Null Hybrid Curves and Some Characterizations of Null Hybrid Bertrand Curves

Abstract

In this paper, we investigate null curves in ${\mathbb{R}}_2^4$, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in ${\mathbb{R}}_2^4$. We first derive Frenet formulas for a null curve in ${\mathbb{R}}_2^3$, the three-dimensional Minkowski space of index 2, by means of spatial hybrid numbers. Next, we apply the Frenet formulas for the associated null spatial hybrid curve corresponding to a null hybrid curve in order to derive the Frenet formulas for this curve in ${\mathbb{R}}_2^4$. This approach is simpler and more efficient than the classical differential geometry methods and enables us to determine a null curve in ${\mathbb{R}}_2^3$ corresponding to the null curve in ${\mathbb{R}}_2^4$. Additionally, we provide an example of a null hybrid curve, demonstrate the construction of its Frenet frame, and calculate the curvatures of the curve. Finally, we introduce null hybrid Bertrand curves, and by using their symmetry properties, we provide some characterizations of these curves.

1. Introduction

In this work, we focus on studying the differential geometry of null curves in ${\mathbb{R}}_2^4$. Bharathi [1] used quaternions and spatial quaternions to derive the Frenet–Serret formulas for a curve in four-dimensional Euclidean space (${\mathbb{E}}^4$). He first re-derived Frenet–Serret formulas in three-dimensional Euclidean space (${\mathbb{E}}^3$) with the aid of spatial quaternions, then used them to derive the Frenet–Serret formulas for a quaternion-valued function corresponding to a Euclidean curve in ${\mathbb{E}}^4$. The elements of ${\mathbb{R}}_2^4$ are identified with hybrid numbers, a concept introduced by Özdemir [2], in a similar manner as the elements of four-dimensional Euclidean space ${\mathbb{E}}^4$ are identified with quaternions. Consequently, it is natural to use hybrid numbers to derive Frenet formulas for a curve in ${\mathbb{R}}_2^4$. Akbıyık [3] introduced hybrid curves and, using a method similar to Bharathi’s, derived Frenet formulas for non-null hybrid curves. Further studies about hybrid numbers can be found in [4,5].

Null curves play a substantial role in physics, particularly in the context of relativity and space-time geometry. Null paths are crucial in physics because they describe the behavior of massless particles that travel at the speed of light. They play a central role in understanding black holes, gravitational waves, and cosmology. They are also of particular interest in the study of singularities of surfaces in different spaces. Consequently, there are many studies in the literature about null curves and Frenet frames [6,7,8,9,10,11,12,13,14].

Frenet equations along null curves in semi-Riemannian space are not unique, since they depend on the parameterization and the screen vector bundle of the curve ($S\left(T{\gamma }^{\perp }\right)$). Bonnor [15] introduced the Cartan frame for null curves in ${\mathbb{R}}_1^4$, which provides a unique Frenet frame with the fewest possible curvatures. Later, Ferrandez et al. [16] generalized this concept to null Cartan curves in ${\mathbb{R}}_1^{(m+2)}$. Furthermore, Duggal and Jin [17] presented Frenet equations for null curves in ${\mathbb{M}}_q^{(m+2)}$, $(m+2)$-dimensional pseudo-Riemannian manifolds of index q. They introduced the type 1 Frenet frame, consisting of two null, $(q-1)$ time-like, and $(m-q+1)$ space-like vector fields; the type 2 Frenet frame, consisting of four null, $(q-2)$ time-like, and $(m-q)$ space-like vector fields; and the type r Frenet frame ($3\le r\le q$), consisting of $2r$ null, $(q-r)$ time-like, and $(m+2-q-r)$ space-like vector fields. They also introduced the natural Frenet equations, which are not affected by the choice of $S\left(T{\gamma }^{\perp }\right)$, generalizing the Cartan equations.

Studies on null quaternionic curves can be found in [18,19], and studies on Bertrand curves and null Bertrand curves in four-dimensional semi-Riemannian space are reported in [20,21,22,23,24,25,26,27]. Moreover, Tuna [28,29], and Kahraman [30] have investigated null quaternionic Bertrand curves.

The paper is structured as follows: In Section 2, we provide some general knowledge about hybrid numbers and their properties, hybrid curves, and null curves in ${\mathbb{M}}_q^{(m+2)}$, which are used throughout this work. In Section 3, we use spatial hybrid numbers to construct a null frame in ${\mathbb{R}}_2^3$. In Section 4, we derive the general Frenet formulas and the Cartan formulas for a null spatial hybrid curve in ${\mathbb{R}}_2^3$ with the aid of spatial hybrids. In Section 5, we introduce the associated null spatial hybrid curve of a null hybrid curve in ${\mathbb{R}}_2^4$, and we utilize the Frenet formulas for the associated null spatial hybrid curve in order to derive the Frenet formulas for this curve in ${\mathbb{R}}_2^4$. We also give an example of a null hybrid curve, construct a Frenet frame along this curve, and calculate its curvatures. In Section 6, we define the null Bertrand hybrid curve and provide some characterizations.

