A study on null cartan curve in Minkowski 3-space

Curves are one of the basic structures of differential geometry. It is safe to report that G. Monge initiated the many important results in Euclidean 3-space curve theory and G. Darboux pioneered the idea of a moving frame.

The curve theory has been one of the most studied subjects because of its many applications area from geometry to the various branch of science. Especially the characteristics of curvature and torsion play an important role in special curve types such as so-called helices. In Euclidean 3-space E³, a general helix or a constant slope curve 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 demonstrated by B. de Saint Venant in 1845 [5,6,11,12]. For nature’s helical structures, helices arise in nano-springs, carbon nano-tubes, helices, 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 shells of the sea [1,2,9]. In fractal geometry, helical structures are used.

In Minkowski 3-space, null Cartan curves are known as the curves whose and Cartan frame contains two null (lightlike) vector fields (see [7] for more information). Null curves of AW(k)-type are studied in the 3-dimensional Lorentzian space by M. Külahcı [8].

In [4] B. Bukcu and M. K. Karacan defined a slant helix according to Bishop frame of the timelike curve and they have given some necessary and sufficent conditions for the slant helix. Ahmad T. Ali and Rafael Lopez gave characterizations of slant helices in terms of the curvature and torsion and discussed the tangent and binormal indicatrices of slant curves in $E_{1}^{3}$ E_1^3 [3,10,13].

F. Gökçelik and I.Gök defined a new kind of slant helix called W-slant helix in 3-dimensional Minkowski space as a curve whose binormal lines make a constant angle with a fixed direction [14].

Preliminaries

Definition 1

The Minkowski 3-space $E_{1}^{3}$ E_1^3is 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 $u \in E_{1}^{3} \ {0}$ u \in E_1^3\backslash \{ 0\}can have one of three properties: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 $u \in E_{1}^{3}$ u \in E_1^3is 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_1^3can 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) = {{d\alpha } \over {dt}}. .

Definition 2

A curve $α : I \to E_{1}^{3}$ \alpha :I \to E_1^3is called a null curve, if its tangent vector α^′ = T is a null vector. A null curve α = α(s) is called a null Cartan curve, if it is parameterized by the pseudo-arc function s defined by(2.2) $s (t) = \int_{0}^{t} \sqrt{‖ α^{″} (u) ‖} du .$ s(t) = \int_0^t \sqrt {\left\| {{\alpha ^{''}}(u)} \right\|} du.

There exists a unique Cartan frame {T,N,B} along a non-geodesic null Cartan curve satisfying the Cartan equations(2.3) $[\begin{matrix} T^{'} \\ N^{'} \\ B^{'} \end{matrix}] = [\begin{matrix} 0 & k & 0 \\ - τ & 0 & k \\ 0 & - τ & 0 \end{matrix}] [\begin{matrix} T \\ N \\ B \end{matrix}],$ \left[ {\matrix{ {{T^{'}}} \cr {{N^{'}}} \cr {{B^{'}}} \cr } } \right] = \left[ {\matrix{ 0 & k & 0 \cr { - \tau } & 0 & k \cr 0 & { - \tau } & 0 \cr } } \right]\left[ {\matrix{ T \cr N \cr B \cr } } \right],where the curvature k(s) = 1 and the torsion τ(s) is an arbitrary function in pseudo-arc parameter s. If τ(s) = 0, the null Cartan curve is called a null Cartan cubic. The Cartan’s frame vectors satisfy the relations(2.4) $\begin{array}{l} 〈 N, N 〉 = 1, 〈 T, T 〉 = 〈 B, B 〉 = 0, \\ 〈 T, B 〉 = - 1, 〈 T, N 〉 = 〈 N, B 〉 = 0 \end{array}$ \matrix{ {\left\langle {N,N} \right\rangle = 1,\left\langle {T,T} \right\rangle = \left\langle {B,B} \right\rangle = 0,}\,\,\,\,\,\,\, \cr {\left\langle {T,B} \right\rangle = - 1,\left\langle {T,N} \right\rangle = \left\langle {N,B} \right\rangle = 0}}and(2.5) $T \times N = - T, N \times B = - B, B \times T = N .$ T \times N = - T,N \times B = - B,B \times T = N.

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

The Frenet frame is created for the non-degenerated curves of three times continuously differentiable. But, at some points on the curve, curvature may vanish. In this case, we need an alternative frame in E³. Bishop introduced a new frame called Bishop frame or parallel transport frame, which is well defined even if the curve has a vanishing second derivative [7].

2.1

The Bishop Frame

The Bishop frame or relatively parallel adapted frame {T,N₁,N₂} of a regular curve in Euclidean 3-space is introduced by R.L. Bishop. It contains the 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 $N_{1}^{'}$ N_1^{'} and $N_{2}^{'}$ N_2^{'} with respect to the arc-length parameter s of the curve are collinear with the tangential vector field T [7].

Remark 1

We can also define N₁and N₂to be relatively parallel, if the normal component $T_{1}^{⊥} = span {N_{1}, N_{2}}$ T_1^ \bot = span\{ {N_1},{N_2}\}of their derivatives $N_{1}^{'}$ N_1^{'} and $N_{2}^{'}$ N_2^{'}is zero, which implies that the mentioned derivatives are collinear with T₁.

