Special Curves According to Bishop Frame in Minkowski 3-Space

The ability to “ride” along a three-dimensional space curve and illustrate the properties of the curve, such as curvature and torsion, would be a great asset to Mathematicians. The classic Serret-Frenet frame provides such ability. The tangent normal, and binormal vector fields are called the Frenet–Serret frame or T NB frame. But, the curve might not be continuous at some points, which is undefined when the second derivative of the curve vanishes [9]. In 1975, Richard Lawrence Bishop first introduced the parallel frame as a new frame which is well defined even if the curve has vanishing second derivative, then the parallel frame came to be called the Bishop frame [8, 9, 10]. Bishop frame contains the tangential vector field T and two normal vector fields N₁ and N₂. The Bishop frame may have applications in the area of Biology and Computer Graphics. For example, it may be possible to compute information about the shape of sequences of DNA using a curve defined by the Bishop frame. It also provides a new way to control virtual cameras in computer animation [12]. Some applications of the Bishop frames in Minkowski spaces can be found in [3, 4].

In differential geometry, a general helix or a curve of constant slope in Euclidean 3-space E³ is defined in such a way that the tangent makes a constant angle with a fixed direction. A classical result stated by M. A. Lancret in 1802 and first proved by B. de Saint Venant in 1845, [5, 7, 16, 18]. For helical structures in nature, helices arise in nano-springs, carbon nano-tubes, DNA double and collagen triple helix, lipid bilayers, bacterial flagella in salmonella and escherichia coli, aerial hyphae in actinomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, helical staircases and seashells [1, 2, 11]. Helical structures are used in fractal geometry, for instance, hyper-helices.

In [7], a slant helix in Euclidean 3-space was defined by the property that the principal normal makes a constant angle with a fixed direction. Moreover, Izumiya and Takeuchi showed that α is a slant helix in E³ if and only if the geodesic curvature of the principal normal of a space curve α is a constant function [15, 17, 19].

Preliminaries

Definition 1

The Minkowski 3-space $E_{1}^{3}$ [E_1^3 is the real vector space E³which is endowed with the standard indefinite flat metric 〈.,.〉 defined by(2.1) $〈 u, v 〉 = - u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3},$ [\left\langle {u,v} \right\rangle = - u_1 v_1 + u_2 v_2 + u_3 v_3 ,\for any two vectors u = (u₁,u₂,u₃) and v = (v₁,v₂,v₃) in $E_{1}^{3}$ [E_1^3 . Since〈.,.〉 is an indefinite metric, an arbitrary vector u ∈ $E_{1}^{3}$ [E_1^3 {0} can have one of three causal characters:i)

it can be space-like, if 〈u,u〉₁> 0,

ii)

time-like, if 〈u,u〉₁< 0 or

iii)

light-like or isotropic or null vector, if 〈u,u〉₁ = 0 but u ≠0.

In particular, the norm (length) of a non-lightlike vector u ∈ $E_{1}^{3}$ [E_1^3 is given by $‖ u ‖ = \sqrt{| 〈 u, u 〉 |} ˙$ [\left\| u \right\| = \sqrt {\left| {\left\langle {u,u} \right\rangle } \right|} .

Given a regular curve $α : I \to E_{1}^{3}$ [\alpha :I \to E_1^3 \can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α^′(t) satisfy 〈α^′(t),α^′(t)〉₁> 0, 〈α^′(t),α^′(t)〉₁< 0 or 〈α^′(t),α^′(t)〉₁= 0 respectively, at any tɛI, where $α^{'} (t) = \frac{d α}{dt}$ [\alpha ^{^' } (t) = \frac{{d\alpha }}{{dt}} .

Definition 2

A spacelike curve $α : I \to E_{1}^{3}$ [\alpha :I \to E_1^3 \is called a pseudo null curve, if its principal normal vector field N and binormal vector filed B are null vector fields satisfying the condition 〈N,B〉 = 1. The Frenet formulae of a non-geodesic pseudo null curve α has the form(2.2) $[\begin{matrix} T^{'} \\ N^{'} \\ B^{'} \end{matrix}] = [\begin{matrix} 0 & k & 0 \\ 0 & - τ & 0 \\ - k & 0 & - τ \end{matrix}] [\begin{matrix} T \\ N \\ B \end{matrix}],$ [\left[ {\begin{array}{*{20}c} {T^' } \\ {N^' } \\ {B^' } \\\end{array}} \right] = \left[ {\begin{array}{*{20}c} 0 & k & 0 \\ 0 & { - \tau } & 0 \\ { - k} & 0 & { - \tau } \\\end{array}} \right]\left[ {\begin{array}{*{20}c} T \\ N \\ B \\\end{array}} \right]{\rm{,}}where the curvature k(s) = 1 and the torsion τ(s) is an arbitrary function in arc-length parameter s of α. The Frenet’s frame vectors satisfy the equations(2.3) $〈 N, B 〉 = 1, 〈 T, N 〉 = 〈 T, B 〉 = 0, 〈 T, T 〉 = 1, 〈 N, N 〉 = 〈 B, B 〉 = 0$ [\left\langle {N,B} \right\rangle = 1{\rm{,}}\left\langle {T,N} \right\rangle = \left\langle {T,B} \right\rangle = 0{\rm{,}}\,\left\langle {T,T} \right\rangle = 1{\rm{,}}\left\langle {N,N} \right\rangle = \left\langle {B,B} \right\rangle = 0and(2.4) $T \times N = N, N \times B = T, B \times T = B .$ [T \times N = N{\rm{,}}N \times B = T{\rm{,}}B \times T = B{\rm{.}}

The frame {T,N,B} is positively oriented, if det (T,N,B) = [T,N,B] = 1 [10].

2.1

The Bishop Frame

The Bishop frame or relatively parallel adapted frame {T,N₁,N₂} of a regular curve in Euclidean 3-space contains a tangential vector field T and two normal vector fields N₁ and N₂, which can be obtained by rotating the Frenet vectors N and B in the normal plane T^⊥ of the curve, in such a way that they become relatively parallel. This means that their derivatives N^′₁ and N^′₂ with respect to the arc-length parameter s of the curve are collinear with the tangential vector field T [10].

Remark 1

We can also define N₁and N₂to be relatively parallel if the normal component T^⊥₁ = span{N₁,N₂} of their derivatives N^′₁and N^′₂is zero, which implies that the mentioned derivatives are collinear with T₁.