2. Preliminary

In this section, we provide general knowledge about hybrid numbers and their properties, hybrid curves, and null curves in $M_q^{m+2}$.

2.1. Hybrid Numbers and Their Properties

Özdemir [2] presented hybrid numbers as a broader concept that generalizes complex, hyperbolic, and dual numbers. Let

$$\begin{array}{cc}\hfill \mathbb{K}=\{q_0+q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}:q_0,q_1,q_3,q_3\in \mathbb{R},& \mathbf{i}\times \mathbf{i}=-1,\mathit{ϵ}\times \mathit{ϵ}=0,\mathbf{h}\times \mathbf{h}=1,\\ & \mathbf{i}\times \mathbf{h}=-\mathbf{h}\times \mathbf{i}=\mathit{ϵ}+\mathbf{i}\}\end{array}$$

(1)

be a set of hybrid numbers. Then, for the hybrid number $Q=q_0+q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$, $S\left(Q\right)=q_0$ is called the scalar part and $V\left(Q\right)=q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$ is called the vector part of Q [2].

It is well known that the geometries of Euclidean, Minkowski, and Galilean planes are described by complex, hyperbolic, and dual numbers, respectively. The corresponding geometry of hybrid numbers is the the most general geometry that includes the combination of the Euclidean, Minkowski, and Galilean plane geometries. This geometry is called a hybrid plane geometry.

Let $Q=q_0+q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$ and $Q^{*}=q_0^{*}+q_1^{*}\mathbf{i}+q_2^{*}\mathit{ϵ}+q_3^{*}\mathbf{h}$ be two hybrid numbers. Then,

• According to Ref. [2], the hybridian product,

  $$Q\times Q^{*}=(q_0+q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h})\times (q_0^{*}+q_1^{*}\mathbf{i}+q_2^{*}\mathit{ϵ}+q_3^{*}\mathbf{h}),$$

  (2)

  can be obtained by using the relations given in (1). The table of multiplication is given in

  Table 1. Products of $\{\mathbf{1},\mathbf{i},\mathit{ϵ},\mathbf{h}\}$.

  ×	1	i	$\mathit{ϵ}$	h
  1	1	i	$\mathit{ϵ}$	h
  i	i	$-\mathbf{1}$	$\mathbf{1}-\mathbf{h}$	$\mathit{ϵ}+\mathbf{i}$
  $\mathit{ϵ}$	$\mathit{ϵ}$	$\mathbf{1}+\mathbf{h}$	0	$-\mathit{ϵ}$
  h	h	$-\mathit{ϵ}-\mathbf{i}$	$\mathit{ϵ}$	1

  :
• According to Ref. [2], the conjugate of Q is defined by

  $$\overline{Q}=S\left(Q\right)-V\left(Q\right)=q_0-q_1\mathbf{i}-q_2\mathit{ϵ}-q_3\mathbf{h}.$$
  It can be shown that for a hybrid number ($Q=q_0+q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$), $Q\times \overline{Q}=\overline{Q}\times Q$.
• According to Ref. [2], the character of Q is defined by

  $$C\left(Q\right)=q_0^2+{(q_1-q_2)}^2-q_2^2-q_3^2=-{\langle {\nu }_Q,{\nu }_Q\rangle }_{R_2^4},$$

  (3)

  where

  $${\nu }_Q=(q_0,(q_1-q_2),q_2,q_3)$$

  (4)

  is the vector representation of a hybrid number (Q) and ${\langle ,\rangle }_{{\mathbb{R}}_2^4}$ is the inner product in ${\mathbb{R}}_2^4$ with a signature of $(-,-,+,+)$. Therefore, the set of hybrid numbers is identified with ${\mathbb{R}}_2^4$. The hybrid number (Q) is called space-like if $C\left(Q\right)<0$, time-like if $C\left(Q\right)>0$, and light-like (null) if $C\left(Q\right)=0$.
• According to Ref. [2], the norm of Q is defined by $\parallel Q\parallel =\sqrt{\left|C\right(Q\left)\right|}$.
• According to Ref. [2], the inner product is defined by the following symmetric bilinear form:

  $$\begin{array}{ccc}\hfill h(Q,Q^{*})& =& {\displaystyle \frac{1}{2}}(Q\times \overline{Q^{*}}+Q^{*}\times \overline{Q})\hfill \end{array}$$

