Osculating curves in 4-dimensional semi-Euclidean space with index 2

Abstract In this paper, we give the necessary and sufficient conditions for non-null curves with non-null normals in 4-dimensional Semi-Euclidian space with indeks 2 to be osculating curves. Also we give some examples of non-null osculating curves in E24 $\mathbb{E}_{2}^{4}$ .


Introduction
In the Euclidian space E 3 , it is well known that to each unit speed curve˛W I ! E 3 ;whose successive derivatives 0 .s/;˛0 0 .s/ and˛0 00 .s/ are linearly independent vectors, one can associate the moving orthonormal Frenet frame fT; N; Bg ; consisting of the tangent, the principal normal and the binormal vector field, respectively. The planes spanned by fT; N g ; fT; Bg and fN; Bg are respectively known as the osculating, rectifying and the normal plane. The rectifying curve in E 3 is defined in [2] as a curve whose position vector (with respect to some chosen origin) always lies in its rectifying plane. It is shown in [1] that there exists a simple relationship between the rectifying curves and centrodes, which play some important roles in mechanics and kinematics. Some characterizations of rectifying curves in Minkowski space-time are given in [6]. It is well-known that the position vector of a curve in E 3 always lies in its osculating plane B ? D Sp fT; N g if and only if its second curvature k 2 .s/ is equal to zero for each s ( [7]). The same property holds for timelike and spacelike curves (with non-null principal normal) in Minkowski 3-space. Osculating curves of first kind and second kind in Euclidian 4-space and Minkowski space time were studied byİlarslan and Nesovic in [4,5].
In the light of the papers in [4,5], in this paper we define the first kind and the second kind osculating curves in 4-dimensional semi-Euclidian space with index 2, by means of the orthogonal complements B ? 2 and B ? 1 of binormal vector fields B 2 and B 1 ; respectively. We restrict our investigation of the first kind and the second kind osculating curves to timelike curves as well as to spacelike curves whose Frenet frame fT; N; B 1 ; B 2 g contains only non-null vector fields. We characterize such osculating curves in terms of their curvature functions and find the necessary and the sufficient conditions for such curves to be the osculating curve.

Preliminaries
2 is the Euclidean 4-space E 4 equipped with indefinite flat metric given by where .x 1 ; x 2 ; x 3 ; x 4 / is a rectangular coordinate system of E 4 2 . Recall that an arbitrary vector v 2 E 4 2 n f0g can be spacelike, timelike or null(lightlike), if respectively holds g.v; v/ > 0 or g.v; v/ < 0 or g.v; v/ D 0. In particular the vector v D 0 is a spacelike. The norm of a vector v is given by jjvjj D p jg.v; v/j and two vectors v and w are said to be orthogonal if g.v; w/ D 0. An arbitrary curve˛.s/ in E 4 2 ; can locally be spacelike, timelike or null (lightlike), if all its velocity vectors˛0.s/ are respectively spacelike, timelike or null. A spacelike or timelike curve˛.s/ has unit speed, if g.˛0.s/;˛0.s// D˙1. Recall that the pseudosphere, the pseudohyperbolic space and lightcone are hyperquadrics in E 4 2 , respectively defined by S 3 where r > 0 is the radius and m 2 E 4 2 is the centre (or vertex) of hyperquadric ( [8]). Let fT; N; B 1 ; B 2 g be the non-null moving Frenet frame along a unit speed non-null curve˛in E 4 2 , consisting of the tangent, principal normal, first binormal and second binormal vector field, respectively. If˛is a non-null curve with non-null vector fields, then fT; N; B 1 ; B 2 g is an orthonormal frame. Accordingly, let us put (1) where the following conditions are satisfied: The curve˛lies fully in E 4 2 if k 3 .s/ ¤ 0 for each s: Let˛be a non-null curve with non-null normals in E 4 2 . We define that˛is the first or the second kind osculating curve in E 4 2 ; if its position vector with respect to some chosen origin always lies in the orthogonal complement B ? 2 or B ? 1 , respectively. The orthogonal complements B ? 1 and B ? 2 are non-degenerate hyperplanes of E 4 2 ; spanned by fT; N; B 2 g and fT; N; B 1 g; respectively.
Consequently, the position vector of the timelike and the spacelike first kind osculating curve (with non-null vector fields N and B 1 /; satisfies the equation and the position vector of the timelike and the spacelike second kind osculating curve (with non-null vector fields N and B 1 /; satisfies the equation˛.

