On k-type pseudo null slant helices due to the Bishop frame in Minkowski 3-space E₁³

1.   Introduction

In the theory of curves, general helix is very important class of curves. One of the most elementary but widely researched topics is helix in the theory of curves. In his famous theorem, Lancret states that helices are characterized by the constant ratio of curvature and torsion. Slant helices as a special class of general helices were firstly defined by Izumiya and Takeuchi ^[1]. All helices (W-curves) have been completely classified in $E_1^3$ by Walrave ^[2]. Several authors introduced different types of helices and investigated their properties. Kula and Yayli studied spherical images of tangent and binormal indicatrices of slant helices and also showed that spherical images are spherical helices ^[3]. Kula et al. characterized slant helices in Euclidean 3-space ^[4]. Also the work ^[5] studies the physical applications of slant helices in the ordinary space.

The notion of k-type slant helices refers to the class of curves having a property that the scalar product of frame's vector field and a fixed axis is constant. The studies about k-type slant helices are as follows: Ergüt et al. studied non-null k-type slant helices in Minkowski 3-space ^[6]. Ali et al. examined k-type partially null and pseudo null curves in Minkowski 4-space $E_1^4$ ^[7]. Pseudo null Darboux helices, null Cartan Darboux helices, k-type pseudo null Darboux helices, and k-type null Cartan helices were discussed in ^[8,9,10]. Qian presented some results of k-type null slant helices in Minkowski space time ^[11]. Recently, Grbovic and Nešovic obtained some results of k-type null Cartan slant helices according to the generalized Bishop frame ^[12].

The vanishing of second derivative of a curve has led to the study of the new frame. First the behaviour of a curve was studied by a new adapted frame which is called Bishop frame or relatively parallel adapted frame ^[13]. This frame is composed of the vectors; the tangential vector field $T$, and two normal vector fields ${N_1}$ and ${N_2}$ which are obtained by rotating the Serret-Frenet vectors $N$ and $B$ in the normal plane ${T^ \bot }$ of the curve, in such a way that they become relatively parallel ^[13]. Bishop frame have been defined for curves in different Euclidean ambient spaces ^{[14,15,16,17]}. There is also interesting study which points out the physical applications of Bishop frame, see ^[18].

In this paper, we study k-type pseudo null slant helices according to two possible forms of the Bishop frame given by Grbovic and Nesovic ^[15]. We show that every pseudo null curve is a k-type pseudo null curve according to the Bishop frame in Minkowski 3-space $E_1^3.$ Then we find the axes of k-type pseudo null slant helices, and determine their causal characters.

2.   Preliminaries

The three dimensional Minkowski space $E_1^3$ is a real vector space ${R^3}$ endowed with the standard indefinite flat metric $\left\langle, \right\rangle$ defined by

where $x = \left({{x_1}, {x_2}, {x_3}} \right)$ and $y = \left({{y_1}, {y_2}, {y_3}} \right)$ are any two vectors in $E_1^3.$ Since this metric is an indefinite metric, an arbitrary vector $x \in E_1^3$ has one of three Lorentzian characters: it is a spacelike vector if $\langle x, x\rangle > 0$ or $x = 0$; timelike $\langle x, x\rangle < 0$ and null (lightlike) $\langle x, x\rangle = 0$ for $x \ne 0$ The pseudo-norm of the arbitrary vector $x \in E_1^3$ is given by $\|x\| = \sqrt{|\langle x, x\rangle|}$. Similarly, an arbitrary curve $\gamma = \gamma(s)$ in $E_1^3$ can locally be spacelike, timelike or null (lightlike) if its velocity vector $\gamma '$ is, respectively, spacelike, timelike or null (lightlike), for every $s \in I \subset E$. The curve $\gamma = \gamma (s)$ is called a unit speed curve if its velocity vector $\gamma '$ is unit one i.e, $\left\| {\gamma '} \right\| = 1$^[19,20].