  (5)

  $$\begin{array}{ccc}& =& q_0{q}_0^{*}+q_1{q}_1^{*}-q_1{q}_2^{*}-q_2{q}_1^{*}-q_3{q}_3^{*}.\hfill \end{array}$$

  (6)

The hybridian product is non-commutative and associative, and we can represent it as a multiplication of two matrices:

$$Q\times Q^{*}=\left[\begin{array}{cccc}{q}_0& -(q_1-q_2)& q_1& q_3\\ q_1& q_0-q_3& 0& q_1\\ q_2& -q_3& q_0+q_3& q_1-q_2\\ q_3& q_2& -q_1& q_0\end{array}\right]\left[\begin{array}{c}{q}_0^{*}\\ q_1^{*}\\ q_2^{*}\\ q_3^{*}\end{array}\right].$$

(7)

${\mathbb{K}}^p=\{H\in \mathbb{K}|H+\overline{H}=0\}$ denotes the set of all spatial hybrid numbers. The inner product is a symmetric bilinear form:

$$h(q,q^{*})=q_1{q}_1^{*}-q_1{q}_2^{*}-q_2{q}_1^{*}-q_3{q}_3^{*}$$

(8)

where $q=q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$ and $q^{*}=q_1^{*}\mathbf{i}+q_2^{*}\mathit{ϵ}+q_3^{*}\mathbf{h}$. This set (${\mathbb{K}}^p$) is identified with the three-dimensional Minkowski space (${\mathbb{R}}_2^3$) by identifying each spatial hybrid number ($q=q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$) with a unique element ($(q_1,q_2,q_3)\in {\mathbb{R}}_2^3$). If q is a spatial hybrid number, then $\overline{q}=-q$.

2.2. Hybrid Curves

Hybrid curves were represented by Akbıyık in [3]. The curve expressed as $\Gamma :t\to \Gamma \left(t\right)=x^0\left(t\right)+x^1\left(t\right)\mathbf{i}+x^2\left(t\right)\mathit{ϵ}+x^3\left(t\right)\mathbf{h},t\in I\subseteq \mathbb{R}$, where $x^i$ represents differentiable functions, is called a hybrid curve. Let $A={\Gamma }^{\prime }\left(t\right)$; then, the curve is called space-like, time-like, or null if $h(A,A)<0$, $h(A,A)>0$, or $h(A,A)=0$, respectively. The $C^{\infty }$ differentiable curve ($\gamma :t\to \gamma \left(t\right)$) given by $\gamma \left(t\right)=x^1\left(t\right)\mathbf{i}+x^2\left(t\right)\mathit{ϵ}+x^3\left(t\right)\mathbf{h},t\in I\subset \mathbb{R}$ is called a spatial hybrid curve. Let $\gamma$ be a spatial hybrid curve whose Frenet frame is $F=\{A\times \overline{C}=a,b,c\}$, where A and C are the tangent and the principal normal to the curve $\Gamma \left(s\right)$, respectively. Then, the $\gamma$ curve is called the associated curve of $\Gamma$.

2.3. Null Curves in $M_q^{m+2}$

Let $\gamma \left(t\right)$, $t\in I\subseteq \mathbb{R}$ be a null curve in $M_q^{m+2}$. If $\gamma \left(t\right)$ is given by $x^i=x^i\left(t\right)$, $i=\{1,2,\cdots$, $m+2\}$, then the tangent vector field expressed as

$$a=\left({\displaystyle \frac{d{x}^1}{dt}},{\displaystyle \frac{d{x}^2}{dt}},\cdots ,{\displaystyle \frac{d{x}^{m+2}}{dt}}\right)$$

(9)

satisfies

$$h\left(a,a\right)=0.$$

(10)

If $h(a^{\prime },a^{\prime })=\mu$ and ${\mu }^2=1$, then $\gamma$ is said to be pseudo-arc length-parameterized and t is called the pseudo-arc parameter [16,31].

Let $T\gamma$ be the tangent bundle of $\gamma$ and

$$T{\gamma }^{\perp }=\{v\in \Gamma \left(T{\mathbb{R}}_q^{m+2}\right):g(v,a)=0\}$$