2.2

The Bishop frame of a pseudo null curve in

E_{1}^{3}

[E_1^3

The Bishop frame {T₁,N₁,N₂} of a pseudo null curve in $E_{1}^{3}$ [E_1^3 is positively oriented pseudo orthonormal frame consisting of the tangential vector field T₁ and two relatively parallel lightlike normal vector fields N₁ and N₂. Bishop vector N₁ (of the first Bishop frame) can be obtained by applying the hyperbolic rotation to the principal normal vector N, while the normal Bishop vector N₂ (of the first Bishop frame) can be obtained by applying the composition of three rotations about two lightlike and one spacelike axis to the binormal vector B [10].

Theorem 1

Let α be a pseudo null curve in $E_{1}^{3}$ [E_1^3 parameterized by arc-length parameter s with the curvature k(s) = 1 and the torsion τ(s). Then the Bishop frame {T₁,N₁,N₂} and the Frenet frame {T,N,B} of α are related by(2.5) $[\begin{matrix} T_{1} \\ N_{1} \\ N_{2} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & \frac{1}{k_{2}} & 0 \\ 0 & 0 & k_{2} \end{matrix}] [\begin{matrix} T \\ N \\ B \end{matrix}],$ [\left[ {\begin{array}{*{20}c} {T_1 } \\ {N_1 } \\ {N_2 } \\\end{array}} \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\frac{1}{{k_2 }}} & 0 \\ 0 & 0 & {k_2 } \\\end{array}} \right]\left[ {\begin{array}{*{20}c} T \\ N \\ B \\\end{array}} \right]{\rm{,}}and the Frenet equations of α according to the Bishop frame read(2.6) $[\begin{matrix} T_{1}^{'} \\ N_{1}^{'} \\ N_{2}^{'} \end{matrix}] = [\begin{matrix} 0 & k_{2} & k_{1} \\ - k_{1} & 0 & 0 \\ - k_{2} & 0 & 0 \end{matrix}] [\begin{matrix} T_{1} \\ N_{1} \\ N_{2} \end{matrix}],$ [\left[ {\begin{array}{*{20}c} {T_1^' } \\ {N_1^' } \\ {N_2^' } \\\end{array}} \right] = \left[ {\begin{array}{*{20}c} 0 & {k_2 } & {k_1 } \\ { - k_1 } & 0 & 0 \\ { - k_2 } & 0 & 0 \\\end{array}} \right]\left[ {\begin{array}{*{20}c} {T_1 } \\ {N_1 } \\ {N_2 } \\\end{array}} \right]{\rm{,}}where k₁(s) = 0 and k₂(s) = c₀e^∫τ(s)ds, $c_{0} \in R_{0}^{+}$ [c_0 \in R_0 + .

Which satisfies the conditions $〈 T_{1}, T_{1} 〉 = 1, 〈 N_{1}, N_{1} 〉 = 〈 N_{2}, N_{2} 〉 = 0, 〈 N_{1}, N_{2} 〉 = 1, 〈 T_{1}, N_{2} 〉 = 〈 T_{1}, N_{1} 〉 = 0 [10] ˙$ [\left\langle {T_1 ,T_1 } \right\rangle = 1{\rm{,}}\left\langle {N_1 ,N_1 } \right\rangle = \left\langle {N_2 ,N_2 } \right\rangle = 0{\rm{, }}\left\langle {N_1 ,N_2 } \right\rangle = 1{\rm{,}}\left\langle {T_1 ,N_2 } \right\rangle = \left\langle {T_1 ,N_1 } \right\rangle = 0\left[ {10} \right]{\rm{.}}

Curves of AW(k)- type

Proposition 1

Let α be a Frenet curve of osculating order 3 in $E_{1}^{3}$ [E_1^3 , by using the Bishop frame of pseudo null curve(2.6), then we have $α^{'} (s) = T_{1} (s), α^{''} (s) = k_{2} N_{1} + k_{1} N_{2}, α^{'''} (s) = - 2 k_{1} k_{2} T_{1} + k_{2}^{'} N_{1} + k_{1}^{'} N_{2}, α^{''''} (s) = (- 3 k_{1}^{'} k_{2} - 3 k_{1} k_{2}^{'}) T_{1} + (k_{2}^{''} - 2 k_{1} k_{2}^{2}) N_{1} + (k_{1}^{''} - 2 k_{1}^{2} k_{2}) N_{2} ˙$ [\alpha ^{^' } \left( s \right) = T_1 \left( s \right){\rm{, }}\alpha ^{^{''} } \left( s \right) = k_2 N_1 + k_1 N_2 {\rm{, }}\alpha ^{^{'''} } \left( s \right) = - 2k_1 k_2 T_1 + k_2^{^' } N_1 + k_1^{^' } N_2 {\rm{,}}\,\alpha ^{^{''''} } \left( s \right) = \left( { - 3k_1^{^' } k_2 - 3k_1 k_2^{^' } } \right)T_1 + \left( {k_2^{^{''} } - 2k_1 k_2^2 } \right)N_1 + \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)N_2 {\rm{.}}

Notation 1

Let us write(3.1) $M_{1} (s) = k_{2} N_{1} + k_{1} N_{2},$ [M_1 \left( s \right) = k_2 N_1 + k_1 N_2 {\rm{,}}(3.2) $M_{2} (s) = k_{2}^{'} N_{1} + k_{1}^{'} N_{2},$ [M_2 \left( s \right) = k_2^{^' } N_1 + k_1^{^' } N_2 {\rm{,}}(3.3) $M_{3} (s) = (k_{2}^{''} - 2 k_{1} k_{2}^{2}) N_{1} + (k_{1}^{''} - 2 k_{1}^{2} k_{2}) N_{2} .$ [M_3 \left( s \right) = \left( {k_2^{^{''} } - 2k_1 k_2^2 } \right)N_1 + \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)N_2 {\rm{.}}

Corollary 1

α^′ (s), α^″ (s), α^″′ (s) and α^″″ (s) are linearly dependent if and only if M₁ (s), M₂ (s) and M₃ (s) are linearly dependent.

