()-type slant helices for partially null and pseudo null curves in Minkowski space

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 Eⁿ and conclude that there are no slant helices with nonzero constant curvatures in the space E⁴. 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, 9, 10, 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 $E_{1}^{4}$ E_1^4 . In this study they called α curve as k-type slant helix if there exists on (non-zero) constant vector field $U \in E_{1}^{4}$ U \in E_1^4 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 E⁴ and we show that there do not exist (1, m) type slant helices in E⁴ [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 $E_{1}^{4}$ E_1^4 .

Preliminaries

Because of the indefiniteness of the Lorentzian metric g in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 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 $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 is called spacelike, timelike or lightlike if its velocity vectors al α^′(s) are spacelike, timelike and lightlike, respectively. In $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 , if u is a unit vector, we know that g(u,u) = ±1 and the norm of a vector u is given by $|| u || = \sqrt{| g (u, u) |}$ ||u|| = \sqrt {|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), B₁(s), B₂(s)}. Here T,N,B₁, B₂ 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,B₁ and B₂ 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 B₁ is lightlike [16, 17].

For partially null curves the second binormal B₂ is the only lightlike vector orthogonal to T and N such that g(B₁, B₂) = 1. The Frenet equations are given as follows (2.1) $[\begin{matrix} T^{'} \\ N^{'} \\ B_{1}^{'} \\ B_{2}^{'} \end{matrix}] = [\begin{matrix} 0 & κ & 0 & 0 \\ - κ & 0 & τ & 0 \\ 0 & 0 & σ & 0 \\ 0 & - τ & 0 & - σ \end{matrix}] [\begin{matrix} T \\ N \\ B_{1} \\ B_{2} \end{matrix}] [16, 17]$ \left[ {\matrix{{T'} \cr {N'} \cr {B'_1} \cr {B'_2} \cr}} \right] = \left[ {\matrix{0 & \kappa & 0 & 0 \cr {- \kappa} & 0 & \tau & 0 \cr 0 & 0 & \sigma & 0 \cr 0 & {- \tau} & 0 & {- \sigma} \cr}} \right]\left[ {\matrix{T \cr N \cr {{B_1}} \cr {{B_2}} \cr}} \right]\;\;\;[16,17] 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 B₁.

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, B₁ is a unit spacelike vector orthogonal to {T,N} and B₂ is the only lightlike vector orthogonal to T and B₁ such that g(N,B₂) = 1. The Frenet equations are given as follows (2.2) $[\begin{matrix} T^{'} \\ N^{'} \\ B_{1}^{'} \\ B_{2}^{'} \end{matrix}] = [\begin{matrix} 0 & κ & 0 & 0 \\ - 0 & 0 & τ & 0 \\ 0 & σ & 0 & - τ \\ - κ & - 0 & - σ & 0 \end{matrix}] [\begin{matrix} T \\ N \\ B_{1} \\ B_{2} \end{matrix}] [16, 17]$ \left[ {\matrix{{T'} \cr {N'} \cr {B'_1} \cr {B'_2} \cr}} \right] = \left[ {\matrix{0 & \kappa & 0 & 0 \cr {- 0} & 0 & \tau & 0 \cr 0 & \sigma & 0 & {- \tau} \cr {- \kappa} & {- 0} & {- \sigma} & 0 \cr}} \right]\left[ {\matrix{T \cr N \cr {{B_1}} \cr {{B_2}} \cr}} \right]\;\;\;[16,17]

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 στ ≠ O [2].

(k,m)-type slant helices for partially null curves in

𝔼_{1}^{4}

{\rm{\mathbb E}}_1^4

In this section, we will define (k,m) type partially null slant helices in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

From (2.1) we have that σ = O. We also suppose that κ, τ ≠ O.