(11)

be the normal bundle of $T\gamma$; then, $dim\left(T{\gamma }^{\perp }\right)=m+1$. Since $g(a,a)=0$, $T\gamma \subset T{\gamma }^{\perp }$, and the rank of $T\gamma$ is 1. Therefore, the decomposition of the tangent bundle into components tangent to $\gamma$ and normal to $\gamma$ is not unique. A vector bundle complementary to $T\gamma$ in $T{\gamma }^{\perp }$ is called a screen vector bundle of $\gamma$, denoted by $S\left(T{\gamma }^{\perp }\right)$. The bundle ($S\left(T{\gamma }^{\perp }\right)$) is an m-dimensional Minkowski space of index $q-1$. A complementary orthogonal bundle to $S\left(T{\gamma }^{\perp }\right)$ in $T{M}_{|\gamma }$ is denoted by $S{\left(T{\gamma }^{\perp }\right)}^{\perp }$. Thus, we have

$${T{M}_q^{m+2}}_{|\gamma }=S\left(T{\gamma }^{\perp }\right){\oplus }_{orth}S{\left(T{\gamma }^{\perp }\right)}^{\perp },$$

(12)

where $S{\left(T{\gamma }^{\perp }\right)}^{\perp }$ contains $T\gamma$ and has a dimension of 2. This decomposition is not unique, since it depends on the choice of $S\left(T{\gamma }^{\perp }\right)$, which is not unique [17].

Theorem 1.

Let γ be a null curve on $M_q^{m+2}$ with $S\left(T{\gamma }^{\perp }\right)$. Then, there is exactly one null transversal bundle ($ntr\left(\gamma \right)$) with respect to $S\left(T{\gamma }^{\perp }\right)$ and $b\in ntr\left(\gamma \right)$ satisfying

$$h(a,b)=1,h(b,b)=h(b,v)=0,\forall v\in S\left(T{\gamma }^{\perp }\right).$$

(13)

Then,

$$T{M}_{|\gamma }=T\gamma \oplus ntr\left(\gamma \right){\oplus }_{orth}S\left(T{\gamma }^{\perp }\right)=T\gamma \oplus tr\left(T\gamma \right),$$

(14)

where $tr\left(\gamma \right)=ntr\left(\gamma \right){\oplus }_{orth}S\left(T{\gamma }^{\perp }\right)$ is called a $transversal$ bundle corresponding to $S\left(T{\gamma }^{\perp }\right)$ [17].

Vector b can be calculated as

$$b={\displaystyle \frac{1}{h(a,v)}}\{v-{\displaystyle \frac{h(v,v)}{2h(a,v)}}a\},v\in S\left(T{\gamma }^{\perp }\right).$$

(15)

Moreover, if $h(a^{\prime },b)=0$ is satisfied, then the curve expressed as $\gamma$: $s\in I\subset \mathbb{R}\to M_2^{m+2}$ is said to be parameterized by a distinguished parameter [17]. The distinguished parameter is denoted by p. Duggal and Jin [17] defined the Frenet frames in $M_2^{m+2}$ along null curves containing two null, one time-like, and $(m-1)$ space-like vector fields. These frames are called the general Frenet frames of type 1. They prove the subsequent Theorem.

Theorem 2

([17]). For a given set (${\kappa }_1,{\kappa }_2,\cdots$, ${\kappa }_{2m}$) of real-valued continuous functions, a fixed point of ${\mathbb{R}}_2^{m+2}$ and an orthonormal basis ($N_0^0,N_1^0,W_1^0\dots$, $W_m^0$), there is exactly one null curve (γ: $x^i=x^i\left(p\right)$) parameterized by a distinguished parameter (p) such that $\gamma \left(0\right)=x_0$. The curvatures of this curve are $\{{\kappa }_1,{\kappa }_2,\cdots$, ${\kappa }_{2m}\}$, and the Frenet frames of type 1 are $\{{\displaystyle \frac{d}{dp}},N,W_1,\cdots$, $W_m\}$, where ${\displaystyle \frac{d}{dp}}=N_0^0$, $N\left(0\right)=N_1^0$ and $W\left(0\right)=W_{\mathbf{i}}^0$ for $\mathbf{i}=1,2,\cdots$, m.

3. Null Frames in the Space of Spatial Hybrid Numbers

Spatial hybrid numbers $a_0$, $b_0$, and $c_0$ are defined as

$$\begin{array}{ccc}\hfill a_0& =& -\mathit{ϵ},\hfill \\ \hfill b_0& =& {\displaystyle \frac{1}{2}}\mathit{ϵ}+\mathbf{i},\hfill \\ \hfill c_0& =& \mathbf{h}.\hfill \end{array}$$

(16)

Then, by using the scalar product of hybrid numbers and the multiplication Table 1, we obtain the following equations:

$$\begin{array}{c}\hfill h(a_0,a_0)=h(b_0,b_0)=h(a_0,c_0)=h(b_0,c_0)=0,\\ \hfill h(a_0,b_0)=1,h(c_0,c_0)=-1.\end{array}$$

The set expressed as $F_0=\{a_0,b_0,c_0\}$ is a frame in ${\mathbb{K}}^p$, where $a_0$ and $b_0$ are null and $c_0$ is a space-like spatial hybrid number. By using the hybridian product, we construct

Table 2. Products of $\{a_0,b_0,c_0\}$.