2.2

The Bishop frame of a null Cartan curve in

E_{1}^{3}

E_1^3

The Bishop frame {T₁,N₁,N₂} of a non-geodesic null Cartan curve in $E_{1}^{3}$ E_1^3 is positively oriented pseudo-orthonormal frame consisting of the tangential vector field T₁, relatively parallel spacelike normal vector field N₁ and relatively parallel lightlike transversal vector field N₂.

Theorem 1

Let α be a null Cartan curve in $E_{1}^{3}$ E_1^3parameterized by pseudo-arc s with the curvature k(s) = 1 and the torsion τ(s). Then the Bishop frame {T₁,N₁,N₂} and the Cartan frame {T,N,B} of α are related by(2.6) $[\begin{matrix} T_{1} \\ N_{1} \\ N_{2} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ - k_{2} & 1 & 0 \\ \frac{k_{2}^{2}}{2} & - k_{2} & 1 \end{matrix}] [\begin{matrix} T \\ N \\ B \end{matrix}],$ \left[ {\matrix{ {{T_1}} \cr {{N_1}} \cr {{N_2}} \cr } } \right] = \left[ {\matrix{ 1 & 0 & 0 \cr { - {k_2}} & 1 & 0 \cr {{{k_2^2} \over 2}} & { - {k_2}} & 1 \cr } } \right]\left[ {\matrix{ T \cr N \cr B \cr } } \right],and the Cartan equations of according to the Bishop frame read(2.7) $[\begin{matrix} T_{1}^{'} \\ N_{1}^{'} \\ N_{2}^{'} \end{matrix}] = [\begin{matrix} k_{2} & k_{1} & 0 \\ 0 & 0 & k_{1} \\ 0 & 0 & - k_{2} \end{matrix}] [\begin{matrix} T_{1} \\ N_{1} \\ N_{2} \end{matrix}],$ \left[ {\matrix{ {T_1^{'}} \cr {N_1^{'}} \cr {N_2^{'}} \cr } } \right] = \left[ {\matrix{ {{k_2}} & {{k_1}} & 0 \cr 0 & 0 & {{k_1}} \cr 0 & 0 & { - {k_2}} \cr } } \right]\left[ {\matrix{ {{T_1}} \cr {{N_1}} \cr {{N_2}} \cr } } \right],where the first Bishop curvature k₁(s) = 1 and the second Bishop curvature satisfies Riccati differential equation $k_{2}^{'} (s) = - \frac{1}{2} k_{2}^{2} (s) - τ (s),$ k_2^{'}\left( s \right) = - {1 \over 2}k_2^2\left( s \right) - \tau \left( s \right),which satisfies the conditions $\begin{array}{l} 〈 N_{1}, N_{1} 〉 = 1, 〈 T_{1}, T_{1} 〉 = 〈 N_{2}, N_{2} 〉 = 0, \\ 〈 T_{1}, N_{2} 〉 = - 1, 〈 T_{1}, N_{1} 〉 = 〈 N_{1}, N_{2} 〉 = 0 [7] . \end{array}$ \matrix{ {\left\langle {{N_1},{N_1}} \right\rangle = 1,\left\langle {{T_1},{T_1}} \right\rangle = \left\langle {{N_2},{N_2}} \right\rangle = 0,}\hfill\cr\,\,{\left\langle {{T_1},{N_2}} \right\rangle = - 1,\left\langle {{T_1},{N_1}} \right\rangle = \left\langle {{N_1},{N_2}} \right\rangle = 0\;\left[ 7 \right].} \hfill}

Curves of AW(k)- type

Proposition 1

Let α be a Frenet curve of osculating order 3, by using the cartan equations of α according to the Bishop frame(2.7), then we have $\begin{array}{l} α^{'} (s) = T_{1} (s), \\ α^{″} (s) = T_{1}^{'} (s) = k_{2} T_{1} + k_{1} N_{1}, \\ α^{'″} (s) = (k_{2}^{'} + k_{2}^{2}) T_{1} + (k_{1}^{'} + k_{1} k_{2}) N_{1} + k_{1}^{2} N_{2}, \\ α^{″″} (s) = (k_{2}^{″} + 3 k_{2} k_{2}^{'} + k_{2}^{3}) T_{1} \\ + (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2} . \end{array}$ \matrix{ {{\alpha ^{'}}\left( s \right) = {T_1}\left( s \right),} \hfill \cr {{\alpha ^{''}}\left( s \right) = T_1^{'}\left( s \right) = {k_2}{T_1} + {k_1}{N_1},} \hfill \cr {{\alpha ^{'''}}\left( s \right) = \left( {k_2^{'} + k_2^2} \right){T_1} + \left( {k_1^{'} + {k_1}{k_2}} \right){N_1} + k_1^2{N_2},} \hfill \cr {{\alpha ^{''''}}\left( s \right) = \left( {k_2^{''} + 3{k_2}k_2^{'} + k_2^3} \right){T_1}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2}.} \hfill}

Notation 1

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

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 ofi)

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),

ii)

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),where(3.6) $M_{1}^{⋆} (s) = \frac{M_{1} (s)}{‖ M_{1} (s) ‖},$ M_1^ \star \left( s \right) = {{{M_1}\left( s \right)} \over {\left\| {{M_1}\left( s \right)} \right\|}},(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) = {{{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)} \over {\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\|}},

iii)

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

iv)

AW (2)-type, 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),

AW (3)-type, 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).