Definition 3

Frenet curves of osculating order 3 are :i)

of type weak AW (2) if they satisfy(3.4) $M_{3} (s) = 〈 M_{3} (s), M_{2}^{⋆} (s) 〉 M_{2}^{⋆} (s),$ [M_3 \left( s \right) = \left\langle {M_3 \left( s \right),M_2^ \star \left( s \right)} \right\rangle M_2^ \star \left( s \right){\rm{,}}

ii)

of type weak AW (3) if they satisfy(3.5) $M_{3} (s) = 〈 M_{3} (s), M_{1}^{⋆} (s) 〉 M_{1}^{⋆} (s),$ [M_3 \left( s \right) = \left\langle {M_3 \left( s \right),M_1^ \star \left( s \right)} \right\rangle M_1^ \star \left( s \right){\rm{,}}where(3.6) $M_{1}^{⋆} (s) = \frac{M_{1} (s)}{‖ M_{1} (s) ‖},$ [M_1^ \star \left( s \right) = \frac{{M_1 \left( s \right)}}{{\left\| {M_1 \left( s \right)} \right\|}}{\rm{,}}(3.7) $M_{2}^{⋆} (s) = \frac{M_{2} (s) - 〈 M_{2} (s), M_{1}^{⋆} (s) 〉 M_{1}^{⋆} (s)}{‖ M_{2} (s) - 〈 M_{2} (s), M_{1}^{⋆} (s) 〉 M_{1}^{⋆} (s) ‖},$ [M_2^ \star \left( s \right) = \frac{{M_2 \left( s \right) - \left\langle {M_2 \left( s \right),M_1^ \star \left( s \right)} \right\rangle M_1^ \star \left( s \right)}}{{\left\| {M_2 \left( s \right) - \left\langle {M_2 \left( s \right),M_1^ \star \left( s \right)} \right\rangle M_1^ \star \left( s \right)} \right\|}}{\rm{,}}

iii)

of type AW (1) if they satisfy(3.8) $M_{3} (s) = 0,$ [M_3 \left( s \right) = 0{\rm{,}}

iv)

of type AW (2) if they satisfy(3.9) ${‖ M_{2} (s) ‖}^{2} M_{3} (s) = 〈 M_{3} (s), M_{2} (s) 〉 M_{2} (s),$ [left\| {M_2 \left( s \right)} \right\|^2 M_3 \left( s \right) = \left\langle {M_3 \left( s \right),M_2 \left( s \right)} \right\rangle M_2 \left( s \right){\rm{,}}

of type AW (3) if they satisfy(3.10) ${‖ M_{1} (s) ‖}^{2} M_{3} (s) = 〈 M_{3} (s), M_{1} (s) 〉 M_{1} (s) .$ [\left\| {M_1 \left( s \right)} \right\|^2 M_3 \left( s \right) = \left\langle {M_3 \left( s \right),M_1 \left( s \right)} \right\rangle M_1 \left( s \right){\rm{.}}

Proposition 2

Suppose that α is a Frenet curve of osculating order 3 in $E_{1}^{3}$ [E_1^3 , then α is AW (1)-type if and only if(3.11) $k_{1}^{''} = 2 k_{1}^{2} k_{2},$ [k_1^{^{''} } = 2k_1^2 k_2 {\rm{,}}(3.12) $k_{2}^{''} = 2 k_{1} k_{2}^{2} .$ [k_2^{^{''} } = 2k_1 k_2^2 {\rm{.}}

Proof

Since α is a curve of AW (1)-type, then α must satisfy (3.8) $M_{3} (s) = (k_{2}^{''} - 2 k_{1} k_{2}^{2}) N_{1} + (k_{1}^{''} - 2 k_{1}^{2} k_{2}) N_{2}, 0 = (k_{2}^{''} - 2 k_{1} k_{2}^{2}) N_{1} + (k_{1}^{''} - 2 k_{1}^{2} k_{2}) N_{2} ˙$ [M_3 \left( s \right) = \left( {k_2^{^{''} } - 2k_1 k_2^2 } \right)N_1 + \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)N_2 {\rm{, }}0 = \left( {k_2^{^{''} } - 2k_1 k_2^2 } \right)N_1 + \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)N_2 {\rm{.}}

Since N₁ and N₂ are linearly independent, then we have $k_{2}^{''} - 2 k_{1} k_{2}^{2} = 0 k_{2}^{''} = 2 k_{1} k_{2}^{2},$ [k_2^{^{''} } - 2k_1 k_2^2 = 0\,k_2^{^{''} } = 2k_1 k_2^2 {\rm{,}} and $k_{1}^{''} - 2 k_{1}^{2} k_{2} = 0 k_{1}^{''} = 2 k_{1}^{2} k_{2},$ [k_1^{^{''} } - 2k_1^2 k_2 = 0\,k_1^{^{''} } = 2k_1^2 k_2 , which completes the proof of the proposition.

Proposition 3

Let α be a Frenet curve of osculating order 3 in $E_{1}^{3}$ [E_1^3 , then α is AW (2)-type if and only if(3.13) $k_{1}^{''} k_{2}^{'} - 2 k_{1}^{2} k_{2} k_{2}^{'} = k_{1}^{'} k_{2}^{''} - 2 k_{1} k_{1}^{'} k_{2}^{2} .$ [k_1^{^{''} } k_2^{^' } - 2k_1^2 k_2 k_2^{^' } = k_1^{^' } k_2^{^{''} } - 2k_1 k_1^{^' } k_2^2 {\rm{.}}

Proof

Suppose that α is a Frenet curve of osculating order 3. From (2.14) and (2.15) we can write $M_{2} (s) = β (s) N_{1} + γ (s) N_{2}, M_{3} (s) = δ (s) N_{1} + η (s) N_{2},$ [M_2 \left( s \right) = \beta \left( s \right)N_1 + \gamma \left( s \right)N_2 {\rm{, }}M_3 \left( s \right) = \delta \left( s \right)N_1 + \eta \left( s \right)N_2 {\rm{,}} where β (s), γ (s), δ (s) and η (s) are differential functions. Since M₂ (s) and M₃ (s) are linearly dependent, then the determinant of the coefficients of N₁ and N₂ is equal to zero and hence one can write (3.14) $| \begin{matrix} β (s) & γ (s) \\ δ (s) & η (s) \end{matrix} | = 0,$ [\left| {\begin{array}{*{20}c} {\beta \left( s \right)} & {\gamma \left( s \right)} \\ {\delta \left( s \right)} & {\eta \left( s \right)} \\\end{array}} \right| = 0{\rm{,}} where (3.15) $β (s) = k_{2}^{'}, γ (s) = k_{1}^{'}, δ (s) = k_{2}^{''} - 2 k_{1} k_{2}^{2}, η (s) = k_{1}^{''} - 2 k_{1}^{2} k_{2} .$ [\beta \left( s \right) = k_2^{^' } {\rm{,}}\;\gamma \left( s \right) = k_1^{^' } {\rm{,}}\,\delta \left( s \right) = k_2^{^{''} } - 2k_1 k_2^2 {\rm{,}}\,\eta \left( s \right) = k_1^{^{''} } - 2k_1^2 k_2 {\rm{.}}

By substituting (3.15) in (3.14), we can obtain (3.13), which completes the proof of the proposition.