×	$a_0$	$b_0$	$c_0$
$a_0$	0	$-1-c_0$	$-a_0$
$b_0$	$-1+c_0$	0	$b_0$
$c_0$	$a_0$	$-b_0$	1

for this frame.

4. Frenet and Cartan Formulas for Null Spatial Hybrid Curves

In this section, we derive Frenet and Cartan formulas for a null curve in ${\mathbb{R}}_2^3$ by using spatial hybrid numbers.

Let $\gamma$: $t\to \gamma \left(t\right)$ be a spatial hybrid curve given by $\gamma \left(t\right)=x^1\left(t\right)\mathbf{i}+x^2\left(t\right)\mathit{ϵ}+x^3\left(t\right)\mathbf{h}$, $t\in I\subset \mathbb{R}$. Let $a={\gamma }^{\prime }\left(t\right)$ and $h(a,a)=0$; then, $\gamma$ is a null spatial hybrid curve. Based on $h\left(a,a\right)=0$, we find $h(a^{\prime },a)=0$, which implies that $a^{\prime }\in S\left(T{\gamma }^{\perp }\right)$, that is, $a^{\prime }$ is a space-like spatial hybrid. Thus, there exists a unique spatial hybrid ($b\in ntr\left(\gamma \right)$) defined by $b={\displaystyle \frac{1}{h(a,v)}}\{v-{\displaystyle \frac{h(v,v)}{2h(a,v)}}a\}$, where v is a space-like spatial hybrid such that

$$h(a,b)=1,h(b,b)=0.$$

(17)

We define spatial hybrid c along $\gamma$ as $c=b\times a$; then, we have

$$\begin{array}{c}\hfill h(a,c)=h(b,c)=0,\\ \hfill h(c,c)=h(c,b\times a)=h(-b,a)=-1.\end{array}$$

The set expressed as $F=\{a,b,c\}$ generates a Frenet frame in ${\mathbb{K}}^p$ along the null spatial hybrid curve ($\gamma$). We derive the derivative formulas of this frame according to the following theorem.

Theorem 3.

Let γ: $t\to \gamma \left(t\right)$ be a null spatial hybrid curve and $F=\{a,b,c\}$ be a Frenet frame along this curve. Then, the derivatives of a, b, and c are given by

$$\left[\begin{array}{c}{a}^{\prime }\\ b^{\prime }\\ c^{\prime }\end{array}\right]=\left[\begin{array}{ccc}\delta & 0& \kappa \\ 0& -\delta & \tau \\ \tau & \kappa & 0\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right],$$

(18)

where $\kappa =-h(a^{\prime },c)$, $\tau =-h(b^{\prime },c)$, and $\delta =h(a^{\prime },b)$ are differentiable functions. κ is the curvature, and τ the torsion of the curve (γ).

Proof. 

Let $h(a^{\prime },b)=\delta$; then, according to $h(a^{\prime },a)=0$ and (17),

$$a^{\prime }=\delta a+r,$$

(19)

where $r\in S\left(T{\gamma }^{\perp }\right)$ is a space-like spatial hybrid. Let $\rho =\parallel r\parallel$ and the curvature function be $\kappa =-\rho$. Then, the unit space-like spatial hybrid (c) can be written as $c=-{\displaystyle \frac{1}{\rho }}r$. According to (19), we find

$$a^{\prime }=\delta a+\kappa c.$$

Also, by using $h(b^{\prime },b)=0,h(b^{\prime },a)=-\delta$ and $h(b^{\prime },c)=-\tau$, we find

$$b^{\prime }=-\delta b+\tau c.$$

Then, by using $h(a,c^{\prime })=-h(a^{\prime },c)=\kappa$ and $h(b,c^{\prime })=-h(b^{\prime },c)=\tau$, we find

$$c^{\prime }=\tau a+\kappa b,$$

which proves the theorem. □

In accordance with [32], Equation (18) is called a general Frenet equation regarding the frame (F) along the null spatial hybrid curve ($\gamma \left(t\right)$).

Note that if $\gamma$: $p\to \gamma \left(p\right)$ is a curve parameterized by a distinguished parameter (p), then the Frenet equations regarding to the Frenet frame F are

$$\left[\begin{array}{c}{a}^{\prime }\\ b^{\prime }\\ c^{\prime }\end{array}\right]=\left[\begin{array}{ccc}0& 0& \kappa \\ 0& 0& \tau \\ \tau & \kappa & 0\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$$

(20)

Now, we obtain the unique frame for the curve expressed as $\gamma$: $p\to \gamma \left(p\right)$ with the minimum number of curvatures.

