(k,m)-type slant helices for partially null and pseudo null curves in Minkowski space 𝔼14{\rm{\mathbb E}}_1^4

In this study we define the notion of (k,m)-type slant helices in Minkowski 4-space and express some characterizations for partially and pseudo null curves in E1.


Introduction
The curve theory has been one of the most studied research area because of having many application area from geometry to the various branch of science. Especially the characterizations on the curvature and torsion play important role to define special curve types such as so-called helices. The curves of this type have drawn great attention in science. Helices appear naturally in structures of DNA, nanosprings. They are also widely used in engineering and architecture.The concept of slant helix defined by Izumiya and Takeuchi [6] based works have been studied in various spaces. For instance in [1] authors extended slant helix concept to E n and conclude that there are no slant helices with non-zero constant curvatures in the space E 4 . The subject is also considered in 3−, 4−, and n−dimensional Eucliedan spaces, respectively in [7,10,12]. Moreover different properties of helices are also considered in [8-11, 13, 18]. On the other hand in A.T. Ali, R. Lopez and M. Turgut extended this study to the k-type slant helix in E 4 1 . In this study they called α curve as k-type slant helix if there exists on (non-zero) constant vector field U ∈ E 4 1 such that V k+d ,U = const, for 0 ≤ k ≤ 3. Here V k+1 shows the Frenet vectors of this curve [2].
One may easily conclude that O-type slant helices are general helices and 1-type slant helices correspond just slant helices. k-type slant helices for partially null and pseudo null curves are also studied. In accordance with above studies, the authors introduced (k, m)-type slant helices in E 4 and we show that there do not exist (1, m) type slant helices in E 4 [15].
In the present work, we define the notion of (k, m)-type slant helices in Minkowski 4-space and express some characterizations for partially and pseudo null curves in E 4 1 .

Preliminaries
Because of the indefiniteness of the Lorentzian metric g in E 4 1 vector v in this space can have one of three causal characters called spacelike (g(u, u) > O or u = O), timelike (g(u, u) < O) and lightlike (null) (g(u, u) = O, u = O), respectively. In accordance with the metric, a curve in E 4 1 is called spacelike, timelike or lightlike if its velocity vectors al α (s) are spacelike, timelike and lightlike, respectively. In E 4 1 , if u is a unit vector, we know that g(u, u) = ±1 and the norm of a vector u is given by ||u|| = |g(u, u)|. In addition a spacelike or timelike curve is said to be arclength parametrized if a α (s) is a unit vector for any s [2].
Suppose that α = α(s) is a spacelike curve with its arclength parameter. The Frenet frame along the curve a can be denoted by {T (s), N(s), B 1 (s), B 2 (s)}. Here T, N, B 1 , B 2 are called tangent, principal normal, the first binormal and the second binormal vector fileds of the curve α, respectively. Because of the indefiniteness of the metric g, the vectors N, B 1 and B 2 have different causal characters. In this study we will assume that T is spacelike, since α is a spacelike curve. Definition 2.1 A spacelike curve called partially null curve if N is spacelike and B 1 is lightlike [16,17].
For partially null curves the second binormal B 2 is the only lightlike vector orthogonal to T and N such that g(B 1 , B 2 ) = 1. The Frenet equations are given as follows where κ, τ and σ are first, second and third curvature of the curve α, respectively. Note that after a null rotation of the ambient space the third curvature σ can be chosen as zero, and τ is determined up to a constant which means that any partially null curve lies in a three dimensional lightlike subspace orthogonal to B 1 . Definition 2.2. A spacelike curve called pesudo null curve if α (s) is a lightlike vector for all s where the normal vector is N = T [16,17].
For the case N is lightlike the curve a lies in the lightlike plane which we omit this trivial case. For the other cases, B 1 is a unit spacelike vector orthogonal to {T, N} and B 2 is the only lightlike vector orthogonal to T and B 1 such that g(N, B 2 ) = 1. The Frenet equations are given as follows In this case the first curvature κ, can take only 0 and 1 values. As is well known the curve is a straight line if curvature vanishes. We will focus on the cases κ = 1 and σ τ = O [2].
From now on, in sake of easinesss, we will use these notations and assume that k i = O, (1 ≤ i ≤ 3).
In virtue of σ = O we get b = O which means that there are no (3,4) type partially null slant helix in E 4 1 .
4 (k, m)-type slant helices for pseudo null curves in E 4 1 In this part, we will focus on the (k, m)-type pesudo null slant helices. Recall that we assume κ = 1 and σ , τ = O.