A spacelike curve $\gamma :I \to E_1^3$ is called a pseudo null curve, if its principal normal vector field $N$ and binormal vector field B are null vector fields satisfying the condition $\langle N, B\rangle = 1$ The Frenet formulae of a non-geodesic pseudo null curve $\gamma = \gamma (s)$ have the form

where the first Frenet curvature $\kappa(s) = 1$ and the second Frenet curvature (torsion) $\tau(s)$ is an arbitrary function of arc-length parameter $s$ of $\gamma$ ^[2]. Also the vector fields of Frenet frame holds the following relations:

and

The Frenet frame $\{ T, N, B\}$ is positively oriented, if $(T, N, B) = [T, N, B] = 1$.

Definition 2.1. The Bishop frame $\left\{T_{1}, N_{1}, N_{2}\right\}$ of a pseudo null curve $\gamma$ in $E_1^3$ is positively oriented pseudo-orthonormal frame consisting of the tangential vector field ${{T_1}}$ and two relatively parallel lightlike normal vector fields ${{N_1}}$ and ${{N_2}}$ ^[15].

The vector fields of The Bishop frame of a pseudo null curve ${{N_2}}$ in $E_1^3$ satisfy the relations ^[15]

and

Theorem 2.1. (^[15]) Let $\gamma$ be a pseudo null curve in $E_1^3$ parameterized by the arc-length $s$ with the curvature ${\kappa _1}(s) = 1$ and the torsion $\tau (s)$:

(ⅰ) Then the Bishop frame $\left\{ {{T_1}, {N_1}, {N_2}} \right\}$ and the Frenet frame $\{ T, N, B\} {\rm{ }}$ of $\gamma$ are related by:

and the Frenet equations of $\gamma$ according to the Bishop frame read

where ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau (s)ds}}, {c_0} \in R_0^ +;$

(ⅱ) Then the Bishop frame $\left\{ {{T_1}, {N_1}, {N_2}} \right\}$ and the Frenet frame $\{ T, N, B\}$ of $\gamma$ are related by:

and the Frenet equations of $\gamma$ according to the Bishop frame read

where ${\kappa _1}(s) = {c_0}{e^{\int \tau (s)ds}}, {c_0} \in R_0^ -$ and ${\kappa _1}(s) = 0$.

3.   On k-type pseudo null slant helices due to the Bishop frame in Minkowski 3-space

In this section, we study k-type pseudo null slant helices framed by the Bishop frame in Minkowski 3-space $E_1^3.$ From Equations (2.7) and (2.8), there are two cases arising from the Bishop curvatures. In the first case, the first Bishop curvature ${\kappa _1}$ vanishes, and the vector field ${{N'}_1}$ is zero vector. In the second case, the second Bishop curvature ${\kappa _2}$ vanishes, and the vector field ${{N'}_2}$ is zero vector. We will examine these cases separately in this section.

Definition 3.1. A pseudo null curve $\gamma$ in $E_1^3$ given by the Bishop frame $\left\{ {{T_1}, {N_1}, {N_2}} \right\}$ is called a 0-type pseudo null slant helix if there exists a non zero fixed direction $V \in E_1^3$ such that satisfies

and a k-type pseudo null slant helices for $k \in\{1, 2\}$ if there exists a non zero fixed direction $V \in E_1^3$ such that hold

The fixed direction $V$ is called axis of the helix.

3.1.   The first case

Theorem 3.1. Every pseudo null curve $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and $\kappa_{2}(s) = c_{0} e^{\int \tau d s}$ is a k-type $\{k = 0, 1, 2\}$ pseudo null slant helix.

Proof. Let us take the pseudo curve $\gamma$ framed by the Bishop frame. According to Definition 3.1, there exists a fixed direction $V \in E_1^3$ such that

The fixed direction $V$ can be decomposed as

where ${\lambda _1}(s)$ and ${\lambda _2}(s)$ are some differential functions in terms of $s$. Differentiating the Eq. (3.3) with respect to $s$ and using (2.6), we have the following system of differential equations

From (3.4), we have

where ${c_1} \in R.$ Using (3.5), we have the axis $V$ as

Differentiating (3.6) and using (2.6) gives $V^{\prime}(s) = 0$ Hence, $V$ is a fixed direction. Thus, $\gamma$ is a 0-type pseudo null slant helix.