Proposition 2

Let α be a Frenet curve of osculating order 3, then α is AW (1)-type if and only if i) k₁is a constant function, and ii) (3.11) $k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} = 0 .$ k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2 = 0.

Proof

Since α is a curve of type AW (1), then α must satisfy (3.8) $\begin{array}{r} M_{3} (s) = (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2}, \\ 0 = (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2} . \end{array}$ \matrix{ \hfill {{M_3}\left( s \right) = \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2},} \cr \hfill {0 = \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2}.}}

Since N₁ and N₂ are linearly independent, therefore $3 k_{1} k_{1}^{'} = 0,$ 3{k_1}k_1^{'} = 0,k₁ is a constant function, and $k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} = 0 .$ k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2 = 0.

Hence the proposition is proved.

Proposition 3

Let α be a Frenet curve of osculating order 3. Then, if α are of AW (2)–type, we have(3.12) $3 {(k_{1}^{'})}^{2} + 3 k_{1}^{'} k_{1} k_{2} = k_{1}^{″} k_{1} + 2 k_{1}^{2} k_{2}^{'} + k_{1}^{'} k_{1} k_{2} + k_{1}^{2} k_{2}^{2} .$ 3{\left( {k_1^{'}} \right)^2} + 3k_1^{'}{k_1}{k_2} = k_1^{''}{k_1} + 2k_1^2k_2^{'} + k_1^{'}{k_1}{k_2} + k_1^2k_2^2.

Proof

Suppose that α is a Frenet curve of osculating order 3. From (3.2) and (3.3) we can write $\begin{matrix} M_{2} (s) = β (s) N_{1} + γ (s) N_{2}, \\ M_{3} (s) = δ (s) N_{1} + η (s) N_{2}, \end{matrix}$ \matrix{ {{M_2}\left( s \right) = \beta \left( s \right){N_1} + \gamma \left( s \right){N_2},} \cr {{M_3}\left( s \right) = \delta \left( s \right){N_1} + \eta \left( s \right){N_2},}} 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, hence one can write (3.13) $| \begin{matrix} β (s) & γ (s) \\ δ (s) & η (s) \end{matrix} | = 0,$ \left| {\matrix{ {\beta \left( s \right)} & {\gamma \left( s \right)} \cr {\delta \left( s \right)} & {\eta \left( s \right)} \cr } } \right| = 0, where (3.14) $\begin{array}{l} β (s) = k_{1}^{'} + k_{1} k_{2}, γ (s) = k_{1}^{2}, \\ δ (s) = k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} and \\ η (s) = 3 k_{1} k_{1}^{'} . \end{array}$ \matrix{ {\beta \left( s \right) = k_1^{'} + {k_1}{k_2},\;\;\gamma \left( s \right) = k_1^2,} \hfill \cr {\delta \left( s \right) = k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2\,{\rm{and}}} \hfill \cr {\;\eta \left( s \right) = 3{k_1}k_1^{'}.} \hfill}

By considering (3.14) in (3.13) we get $3 {(k_{1}^{'})}^{2} + 3 k_{1}^{'} k_{1} k_{2} = k_{1}^{″} k_{1} + 2 k_{1}^{2} k_{2}^{'} + k_{1}^{'} k_{1} k_{2} + k_{1}^{2} k_{2}^{2} .$ 3{\left( {k_1^{'}} \right)^2} + 3k_1^{'}{k_1}{k_2} = k_1^{''}{k_1} + 2k_1^2k_2^{'} + k_1^{'}{k_1}{k_2} + k_1^2k_2^2.

Hence the proposition is proved.

Proposition 4

Let α be a Frenet curve of osculating order 3, then α is of type AW (3) if and only if k₁ (s) is a constant function

Proof

Suppose that α is a Frenet curve of order 3. From (3.1) and (3.3) we can write $\begin{matrix} M_{1} (s) = β (s) N_{1} + γ (s) N_{2}, \\ M_{3} (s) = δ (s) N_{1} + η (s) N_{2}, \end{matrix}$ \matrix{ {{M_1}\left( s \right) = \beta \left( s \right){N_1} + \gamma \left( s \right){N_2},} \cr {{M_3}\left( s \right) = \delta \left( s \right){N_1} + \eta \left( s \right){N_2},}} 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, one can write (3.15) $| \begin{matrix} β (s) & γ (s) \\ δ (s) & η (s) \end{matrix} | = 0,$ \left| {\matrix{ {\beta \left( s \right)} & {\gamma \left( s \right)} \cr {\delta \left( s \right)} & {\eta \left( s \right)} \cr } } \right| = 0, where (3.16) $\begin{array}{l} β (s) = k_{1}, γ (s) = 0, \\ δ (s) = k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} and \\ η (s) = 3 k_{1} k_{1}^{'} . \end{array}$ \matrix{ {\beta \left( s \right) = {k_1},\;\;\gamma \left( s \right) = 0,} \hfill \cr {\delta \left( s \right) = k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2\;{\rm{and}}} \hfill \cr {\;\eta \left( s \right) = 3{k_1}k_1^{'}.} \hfill}

By substituting (3.16) in (3.15) we get $k_{1}^{2} k_{1}^{'} = 0 .$ k_1^2k_1^{'} = 0.

For $k_{1}^{2} k_{1}^{'}$ k_1^2k_1^{'} to be zero, k₁ (s) has to be a constant function. Hence the proposition is proved.

