circular helix in , then there is not circular helix of type AW(2).(iii)If is an isotropic circle in , then is of type AW(2) curve.

Theorem 17. Let be a curve (of osculating order 3) in the equiform geometry of the Galilean space . If is of type AW(3), then

Proof. Since curve is of type AW(3), (31) holds on . So substituting (19) and (21) into (31), we have (39). The converse statement is trivial. Hence our theorem is proved.

5. Bertrand Curves of AW()-Type in the Equiform Geometry of

This section characterizes the curvatures of AW-type Bertrand curves in the equiform geometry of the Galilean space . 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 .

Definition 18. A curve with is called a Bertrand curve if there exists a curve such that the principal normal lines of and at are equal. In this case is called a Bertrand mate of [11].

The curve is 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.

By definition, for a Bertrand pair , there exists a functional relation such that Let be a Bertrand mate in the equiform geometry of the Galilean space . Then we can write

Theorem 19. Let be Bertrand mate in the equiform geometry of the Galilean space . Then the function defined by relation (41) is a constant, and the equiform curvature .

Proof. Let and be the Frenet frames according to the equiform geometry of the Galilean space along and , respectively. Since is a Bertrand mate, from (41) we can write By differentiation of (42) with respect to , we obtain where and are parameters on and , respectively; . By using relation (17) we have Since is parallel to , then Since is parallel to , then Thus, from (45) and (46), we have Substituting (44) into (47), we obtain Thus, from (48), we get that is constant, and . Hence, the proof is completed.

Theorem 20. Let be Bertrand mate in the equiform geometry of the Galilean space . Then angle measurement of this curve between tangent vectors at corresponding points is constant.

Proof. If we show , then the proof is complete: Since is parallel to and , then Since is parallel to and , then Since is Bertrand mate in the equiform geometry of the Galilean space , from Theorem 19 we have So, substituting (50), (51), and (52) into (49), we have Hence, the proof is completed.

Theorem 21. Let be a curve in the equiform geometry of the Galilean space . Then is a Bertrand curve if and only if is a curve with constant torsion .

Proof. Deneote the Frenet frames of and by and , respectively. Let angle between and which is tangent vector of be . As is a linearly dependent set, we can write If we differentiate (54) and consider is a linearly dependent set, we can easily see that a constant function. Since and are Bertrand curve mates, we have If Differentiating (55) with respect to and with the help of Theorem 19, we get If we consider (54) and (56), we obtain Taking , we get This means that is constant. The converse statement is trivial. Hence, theorem is proved.

Corollary 22. Let be Bertrand curve in the equiform geometry of the Galilean space . Then is a circular helix in .

Proof. Since is Bertrand curve in the equiform geometry of the Galilean space , from Theorems 19 and 21 we have Thus, is a circular helix in . Hence, theorem is proved.

Theorem 23. Let be Bertrand curve in the equiform geometry of the Galilean space . Then is a weak AW(2)-type or AW(3)-type curve.

Proof. Now suppose that is Bertrand curve in the equiform geometry of the Galilean space . 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 be Bertrand curve in the equiform geometry of the Galilean space .Then is not a weak AW(3)-type or AW(2)-type curve.

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

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.