Theorem 4.

Let $\gamma \left(p\right)$ be a null spatial hybrid curve such that $B=\{{\gamma }^{\prime }\left(p\right),{\gamma }^{″}\left(p\right),{\gamma }^{\prime \prime \prime }\left(p\right)\}$ is a basis for $T\gamma$; then, there is exactly one frame ($CF=\{l,n,u\}$) such that

$$\left[\begin{array}{c}{l}^{\prime }\\ n^{\prime }\\ u^{\prime }\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& \sigma \\ \sigma & 1& 0\end{array}\right]\left[\begin{array}{c}l\\ n\\ u\end{array}\right].$$

(21)

Proof. 

Let $\gamma$ be a null spatial hybrid curve, and let $l={\gamma }^{\prime }$ and $u={\gamma }^{″}$ so that ${\kappa }_1=1$. Since $h({\gamma }^{\prime },{\gamma }^{\prime })=0$, we find that $h({\gamma }^{\prime },{\gamma }^{″})=0$; then, by differentiating this equation and using the fact that ${\gamma }^{″}$ is space-like, we find $h({\gamma }^{\prime },{\gamma }^{‴})=1$. Then, according to (15), we obtain

$$n={\gamma }^{‴}-\tau {\gamma }^{\prime },$$

where we define $\tau ={\displaystyle \frac{1}{2}}h({\gamma }^{‴},{\gamma }^{‴})$. Therefore, we find $l^{\prime }={\gamma }^{″}=u$, $u^{\prime }={\gamma }^{‴}=\sigma l+n$, and since $h(n^{\prime },n)=0$, $h(n^{\prime },l)=-h(n,l^{\prime })=-h(n,u)=0$ and $h(n^{\prime },u)=h(n,u^{\prime })=h(n,\sigma l+n)=\sigma$, we obtain

$$\begin{array}{c}\hfill n^{\prime }=h(n^{\prime },n)l+h(n^{\prime },l)n-h(n^{\prime },u)u=\sigma u.\end{array}$$

□

In accordance with Bonnor [15], Equation (21) is called the Cartan Frenet equation. It can be seen from the Frenet equation that $\kappa =1$ and $\tau =\sigma$.

5. General Frenet Frames for Null Hybrid Curves

We identify the set of hybrid numbers ($\mathbb{K}$) with the semi-Riemannian space (${\mathbb{R}}_2^4$) by identifying each hybrid number ($H=q_0+q_1\mathbf{i}+q_2\mathit{ϵ}+q_3\mathbf{h}$) with a unique element ($(q_0,q_1,q_2,q_3)\in {\mathbb{R}}_2^4$). Let

$$\Gamma :t\to \Gamma \left(t\right)=x^0\left(t\right)+x^1\left(t\right)\mathbf{i}+x^2\left(t\right)\mathit{ϵ}+x^3\left(t\right)\mathbf{h},t\in I\subseteq \mathbb{R},$$

be a null curve in $\mathbb{K}$ such that $A=span\{\Gamma \}$. Let $A\left(t\right)={\Gamma }^{\prime }\left(t\right)$; then, $h(A,A)=0$. Since $A^{\prime }\in T{\Gamma }^{\perp }$, we can write

$$A^{\prime }=\varphi A+R_1,$$

(22)

where $R_1\in S\left(T{\Gamma }^{\perp }\right)$. Since $S\left(T{\Gamma }^{\perp }\right)$ is a Lorentzian space, $R_1$ may be space-like, time-like, or null. In this study, we assume that it is non-null. We define ${\rho }_1=\parallel R_1\parallel$ and the first curvature as $K_1=\eta {\rho }_1$, where $\eta \in \{1,-1\}$, depending on the causality of $R_1$. We set $C_1={\displaystyle \frac{R_1}{{\rho }_1}}$; then, $C_1$ is a unit hybrid with $h(C_1,C_1)=\eta$. Therefore, we can write Equation (22) as

$$A^{\prime }=\varphi A+\eta K_1{C}_1.$$

(23)

Based on $h(C_1,A)=0$, we find

$$C_1\times \overline{A}+A\times {\overline{C}}_1=0,$$

which implies that $A\times {\overline{C}}_1$ is a spatial hybrid. If we define

$$A\times {\overline{C}}_1=a,$$

(24)

we find

$$h(a,a)=(A\times {\overline{C}}_1)\times \left(\overline{A\times {\overline{C}}_1}\right)=A\times {\overline{C}}_1\times C_1\times \overline{A}=\eta A\times \overline{A}=0.$$

Lemma 1.

