Motions of Curves in the Pseudo-Galilean Space G 1 3

In mathematical modeling of many nonlinear events of the natural and the applied sciences such as dynamics of vortex filaments, motions of interfaces, shape control of robot arms, propagation of flame fronts, image processing, supercoiled DNAs, magnetic fluxes, deformation of membranes, and dynamics of proteins, the motions of space curves are being used. The evolutions of these nonlinear phenomena are described by the differential equations which characterize the motions of curves as a family. The motions of curves have been widely investigated by many authors in different geometries. In 1992 Nakayama and others explained that the close relation between the integrable evolution equations and the motions of curves is based on the equivalence of Frenet equations and the inverse scattering problem at zero eigenvalue [1], so that they identified the evolution equations that govern the 2D and 3D motions of the curves. They also studied the motions of the plane curves in which the curvature obeys the mKDV equation and its hierarchy [2]. Langer and Perline [3] gave the generalization of the motions of curves to n-dimensional Euclidean space. Many well-known integrable equations or their hierarchies related to the motions of space curves can be found in subsequent studies [4–11]. The subject of the curve flows in the pseudo-Galilean space, which is a real Cayley-Klein space with projective signature, is a virgin area to be searched. Inelastic flows of curves in the Galilean and the pseudo-Galilean spaces are studied at [12, 13]. Yoon [14] examined the inextensible flows of curves in the equiform geometry of the Galilean 3-space. Şahin [15] derived the intrinsic equations for a generalized relaxed elastic line on an oriented surface in the Galilean space. In this study we investigate the motions of curves in the pseudo-Galilean 3-space and in its equiform geometry without any constraints. The first section gives the main definitions and theorems of the pseudo-Galilean 3-space. Next we define the evolution of a one-parameter family of smooth admissible curves in the pseudo-Galilean 3-space andfind the flow equations of the curve evolution with use of the Frenet equations.Then we consider some particular cases where the flow of the intrinsic quantities κ and τ induces the inviscid Burgers’ equation. Finally we study the curve evolution in the equiform geometry of the pseudo-Galilean 3-space regarding the relations between the Frenet vectors of these spaces.


Introduction
In mathematical modeling of many nonlinear events of the natural and the applied sciences such as dynamics of vortex filaments, motions of interfaces, shape control of robot arms, propagation of flame fronts, image processing, supercoiled DNAs, magnetic fluxes, deformation of membranes, and dynamics of proteins, the motions of space curves are being used.The evolutions of these nonlinear phenomena are described by the differential equations which characterize the motions of curves as a family.
The motions of curves have been widely investigated by many authors in different geometries.In 1992 Nakayama and others explained that the close relation between the integrable evolution equations and the motions of curves is based on the equivalence of Frenet equations and the inverse scattering problem at zero eigenvalue [1], so that they identified the evolution equations that govern the 2D and 3D motions of the curves.They also studied the motions of the plane curves in which the curvature obeys the mKDV equation and its hierarchy [2].Langer and Perline [3] gave the generalization of the motions of curves to -dimensional Euclidean space.Many well-known integrable equations or their hierarchies related to the motions of space curves can be found in subsequent studies [4][5][6][7][8][9][10][11].
The subject of the curve flows in the pseudo-Galilean space, which is a real Cayley-Klein space with projective signature, is a virgin area to be searched.Inelastic flows of curves in the Galilean and the pseudo-Galilean spaces are studied at [12,13].Yoon [14] examined the inextensible flows of curves in the equiform geometry of the Galilean 3-space.S ¸ahin [15] derived the intrinsic equations for a generalized relaxed elastic line on an oriented surface in the Galilean space.
In this study we investigate the motions of curves in the pseudo-Galilean 3-space and in its equiform geometry without any constraints.The first section gives the main definitions and theorems of the pseudo-Galilean 3-space.Next we define the evolution of a one-parameter family of smooth admissible curves in the pseudo-Galilean 3-space and find the flow equations of the curve evolution with use of the Frenet equations.Then we consider some particular cases where the flow of the intrinsic quantities  and  induces the inviscid Burgers' equation.Finally we study the curve evolution in the equiform geometry of the pseudo-Galilean 3-space regarding the relations between the Frenet vectors of these spaces.
The scalar product of two vectors u = ( 1 ,  2 ,  3 ) and k This scalar product leaves invariant the pseudo-Galilean norm of the vector u = ( 1 ,  2 ,  3 ) defined by Let  be a spatial curve given first by where (), (), () ∈  3 .Then the curve () is said to be admissible if ẋ () ̸ = 0 [16].For an admissible curve  in G 1 3 parameterized by the arc length  =  with differential form  = , given as where (), () ∈  3 , the curvature () and the torsion () are defined by respectively.The pseudo-Galilean Frenet frame of the admissible curve () parameterized by the arc length has the form where t, n, and b are called the tangent vector, principal normal vector, and binormal vector fields of the curve , respectively.Here  = +1 or −1 is chosen by the criterion det(t, n, b) = 1.If n is a space-like or time-like vector, then the curve () given by ( 6) is time-like or space-like, respectively.
Then the Frenet equations of the curve () are given by where t, n, and b are mutually orthogonal vectors [17,18].

