GENERALIZED OSCULATING CURVES OF TYPE (n-3) IN THE n-DIMENSIONAL EUCLIDEAN SPACE

In this paper, we give a generalization of the osculating curves to the n-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the (n − 3)th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type (n−3)”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in En. In particular, we characterize the osculating curves in En in terms of their curvature functions. Finally, we show that the ratio of the (n− 1)th and (n − 2)th curvatures of the osculating curve is the solution of an (n − 2)th order linear nonhomogeneous differential equation.


Introduction
Curve theory is a popular research interest in classical differential geometry and osculating curves are a known example in this field. There are many studies on osculating curves in the Euclidean 3-space E 3 . The significant property of these curves is that the position vector of osculating curves always lie in their osculating planes. The osculating curve α : I → E 3 is defined by α (s) = λ (s) T (s) + µ (s) N (s) , for some differentiable functions λ and µ of s ∈ I ⊂ R, where T (s) is the tangent vector field and N (s) is the normal vector field. Similar curves are present in curve theory such as normal curves, where the position vector always lies in the normal plane, and the rectifying curves, where the position vector always lies in the GENERALIZED OSCULATING CURVES OF TYPE (N-3) 213 rectifying plane [1]. B. Y. Chen has studied rectifying curves in his paper "When does the position vector of a space curve always lie in its rectifying plane?". Since this study, normal, rectifying and osculating curves have been studied from different perspectives. Some of the studies in the literature have been listed below.
Chen and Dillen have studied rectifying curves in [2], where they established a relationship between rectifying curves and centrodes in mechanics. They also show that rectifying curves satisfy the equality case of a general inequality in their study. The characterization of the rectifying curve in Euclidean 4−space and Minkowski 3−space are given in [3], [4] and [5]. Cambie et al. investigated rectifying curves in an arbitrary dimensional Euclidean space using conditions on their curvature [6]. Additionally, there are some papers on spacelike, timelike and null normal curves in Minkowski space [7], [8]. Characterizations of an osculating curve in the 3-dimensional Euclidean space has been given in [9] and a specific osculating curve has been defined in the Euclidean 4-space. Normal, osculating and rectifying curves have been defined in the Euclidean and semi Euclidean space by using quaternion algebra in [10], [11], [16] and [25]. Bi-null curves of these types have also been analyzed in R 6 3 and R 5 2 in [12,13]. Several studies in the literature on the topic of interest of this study can be found in [14-18, 26, 27].
In this paper, using similar methods to those used in [6] and the definition of an osculating curve as stated in [9], we investigate the properties of a generalized form of osculating curves in the n−dimensional Euclidean space which are independent of the (n − 3)th binormal vector. We call this osculating curve "a generalized osculating curve of type (n − 3)". Firstly, basic concepts of curve theory in E n are given as preliminaries. Then, the characterization of the osculating curves is given in E n . The necessary and sufficient condition for a curve to be an osculating curve in the n-dimensional Euclidean space is also obtained. Additionally, using this necessary and sufficient condition, we show that if a curve is an osculating curve in the n-dimensional Euclidean space, its curvatures define a differential equation. Finally, we state the existence and uniqueness of the solution of this differential equation and propose a general form for the general solution of the equation.

Preliminaries
Basic concepts of curve theory in the n-dimensional Euclidean space E n are given in this section. Let α : I ⊂ R → E n , s ∈ I → α (s) be an arclength parameterized, n times continuously differentiable curve. The curve α is called a unit speed curve if ⟨α, α⟩ = 1, where ⟨ , ⟩ is the function that shows the standart inner product in the n-dimensional Euclidean space E n given by x i y i for each X = (x 1 , x 2 , ..., x n ) and Y = (y 1 , y 2 , ..., y n ) ∈ E n . The norm of X is given by ∥X∥ = ⟨X, X⟩. On the other hand, if ∥X∥ = 1, then X is an unit vector.

214Ö. BEKTAŞ, Z. BEKIRYAZICI
Also, if the curve α in E n is an arclength parameterized curve, then dα ds = 1. The Serret Frenet formulas for E n are given as the following equations (see [19]): where κ 1 , κ 2 , κ 3 ,. . . ,κ n−1 are the curvature functions of the curve and are positive. For more information on curve theory, the reader is advised to see the liteature [20][21][22].