Definition 3.1

Let α be a regular unit speed curve in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 with Frenet frame {V₁, V₂, V₃, V₄}. We call α is a (k,m) type slant helix if there exists a non-zero constant vector field $U \in 𝔼_{1}^{4}$ U \in {\rm{\mathbb E}}_1^4 satisfies 〈V_k, U〉 = α and 〈V_m, U〉 = b(a,b constant) for 1 ≤ k,m ≤ 4, k ≠ m. The constant vector U is on axis of (k,m)-type slant helix. We decompose U with respect to Frenet frame {T,N,B₁, B₂} as U = u₁T + u₂N + u₃B₁ + u₄B₂, where u_i = u_i(s) are differentiable functions of s. Here we denote V₁ = T,V₂ = N,V₃ = B₁, V₄ = B₂. From now on, in sake of easinesss, we will use these notations and assume that k_i ≠ O, (1 ≤ i ≤ 3).

Theorem 3.1

There are no (1, 2) type partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (1, 2) type slant helix. Then for a constant vector field U. g(T,U) = a is constant and g(N,U) = b is constant. Differentiating this equation and using Frenet equations, we obtain κg(N,U) = O means that U is orthogonal to N. Hence there are no (1, 2) type partially null slant helices in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Theorem 3.2

Let α be a (1, 3) partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then α is a general helix.

Proof

Assume that α is a (1, 3) partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we may write 〈T, U〉 = const = a〈B, U〉 = const = b. Also taking account Theorem 3.1 we decompose U as follows (3.1) $U = α T + b B_{1} + u_{1} B_{2}$ U = \alpha T + b{B_1} + {u_1}{B_2}(3.2) $0 = U^{'} = a (κ N) + b (σ B_{1}) + u_{1}^{'} B_{2} + u_{1} (- τ N - σ B_{2})$ 0 = U' = a(\kappa N) + b(\sigma {B_1}) + u_1'{B_2} + {u_1}(- \tau N - \sigma {B_2})(3.3) $0 = (a κ - u_{1} τ) N + (b σ) B_{1} + (u_{1}^{'} - u_{1} σ) B_{2}$ 0 = (a\kappa - {u_1}\tau)N + (b\sigma){B_1} + (u_1' - {u_1}\sigma){B_2}(3.4) $a κ - u_{1} τ = 0$ a\kappa - {u_1}\tau = 0(3.5) $- (u_{1}^{'} - u_{1} σ) = 0$ - (u_1' - {u_1}\sigma) = 0

From Definition 2.1 we have chosen σ = O which means that $u_{1}^{'} = 0$ {u'_1} = 0 hence u₁ = constant. From (3.3) we get $\frac{κ}{τ} = constant$ {\kappa \over \tau} = constant . Then this completes the proof.

Theorem 3.3

Let α be a (1, 4) partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then α is a general helix.

Proof

Assume that α is a (1, 4) type slant helix. Then we may write (3.6) $U = aT + u_{1} B_{1} + b B_{2}$ U = aT + {u_1}{B_1} + b{B_2}(3.7) $0 = U^{'} = a (κ N) + u_{1}^{'} B_{1} + u_{1} (σ B_{1}) + b (- τ N - σ b_{2})$ 0 = U' = a(\kappa N) + u_1'{B_1} + {u_1}(\sigma {B_1}) + b(- \tau N - \sigma {b_2})(3.8) $0 = (a κ - b τ) N + (u_{1}^{'} + u_{1} σ) B_{1} + b σ b_{2}$ 0 = (a\kappa - b\tau)N + (u_1' + {u_1}\sigma){B_1} + b\sigma {b_2}

Theorem 3.4

There are no (2, 3) partially null type slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (2, 3) partially null type slant helix in type slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we may write (3.9) $U = u_{1} T + aN + b B_{1} + u_{2} B_{2}$ U = {u_1}T + aN + b{B_1} + {u_2}{B_2}(3.10) $0 = U^{'} = u_{a} (κ N) + u_{1}^{'} T + a (- κ T + τ B_{1}) + b (σ B_{1}) + u_{2}^{'} B_{2} + u_{2} (- τ N - σ B_{2})$ 0 = U' = {u_a}(\kappa N) + u_1'T + a(- \kappa T + \tau {B_1}) + b(\sigma {B_1}) + u_2'{B_2} + {u_2}(- \tau N - \sigma {B_2})(3.11) $0 = U^{'} = (u_{1}^{'} - a κ) T + (u_{1} κ - u_{2} τ) N + (a τ + b σ) B_{1} + (u_{2}^{'} - u_{2} σ) B_{2}$ 0 = U' = (u_1' - a\kappa)T + ({u_1}\kappa - {u_2}\tau)N + (a\tau + b\sigma){B_1} + (u_2' - {u_2}\sigma){B_2}