Proposition 4

Let α be a Frenet curve of osculating order 3 in $E_{1}^{3}$ [E_1^3 , then α is AW (3)-type if and only if(3.16) $k_{1}^{''} k_{2} = k_{1} k_{2}^{''} .$ [k_1^{^{''} } k_2 = k_1 k_2^{^{''} } {\rm{.}}

Proof

Suppose that α is a Frenet curve of osculating order 3, then $M_{1} (s) = β (s) N_{1} + γ (s) N_{2}, M_{3} (s) = δ (s) N_{1} + η (s) N_{2},$ [M_1 \left( s \right) = \beta \left( s \right)N_1 + \gamma \left( s \right)N_2 {\rm{,}}\,M_3 \left( s \right) = \delta \left( s \right)N_1 + \eta \left( s \right)N_2 {\rm{,}} where β (s), γ (s), δ (s) and η (s) are differential functions. Since M₂ (s) and M₃ (s) are linearly dependent, then the determinant of the coefficients of N₁ and N₂ is equal to zero and hence we can write (3.17) $β (s) η (s) - γ (s) δ (s) = 0$ [\beta \left( s \right)\eta \left( s \right) - \gamma \left( s \right)\delta \left( s \right) = 0 where $β (s) = k_{2}, γ (s) = k_{1}, δ (s) = k_{2}^{''} - 2 k_{1} k_{2}^{2}, η (s) = k_{1}^{''} - 2 k_{1}^{2} k_{2} ˙$ [\beta \left( s \right) = k_2 {\rm{,}}\;\;\gamma \left( s \right) = k_1 {\rm{,}}\,\delta \left( s \right) = k_2^{^{''} } - 2k_1 k_2^2 {\rm{,}}\,\;\eta \left( s \right) = k_1^{^{''} } - 2k_1^2 k_2 {\rm{.}}

Considering these equations in (3.17), we get (3.16), which completes the proof of the proposition.

Proposition 5

Let α be a Frenet curve of osculating order 3 in $E_{1}^{3}$ [E_1^3 , then α is of weak AW (2)−type if and only if(3.18) $(k_{2}^{''} - 2 k_{1} k_{2}^{2}) = \frac{p}{p^{2} + q^{2}} [(k_{2}^{''} - 2 k_{1} k_{2}^{2}) q + (k_{1}^{''} - 2 k_{1}^{2} k_{2}) p],$ [\left( {k_2^{^{''} } - 2k_1 k_2^2 } \right) = \frac{p}{{p^2 + q^2 }}\left[ {\left( {k_2^{^{''} } - 2k_1 k_2^2 } \right)q + \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)p} \right]{\rm{,}}(3.19) $(k_{1}^{''} - 2 k_{1}^{2} k_{2}) = \frac{q}{p^{2} + q^{2}} [(k_{2}^{''} - 2 k_{1} k_{2}^{2}) q + (k_{1}^{''} - 2 k_{1}^{2} k_{2}) p],$ [\left( {k_1^{^{''} } - 2k_1^2 k_2 } \right) = \frac{q}{{p^2 + q^2 }}\left[ {\left( {k_2^{^{''} } - 2k_1 k_2^2 } \right)q + \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)p} \right]{\rm{,}}where $p = (k_{2}^{2} + k_{1}^{2}) k_{2}^{'} - {(k_{1} k_{2})}^{'} k_{2} and q = (k_{2}^{2} + k_{1}^{2}) k_{1}^{'} - {(k_{1} k_{2})}^{'} k_{1} ˙$ [p = \left( {k_2^2 + k_1^2 } \right)k_2^{^' } - \left( {k_1 k_2 } \right)^{^' } k_2 \;{\rm{and}}\;q = \left( {k_2^2 + k_1^2 } \right)k_1^{^' } - \left( {k_1 k_2 } \right)^{^' } k_1 {\rm{.}}

Proposition 6

Let α be a Frenet curve of osculating order 3 in $E_{1}^{3}$ [E_1^3 , then α is of weak AW (3)−type if and only if(3.20) $k_{2}^{''} - 2 k_{1} k_{2}^{2} = \frac{k_{2}}{k_{2}^{2} + k_{1}^{2}} [k_{1} k_{2}^{''} + k_{1}^{''} k_{2} - 4 k_{1}^{2} k_{2}^{2}],$ [k_2^{^{''} } - 2k_1 k_2^2 = \frac{{k_2 }}{{k_2^2 + k_1^2 }}\left[ {k_1 k_2^{^{''} } + k_1^{^{''} } k_2 - 4k_1^2 k_2^2 } \right],(3.21) $k_{1}^{''} - 2 k_{1}^{2} k_{2} = \frac{k_{1}}{k_{2}^{2} + k_{1}^{2}} [k_{1} k_{2}^{''} + k_{1}^{''} k_{2} - 4 k_{1}^{2} k_{2}^{2}] .$ [k_1^{^{''} } - 2k_1^2 k_2 = \frac{{k_1 }}{{k_2^2 + k_1^2 }}\left[ {k_1 k_2^{^{''} } + k_1^{^{''} } k_2 - 4k_1^2 k_2^2 } \right].

The Slant helices according to Bishop frame of the pseudo null curve in Minkowski 3-space

Definition 4

Helix is a curve whose tangent lines make a constant angle with a fixed direction. Helices are characterized by the fact that the ratio $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }}is constant along the curve.

Definition 5

A unit speed curve α is called a slant helix if there exists a non-zero constant vector field U ɛ $E_{1}^{3}$ [E_1^3 such that the function 〈N (s),U〉 is constant.

It is important to point out, in contrast to what happens in E³, we cannot define the angle between two vectors (except that both vectors are of time-like). For this reason, we avoid to say about the angle between the normal vector field N (s) and U [14].

Theorem 2

Let α be a pseudo null curve in $E_{1}^{3}$ [E_1^3 , then α is a general helix if and only if $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }}is constant.

Proof

Let α be a general helix. The slope axis of the curve α is shown as sp{U} . note that (4.1) $〈 T, U 〉 = c (cisconstant) .$ [\left\langle {T,U} \right\rangle = c\;\;{\rm{(cisconstant)}}{\rm{.}}

If we differentiate both sides of the equation (4.1), then we have (4.2) $〈 T^{'}, U 〉 = 0 ˙$ [\left\langle {T^{^' } ,U} \right\rangle = 0.

By using (2.10) and (4.44) (4.3) $〈 k_{2} N_{1} + k_{1} N_{2}, U 〉 = 0, k_{2} cos θ + k_{1} sin θ = 0, \frac{k_{1}}{k_{2}} = - cot θ (constant),$ [\frac{{k_1 }}{{k_2 }} = - \cot \theta \;\;{\rm{(constant),}} as desired.

Theorem 3

Let α be a pseudo null curve in $E_{1}^{3}$ [E_1^3 , then α is a slant helix if and only if $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }}is constant.