Proposition 5

Let α be a Frenet curve of osculating order 3, then α is of weak AW (2)–type if and only ifi)

k₁ (s) is a constant function,

ii)

(3.17) $k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} = 0 and k_{2} (s) = \frac{2}{s - 2 c},$ k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2 = 0\;and\;{k_2}(s) = {2 \over {s - 2c}},where s and c are arc-length parameter and constant respectively.

Proof

Since α is of weak AW (2)–type, it must satisfy (3.4), by using (3.4), (3.6), (3.7) and (3.1)(3.18) $M_{1}^{⋆} (s) = \frac{M_{1} (s)}{‖ M_{1} (s) ‖} = \frac{k_{1} N_{1}}{\sqrt{k_{1}^{2}}} = N_{1},$ M_1^ \star \left( s \right) = {{{M_1}\left( s \right)} \over {\left\| {{M_1}\left( s \right)} \right\|}} = {{{k_1}{N_1}} \over {\sqrt {k_1^2} }} = {N_1},(3.19) $\begin{array}{l} M_{2}^{⋆} (s) = \frac{M_{2} (s) - 〈 M_{2} (s), M_{1}^{⋆} (s) 〉 M_{1}^{⋆} (s)}{‖ M_{2} (s) - 〈 M_{2} (s), M_{1}^{⋆} (s) 〉 M_{1}^{⋆} (s) ‖}, \\ M_{2}^{⋆} (s) = \frac{(k_{1}^{'} + k_{1} k_{2}) N_{1} + k_{1}^{2} N_{2} - (k_{1}^{'} + k_{1} k_{2}) N_{1}}{‖ (k_{1}^{'} + k_{1} k_{2}) N_{1} + k_{1}^{2} N_{2} - (k_{1}^{'} + k_{1} k_{2}) N_{1} ‖}, \\ M_{2}^{⋆} (s) = N_{2} . \end{array}$ \matrix{ {M_2^ \star \left( s \right) = {{{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)} \over {\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\|}},} \hfill \cr {M_2^ \star \left( s \right) = {{\left( {k_1^{'} + {k_1}{k_2}} \right){N_1} + k_1^2{N_2} - \left( {k_1^{'} + {k_1}{k_2}} \right){N_1}} \over {\left\| {\left( {k_1^{'} + {k_1}{k_2}} \right){N_1} + k_1^2{N_2} - \left( {k_1^{'} + {k_1}{k_2}} \right){N_1}} \right\|}},} \hfill \cr {M_2^ \star \left( s \right) = {N_2}.} \hfill}

Since α is of weak AW (2)–type, then it must satisfy $\begin{array}{l} M_{3} (s) & = 〈 M_{3} (s), M_{2}^{⋆} (s) 〉 M_{2}^{⋆} (s), \\ = 〈 (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2}, N_{2} 〉 N_{2}, \\ = 0 . \end{array}$ \matrix{ {{M_3}\left( s \right)} \hfill & { = \left\langle {{M_3}\left( s \right),M_2^ \star \left( s \right)} \right\rangle M_2^ \star \left( s \right),} \hfill \cr {} \hfill & { = \left\langle {\left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2},{N_2}} \right\rangle {N_2},} \hfill \cr {} \hfill & { = 0.} \hfill}

Therefore $(k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2} = 0,$ \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2} = 0, then (3.20) $\begin{matrix} k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} & = 0 and \\ 3 k_{1} k_{1}^{'} & = 0 . \end{matrix}$ \matrix{ {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \hfill & { = 0\;{\rm{and}}} \hfill \cr{\kern 50pt}{\kern 50pt}{\kern 10pt}{3{k_1}k_1^{'}} \hfill & { = 0.} \hfill}

For the $k_{1}^{2} k_{1}^{'}$ k_1^2k_1^{'} to be zero, k₁ (s) has to be a constant function.

Then (3.20) turns to (3.21) $2 k_{1} k_{2}^{'} + k_{1} k_{2}^{2} = 0,$ 2{k_1}k_2^{'} + {k_1}k_2^2 = 0, by integrating the above equation (3.22) $\begin{matrix} 2 k_{2}^{'} + k_{2}^{2} & = 0, \\ \frac{k_{2}^{'}}{k_{2}^{2}} & = - \frac{1}{2}, \\ \int \frac{k_{2}^{'}}{k_{2}^{2}} ds & = - \int \frac{1}{2} ds, \end{matrix}$ \matrix{ {2k_2^{'} + k_2^2} \hfill & { = 0,} \hfill \cr{\kern 33pt} {{{k_2^{'}} \over {k_2^2}}} \hfill & { = - {1 \over 2},} \hfill \cr{\kern 9pt} {\int {{k_2^{'}} \over {k_2^2}}ds} \hfill & { = - \int {1 \over 2}ds,} \hfill} let (3.23) $k_{2} (s) = u,$ {k_2}\left( s \right) = u,(3.24) $k_{2}^{'} ds = du,$ k_2^{'}ds = du, therefore (3.22) turns to $\begin{matrix} \int \frac{du}{u^{2}} = - \int \frac{1}{2} ds, \\ - \frac{1}{u} = - \frac{1}{2} + c, \end{matrix}$ \matrix{ {\int {{du} \over {{u^2}}} = - \int {1 \over 2}ds,} \cr { - {1 \over u} = - {1 \over 2} + c,}} by simplifying the above equation and using (3.23) we get $k_{2} (s) = \frac{2}{s - 2 c} .$ {k_2}(s) = {2 \over {s - 2c}}.