Let Γ be a null hybrid curve with ${\Gamma }^{\prime }=A$ and $A^{\prime }=\varphi A+\eta K_1{C}_1$, where A is null and $C_1$ is a space-like or timel-ike hybrid with $h(C_1,C_1)=\eta$, $\eta \in \{-1,1\}$. Let γ be its associated curve with a Frenet frame ($F=\{A\times {\overline{C}}_1=a,b,c\}$), then the set expressed as ${\mathfrak{F}}_1=\{A,B,C_1,D_1\}$ where $B=b\times C_1$ and $D_1=c\times C_1$, is an orthonormal set.

Proof. 

From Equation (24), we find

$$A=\eta a\times C_1.$$

(25)

We define

$$B=b\times C_1$$

(26)

and

$$D_1=c\times C_1.$$

(27)

Then by using Equations (25)–(27) we find

$$\begin{array}{ccc}\hfill h(B,B)& =& h(b\times C_1,b\times C_1)=(b\times C_1)\times \left(\overline{b\times C_1}\right)=b\times C_1\times \overline{C_1}\times \overline{b}=\eta h(b,b)=0,\hfill \\ \hfill h(D_1,D_1)& =& h(c\times C_1,c\times C_1)=(c\times C_1)\times \left(\overline{c\times C_1}\right)=c\times C_1\times {\overline{C}}_1\times \overline{c}=\eta h(c,c)=-\eta ,\hfill \\ \hfill h(A,D_1)& =& {\displaystyle \frac{1}{2}}\{(\eta a\times C_1)\times \left(\overline{c\times C_1}\right)+\eta (c\times C_1)\times \left(\overline{a\times C_1}\right)\}={\displaystyle \frac{1}{2}}(a\times \overline{c}+c\times \overline{a})\hfill \\ & =& h(c,a)=0,\hfill \\ \hfill h(C_1,D_1)& =& h(C_1,c\times C_1)={\displaystyle \frac{1}{2}}\{C_1\times \left(\overline{c\times C_1}\right)+(c\times C_1)\times {\overline{C}}_1\}={\displaystyle \frac{1}{2}}(\eta \overline{c}+\eta c)=0,\hfill \\ \hfill h(B,A)& =& h(b\times C_1,\eta a\times C_1)={\displaystyle \frac{1}{2}}\{b\times C_1)\times \left(\eta \overline{a\times C_1}\right)+(\eta a\times C_1)\times \left(\overline{b\times C_1}\right\}\hfill \\ & =& h(b,a)=1.\hfill \end{array}$$

Therefore, the set expressed as ${\mathfrak{F}}_1=\{A,B,C_1,D_1\}$ is an orthonormal frame along the null hybrid curve ($\Gamma \left(t\right)$) that contains two null hybrids: one space-like hybrid and one time-like hybrid. □

Theorem 5.

Let $\Gamma \left(t\right)$ be a null hybrid curve and $\mathfrak{F}=\{A,B,C=\eta C_1,D=-\eta D_1\}$ be a frame along this curve as constructed in Lemma 1. Let $\gamma \left(t\right)$ be an associated null spatial hybrid curve of Γ and $F=\{a,b,c;\kappa ,\tau \}$ be a Frenet frame along this curve. Then, the derivatives of the frame ($\mathfrak{F}$) are given by

$$\left[\begin{array}{c}{A}^{\prime }\\ B^{\prime }\\ C^{\prime }\\ D^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\varphi & 0& K_1& 0\\ 0& -\varphi & K_2& K_3\\ -\eta K_2& -\eta K_1& 0& K_4\\ \eta K_3& 0& -K_4& 0\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\\ D\end{array}\right],$$

(28)

where $\varphi =h(A^{\prime },B)$, $K_1=\eta h(A^{\prime },C)$, $K_2=\eta h(B^{\prime },C)$, $K_3=K_2-\eta \tau$ and $K_4=\eta (C^{\prime },D)$.

Proof. 

Let $\Gamma \left(s\right)$ be a null hybrid curve, and let $\gamma \left(s\right)$ be its associated null spatial hybrid curve with a Frenet frame (F). Let $A^{\prime }=\varphi A+\eta K_1{C}_1$; then, by differentiating (25) and using (18) and (27), we find

$$A^{\prime }=\eta a^{\prime }\times C_1+\eta a\times C_1^{\prime }=\eta (\delta a+\kappa c)\times C_1+\eta a\times C_1^{\prime }=\delta A+\eta \kappa D_1+\eta a\times C_1^{\prime }.$$

(29)

Differentiating (27) and using (18) and (26), we find

$$D_1^{\prime }=c^{\prime }\times C_1+c\times C_1^{\prime }=(\tau a+\kappa b)\times C_1+c\times C_1^{\prime }=\eta \tau A+\kappa B+c\times C_1^{\prime }.$$