Proof

Let α be a slant helix in $E_{1}^{3}$ [E_1^3 and 〈N (s),U〉 is constant. Then α is a slant helix, from the definition we have (4.4) $〈 N (s), U 〉 = c (cisaconstant),$ [\left\langle {N\left( s \right),U} \right\rangle = c\;\;{\rm{(cisaconstant),}} where U is a constant vector in $E_{1}^{3}$ [E_1^3 . By differentiating (4.4) and using (2.6) $〈 N^{'} (s), U 〉 = 0, - k_{1} 〈 T_{1}, U 〉 = 0, k_{1} \neq 0 .$ [\left\langle {N^{^' } \left( s \right),U} \right\rangle = 0{\rm{,}}\, - k_1 \left\langle {T_1 ,U} \right\rangle = 0,\;k_1 \ne 0.

Hence (4.5) $〈 T_{1}, U 〉 = 0,$ [\left\langle {T_1 ,U} \right\rangle = 0{\rm{,}}uɛsp{N₁,N₂}, therefore u=cosθ N₁ + sinθ N₂. U is a linear combination of N₁ and N₂. By differentiating (4.5) and using (2.6) $〈 T^{'}, U 〉 = 0, k_{2} cos θ + k_{1} sin θ = 0, \frac{k_{1}}{k_{2}} = - cot θ,$ [\left\langle {T^{^' } ,U} \right\rangle = 0{\rm{,}}\,k_2 \cos \theta + k_1 \sin \theta = 0{\rm{,}}\,\frac{{k_1 }}{{k_2 }} = - \cot \theta {\rm{,}} as desired.

Theorem 4

Let α be a pseudo null curve in $E_{1}^{3}$ [E_1^3 , then α is a slant helix if and only if(4.6) $\det (N_{1}^{'}, N_{1}^{''}, N_{1}^{'''}) = 0 .$ [det(N_1^' ,N_1^{^{''} } ,N_1^{^{'''} } ) = 0.

Proof

(⇒) Suppose that $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }} be constant. We have equalities as $N_{1}^{'} = - k_{1} T, N_{1}^{''} = - k_{1}^{'} T - k_{1} k_{2} N_{1} - k_{1}^{2} N_{2}, N_{1}^{'''} = (2 k_{1}^{2} k_{2} - k_{1}^{''}) T + (- 2 k_{1}^{'} k_{2} - k_{1} k_{2}^{'}) N_{1} - 3 k_{1} k_{1}^{'} N_{2} ˙$ [N_1^' = - k_1 T{\rm{,}}\,N_1^{''} = - k_1^{^' } T - k_1 k_2 N_1 - k_1^2 N_2 {\rm{,}}\,N_1^{^{'''} } = \left( {2k_1^2 k_2 - k_1^{^{''} } } \right)T + \left( { - 2k_1^{^' } k_2 - k_1 k_2^{^' } } \right)N_1 - 3k_1 k_1^{^' } N_2 {\rm{.}}

So we get $\det (N_{1}^{'}, N_{1}^{''}, N_{1}^{'''}) = | \begin{matrix} - k_{1} & 0 & 0 \\ - k_{1}^{'} & - k_{1} k_{2} & - k_{1}^{2} \\ (2 k_{1}^{2} k_{2} - k_{1}^{''}) & - (2 k_{1}^{'} k_{2} + k_{1} k_{2}^{'}) & - 3 k_{1} k_{1}^{'} \end{matrix} |, \det (N_{1}^{'}, N_{1}^{''}, N_{1}^{'''}) = - k_{1}^{3} k_{2}^{2} {(\frac{k_{1}}{k_{2}})}^{'} ˙$ [det(N_1^' ,N_1^{^{''} } ,N_1^{^{'''} } ) = \left| {\begin{array}{*{20}c} { - k_1 } & 0 & 0 \\ { - k_1^{^' } } & { - k_1 k_2 } & { - k_1^2 } \\ {\left( {2k_1^2 k_2 - k_1^{^{''} } } \right)} & { - \left( {2k_1^{^' } k_2 + k_1 k_2^{^' } } \right)} & { - 3k_1 k_1^{^' } } \\\end{array}} \right|{\rm{,}}\,det(N_1^' ,N_1^{^{''} } ,N_1^{^{'''} } ) = - k_1^3 k_2^2 \left( {\frac{{k_1 }}{{k_2 }}} \right)^{^' } {\rm{.}} Since α is a slant helix and $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }} is constant. Hence, we have $\det (N_{1}^{'}, N_{1}^{''}, N_{1}^{'''}) = 0, but k_{2} \neq 0 .$ [det(N_1^' ,N_1^{^{''} } ,N_1^{^{'''} } ) = 0,\;{\rm{but}}\;k_2 \ne 0. (⇐) Suppose that det(N^′₁,N^″₁,N^″′₁) = 0, then it is clear that the $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }} is constant, since ${(\frac{k_{1}}{k_{2}})}^{'}$ [\left( {\frac{{k_1 }}{{k_2 }}} \right)^' is zero. Hence the theorem is proved.

Theorem 5

Let α be a pseudo null curve in $E_{1}^{3}$ [E_1^3 , then α is a slant helix if and only if(4.7) $\det (N_{2}^{'}, N_{2}^{''}, N_{2}^{'''}) = 0 .$ [det(N_2^' ,N_2^{^{''} } ,N_2^{^{'''} } ) = 0.

Proof

(⇒) Suppose that $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }} be constant. From (2.6) we have $N_{2}^{'} = - k_{2} T,$ [N_2^' = - k_2 T{\rm{,}} therefore $N_{2}^{''} = - k_{2}^{'} T - k_{2}^{2} N_{1} - k_{1} k_{2} N_{2}, N_{2}^{'''} = (2 k_{1} k_{2}^{2} - k_{2}^{''}) T + (- k_{2}^{2} - 2 k_{2} k_{2}^{'}) N_{1} + (- k_{1} k_{2} - k_{1}^{'} k_{2} - k_{1} k_{2}^{'}) N_{2} ˙$ [N_2^{''} = - k_2^{^' } T - k_2^2 N_1 - k_1 k_2 N_2 {\rm{,}}\,N_2^{^{'''} } = \left( {2k_1 k_2^2 - k_2^{^{''} } } \right)T + \left( { - k_2^2 - 2k_2 k_2^{^' } } \right)N_1 + \left( { - k_1 k_2 - k_1^{^' } k_2 - k_1 k_2^{^' } } \right)N_2 {\rm{.}}