Osculating curves of type (n-3) in the n−dimensional Euclidean space
In this section, generalizations of several fundamental definitions, theorems, and results to generalized osculating curves of type (n-3) in the n-dimensional Euclidean space are given. All of the mentions to osculating curves in our study refer to the generalized osculating curves of type (n-3) from this point.
be an arclength parameterized, n times continuously differentiable curve. In E n , a curve for which the position vector always lies in the orthogonal complement where ⟨ , ⟩ denotes the standard scalar product in E n . Thus B ⊥ n−3 (s) is a (n − 1)dimensional subspace of E n , spanned by the tangent, the principal normal, the first binormal, second binormal,...,(n − 4) th binormal and, (n − 2) th binormal vector fields T ,N ,B 1 (s),...,B n−4 (s),B n−2 (s) respectively. Therefore, the position vector of an osculating curve with respect to a specific origin is given as Theorem 1. Let α (s) be a unit speed curve in E n with nonzero curvatures. Then α (s) is congruent to a osculating curve in E n if and only if Proof. Let α be an arclength parameterized osculating curve in the n-dimensional Euclidean space. The derivative of (2) with respect to s for both sides of the equation is . Implementing the Serret Frenet formulas for the n-dimensional Euclidean space and rearranging the terms of the right hand side, we get Using the equality of both sides, we get the following expressions for the coefficients of T (s), N (s), B i (s) for i = 2, 3, . . . , n − 2: Starting from (9), we integrate these expressions with respect to s to obtain the coefficient functions

216Ö. BEKTAŞ, Z. BEKIRYAZICI
Similarly, the integrations of (7) and (8) yield and On the other hand, for i = n − 4, and n − 5, we get the following equations: − c κ n−5 (s) Similarly, Γ n−5,0 (s), Γ n−5,1 (s), Γ n−5,2 (s), and Γ n−5,3 (s) are defined as then we get Altogether, the following expression can be defined for the functions defined above: Thus we get the following coefficient functions for i = 1, and i = 2 and 218Ö. BEKTAŞ, Z. BEKIRYAZICI Substituting (16) and (17) into (3), we obtain the relations below for c ∈ R − 0. Conversely, consider an arbitrary unit speed curve in E n for which the curvature functions satisfy the relation (18). Then, we consider the the vector X ∈ E n defined by It can be seen that X ′ (s) = 0 through the relations (1) and (18). Thus, X is a constant vector. This implies that α is congruent to an osculating curve. Hence, the proof is complete. □ Theorem 2. Let α (s) be a unit speed osculating curve in E n with nonzero curvatures. Then the following hold: i) The tangential, the principal normal, the first, the second, ..., the i-th,..., the (n − 5)th, and (n − 4)th binormal components of the position vector of the curve are respectively given by . . .
ii) The (n − 2)th binormal component of the position vector of the curve is a nonzero constant. Conversely, if α (s) is a unit speed curve in E n with non-zero curvatures and one of the statements (i), (ii) holds, then α (s) is an osculating curve or is congruent to an osculating curve in E n .
Proof. By using the relations (2)  Let α (s) be a unit speed osculating curve in E n with nonzero curvatures. The differential equation where c ∈ R − {0} ., n > 4, i = 1, 2, . . . , n − 1 with the initial conditions for s 0 ∈ I ⊂ R has a unique solution on an open interval I ⊂ R if the functions satisfying the condition W κ n−1 κ n−2 1 , κ n−1 κ n−2 2 , . . . , κ n−1 κ n−2 n−2 There are several methods in the literature for analyzing the solutions of higher order linear differential equations. For instance, the variation of parameters method proposes a particular solution for the nonhomogeneous differential equation in the form of

Conclusion
In this paper, we have investigated some concepts of osculating curves, defined on 3-and 4-dimensional Euclidean spaces, on the n-dimensional Euclidean space. This generalization of osculating curves to E n has been called "generalized osculating curve of type (n − 3). A total of n − 2 generalizations of osculating curves to E n can be found by using the other binormal vectors. However, we have used the (n − 3)th binormal vector for the generalization since the relations between the curvatures provide meaningful results. Since the differential equation that gives the relation between curvature functions of the osculating curve in the n-dimensional Euclidean space is a higher order differential equation, we have invesitaged the existence and uniquness of a general solution for the initial value problem of order n − 2. The differential equation of order n-2 is a linear differential equation with variable coefficients. Several methods in the literature can be used for analyzing the particular solution of the higher order differential equation.

Author Contribution Statements
The authors jointly worked on the results and they read and approved the final manuscript.

Declaration of Competing Interests
The authors declare that they have no competing interest.