Let us show that pseudo null curve is also a 1-type pseudo null slant helix. According to Definition 3.1, there exists a fixed direction $V \in E_1^3$ such that

The fixed direction $V$ is decomposed as follows:

where ${\lambda _1}(s)$ and ${\lambda _2}(s)$ are some differential functions in terms of $s$. Differentiating the Eq. (3.8) with respect to $s$ and using (2.6), we have the following system of differential equations

From (3.9), we have

where $c \in R.$

Using (3.10), then we have

Differentiating (3.11) and using (2.6), then we arrive at ${V^\prime }(s) = 0$. Hence, $V$ is a fixed direction. Therefore, $\gamma$ is a 1-type pseudo null slant helix.

Let us show that pseudo null curve is also a 2-type pseudo null slant helix. Due to Definition 3.1, there exists a fixed direction $V \in E_1^3$ such that

The fixed direction $V$ is written as

where ${\lambda _1}(s)$ and ${\lambda _2}(s)$ are some differential functions in terms of $s$. Differentiating the Eq. (3.13) with respect to $s$ and using (2.6), we have the following differential equation system

From (3.14), we get

Using (3.15), the axis $V$ is obtained as

From (3.16) and (2.6), we find ${V^\prime }(s) = 0$. So, $V$ is a fixed direction.

As a result, every pseudo null curve according to the Bishop frame with the Bishop curvatures ${\kappa _1}(s) = 0$ and $\kappa_{2}(s) = c_{0} e^{\int \tau d s}$ is a k-type pseudo null slant helix.

Corollary 3.1. An axis of the 0-type null Cartan slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau ds}}$ is given by

where $c \in R$ and ${c_1} \in R.$

Corollary 3.2. An axis of the 1-type null Cartan slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau ds}}$ is given by

where $c \in R$.

Corollary 3.3. An axis of the 2-type null Cartan slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau ds}}$ is given by

where $c \in R$.

Corollary 3.4. The causal character of the axis $V$ of the 0-type pseudo null slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau ds}}$ is either spacelike or null.

Corollary 3.5. The causal character of the axis $V$ of the 1-type pseudo null slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau ds}}$ is

(ⅰ) spacelike if

(ⅱ) timelike if

(ⅲ) null if

Corollary 3.6. The causal character of the axis $V$ of the 2-type pseudo null slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = 0$ and ${\kappa _2}(s) = {c_0}{e^{\int \tau ds}}$ is null.

Example 3.1. Let us consider a pseudo null curve in $E_1^3$ given by (Figure 1)

According to the statement (ⅱ) of Theorem 2.1, the Bishop curvatures of $\gamma$ are

The Bishop frame of $\gamma$ is computed as

By Theorem 3.1, the pseudo null curve $\gamma$ holds 0, 1, 2-type slant helices whose axes are, respectively, calculated as follows:

all of these axes satisfy the Eqs. (3.2), (3.7), and (3.12).

3.2.   The second case

Theorem 3.2. Every pseudo null curve $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}$ and ${\kappa _2}(s) = 0$ is a k-type $\{k = 0, 1, 2\}$ pseudo null slant helix.

Proof. Let us take the pseudo curve $\gamma$ framed by the Bishop frame. According to Definition 3.1, there exists a fixed direction $V \in E_1^3$ such that

The fixed direction $V$ can be decomposed as

where ${\lambda _1}(s)$ and ${\lambda _2}(s)$ are some differential functions in terms of $s$. Differentiating the Eq. (3.18) with respect to $s$ and using (2.8), we have the following system of differential equations

From (3.19), we have

where ${c_1} \in R.$ Using (3.20), we arrive at

Using (2.8) in the differentiation of (3.21), then we find ${V^\prime }(s) = 0$. Hence, $V$ is a fixed direction. Thus, $\gamma$ is a 0-type pseudo null slant helix.