So we get $\begin{array}{l} \det (N_{2}^{'}, N_{2}^{″}, N_{2}^{' ″}) = & | \begin{matrix} - k_{2} & 0 & 0 \\ - k_{2}^{'} & - k_{2}^{2} & - k_{1} k_{2} \\ (\begin{matrix} k_{1} k_{2}^{2} \\ - k_{2}^{″} \end{matrix}) & - (\begin{matrix} 2 \\ 2 \\ + 2 k_{2} k_{2}^{'} \end{matrix}) & - (\begin{matrix} {k_{2}}_{1} \\ + k_{1}^{'} k_{2} \\ + k_{1} k_{2}^{'} \end{matrix}) \end{matrix} |, \\ \det (N_{2}^{'}, N_{2}^{″}, N_{2}^{' ″}) = & - k_{2}^{5} {(\frac{k_{1}}{k_{2}})}^{'} . \end{array}$ [\begin{array}{l} \user1{det}\left( {N_2^' ,N_2^{''} ,N_2^{'''} } \right) = \left| {\begin{array}{*{20}c} { - k_2 } & 0 & 0 \\ { - k_2^' } & { - k_2^2 } & { - k_1 k_2 } \\ {\left( {\begin{array}{*{20}c} {k_1 k_2^2 } \\ { - k_2^{''} } \\\end{array}} \right)} & { - \left( {\begin{array}{*{20}c} 2 \\ 2 \\ { + 2k_2 k_2^' } \\\end{array}} \right)} & { - \left( {\begin{array}{*{20}c} {{}_1k_2 } \\ { + k_1^' k_2 } \\ { + k_1 k_2^' } \\\end{array}} \right)} \\\end{array}} \right|, \\ \user1{det}\left( {N_2^' N_2^{''} ,N_2^{'''} } \right) = - k_2^5 \left( {\frac{{k_1 }}{{k_2 }}} \right)^' \\ \end{array}

Since α is a slant helix and $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }} is constant. Hence, we have $\det (N_{2}^{'}, N_{2}^{''}, N_{2}^{'''}) = 0, but k_{2} \neq 0 .$ [det(N_2^' ,N_2^{^{''} } ,N_2^{^{'''} } ) = 0,\;but\;k_2 \ne 0. (⇐) Suppose that det(N^′₂,N^″₂,N^″′₂) = 0, then it is clear that the $\frac{k_{1}}{k_{2}}$ [\frac{{k_1 }}{{k_2 }} is constant, since ${(\frac{k_{1}}{k_{2}})}^{'}$ [\left( {\frac{{k_1 }}{{k_2 }}} \right)^' is zero. Hence the theorem is proved.

From (2.6)(4.8) $α (s) = T, D_{T} T = k_{2} N_{1} + k_{1} N_{2}, D_{T} N_{1} = - k_{1} T_{1}, D_{T} N_{2} = - k_{2} T_{1},$ [\alpha \left( s \right) = T{\rm{,}}\,D_T T = k_2 N_1 + k_1 N_2 {\rm{,}}\,\,D_T N_1 = - k_1 T_1 {\rm{,}}\,D_T N_2 = - k_2 T_1 {\rm{,}}

Theorem 6

Let $α : I \to E_{1}^{3}$ [\alpha :I \to E_1^3 \be a unit speed pseudo null curve on M₁then α is a general slant helix if and only if(4.9) $D_{T} (D_{T} D_{T} N_{1}) + 3 k_{1}^{'} D_{T} T = (k_{1}^{''} - 2 k_{1}^{2} k_{2}) \frac{1}{k_{1}} D_{T} N_{1} .$ [D_T \left( {D_T D_T N_1 } \right) + 3k_1^{^' } D_T T = \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)\frac{1}{{k_1 }}D_T N_1 {\rm{.}}

Proof

(⇒) Suppose that α is a general slant helix. Then, from (4.8), we have (4.10) $D_{T} N_{1} = - k_{1} T_{1}, D_{T} (D_{T} N_{1}) = - k_{1}^{'} T_{1} - k_{1} k_{2} N_{1} - k_{1}^{2} N_{2}, D_{T} (D_{T} D_{T} N_{1}) = (k_{1}^{2} k_{2} - k_{1}^{''}) T_{1} - k_{1}^{'} D_{T} T - (k_{1}^{'} k_{2} + k_{1} k_{2}^{'}) N_{1}$ [D_T N_1 = - k_1 T_1 {\rm{,}}\,D_T \left( {D_T N_1 } \right) = - k_1^{^' } T_1 - k_1 k_2 N_1 - k_1^2 N_2 {\rm{,}}\,D_T \left( {D_T D_T N_1 } \right) = \left( {k_1^2 k_2 - k_1^{^{''} } } \right)T_1 - k_1^{^' } D_T T - \left( {k_1^{^' } k_2 + k_1 k_2^{^' } } \right)N_1 \(4.11) $- k_{1} k_{2} D_{T} N_{1} - 2 k_{1} k_{1}^{'} N_{2} .$ [ - k_1 k_2 D_T N_1 - 2k_1 k_1^{^' } N_2 {\rm{.}}

Since α is a general helix (4.12) $\frac{k_{1}}{k_{2}} = c c isconstant,$ [\frac{{k_1 }}{{k_2 }} = c\;\;c{\rm{isconstant,}} by differentiating (4.12), we get (4.13) ${(k_{1} k_{2})}^{'} = 2 k_{1}^{'} k_{2},$ [\left( {k_1 k_2 } \right)^{^' } = 2k_1^{^' } k_2 {\rm{,}} but (4.14) $D_{T} N_{1} = N_{1}^{'} = - k_{1} T, T = - \frac{1}{k_{1}} D_{T} N_{1} .$ [D_T N_1 = N_1^{^' } = - k_1 T{\rm{,}}\,T = - \frac{1}{{k_1 }}D_T N_1 {\rm{.}}