Hence the proposition is proved.

Proposition 6

Let α be a Frenet curve of osculating order 3, then α is of weak AW (3)–type if and only if k₁ (s) is a constant function.

Proof

Since α is of weak AW (3)–type, by using (3.3) and (3.18) $\begin{array}{l} M_{3} (s) & = 〈 M_{3} (s), M_{1}^{⋆} (s) 〉 M_{1}^{⋆} (s), \\ = (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} \end{array}$ \matrix{ {{M_3}\left( s \right)} \hfill {\ = \left\langle {{M_3}\left( s \right),M_1^ \star \left( s \right)} \right\rangle M_1^ \star \left( s \right),} \hfill \cr {} \hfill {\kern 35pt} {\ = \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1}} \hfill}

Therefore $\begin{matrix} (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2} & = (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} \\ k_{1} k_{1}^{'} & = 0 . \end{matrix}$ \matrix{ {\left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2}} \hfill \ { = \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1}} \hfill \cr{\kern 217pt} {{k_1}k_1^{'}} \hfill { = 0.} \hfill}

For the $k_{1}^{2} k_{1}^{'}$ k_1^2k_1^{'} to be zero, k₁ (s) has to be a constant function. Hence the proposition is proved.

The helices according to Bishop frame of null cartan curve in Minkowski 3-space

Definition 4

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

Theorem 2

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

Proof

Let α be a general helix in $E_{1}^{3}$ E_1^3 and 〈T, U〉 is constant, then α is a general helix, from the definition we have (4.1) $〈 T, U 〉 = c c is constant,$ \left\langle {T,U} \right\rangle = c\;\;\;{\rm{c}}\,{\rm{is}}\,{\rm{constant}}, by differentiating the above equation (4.2) $\begin{array}{r} 〈 T^{'}, U 〉 + 〈 T, U^{'} 〉 = 0, \\ 〈 T^{'}, U 〉 = 0, \\ k_{2} cos θ + k_{1} sin θ = 0, \\ \frac{k_{1}}{k_{2}} = - cot θ (constant), \end{array}$ \matrix{ \hfill {\left\langle {{T^{'}},U} \right\rangle + \left\langle {T,{U^{'}}} \right\rangle = 0,} \cr \hfill {\left\langle {{T^{'}},U} \right\rangle = 0,} \cr \hfill {{k_2}\cos \,\,\theta + {k_1}\sin \,\,\theta = 0,} \cr \hfill {{{{k_1}} \over {{k_2}}} = - \cot \,\,\theta \;\;\;{\rm{(constant),}}}} as disered.

Theorem 3

Suppose that α is a null cartan curve in $E_{1}^{3}$ E_1^3 , then α is a general helix if and only if(4.3) $\det (T_{1}^{'}, T_{1}^{″}, T_{1}^{'″}) = k_{1}^{2} (k_{1} k_{2}^{″} - k_{2} k_{1}^{″}) .$ det(T_1^{'},T_1^{''},T_1^{'''}) = k_1^2\left( {{k_1}k_2^{''} - {k_2}k_1^{''}} \right).

Proof

(⇒) Let $\frac{k_{1}}{k_{2}}$ {{{k_1}} \over {{k_2}}} be constant. We have equalities as $\begin{array}{l} T_{1}^{'} (s) & = k_{2} T_{1} + k_{1} N_{1}, \\ T_{1}^{″} (s) & = (k_{2}^{'} + k_{2}^{2}) T_{1} + (k_{1}^{'} + k_{1} k_{2}) N_{1} + k_{1}^{2} N_{2}, \\ T_{1}^{'″} (s) & = (k_{2}^{″} + 3 k_{2} k_{2}^{'} + k_{2}^{3}) T_{1} \\ + (k_{1}^{″} + 2 k_{1} k_{2}^{'} + k_{1}^{'} k_{2} + k_{1} k_{2}^{2}) N_{1} + 3 k_{1} k_{1}^{'} N_{2} . \end{array}$ \matrix{ {T_1^{'}\left( s \right)} \hfill \ { = {k_2}{T_1} + {k_1}{N_1},} \hfill \cr {T_1^{''}\left( s \right)} \hfill \ { = \left( {k_2^{'} + k_2^2} \right){T_1} + \left( {k_1^{'} + {k_1}{k_2}} \right){N_1} + k_1^2{N_2},} \hfill \cr {T_1^{'''}\left( s \right)} \hfill \ { = \left( {k_2^{''} + 3{k_2}k_2^{'} + k_2^3} \right){T_1}} \hfill \cr {} \hfill {\kern 37pt} { + \left( {k_1^{''} + 2{k_1}k_2^{'} + k_1^{'}{k_2} + {k_1}k_2^2} \right){N_1} + 3{k_1}k_1^{'}{N_2}.} \hfill}