Motions of Curves in the Pseudo-Galilean Space G 1 3
In this section we study the curve evolution in the pseudo-Galilean 3-space by using the Frenet frame structure to obtain some related integrable equations.
Let us consider a one-parameter family of smooth admissible curves r(, ) in the pseudo-Galilean space G 1  3 where  denotes the time or the scale and  parameterizes each curve of the family.We assume that this family r(, ) evolves according to the flow equation where denote the length along the curve.The arc length parameter  is given by From (10) we can express the Frenet vectors and the intrinsic quantities as respectively.Now we will derive the flow equations for the Frenet frame {t, n, b}, the metric , the curvature , and the torsion  for the curve evolution r(, ) satisfying (12).Since  2 = ⟨r/, r/⟩ taking the derivatives of both sides and using (11) and (15) So the flow of the metric equals It is important to notice that the variables  and  are independent but  and  are not.As a consequence, we have We  (30) Therefore, we have the following theorem.
Theorem 1.Let r = r(, ) be a one-parameter family of smooth admissible curves in the pseudo-Galilean space G 1 3 .If r evolves according to (11), then, the Frenet frame {t, n, b} of r is not time dependent and the intrinsic quantities  and  of r satisfy the equations where  is the arc length parameter of r.
Remark 2. Burgers' equations describe various kinds of phenomena such as a mathematical model of turbulence and the approximate theory of flow through a shock wave traveling in viscous fluid.The inviscid Burgers' equation is a model for the nonlinear wave propagation, especially in fluid mechanics.It takes the form where (, ) is a solution of the equation.
From Remark 2, if we choose the curvature  =  or the torsion  =  in (30), then we have that the intrinsic quantities  and  evolve according to the inviscid Burgers' equation.So, we obtain the following corollary.Corollary 3. Let r = r(, ) be a curve evolution in the pseudo-Galilean space G 1  3 with the intrinsic quantities  and  given by (11).If one sets  =  or  = , then the intrinsic quantities  and  satisfy the inviscid Burgers' equation.

Inextensible Curve Flows in the Pseudo-Galilean Space.
In this section, we investigate some properties of the inextensible flows in the pseudo-Galilean space G 1 3 .
Definition 4. A curve evolution r(, ) and its flow r/ in the pseudo-Galilean space G According to Definition 4 and (11), in case the family of curves r(, ) is inextensible, from (18) we get for some single variable function .Therefore, we have the following corollary.
Corollary 5.The curve evolution r(, ) which is given by ( 11) is inextensible if and only if / = 0.
If we now restrict ourselves to the arc length parameterized admissible curves that undergo purely inextensible deformations, that is, (, ) = () = 1 and / = 0, then the local coordinate  corresponds to the arc length parameter .Thus the flow of the curve is expressed as and the flow of the Frenet frame {t, n, b} with the intrinsic quantities  and  is given by So, we get the following corollary.
Corollary 6.Let r = r(, ) be a curve evolution in the pseudo-Galilean space G 1 3 with its flow r/ given by (11).If the curve flow r(, ) is inextensible, then the Frenet vectors {t, n, b}, the curvature , and the torsion  are not time dependent.

Motions of Curves in
the Equiform Geometry of G Let  :  → G 3 be an admissible curve with the arc length parameter .We define the equiform invariant parameter of  by where  = 1/ is the radius of the curvature of the curve .It follows that We then have the new equiform invariant Frenet equations as where κ is called the equiform curvature and τ is called the equiform torsion of the curve  [12].These are related to the curvature  and torsion  by the equations Also the equiformly invariant Frenet vectors T, N, and B are related to the pseudo-Galilean Frenet vectors t, n, and b as The equiformly invariant arc length parameter of the curve evolution r(, ) can be defined as a function of  by So the operator / is equal to (/).The flow of the curve evolution r(, ) can be expressed in the form where , , and  are arbitrary functions.The preceding flow of r(, ) is related to flow (11) in the pseudo-Galilean space G 1 3 as r  = t + n + b, with  = ,  = , and  = .Then using the formulas in Section 3 we obtain the flow of the metric or The partial derivatives / and / do not commute in general while the partials / and / commute: Using ( 41) and (29) the flow equation of the equiformly invariant tangent vector field T is calculated as Similarly, we can write the flows of the equiformly invariant principal normal and binormal vector fields, the equiform curvature, and the equiform torsion, respectively, as follows: where  is the equiform invariant parameter and  is an arbitrary function.
Remark 9. Viscous Burgers' equation can be regarded as a one-dimensional analog of the Navier-Stokes equations which model the behavior of viscous fluids.It is given by the equation where (, ) is a solution of the equation.
we can compute the flow of the metric  as = const.)into a segment of length proportional to the original one with the coefficient of proportionality  23 .Other line elements (, , ), which lie on an isotropic plane ( = 0), are matched into proportional ones with the coefficient  12 .So, all line segments are matched into proportional ones with the same coefficient of proportionality if and only if  12 =  23 .Then we obtain a subgroup  7 ⊂  8 which preserves length ratio of segments and angles between planes and lines, respectively.This group is called the group of equiform transformations of the pseudo-Galilean space.