By substituting (4.13) and (4.14) in (4.11) we get (4.15) $D_{T} (D_{T} D_{T} N_{1}) = (k_{1}^{''} - 2 k_{1}^{2} k_{2}) (\frac{1}{k_{1}} D_{T} N_{1}) - 3 k_{1}^{'} D_{T} T, D_{T} (D_{T} D_{T} N_{1}) + 3 k_{1}^{'} D_{T} T = (k_{1}^{''} - 2 k_{1}^{2} k_{2}) (\frac{1}{k_{1}} D_{T} N_{1}) .$ [D_T \left( {D_T D_T N_1 } \right) = \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)\left( {\frac{1}{{k_1 }}D_T N_1 } \right) - 3k_1^{^' } D_T T{\rm{,}}\,D_T \left( {D_T D_T N_1 } \right) + 3k_1^{^' } D_T T = \left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)\left( {\frac{1}{{k_1 }}D_T N_1 } \right){\rm{.}} (⇐) We will now show that pseudo null curve α is a slant helix. By differentianting (4.14) covariantly $T = - \frac{1}{k_{1}} D_{T} N_{1}$ [T = - \frac{1}{{k_1 }}D_T N_1 (4.16) $D_{T} T = \frac{k_{1}^{'}}{k_{1}^{2}} D_{T} N_{1} - \frac{1}{k_{1}} D_{T} D_{T} N_{1},$ [D_T T = \frac{{k_1^{^' } }}{{k_1^2 }}D_T N_1 - \frac{1}{{k_1 }}D_T D_T N_1 {\rm{,}}(4.17) $D_{T} D_{T} T = {(\frac{k_{1}^{'}}{k_{1}^{2}})}^{'} D_{T} N_{1} + \frac{2 k_{1}^{'}}{k_{1}^{2}} D_{T} D_{T} N_{1} - \frac{1}{k_{1}} D_{T} D_{T} D_{T} N_{1} .$ [D_T D_T T = \left( {\frac{{k_1^{^' } }}{{k_1^2 }}} \right)^{^' } D_T N_1 + \frac{{2k_1^{^' } }}{{k_1^2 }}D_T D_T N_1 - \frac{1}{{k_1 }}D_T D_T D_T N_1 {\rm{.}}

By substituting (4.15) in (4.17) we get (4.18) $D_{T} D_{T} T = [{(\frac{k_{1}^{'}}{k_{1}^{2}})}^{'} - \frac{1}{k_{1}^{2}} (k_{1}^{''} - 2 k_{1}^{2} k_{2})] D_{T} N_{1} + \frac{2 k_{1}^{'}}{k_{1}^{2}} D_{T} D_{T} N_{1} + \frac{3 k_{1}^{'}}{k_{1}} D_{T} T,$ [D_T D_T T = \left[ {\left( {\frac{{k_1^{^' } }}{{k_1^2 }}} \right)^{^' } - \frac{1}{{k_1^2 }}\left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)} \right]D_T N_1 \, + \frac{{2k_1^{^' } }}{{k_1^2 }}D_T D_T N_1 + \frac{{3k_1^{^' } }}{{k_1 }}D_T T{\rm{,}} by substituting (4.8) and (4.10) in (4.18) we get (4.19) $D_{T} D_{T} T = [{(\frac{k_{1}^{'}}{k_{1}^{2}})}^{'} - \frac{1}{k_{1}^{2}} (k_{1}^{''} - 2 k_{1}^{2} k_{2})] D_{T} N_{1} - 2 {(\frac{k_{1}^{'}}{k_{1}})}^{2} T + \frac{k_{1}^{'} k_{2}}{k_{1}} N_{1} + k_{1}^{'} N_{2} .$ [D_T D_T T = \left[ {\left( {\frac{{k_1^{^' } }}{{k_1^2 }}} \right)^{^' } - \frac{1}{{k_1^2 }}\left( {k_1^{^{''} } - 2k_1^2 k_2 } \right)} \right]D_T N_1 \, - 2\left( {\frac{{k_1^{^' } }}{{k_1 }}} \right)^2 T + \frac{{k_1^{^' } k_2 }}{{k_1 }}N_1 + k_1^{^' } N_2 {\rm{.}}

From (4.8)(4.20) $D_{T} T = k_{2} N_{1} + k_{1} N_{2}, D_{T} D_{T} T = k_{2}^{'} N_{1} + k_{2} D_{T} N_{1} + k_{1}^{'} N_{2} - k_{1} k_{2} T_{1} .$ [D_T T = k_2 N_1 + k_1 N_2 {\rm{,}}\,D_T D_T T = k_2^{^' } N_1 + k_2 D_T N_1 + k_1^{^' } N_2 - k_1 k_2 T_1 {\rm{.}}

By comparing (4.19) and (4.20)(4.21) $\frac{k_{1}^{'} k_{2}}{k_{1}} = k_{2}^{'}, \frac{k_{1}^{'}}{k_{1}} = \frac{k_{2}^{'}}{k_{2}},$ [\frac{{k_1^{^' } k_2 }}{{k_1 }} = k_2^{^' } {\rm{,}}\,\frac{{k_1^{^' } }}{{k_1 }} = \frac{{k_2^{^' } }}{{k_2 }}{\rm{,}} by integrating (4.21) we get $\frac{k_{1}}{k_{2}} = e^{c} (constant) ˙$ [\frac{{k_1 }}{{k_2 }} = e^c \;\;{\rm{(constant)}}{\rm{.}}

Hence α is a general slant helix.

Theorem 7

Let $α : I \to E_{1}^{3}$ [\alpha :I \to E_1^3 \be a unit speed pseudo null curve on M₁, then α is a general slant helix if and only if(4.22) $D_{T} (D_{T} D_{T} N_{2}) + 3 k_{2}^{'} D_{T} T = (\frac{k_{2}^{''}}{k_{2}} - 2 k_{1} k_{2}) D_{T} N_{2} .$ [D_T \left( {D_T D_T N_2 } \right) + 3k_2^{^' } D_T T = \left( {\frac{{k_2^{^{''} } }}{{k_2 }} - 2k_1 k_2 } \right)D_T N_2 {\rm{.}}

Proof

(⇒) Suppose that α is a general slant helix. Then, from (4.8), we have (4.23) $D_{T} N_{2} = - k_{2} T_{1}, D_{T} D_{T} N_{2} = - k_{2}^{'} T_{1} - k_{2}^{2} N_{1} - k_{1} k_{2} N_{2}, D_{T} (D_{T} D_{T} N_{2}) = (k_{2}^{2} k_{1} - k_{2}^{''}) T_{1} - k_{2}^{'} D_{T} T - 2 k_{2} k_{2}^{'} N_{1}$ [D_T N_2 = - k_2 T_1 {\rm{,}}\,D_T D_T N_2 = - k_2^{^' } T_1 - k_2^2 N_1 - k_1 k_2 N_2 {\rm{,}}\,D_T \left( {D_T D_T N_2 } \right) = \left( {k_2^2 k_1 - k_2^{^{''} } } \right)T_1 - k_2^{^' } D_T T - 2k_2 k_2^{^' } N_1 (4.24) $- (k_{1}^{'} k_{2} + k_{1} k_{2}^{'}) N_{2} - k_{1} k_{2} D_{T} N_{2} .$ [ - \left( {k_1^{^' } k_2 + k_1 k_2^{^' } } \right)N_2 - k_1 k_2 D_T N_2 {\rm{.}}

