Some Characterizations of Osculating Curves in the Lightlike Cone

In this paper, we give the first kind and second kind osculating curves in the lightlike cone. In addition we characterize osculating curves in terms of their curvature functions.

As is well-known, semi-Riemannian manifold has played a key role in the development of general relativity.
Because the physical events space is represented by a semi-Riemannian manifold.A semi-Riemannian manifold has three causal types of submanifolds; spacelike, timelike and lightlike depending on the character of the induced metric on the tanget space [1].Due to the degeneracy of the metric, the study of lightlike submanifold have attracted the attention of many scientist.
Researchers use lightlike hypersurfaces in order to show a class of lightlike hypersurfaces came from the physically significant homegeneous spacetime manifolds of general relativity [2].
On the other hand in special relativity, a lightlike cone is the surface describing the temporal evolution of flash of light in Minkowski spacetime.Many studies have been made on curves in the lightlike cone by many mathmaticians.For example, in [3], Liu studied curves in the lightlike cone and in [4], Liu and Mong gave representation formulas of curves in a Two and Three Dimensional Lightlike Cone.Furthermore, in [5], Külahcı and others (the authors) studied AW(k)-type curves in the 3-dimensional null cone.
Another research area is the characterizations of osculating curves [6,7].In this paper we are concerned with osculating curves in the lightlike cone.In the following, we use the notations and copcepts from [3,4] unless otherwise stated.
Let E m q be the m-dimensional pseudo-Euclidean space with the metric respectively, a Riemannian metric, a degenerate quadratic form) on M , then M is called timelike (respectively, spacelike , degenerate) submanifold of E m q .Let c be a fixed point in E m q and r > 0 be a constant.The pseudo -Riemannian sphere is defined by the pseudo-Riemannian hyperbolic space is defined by the pseudo-Riemannian null cone (quadratic cone) is defined by It is well known that S n q (c, r) is a complete pseudo-Riemannian hypersurface of signature (n−q, q), q ≥ 1 in E n+1 q with constant sectional curvature r −2 ; H n q (c, r) is a complete pseudo-Riemannian hypersurface of signature (n − q, q), q ≥ 1 in E n+1 q+1 with constant sectional curvature −r −2 ; Q n q (c) is a degenerate hypersurface in E n+1 q .The spaces E n q , S n q (c, r), H n q (c, r) and Q n q (c) are called pseudo-Riemannian space form.The point c is called the center of S n q (c, r), H n q (c, r) and Q n q (c).When c = 0 and q = 1, we simply denote Q n 1 (0) by Q n and call it the lightlike cone(simply the light cone).
1 is called a Frenet curve, if for all t ∈ I, the vector fields x(t), x ′ (t), x ′′ (t), ..., x (n) (t), x (n+1) (t) are linearly independent and the vector fields x(t), x ′ (t), x ′′ (t), ..., x (n) (t), x (n+1) (t), x (n+2) (t) are linearly independent, and the vector fields dt n .Since x, x = 0 and x, dx = 0, dx(t) is spacelike.Then the induced arc lenght(or simply the arc lenght) s of the curve x(t) can be defined by If we take the arc lenght s of the curve x(t) as the parameter and denote ds is a spacelike unit tangent vector field of the curve x(s).Now we choose the vector y(s), the spacelike normal space V n−1 of the curve x(s) such that they satisfy the following conditions: x(s), y(s 1 , From the above explanations, we have the following remark.
Consequently the position vector of the osculating curve of the first and second kind satisfies the equations respectively some differantiable functions θ(s), µ(s) and γ(s).Since < x(s), y(s) >= 1, the equation (2.1.3)easily implies a contradiction.Hence we can say that there isn't a osculating curve of the first kind in the lightlike cone 3. Osculating Curves of the Second Kind in the Lightlike Cone Q 3 In this section, we characterize osculating curves of the second kind in the lightlike cone Q 3 ⊂ E 4  1 by using the components of their position vectors and the curvature functions.
Theorem 3.1.Let x(s) be a unit speed curve with the non-zero cone curvature functions κ(s), τ (s), λ(s) and if κ = 1 2 ( τ λ ) 2 , then x(s) is congruent to an osculating curve of the second kind if and only if Exposing the inner product y, x, α, β of the both side of (3.1.1),respectively, we have Using (3.1.2) and making necessary calculations, we get, for Thus by using (3.1.3),x(s) can be written as osculating curve of the second kind as follows: {A.e τ (s) Conserversely, let x(s) be unit speed curve with the cone curvatures κ(s), τ (s), λ(s) and assume that x(s) holds (3.1.4).From (2.1.4),we can write In particular, assume that the curvature functions κ(s) and τ (s), λ(s) of rectifying curve x(s) in Q 3 are constant and different from zero and let be κ = 1 2 ( τ λ ) 2 .Then equation (3.1.4)easily implies a contradiction.Hence we can say the following theorem. 2,so that the curvature functions are non-zero.But if any two of the curvature functions are constant, we can think the following statement: Theorem 3.3.Let x(s) be unit speed curve in the lightlike cone Q 3 , with curvatures κ(s), τ (s) and λ(s).Then x(s) is congruent to an osculating curve if and only if a

Theorem 3.2. There is not an osculating curve lying fully in the lightlike cone
By solving this equation, we can have Only if τ = 0, the distance function ρ is cons tan t and b) the second binormal component, the pirincipal normal component and the tangential component of the position vector of the curve is as follows respectively: .
In addition, if τ = 0, tangential component of the position vector of the curve is constant.c) the normal component of the position vector of the curve of the second kind d) the curvatures κ(s), τ (s) and λ(s) satisfy the following equality Proof: (a) Let x(s) be unit speed osculating curve of the second kind with curvatures κ(s), τ (s), λ(s).The position vector of x(s) holds (2.1.4).Also the functions θ(s), µ(s), γ(s) holds (3.1.3).
Differentiating x(s), x(s) = γ(s) with respect to s, we can have . Thus,

2
Curves in the lightlike cone Q n+1 40 3 Osculating Curves of the Second Kind in the Lightlike Cone Q 3 42 1.Introduction