Timelike and spacelike first kind osculating curves in E 4 2
In this section we show that a non-null curve with non-null normals is the first kind osculating curve if and only if it lies fully in non-degenerate hyperplane of E 4 2 : In relation to that we give the following theorem. Proof. First assume that˛is the first kind osculating curve in E 4 2 . Then its position vector satisfies relation (4). Differentiating relation (4) with respect to s and using Frenet equations (2), we easily find k 3 .s/ D 0.
Conversely, assume that the third curvature k 3 .s/ D 0 for each s. Let us decompose the position vector ofw ith respect to the orthonormal frame fT; N; B 1 ; B 2 g by Since k 3 .s/ D 0, relation (2) implies that B 2 is a constant vector and g.˛; B 2 / D constant. Substituting this in (6), we conclude that˛is congruent to the first kind osculating curve. This completes the proof of the theorem. In this section, we characterize non-null second kind osculating curves in E 4 2 with non-null vector fields N and B 1 in terms of their curvatures. Let˛D˛.s/ be the unit speed non-null second kind osculating curve in E 4 2 with non-null vector fields N and B 1 and non-zero curvatures k 1 .s/; k 2 .s/ and k 3 .s/: By definition, the position vector of the curve˛satisfies the equation (5), for some differentiable functions a.s/, b.s/ and c.s/: Differentiating equation (5) with respect to s and using the Frenet equations (2), we obtain It follows that a and therefore where c 0 2 R 0 : In this way functions a.s/, b.s/ and c.s/ are expressed in terms of curvature functions k 1 .s/; k 2 .s/ and k 3 .s/ of the curve˛: Moreover, by using the first equation in (7) and relation (8), we easily find that the curvatures k 1 .s/; k 2 .s/ and k 3 .s/ satisfy the equation In this way, we obtain the following theorem.
where 2 3 D˙1; c 0 2 0 : Proof. First assume that˛.s/ is congruent to the second kind osculating curve in E 4 2 . By using (8) and the first equation in relation (7), we easily find that relation (9) holds.
Conversely, assume that equation (9) is satisfied. Let us consider the vector X 2 E 4 2 given by N.s/ c 0 B 2 .s/ By using relations (2) and (9) we easily find X 0 .s/ D 0, which means that X is a constant vector. Consequently,į s congruent to the second kind osculating curve.
Recall that a unit speed non-null curve in E 4 2 is called a W-curve, if it has constant curvature functions (see [9]). The following theorem gives the characterization of a non-null W-curves in E 4 2 in terms of osculating curves. We easily obtain the Frenet vectors and curvatures as follows: Theorem 4.5. Let˛.s/ be a unit speed non-null curve with non-null vector fields N , B 1 and B 2 with curvatures k 1 .s/; k 2 .s/ and k 3 .s/ ¤ 0 lying fully in E 4 2 . If˛is the second kind osculating curve, then the following statements hold: i) The tangential and the principal normal component of the position vector˛are respectively given by ii) The second binornmal component of the position vector˛is a non-zero constant, i.e. h˛.s/; Conversely, if˛.s/is a unit speed non-null curve with non-null vector fields N , B 1 and B 2 , lying fully in E 4 2 and one of the statements (i) or (ii) hold, then˛is congruent to the second kind osculating curve.
Proof. First assume that˛is congruent to the second kind osculating curve in E 4 2 : By using relation (4) and (8), the position vector of˛can be written as Relation (12) easily implies that relations (10) and (11) are satisfied, which proves statements (i) and (ii). Conversely, assume that the statement (i) holds. By taking the derivative of the equations h˛.s/; N.s/i D ;with respect to s and using (2) we get h˛; B 1 i D 0; which means that˛is congruent to the second kind osculating curve.
If the statement (ii) holds, in a similar way we conclude that˛is congruent to the second kind osculating curve. where .s/ D R k 1 .s/ds, c 0 2 R 0 and c 1 ; c 2 2 R: Conversely, if˛.s/ is a unit speed non-null curve with non-null vector fields N; B 1 , lying fully in E 4 2 and one of the statements (i) or (ii) holds, then˛is congruent to the second kind osculating curve.
Proof. Let us first assume that˛.s/ is the unit speed non-null second kind osculating curve with non-null Frenet vector fields. From Theorem 4.1, the curvature functions of˛satisfy the equation Conversely, if relation (13) holds, by taking the derivative of relation (13) two times with respect to s, we obtain that relation (9) is satisfied. Hence Theorem 4.1 implies that˛is congruent to the first kind osculating curve. In a similar way the statement (ii) can be proved.