Since α is a general helix $\frac{k_{1}}{k_{2}} = c cisconstant,$ [\;\frac{{k_1 }}{{k_2 }} = c\;{\rm{cisconstant}}, by differentiating the above equation we get (4.25) ${(k_{1} k_{2})}^{'} = 2 k_{1} k_{2}^{'},$ [\left( {k_1 k_2 } \right)^{^' } = 2k_1 k_2^{^' } {\rm{,}} but (4.26) $T = - \frac{1}{k_{2}} D_{T} N_{2} .$ [T = - \frac{1}{{k_2 }}D_T N_2 {\rm{.}}

By substituting (4.25) and (4.26) in (4.24) we get (4.27) $D_{T} (D_{T} D_{T} N_{2}) = (\frac{k_{2}^{''}}{k_{2}} - 2 k_{1} k_{2}) D_{T} N_{2} - 3 k_{2}^{'} D_{T} T, D_{T} (D_{T} D_{T} N_{2}) + 3 k_{2}^{'} D_{T} T = (\frac{k_{2}^{''}}{k_{2}} - 2 k_{1} k_{2}) D_{T} N_{2} .$ [D_T \left( {D_T D_T N_2 } \right) = \left( {\frac{{k_2^{^{''} } }}{{k_2 }} - 2k_1 k_2 } \right)D_T N_2 - 3k_2^{^' } D_T T{\rm{,}}\,D_T \left( {D_T D_T N_2 } \right) + 3k_2^{^' } D_T T = \left( {\frac{{k_2^{^{''} } }}{{k_2 }} - 2k_1 k_2 } \right)D_T N_2 {\rm{.}} (⇐) We will show that pseudo null curve α is a slant helix. So by differentianting (4.26) covariantly we get (4.28) $D_{T} T = \frac{k_{2}^{'}}{k_{2}^{2}} D_{T} N_{2} - \frac{1}{k_{2}} D_{T} D_{T} N_{2}, D_{T} D_{T} T = {(\frac{k_{2}^{'}}{k_{2}^{2}})}^{'} D_{T} N_{2} + \frac{2 k_{2}^{'}}{k_{2}^{2}} D_{T} D_{T} N_{2}$ [D_T T = \frac{{k_2^{^' } }}{{k_2^2 }}D_T N_2 - \frac{1}{{k_2 }}D_T D_T N_2 {\rm{,}}\,D_T D_T T = \left( {\frac{{k_2^{^' } }}{{k_2^2 }}} \right)^{^' } D_T N_2 + \frac{{2k_2^{^' } }}{{k_2^2 }}D_T D_T N_2 (4.29) $- \frac{1}{k_{2}} D_{T} D_{T} D_{T} N_{2} .$ [ - \frac{1}{{k_2 }}D_T D_T D_T N_2 {\rm{.}}

By substituting (4.27) in (4.29) we get (4.30) $D_{T} D_{T} T = [{(\frac{k_{2}^{'}}{k_{2}^{2}})}^{'} - (\frac{k_{2}^{''}}{k_{2}} - 2 k_{1} k_{2}) \frac{1}{k_{2}}] D_{T} N_{2} + \frac{2 k_{2}^{'}}{k_{2}^{2}} D_{T} D_{T} N_{2} + \frac{3 k_{2}^{'}}{k_{2}} D_{T} T,$ [D_T D_T T = \left[ {\left( {\frac{{k_2^{^' } }}{{k_2^2 }}} \right)^{^' } - \left( {\frac{{k_2^{^{''} } }}{{k_2 }} - 2k_1 k_2 } \right)\frac{1}{{k_2 }}} \right]D_T N_2 \, + \frac{{2k_2^{^' } }}{{k_2^2 }}D_T D_T N_2 + \frac{{3k_2^{^' } }}{{k_2 }}D_T T{\rm{,}} by substituting (4.8) and (4.23) in (4.30) we get (4.31) $D_{T} D_{T} T = [{(\frac{k_{2}^{'}}{k_{2}^{2}})}^{'} - (\frac{k_{2}^{''}}{k_{2}} - 2 k_{1} k_{2}) \frac{1}{k_{2}}] D_{T} N_{2} - 2 {(\frac{k_{2}^{'}}{k_{2}})}^{2} T_{1} + k_{2}^{'} N_{1} + \frac{k_{1} k_{2}^{'}}{k_{2}} N_{2} .$ [D_T D_T T = \left[ {\left( {\frac{{k_2^{^' } }}{{k_2^2 }}} \right)^{^' } - \left( {\frac{{k_2^{^{''} } }}{{k_2 }} - 2k_1 k_2 } \right)\frac{1}{{k_2 }}} \right]D_T N_2 \, - 2\left( {\frac{{k_2^{^' } }}{{k_2 }}} \right)^2 T_1 + k_2^{^' } N_1 + \frac{{k_1 k_2^{^' } }}{{k_2 }}N_2 {\rm{.}}

From (4.8)(4.32) $D_{T} T = k_{2} N_{1} + k_{1} N_{2}, D_{T} D_{T} T = k_{2}^{'} N_{1} + k_{2} D_{T} N_{1} + k_{1}^{'} N_{2} - k_{1} k_{2} T_{1} .$ [D_T T = k_2 N_1 + k_1 N_2 {\rm{,}}\,D_T D_T T = k_2^{^' } N_1 + k_2 D_T N_1 + k_1^{^' } N_2 - k_1 k_2 T_1 {\rm{.}}

By comparing (4.31) and (4.32)(4.33) $\frac{k_{1} k_{2}^{'}}{k_{2}} = k_{1}^{'} \frac{k_{1}^{'}}{k_{1}} = \frac{k_{2}^{'}}{k_{2}},$ [\frac{{k_1 k_2^{^' } }}{{k_2 }} = k_1^{^' } \,\frac{{k_1^{^' } }}{{k_1 }} = \frac{{k_2^{^' } }}{{k_2 }}{\rm{,}} by integrating (4.33) we get $\frac{k_{1}}{k_{2}} = e^{c} constant ˙$ [\frac{{k_1 }}{{k_2 }} = e^c \;\;{\rm{constant}}{\rm{.}}

Hence α is a general slant helix, the theorem is now proved.

eISSN:: 2444-8656
Langue:: Anglais

Périodicité:: Volume Open
Sujets de la revue:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

RSS Feed de la revue

Special Curves According to Bishop Frame in Minkowski 3-Space

Publié en ligne: 31 mars 2020

Pages: 237 - 248

Reçu: 02 mai 2019

Accepté: 12 juil. 2019

DOI: https://doi.org/10.2478/amns.2020.1.00021

Mots clésBishop frame, Pseudo null curve, slant helices

© 2020 Muhammad Abubakar Isah et al., published by Sciendo

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

Mots clés
Bishop frame, Pseudo null curve, slant helices