Taking into account of σ = O we get a = O which is a contradiction. Hence there are no (2, 3) type slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Theorem 3.5

There are no (2, 4) type partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (2, 4) partially null type slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we may write (3.12) $U = u_{1} T + aN + u_{2} B_{1} + b B_{2}$ U = {u_1}T + aN + {u_2}{B_1} + b{B_2}(3.13) $\begin{array}{l} 0 = U^{'} = u_{1}^{'} T + u_{1} (κ N) + a (- κ T + τ B_{1}) + u_{2}^{'} B_{1} + u_{2} (σ B_{1}) + b (- τ N - σ B_{2}) \\ = (u_{1}^{'} 0 a κ) T + (u_{1} κ - b τ) N + (a τ + u_{2}^{'} + u_{2} σ) B_{1} + (- b σ) B_{2} \end{array}$ \matrix{{0 = U' = u_1'T + {u_1}(\kappa N) + a(- \kappa T + \tau {B_1}) + u_2'{B_1} + {u_2}(\sigma {B_1}) + b(- \tau N - \sigma {B_2})} \hfill \cr {\,\,\,\,\, = (u_1'0a\kappa)T + ({u_1}\kappa - b\tau)N + (a\tau + u_2' + {u_2}\sigma){B_1} + (- b\sigma){B_2}} \hfill \cr}

Noting that σ = O we get b = O which means that there are no (2, 4) type partially null type slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Theorem 3.6

There are no (3, 4) type partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (3, 4) type partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we may write (3.14) $U = u_{1} T + u_{2} N + a B_{1} + b B_{2}$ U = {u_1}T + {u_2}N + a{B_1} + b{B_2}(3.15) $0 = U^{'} = u_{1}^{'} T + u_{1} (κ N) + u_{2}^{'} N + u_{2} (- κ T + τ B_{1}) + a (σ B_{1}) + b (- τ N - σ B_{2})$ 0 = U' = u_1'T + {u_1}(\kappa N) + u_2'N + {u_2}(- \kappa T + \tau {B_1}) + a(\sigma {B_1}) + b(- \tau N - \sigma {B_2})(3.16) $0 = (u_{1}^{'} - u_{2} κ) T + (u_{1} κ + u_{2}^{'} - b τ) N + (u_{2} τ + a σ) B_{1} + (u_{2} τ + a σ) B_{1} + (- b σ) B_{2}$ 0 = (u_1' - {u_2}\kappa)T + ({u_1}\kappa + u_2' - b\tau)N + ({u_2}\tau + a\sigma){B_1} + ({u_2}\tau + a\sigma){B_1} + (- b\sigma){B_2}

In virtue of σ = O we get b = O which means that there are no (3, 4) type partially null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

(k,m)-type slant helices for pseudo null curves in

𝔼_{1}^{4}

{\rm{\mathbb E}}_1^4

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.

Theorem 4.1

There are no (1, 2) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (1, 2) type pseudo null slant helix. Then for a constant vector field U. g(T,U) = a is constant and g(N,U) = b is constant. Differentiating this equation and using Frenet equations, we obtain κg(N,U) = O means that U is orthogonal to N. Hence there are no (1, 2) type pseudo null slant helices in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Theorem 4.2