So we get $\begin{matrix} \det (T_{1}^{'}, T_{1}^{″}, T_{1}^{'″}) = | \begin{matrix} k_{2} & k_{1} & 0 \\ (k_{2}^{'} + k_{2}^{2}) & (k_{1}^{'} + k_{1} k_{2}) & k_{1}^{2} \\ (\begin{matrix} k_{2}^{″} + 3 k_{2} k_{2}^{'} \\ + k_{2}^{3} \end{matrix}) & (\begin{matrix} k_{1}^{″} + 2 k_{1} k_{2}^{'} \\ + k_{1}^{'} k_{2} + k_{1} k_{2}^{2} \end{matrix}) & 3 k_{1} k_{1}^{'} \end{matrix} |, \\ \det (T_{1}^{'}, T_{1}^{″}, T_{1}^{'″}) = - k_{1}^{2} k_{2}^{3} {(\frac{k_{1}}{k_{2}})}^{'} + 3 k_{1} k_{1}^{'} k_{2}^{2} {(\frac{k_{1}}{k_{2}})}^{'} + k_{1}^{2} (k_{1} k_{2}^{″} - k_{2} k_{1}^{″}) . \end{matrix}$ \matrix{ {det \,(T_1^{'},T_1^{''},T_1^{'''}) = \left| {\matrix{ {{k_2}} & {{k_1}} & 0 \cr {\left( {k_2^{'} + k_2^2} \right)} & {\left( {k_1^{'} + {k_1}{k_2}} \right)} & {k_1^2} \cr {\left( {\matrix{ {k_2^{''} + 3{k_2}k_2^{'}} \cr { + k_2^3} \cr } } \right)} & {\left( {\matrix{ {k_1^{''} + 2{k_1}k_2^{'}} \cr { + k_1^{'}{k_2} + {k_1}k_2^2} \cr } } \right)} & {3{k_1}k_1^{'}} \cr } } \right|,\,\,\,\,\,\,\,\,\,\,} \cr {det (T_1^{'},T_1^{''},T_1^{'''}) = - k_1^2k_2^3{{\left( {{{{k_1}} \over {{k_2}}}} \right)}^{'}} + 3{k_1}k_1^{'}k_2^2{{\left( {{{{k_1}} \over {{k_2}}}} \right)}^{'}} + k_1^2\left( {{k_1}k_2^{''} - {k_2}k_1^{''}} \right).}}

Since α is a general helix, and $\frac{k_{1}}{k_{2}}$ {{{k_1}} \over {{k_2}}} is constant. Hence, we have $\det (T_{1}^{'}, T_{1}^{″}, T_{1}^{'″}) = k_{1}^{2} (k_{1} k_{2}^{″} - k_{2} k_{1}^{″}), but k_{2} \neq 0 .$ det (T_1^{'},T_1^{''},T_1^{'''}) = k_1^2\left( {{k_1}k_2^{''} - {k_2}k_1^{''}} \right),\;{\rm{but}}\;{k_2} \ne 0.

(⇐) Suppose that $\det (T_{1}^{'}, T_{1}^{″}, T_{1}^{'″}) = k_{1}^{2} (k_{1} k_{2}^{″} - k_{2} k_{1}^{″})$ det (T_1^{'},T_1^{''},T_1^{'''}) = k_1^2\left( {{k_1}k_2^{''} - {k_2}k_1^{''}} \right) , then it is clear that the $\frac{k_{1}}{k_{2}}$ {{{k_1}} \over {{k_2}}} is constant, since ${(\frac{k_{1}}{k_{2}})}^{'}$ {\left( {{{{k_1}} \over {{k_2}}}} \right)^{'}} is zero. Hence the theorem is proved.

Theorem 4

Let α be a null cartan curve in $E_{1}^{3}$ E_1^3 , then α is a general helix if and only if(4.4) $\det (N_{1}^{'}, N_{1}^{″}, N_{1}^{'″}) = 0 .$ det(N_1^{'},N_1^{''},N_1^{'''}) = 0.

Proof

(⇒) Suppose that $\frac{k_{1}}{k_{2}}$ {{{k_1}} \over {{k_2}}} be constant. We have equalities as $\begin{array}{l} N_{1}^{'} = k_{1} N_{2}, \\ N_{1}^{″} = (k_{1}^{'} - k_{1} k_{2}) N_{2}, \\ N_{1}^{'″} = (k_{1}^{″} - 2 k_{1}^{'} k_{2} - k_{1} k_{2}^{'} + k_{1} k_{2}^{2}) N_{2} . \end{array}$ \matrix{ {N_1^{'} = {k_1}{N_2},} \hfill \cr {N_1^{''} = \left( {k_1^{'} - {k_1}{k_2}} \right){N_2},} \hfill \cr {N_1^{'''} = \left( {k_1^{''} - 2k_1^{'}{k_2} - {k_1}k_2^{'} + {k_1}k_2^2} \right){N_2}.} \hfill}

So we get $\begin{array}{l} \det (N_{1}^{'}, N_{1}^{″}, N_{1}^{'″}) = | \begin{matrix} 0 & 0 & k_{1} \\ 0 & 0 & - k_{1}^{2} \\ 0 & 0 & (k_{1}^{″} - 2 k_{1}^{'} k_{2} - k_{1} k_{2}^{'} + k_{1} k_{2}^{2}) \end{matrix} |, \\ \det (N_{1}^{'}, N_{1}^{″}, N_{1}^{'″}) = 0 . \end{array}$ \matrix{ {det (N_1^{'},N_1^{''},N_1^{'''}) = \left| {\matrix{ 0 & 0 & {{k_1}} \cr 0 & 0 & { - k_1^2} \cr 0 & 0 & {\left( {k_1^{''} - 2k_1^{'}{k_2} - {k_1}k_2^{'} + {k_1}k_2^2} \right)} \cr } } \right|,} \hfill \cr {det (N_1^{'},N_1^{''},N_1^{'''}) = 0.} \hfill}

(⇐) Suppose that $\det (N_{1}^{'}, N_{1}^{″}, N_{1}^{'″}) = 0$ det(N_1^{'},N_1^{''},N_1^{'''}) = 0 , then it is clear that the $\frac{k_{1}}{k_{2}}$ {{{k_1}} \over {{k_2}}} is constant, since ${(\frac{k_{1}}{k_{2}})}^{'}$ {\left( {{{{k_1}} \over {{k_2}}}} \right)^{'}} is zero. Hence the theorem is proved.

Theorem 5

Let α be a null cartan curve in $E_{1}^{3}$ E_1^3 , then α is a general helix if and only if(4.5) $\det (N_{2}^{'}, N_{2}^{″}, N_{2}^{'″}) = 0 .$ det(N_2^{'},N_2^{''},N_2^{'''}) = 0.

From (2.7)(4.6) $\begin{array}{l} α (s) = T, \\ D_{T} T = k_{2} T_{1} + k_{1} N_{1}, \\ D_{T} N_{1} = k_{1} N_{2}, \\ D_{T} N_{2} = - k_{2} N_{2} . \end{array}$ \matrix{\,\,\,\,{\alpha \left( s \right) = T,} \hfill \cr {\,\,\,{D_T}T = {k_2}{T_1} + {k_1}{N_1},} \hfill \cr {{D_T}{N_1} = {k_1}{N_2},} \hfill \cr {{D_T}{N_2} = - {k_2}{N_2}.} \hfill}

Theorem 6

Let $α : I \to E_{1}^{3}$ \alpha :I \to E_1^3be a unit speed null cartan curve with the cartan frame apparatus {T, N₂, N₂, k₁, k₂}, then α is a general helix if and only if(4.7) $D_{T} (D_{T} D_{T} N_{1}) + k_{1} k_{2} D_{T} N_{2} = (k_{1}^{″} - 3 k_{1}^{'} k_{2}) \frac{1}{k_{1}} D_{T} N_{1} .$ {D_T}\left( {{D_T}{D_T}{N_1}} \right) + {k_1}{k_2}{D_T}{N_2} = \left( {k_1^{''} - 3k_1^{'}{k_2}} \right){1 \over {{k_1}}}{D_T}{N_1}.

Proof

(⇒) Suppose that α is a general helix. Then, from (4.6), we have (4.8) $\begin{matrix} D_{T} N_{1} = k_{1} N_{2}, \\ D_{T} (D_{T} N_{1}) = k_{1}^{'} N_{2} - k_{1} k_{2} N_{2}, \end{matrix}$ \matrix{ {{D_T}{N_1} = {k_1}{N_2},}{\kern 28pt}\cr {{D_T}\left( {{D_T}{N_1}} \right) = k_1^{'}{N_2} - {k_1}{k_2}{N_2},}}(4.9) $D_{T} (D_{T} D_{T} N_{1}) = k_{1}^{″} N_{2} - (k_{1}^{'} k_{2} + k_{1} k_{2}^{'}) N_{2} - k_{1}^{'} k_{2} N_{2} - k_{1} k_{2} D_{T} N_{2} .$ {D_T}\left( {{D_T}{D_T}{N_1}} \right) = k_1^{''}{N_2} - \left( {k_1^{'}{k_2} + {k_1}k_2^{'}} \right){N_2} - k_1^{'}{k_2}{N_2} - {k_1}{k_2}{D_T}{N_2}.

Since α is a general helix (4.10) $\frac{k_{1}}{k_{2}} = c c is constant,$ {{{k_1}} \over {{k_2}}} = c\;\;\;c\,{\rm{is}}\,{\rm{constant}}, by differentiating (4.10) we get (4.11) ${(k_{1} k_{2})}^{'} = 2 k_{1}^{'} k_{2},$ {\left( {{k_1}{k_2}} \right)^{'}} = 2k_1^{'}{k_2}, but (4.12) $\begin{matrix} D_{T} N_{1} = k_{1} N_{2}, \\ N_{2} = \frac{1}{k_{1}} D_{T} N_{1} . \end{matrix}$ \matrix{ {{D_T}{N_1} = {k_1}{N_2},} \cr {{N_2} = {1 \over {{k_1}}}{D_T}{N_1}.}}