Let us show that pseudo null curve is also a 1-type pseudo null slant helix. According to Definition 3.1, there exists a fixed direction $V \in E_1^3$ such that

The fixed direction $V$ is decomposed by

where ${\lambda _1}(s)$ and ${\lambda _2}(s)$ are some differential functions in terms of $s$. If we differentiate the Eq. (3.23) and use Eq. (2.8), then we obtain the following system

By (3.24), we find

Using (3.25), then the axis $V$ is as

Using (2.8) in the differentiation of (3.26) gives ${V^\prime }(s) = 0$. From here, $\gamma$ is a 1-type pseudo null slant helix.

Consider pseudo null curve is also a 2-type pseudo null slant helix. According to Definition 3.1, there exists a fixed direction $V \in E_1^3$ such that

The fixed direction $V$ is decomposed by

where ${\lambda _1}(s)$ and ${\lambda _2}(s)$ are some differential functions in terms of $s$. If we differentiate the Eq. (3.28) and use the Eq. (2.8), then we obtain following differential equation system

By (3.29), we get

Using (3.30), then the axis $V$ is as

where $c \in R.$ Using (2.8) in the differentiation of (3.31) gives ${V^\prime }(s) = 0$. Hence, $V$ is a fixed direction.

Consequently, every pseudo null curve according to the Bishop frame with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}$ and ${\kappa _2}(s) = 0$ is a k-type ${\rm{\{ }}k{\rm{ = 0, 1, 2\} }}$ pseudo null slant helix.

Corollary 3.7. An axis of the 0-type null Cartan slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}$ and ${\kappa _2}(s) = 0$ is given by

where $c, {c_1} \in R$.

Corollary 3.8. An axis of the 1-type null Cartan slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}$ and ${\kappa _2}(s) = 0$ is given by

where $c \in R$.

Corollary 3.9. An axis of the 2-type null Cartan slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}$ and ${\kappa _2}(s) = 0$ is given by

where $c \in R$.

Corollary 3.10. The causal character of the axis $V$ of the 0-type pseudo null slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}$ and ${\kappa _2}(s) = 0$ is either spacelike or null.

Corollary 3.11. The causal character of the axis $V$ of the 1-type pseudo null slant helix $\gamma$ in $E_1^3$ with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}\ $ and ${\kappa _2}(s) = 0$ is null.

Corollary 3.12. The causal character of the axis $V$ of the 2-type pseudo null slant helix $\gamma$ in $E_1^3$with the Bishop curvatures ${\kappa _1}(s) = {c_0}{e^{\int \tau ds}}\ $ and ${\kappa _2}(s) = 0$ is

(ⅰ) spacelike if

(ⅱ) timelike if

(ⅲ) null if

Example 3.2. Let us consider a pseudo null curve in $E_1^3$ given by (Figure 2)

According to the statement (ⅱ) of Theorem 2.1, the Bishop curvatures of $\gamma$ are

The Bishop frame of $\gamma$ is computed as

By Theorem 3.2, the pseudo null curve $\gamma$ holds 0, 1, 2-type slant helices whose axes are, respectively, calculated as follows:

all of these axes satisfy the Eqs. (3.17), (3.22), and (3.26).

4.   Conclusion

In this study, we examine k-type pseudo null slant helices due to the Bishop frame given by Grbovic and Nešovic ^[15] under two different cases. We show that every pseudo null curve framed by the Bishop frame is a $k-$type pseudo null slant helix. We find parameter equation of axis $V$ of all k-type pseudo null slant helices in terms of the Bishop frame's vector fields. Finally, we determine the causal characters of the axes in two possible cases.

Acknowledgments

We would like to express our thanks to the anonymous referees who help us improved this paper.

Conflict of interest

All authors declare that there is no conflict of interest in this paper.