There are no (1, 3) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (1, 3) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we may write 〈T,U〉 = const = a〈B,U〉 = const = b. Also taking account Theorem 3.1 we decompose U as follows (4.1) $U = aT + b B_{1} + u_{1} B_{2}$ U = aT + b{B_1} + {u_1}{B_2}(4.2) $0 = U^{'} = a (N) + b (σ N - τ B_{2}) + u_{1}^{'} B_{2} + u_{1} (- T - σ B_{1})$ 0 = U' = a(N) + b(\sigma N - \tau {B_2}) + u_1'{B_2} + {u_1}(- T - \sigma {B_1})(4.3) $0 = (- u_{1}) T + (a + b σ) N + (- u_{1} σ) B_{1} + (u_{1}^{'} - b τ) B_{2}$ 0 = (- {u_1})T + (a + b\sigma)N + (- {u_1}\sigma){B_1} + (u_1' - b\tau){B_2}(4.4) $u_{1} = 0$ {u_1} = 0(4.5) $(a + b σ) = 0$ (a + b\sigma) = 0(4.6) $u_{1}^{'} + b τ = 0$ u_1' + b\tau = 0

From (3.25) and (3.27) we conclude b = O which is a contradiction. Hence there do not exist (1, 3) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Theorem 4.3

There are no (1, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that a is a (1, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we decompose U as follows (4.7) $U = aT + u_{1} B_{1} + b B_{2}$ U = aT + {u_1}{B_1} + b{B_2}(4.8) $0 = U^{'} = a (N) + b (- T - σ B_{1}) + u_{1}^{'} B_{1} + u_{1} (σ N - τ B_{2})$ 0 = U' = a(N) + b(- T - \sigma {B_1}) + u_1'{B_1} + {u_1}(\sigma N - \tau {B_2})(4.9) $0 = (- b) T + (a + u_{1} σ) N + (u_{1}^{'} - b σ) B_{1} + (- u_{1} τ) B_{2}$ 0 = (- b)T + (a + {u_1}\sigma)N + (u_1' - b\sigma){B_1} + (- {u_1}\tau){B_2}

Using (3.25) we get b = O which is a contradiction. Hence there do not exist (1, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

From the theorem proved above we conclude the following corollary.

Corollary 4.1

There do not exist (1, k) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 for 2 ≤ k ≤ 4.

Theorem 4.4

Let α be a (2, 3) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 if and only $- (a σ \frac{\int τ ds}{τ}) - b \int τ ds = 0$ - \left({a\sigma {{\int \tau ds} \over \tau}} \right) - b\int \tau ds = 0

Proof

Assume that α is a (2, 3) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we decompose U as follows (4.10) $U = u_{1} T + aN + b B_{1} + u_{2} B_{2}$ U = {u_1}T + aN + b{B_1} + {u_2}{B_2}(4.11) $0 = U^{'} = u_{1}^{'} T + u_{1} (N) + a (τ B_{1}) + b (σ N - τ B_{2}) + u_{2}^{'} B_{2} + u_{2} (- T - σ B_{1})$ 0 = U' = u_1'T + {u_1}(N) + a(\tau {B_1}) + b(\sigma N - \tau {B_2}) + u_2'{B_2} + {u_2}(- T - \sigma {B_1})(4.12) $0 = (u_{1}^{'} - u_{2}) T + (u_{1} + b σ) N + (a τ - u_{2} σ) B_{1} + (u_{2}^{'} - b τ) B_{2}$ 0 = (u_1' - {u_2})T + ({u_1} + b\sigma)N + (a\tau - {u_2}\sigma){B_1} + (u_2' - b\tau){B_2}

Hence we get (4.13) $u_{1}^{'} - u_{2} = 0$ u_1' - {u_2} = 0(4.14) $u_{1} + b σ = 0$ {u_1} + b\sigma = 0(4.15) $a τ - u_{2} σ = 0$ a\tau - {u_2}\sigma = 0(4.16) $(u_{2}^{'} - b τ) = 0$ (u_2' - b\tau) = 0 Using (4.17) we get u₂ = b ∫ τds and taking into account of (4.15) with (4.14) we get $u_{1} = - a σ \frac{\int τ ds}{τ}$ {u_1} = - a\sigma {{\int \tau ds} \over \tau} . Considering these facts in (4.13) we conclude the desired proof.