By substituting (4.11) and (4.12) in (4.9) we get (4.13) $\begin{matrix} D_{T} (D_{T} D_{T} N_{1}) & = (k_{1}^{″} - 3 k_{1}^{'} k_{2}) (\frac{1}{k_{1}} D_{T} N_{1}) - k_{1} k_{2} D_{T} N_{2}, \\ D_{T} (D_{T} D_{T} N_{1}) + k_{1} k_{2} D_{T} N_{2} & = (k_{1}^{″} - 3 k_{1}^{'} k_{2}) (\frac{1}{k_{1}} D_{T} N_{1}) . \end{matrix}$ \matrix{ {\kern 78pt}{{D_T}\left( {{D_T}{D_T}{N_1}} \right)} \hfill { = \left( {k_1^{''} - 3k_1^{'}{k_2}} \right)\left( {{1 \over {{k_1}}}{D_T}{N_1}} \right) - {k_1}{k_2}{D_T}{N_2},} \hfill \cr {{D_T}\left( {{D_T}{D_T}{N_1}} \right) + {k_1}{k_2}{D_T}{N_2}} \hfill { = \left( {k_1^{''} - 3k_1^{'}{k_2}} \right)\left( {{1 \over {{k_1}}}{D_T}{N_1}} \right).} \hfill} (⇐) We will show that null cartan curve α is a general helix. By differentianting (4.12) covariently (4.14) $\begin{matrix} N_{2} & = \frac{1}{k_{1}} D_{T} N_{1}, \\ D_{T} N_{2} & = - \frac{k_{1}^{'}}{k_{1}^{2}} D_{T} N_{1} + \frac{1}{k_{1}} D_{T} D_{T} N_{1}, \end{matrix}$ \matrix{ {\kern 20pt}{{N_2}} \hfill \ { = {1 \over {{k_1}}}{D_T}{N_1},} \hfill \cr {{D_T}{N_2}} \hfill \ { = - {{k_1^{'}} \over {k_1^2}}{D_T}{N_1} + {1 \over {{k_1}}}{D_T}{D_T}{N_1},} \hfill}(4.15) $D_{T} D_{T} N_{2} = {(- \frac{k_{1}^{'}}{k_{1}^{2}})}^{'} D_{T} N_{1} - \frac{2 k_{1}^{'}}{k_{1}^{2}} D_{T} D_{T} N_{1} + \frac{1}{k_{1}} D_{T} D_{T} D_{T} N_{1} .$ {D_T}{D_T}{N_2} = {\left( { - {{k_1^{'}} \over {k_1^2}}} \right)^{'}}{D_T}{N_1} - {{2k_1^{'}} \over {k_1^2}}{D_T}{D_T}{N_1} + {1 \over {{k_1}}}{D_T}{D_T}{D_T}{N_1}.

By substituting (4.8) and (4.13) in (4.15) we get (4.16) $\begin{matrix} D_{T} D_{T} N_{2} & = [{(- \frac{k_{1}^{'}}{k_{1}^{2}})}^{'} + (k_{1}^{″} - 3 k_{1}^{'} k_{2}) \frac{1}{k_{1}^{2}}] D_{T} N_{1} \\ - \frac{2 {(k_{1}^{'})}^{2}}{k_{1}^{2}} N_{2} - (\frac{2 k_{1}^{'}}{k_{1}} + k_{2}) D_{T} N_{2} . \end{matrix}$ \matrix{ {{D_T}{D_T}{N_2}} \ { = \left[ {{{\left( { - {{k_1^{'}} \over {k_1^2}}} \right)}^{'}} + \left( {k_1^{''} - 3k_1^{'}{k_2}} \right){1 \over {k_1^2}}} \right]{D_T}{N_1}} \cr{\kern 20pt} { - {{2{{\left( {k_1^{'}} \right)}^2}} \over {k_1^2}}{N_2} - \left( {{{2k_1^{'}} \over {{k_1}}} + {k_2}} \right){D_T}{N_2}.}}

From (4.6)(4.17) $\begin{matrix} D_{T} N_{2} = - k_{2} N_{2}, \\ D_{T} (D_{T} N_{2}) = - k_{2}^{'} N_{2} - k_{2} D_{T} N_{2} . \end{matrix}$ \matrix{{{D_T}{N_2} = - {k_2}{N_2},}{\kern 35pt} \cr {{D_T}\left( {{D_T}{N_2}} \right) = - k_2^{'}{N_2} - {k_2}{D_T}{N_2}.}} By comparing (4.16) and (4.17) $- (\frac{2 k_{1}^{'}}{k_{1}} + k_{2}) = - k_{2},$ - \left( {{{2k_1^{'}} \over {{k_1}}} + {k_2}} \right) = - {k_2}, by integrating the above equation we get $k_{1} = 1,$ {k_1} = 1, to find k₂, by comparing (4.16) and (4.17) we have $- \frac{2 {(k_{1}^{'})}^{2}}{k_{1}^{2}} = - k_{2}^{'} .$ - {{2{{\left( {k_1^{'}} \right)}^2}} \over {k_1^2}} = - k_2^{'}.

But k₁ = 1, therefore $k_{2}^{'} = 0,$ k_2^{'} = 0, which means k₂ is a constant function. $\frac{k_{1}}{k_{2}} is constant .$ {{{k_1}} \over {{k_2}}}\,{\rm{is\,constant}}. Hence α is a general helix.

Theorem 7

Let $α : I \to E_{1}^{3}$ \alpha :I \to E_1^3be a unit speed null cartan curve with the cartan frame apparatus {T, N₂, N₂, k₁, k₂}, then α is a general helix if and only if(4.18) $D_{T} (D_{T} D_{T} N_{2}) = (k_{2}^{″} - 3 k_{2} k_{2}^{'} + k_{2}^{3}) (\frac{1}{k_{2}} D_{T} N_{2}) .$ {D_T}\left( {{D_T}{D_T}{N_2}} \right) = \left( {k_2^{''} - 3{k_2}k_2^{'} + k_2^3} \right)\left( {{1 \over {{k_2}}}{D_T}{N_2}} \right).

The above theorem can be proven analogously, so we skip its proof.

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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A study on null cartan curve in Minkowski 3-space

Published Online: Mar 31, 2020

Page range: 413 - 424

Received: May 02, 2019

Accepted: Aug 30, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00039

KeywordsNull cartan curve, AW(k)-type curve, helix

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

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

Keywords
Null cartan curve, AW(k)-type curve, helix