Second Binormal Motions of Inextensible Curves in 4-dimensional Galilean Space

In our study, we give the associated evolution equations for curvature and torsion as a system of partial differential equations. In addition, we study second binormal motions of inextensible curves in 4-dimensional Galilean space.


Introduction
Galilean 3-space G 3 is simply defined as a Klein geometry of the product space RxE 2 whose symmetry group is Galilean transformation group which has an important place in classical and modern physics. It is well known that the idea of world lines originates in physics and was pioneered by Einstein. In the recent time, the study of the motion of inextensible curves has been arisen in a number of diverse engineering applications and has many applications in computer vision, snake-like, robots. The flow of a curve is said to be inextensible if the arc length is preserved. Nowadays, many important and intensive studies are seen about inextensible curve flows in the different space. Physically, inextensible curve flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length, for example, or of a piece of paper carried by the wind, can be described by inextensible curve and surface flows. Such motions arise quite naturally in a wide range of a physical applications. For example, both Chirikjian and Burdick [1] used novel and efficient kinematic modeling techniques for "hyper-redundant" robots that to determining the time varying backbone curve behavior and Mochiyama et al.
Inextensible flows of curves are studied surfaces in R 3 by Kwon in [2]. In addition, many researchers have studied on inextensible flows such as [3], [4] and [8]. In [7] and [19], the authors studied inextensible flows in Minkowski space-time E 4 1 . In [10] is obtained a theoretical framework for controlling a manipulator with hyper degrees of freedom in which the shape control of hyper-redundant, or snake-like robots. In [15], the generalization of Bertrand curves in Galilean 4-space is introduced and the characterization of the generalized Bertrand curves is obtained. In [16], the author constructed Frenet-Serret frame of a curve in the Galilean 4-space and obtained the mentioned curve's Frenet-Serret equations. Inextensible curve and surface flows also arise in the context of many problems in computer vision [6], [12] and computer animation [9], and even structural mechanics [17]. Papers in [5], [11], [13], [14] and [18] are obtained some new characterizations of the inextensible curve flows curves. By drawing inspiration from them, in this paper, we consider the second binormal motions of inextensible curves in Galilean 4-space. We give the necessary and sufficient conditions to be inextensible flows and we find the evolution equations for the inextensible curve flows in 4-dimensional Galilean Space.

Geometric Preliminaries
The scalar product of two vectors The Galilean cross product in G 4 for the vectors u = (u 1 , where e i 1 ≤ i ≤ 4, are the standard basis vectors. The norm of vector U = (u 1 , u 2 , u 3 , u 4 ) is defined by Let α : I ⊂ R → G 4 , α(s) = (s, y(s), z(s), w(s)) be a curve parametrized by arclength s in G 4 . .The first vector of the Frenet-Serret frame,that is,the tangent vector of α is defined by Since t is a unit vector, we can express < t, t > = 1.
The vector function t gives us the rotation measurement of the curve α. The real valued function is called the first curvature of the curve α. We assume that κ(s) = 0 for all s ∈ l. Similar to space G 3 , the principal vector is defined by By the aid of the differentiation of the principal normal vector given in (2.7), define the second curvature function that is defined by This real valued function is called torsion of the curve α. The third vector field, namely, binormal vector field of the curve α,is defined by Thus the vector b(s) is perpendicular to both t and n. The fourth unit vector is defined by Here the coefficient µ is taken ±1 to make +1 determinant of the matrix [t, n, b, e] . The third curvature of the curve α by the Galilean inner product is defined by (2.8) Here, as well known, the set {t, n, b, e, κ, τ, σ } is called the Frenet-Serret apparatus of the curve α. We know that the vectors {t, n, b, e} are mutually orthogonal vectors satisfying  where √ g = γ( σ ,t) . Then the element of the arc-length is d s = g( u,t)d u, and the operator ∂ ∂ s satisfies the following Hence, the arclength of curveγ( s,t) is preserved. The second binormal motions of the curves can be expressed by the velocity vector field where {t, n, b, e}is the orthonormal Frenet Frame to the curve C t and v is the velocity vector in the direction of second binormal vector e and it is a function of curvature k( s,t),torsion τ( s,t). Theorem 3.1 The time evolution equations of the curvature and torsion for the inextensible timelike curveĈ t are given by: and Proof. Take theû derivative of (3.2), then Since the derivatives with respect to u and t commute, then Substituting from (3.7) and (3.5) into (3.6), then Take the u derivative of the second equation of (3.8), then Taking the t derivative of (3.9), then we have t ut = √ g(k t n + kn t ). (3.10) Since t ut = t tu , we obtain k t = vσ τ.
On the other hand, we write ,then we rewrite the equation of (3.11) as following Taking the u derivative of (3.12), then we have Since Taking the t derivative of (3.14), then we have Taking the u derivative of (3.19), then we have (3.20) Taking the t derivative of (3.21), then we have b ut = √ g(−τ t n − τn t + σ t e + σ e t ). where τ second curvature function, σ third curvature function, K = ψσ + φ s τ , ψ = −2v s σ − vσ K and φ = v ss − vσ 2 K . Hence, the theorem holds.
Theorem 3.2 The time evolution of the Serret Frenet frame for the curve can be given in matrix form: