OSCULATING CURVES IN THE GALILEAN 4-SPACE

In this paper, we study osculating curves and equiform osculating curves in the 4-dimensional Galilean space G4 and characterize such curves in terms of their curvature functions and their equiform curvature functions, respectively. AMS Subject Classification: 53A35, 53C44


Introduction
In the Euclidean space E 3 , there exist three classes of curves, called rectifying, normal, and osculating curves, which satisfy the Cesaro's fixed point condition (see [5]).Namely, rectifying, normal, and osculating planes of such curves always contain a particular point.It is well-known that if all the normal or osculating planes of a curve in E 3 pass through a particular point, then the curve lies in a sphere or is a planner curve, respectively.It is also known that if all rectifying planes of a non-planar curve in E 3 pass through a particular point, then the ratio of torsion and curvature of such curve is a non-constant linear function (see [2]).Moreover, İlarslan and Nešoviċ (see [4]) gave some characterizations for osculating curves in E 3 , and they also constructed osculating curves in E 4 as a curve whose position vector all the time lies in the orthogonal complement of its first binormal vector field.As the results, they classified the osculating curves in terms of their curvature functions and gave the necessary and the sufficient conditions of osculating curves for arbitrary curves in E 4 .
On the other hand, the equiform geometry of the Cayley-Klein space is defined by requesting that similarity group of the space preserves angles between planes and lines.Cayley-Klein geometries are studied for many years.However, they recently have become interesting again because of their importance for other fields, like soliton theory (see [7]) A Galilean space is one of the Cayley-Klein spaces and it has been largely developed by Röschel (see [6]).A Galilean space may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane.The limit transition corresponds to the limit transition from the special relatively theory to classical mechanics.
In this paper, we study osculating curves and equiform osculating curves in the 4-dimensional Galilean space G 4 and characterize such curves in terms of their curvature functions and their equiform curvature functions, respectively.

Preliminaries
The 3-dimensional Galilean space G 3 is the Cayley-Klein space equipped with the projective metric of signature (0, 0, +, +).The absolute figure of the Galilean space consists of an ordered triple {w, f, I}, where w is the ideal (absolute) plane, f is the line (absolute line) in w and I is the fixed elliptic involution of points of f .
The study of mechanics of plane-parallel motions reduces to the study of a geometry of the 3-dimensional space with coordinates {x, y, t}, given by the motion formula.It is explained that the 4-dimensional Galilean geometry, which studies all properties invariant under motions of objects in the space, is even complex.
In addition, it is started that this geometry can be described more precisely as the study of those properties of the 4-dimensional space with coordinates which are invariant under the general Galilean transformations as follows [8]: The Galilean scalar product in G 4 can be written as where x = (x 1 , x 2 , x 3 , x 4 ) and y = (y 1 , y 2 , y 3 , y 4 ) are vectors in G 4 .It leaves invariant the Galilean norm of the vector x, defined by The Galilean cross product of x, y and z on G 4 is defined by , where e 2 = (0, 1, 0, 0), e 3 = (0, 0, 1, 0), and e 4 = (0, 0, 0, 1).A curve α : where s is a Galilean invariant arc-length of α.
On the other hand, the Frenet vectors of α(s) in G 4 are defined by where κ 1 (s), κ 2 (s), and κ 3 (s) are the first, second and third curvature functions, respectively, given by The vectors t, n, b 1 , b 2 are called the tangent, principal normal, first binormal, and second binormal vectors of α, respectively.If the curvature functions κ 1 , κ 2 and κ 3 of α are constants, then a curve α is called a W -curve.For their derivatives the following Frenet formula satisfies (cf.[3]) Now, we define osculating curves in the Galilean space G 4 .Let α be a unit speed curve in G 4 .If its position vector always lies in the orthogonal complement b ⊥ 1 or b ⊥ 2 of b 1 or b 2 , then a curve α is called an osculating curve in G 4 .Consequently, an osculating curve can be expressed as for some smooth functions λ(s), µ(s) and ν(s).
In this paper, we deal with an osculating curve generating by the tangent vector, the principal normal vector and the second binormal vector of the curve α in G 4 .

Osculating Curves in G 4
In this section, we characterize osculating curves in G 4 in terms of their curvatures.
Theorem 1.Let α be a unit speed curve in G 4 with non-zero curvatures κ 1 , κ 2 and κ 3 .Then α is an osculating curve if and only if where c, d = 0 are constant.
Proof.Let α = α(s) be a unit speed osculating curve and κ 1 (s), κ 2 (s), and κ 3 (s) be non-zero curvatures of α.Then, the position vector α(s) of the curve α satisfies the following equation: for some smooth functions λ(s), µ(s) and ν(s).Differentiating the above equation with respect to s with the Frenet formulae (2.2), we obtain From this, we get Using the second equation in (3.2) and (3.3), we obtain the curvatures κ 1 , κ 2 and κ 3 satisfying the equation Conversely, assume that the curvatures κ 1 (s), κ 2 (s) and κ 3 (s) of a unit speed curve α in G 4 satisfy equation (3.4).Let us consider the vector x ∈ G 4 , given by Then, we can easily find ẋ(s) = 0, that is, x is a constant vector.Thus, α is an osculating curve.
From (3.1), we have the following: Theorem 2. None of a unit speed W -curve with non-zero curvatures κ 1 , κ 2 and κ 3 in G 4 is an osculating curve .
Remark 3. The above theorem gives the opposite result from the case of the Euclidean space (see [4]).
Theorem 4. Let α be a unit speed osculating curve in G 4 with non-zero curvatures κ 1 , κ 2 and κ 3 .Then the following statements hold: Conversely, if α is a unit speed curve in G 4 with non-zero curvatures κ 1 , κ 2 and κ 3 with one of equations in (3.5), then α is an osculating curve .where s is the arc-length parameter of α and ρ the radius of curvature of this curve.Therefore, σ is an equiform invariant parameter of α (see [4]).
From now on, we define the Frenet formula of the curve α with respect to the equiform invariant parameter σ in G 4 .
The vector T = dα dσ is called a tangent vector of the curve α.From (2.1) and (2.2), we get We define the principal normal vector, the first binormal vector and the second binormal vector by Then, we easily show that {T, N, B 1 , B 2 } is an equiform invariant tetrahedron of the curve α.
On the other hand, the derivations of these vectors with respect to σ are given by where c is non-zero constant.In this way, the functions λ(σ), µ(σ), and ν(σ) are expressed in terms of the equiform curvatures K 1 , K 2 , and K 3 of the curve α.Moreover, by using the first equation in (5.1) and relation (5.2), we easily find that equiform curvature functions K 1 , K 2 , and K 3 satisfy the equation Conversely, assume that the equiform curvatures K 1 , K 2 , and K 3 of an arbitrary unit speed curve α in G 4 satisfy the euqation (5.3).Let us consider the vector X ∈ G 4 given by From the relations (4.4) and (5.3), we have X ′ (σ) = 0, which means that X is a constant vector.This implies that α is congruent to an osculating curve.In this process, we have the following theorem.
Theorem 6.Let α(σ) be a unit speed curve in equiform geometry in G 4 with non-zero equiform curvatures K 1 , K 2 , and K 3 .Then α(σ) is congruent to an osculating curve if and only if From (5.3), we obtain the following theorem.
Theorem 7.There are no osculating curves lying fully in G 4 with nonzero constant equiform curvatures K 2 and K 3 .Theorem 8. Let α(σ) be unit speed curve in equiform geometry of G 4 with non-zero equiform curvatures K 1 , K 2 , and K 3 .Then α(σ) is congruent to an osculating curve if Proof.Suppose that K 1 = constant = 0.By using the equation (5.3), we find differential equation