Corollary 4.2

An axis of a (2, 3) type pseudo null slant helix is the vector given by $D = [σ \frac{\int τ ds}{τ}] T + N + \frac{\int τ ds}{τ} B_{1} + [b \int τ ds] B_{2}$ D = \left[ {\sigma {{\int \tau ds} \over \tau}} \right]T + N + {{\int \tau ds} \over \tau}{B_1} + \left[ {b\int \tau ds} \right]{B_2}

Theorem 4.5

There are no (2, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (2, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we decompose U as follows (4.17) $U = u_{1} T + aN + u_{2} B_{1} + b B_{2}$ U = {u_1}T + aN + {u_2}{B_1} + b{B_2}(4.18) $0 = U^{'} = u_{1}^{'} T + u_{1} (N) + a (τ B_{1}) + b (- T - σ B_{1}) + u_{2}^{'} B + u_{2} (σ N - τ B_{2})$ 0 = U' = {u'_1}T + {u_1}(N) + a(\tau {B_1}) + b(- T - \sigma {B_1}) + u_2'B + {u_2}(\sigma N - \tau {B_2})(4.19) $0 = (u_{1}^{'} - b) T + (u_{1} + u_{2} σ) N + (u_{2}^{'} + a τ - b σ) B_{1} + (- u_{2} τ) B_{2}$ 0 = (u_1' - b)T + ({u_1} + {u_2}\sigma)N + (u_2' + a\tau - b\sigma){B_1} + (- {u_2}\tau){B_2}(4.20) $u_{1}^{'} - b = 0$ u_1' - b = 0(4.21) $u_{1} + u_{2} σ = 0$ {u_1} + {u_2}\sigma = 0(4.22) $u_{2}^{'} + a τ - b σ = 0$ u_2' + a\tau - b\sigma = 0(4.23) $- u_{2} τ = 0$ - {u_2}\tau = 0

From (4.23) we conclude that u₂ = O and considering this fact in (4.20) we also see that u₁ = O which means a contradiction. Hence there do not exist (2, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Theorem 4.6

There are no (3, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 .

Proof

Assume that α is a (3, 4) type pseudo null slant helix in $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4 . Then we decompose U as follows (4.24) $U = u_{1} T + u_{2} N + a B_{1} + b B_{2}$ U = {u_1}T + {u_2}N + a{B_1} + b{B_2}(4.25) $0 = U^{'} = u_{1}^{'} T + u_{1} (N) + u_{2}^{'} N + u_{2} (τ B_{1}) + a (σ N - τ B_{2}) + b (- T - σ B_{1})$ 0 = U' = u_1'T + {u_1}(N) + u_2'N + {u_2}(\tau {B_1}) + a(\sigma N - \tau {B_2}) + b(- T - \sigma {B_1})(4.26) $9 = (u_{1}^{'} - b) T + (u_{1} + u_{2}^{'} + a σ) N + (u_{2} τ - b σ) B_{1} + (- a τ) B_{2}$ 9 = (u_1' - b)T + ({u_1} + u_2' + a\sigma)N + ({u_2}\tau - b\sigma){B_1} + (- a\tau){B_2}(4.27) $u_{1}^{'} - b = 0$ u_1' - b = 0(4.28) $u_{1} + u_{2}^{'} + a σ = 0$ {u_1} + u_2' + a\sigma = 0(4.29) $u_{2} τ - b σ = 0$ {u_2}\tau - b\sigma = 0(4.30) $- a τ = 0$ - a\tau = 0

Taking into account of (3.55) we see that a = O and this is a contradiction.

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

Published Online: May 29, 2020

Page range: 515 - 520

Received: Apr 22, 2019

Accepted: Jun 13, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00048

KeywordsDifference equations, rational difference equations

© 2019 Mehmet Bektaş et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

(k,m)-type slant helices for partially null and pseudo null curves in Minkowski space $𝔼_{1}^{4}$ {\rm{\mathbb E}}_1^4

Keywords
Difference equations, rational difference equations