Normal Curves in 4-Dimensional Galilean Space G4

In this article, first, we give the definition of normal curves in 4-dimensional Galilean space G4. Second, we state the necessary condition for a curve of curvatures τ(s) and σ(s) to be a normal curve in 4-dimensional Galilean space G4. Finally, we give some characterizations of normal curves with constant curvatures in G4.


INTRODUCTION
Galilean geometry is one of the Cayley-Klein geometries whose motions are the Galilean transformations of classical kinematics [1]. The Galilean transformation group 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. The world line of a particle is just the curve in space-time which indicates its trajectory [2].
In Euclidean 3-space E 3 , there are three types of curves, namely, osculating, rectifying, and normal curves [3]. The osculating curve in E 3 is defined as a curve whose position vector always lies in its osculating plane, which is spanned by the tangent vector T and the normal vector N [3]. The rectifying curve in E 3 is defined as a curve whose position vector always lies in its rectifying plane, which is spanned by the tangent vector T and the binormal vector B. Many researchers have investigated rectifying curves in Euclidean, Lorentz-Minkowski, and Galilean space, as can be seen in [4][5][6]. Similarly, a normal curve in E 3 is defined as a curve whose position vector always lies in its normal plane, which is spanned by the normal vector N and the binormal vector B of the curve. Normal curves in n-dimensional Euclidean space was studied by Ozcan Bekats [7], framed normal curves in Euclidean space was studied by B.D. Yazici, S. O.Karakus, and M. Tosun [8], and normal curves on a smooth immersed surface was investigated by A.A. Shaikh, M.S. Lone, and P.R. Ghosh [9].
There are also many studies related to normal curves in non-Euclidean spaces, for example, normal curves and their characterizations in Lorentzian n-space was studied by Ozgür Boyacıoglu Kalkan [10] and to the classification of normal and osculating curves in 3-dimensional Sasakian space was studied by M. Kulahck, M. Bahatas, and A. Bilici [11].
In the present study, we considered a curve in Galilean 4-space G 4 whose position vector satisfies the equation α(s) = λ(s)N(s) + µ 1 (s)B 1 (s) + µ 2 (s)B 2 (s) for differentiable functions λ(s), µ 1 (s), and µ 2 (s). N(s), B 1 (s), and B 2 (s) are normal, first binormal, and second binormal vectors of the curve in Galilean space G 4 . In the first part of the study, the necessary condition for a curve to be a normal curve was obtained; then, we considered a special case when the curvatures are constant and got the position vector of the normal curve in G 4 . At the end of the study, it can be seen that the normal curve in G 4 lies on a sphere if τ σ = constant, where τ and σ are the second and the third curvatures of the normal curve α(s).
If the curve is parameterized by Galilean arc-length s, it is defined by α(s) = (s, y(s), z(s), w(s)).
The Frenet frame for the parameterized curve α(s) = (s, y(s), z(s), w(s)) in G 4 is denoted by the following vectors where T(s), N(s), B 1 (s) and B 2 (s) are the tangent, normal, the first binormal, and the second binormal vectors of α(s). k(s) and τ (s) are the first and second curvatures, which are given by the following equations The third curvature of the parameterized curve α(s) is denoted by σ (s) =< B < The derivatives of the Frenet equations are defined by [26].

NORMAL CURVES IN G 4
In the following section, we will define the normal curves in Galilean 4-dimensional space and prove that there are no congruent curves to the normal curve α(s); finally, we will provide some characterizations of the normal curves in G 4 .
By differentiating the second equation of the Equation (3.20) and substituting the first and the third Equation of (3.20), we obtain the following differential equation By solving the ordinary differential Equation (3.21), we obtain µ 1 (s) = c 1 cos τ 2 + σ 2 s + c 2 sin τ 2 + σ 2 s, where c 1 , c 2 , c 3 and c 4 are constants.
In the following corollary, we give some characterizations for the curve to be a normal curve.

Corollary 3.
Let α(s) be a normal curve in Galilean 4-space G 4 with non-zero constant curvatures τ and σ . The following statements are satisfied.
In the following theorem, we prove that, if α(s) is a normal curve, there are no curves which are congruent to α(s).

Theorem 2.
Let α(s) be a normal curve in Galilean space G 4 with non-zero constant curvatures τ and σ . Then, there are no curves which are congruent to α(s).