(30)

Differentiating (26) and using (18) and (27), we find

$$B^{\prime }=b^{\prime }\times C_1+b\times C_1^{\prime }=(-\delta b+\tau c)\times C_1+b\times C_1^{\prime }=-\delta B+\tau D_1+b\times C_1^{\prime }.$$

(31)

Now, let

$$C_1^{\prime }={\alpha }_{1}A+{\alpha }_{2}B+{\alpha }_3{C}_1+{\alpha }_4{D}_1.$$

Since

$$\begin{array}{ccc}& h(C_1^{\prime },A)& =-h(C_1,A^{\prime })=-h(C_1,\eta K_1{C}_1)=-K_1={\alpha }_2,\hfill \\ & h(C_1^{\prime },B)& ={\alpha }_1,\hfill \\ & h(C_1^{\prime },C_1)& =0=\eta {\alpha }_3,\hfill \\ & h(C_1^{\prime },D_1)& =-\eta {\alpha }_4,\hfill \end{array}$$

we obtain

$$C_1^{\prime }={\alpha }_{1}A-K_{1}B+{\alpha }_4{D}_1.$$

(32)

By using Equations (25)–(27), we obtain

$$C_1^{\prime }={\alpha }_1\eta a\times C_1-K_{1}b\times C_1+{\alpha }_{4}c\times C_1.$$

(33)

If we multiply (32) by a and use (25)–(27) and Table 2, we obtain

$$a\times C_1^{\prime }=0-K_1(-1-c)\times C_1+{\alpha }_4(-a)\times C_1=K{C}_1+K_1{D}_1-\eta {\alpha }_{4}A.$$

(34)

By substituting this equation in (29), we obtain

$$A^{\prime }=(\delta -{\alpha }_4)A+\eta K_1{C}_1+\eta (K_1+\kappa )D_1.$$

(35)

Then, according to (22), we find that $\varphi =\delta -{\alpha }_4$ and

$$K_1+\kappa =0.$$

(36)

If we multiply (32) by b and use (25), (26), and Table 2, we obtain

$$b\times C_1^{\prime }={\alpha }_1\eta (-1+c)\times C_1+{\alpha }_{4}b\times C_1=-\eta {\alpha }_1{C}_1+\eta {\alpha }_1{D}_1+{\alpha }_{4}B.$$

(37)

By substituting (37) in (31), we obtain

$$B^{\prime }=-\varphi A-\eta {\alpha }_1{C}_1+(\eta {\alpha }_1+\tau )D_1.$$

(38)

If we multiply (32) by c and use Equations (25) and (26) and Table 2, we obtain

$$c\times C_1^{\prime }={\alpha }_1\eta a\times C_1+K_{1}b\times C+{\alpha }_4{C}_1={\alpha }_{1}A+K_{1}B+{\alpha }_4{C}_1.$$

(39)

By substituting (39) in (30) and using (36), we obtain

$$D_1^{\prime }=({\alpha }_1+\eta \tau )A+{\alpha }_4{C}_1.$$

(40)

If we substitute the following in Equations (23), (32) ,(38), and (40),

$$\begin{array}{ccccccccccc}\hfill \eta C_1=C& ,\hfill & \hfill -\eta D_1=D& ,\hfill & \hfill {\alpha }_1=-K_2& ,\hfill & \hfill {\alpha }_1+\eta \tau =-K_3& \hfill & \hfill and& \hfill & {\alpha }_4=K_4\end{array}$$

(41)

we obtain Equation (28). □

The frame expressed as $\mathfrak{F}=\{A,B,C,D\}$ is called the general Frenet frame of type 1 along the null hybrid curve ($\Gamma$) with respect to $S\left(T{\Gamma }^{\perp }\right)$, and Equation (28) is called the general Frenet equation of type 1.

$$\begin{array}{ccccccccc}\hfill K_1=-\kappa & ,\hfill & \hfill K_2=\eta h(B^{\prime },C)& ,\hfill & \hfill K_3=K_2-\eta \tau ,& \hfill & \hfill and& \hfill & \hfill K_4=\eta (C^{\prime },D)=\delta -\varphi \end{array}$$

are the curvatures of $\Gamma$. $\kappa$ and $\tau$ are the curvature and torsion of its associated curve ($\gamma$), respectively. These formulas provide the relationship between the curvatures of a null hybrid curve in ${\mathbb{R}}_2^4$ and its associated null spatial hybrid curve in ${\mathbb{R}}_2^3$.

Corollary 1.

If the null hybrid curve ($\Gamma \left(p\right)$) is parameterized by a distinguished parameter (p), then the Frenet equations are

$$\left[\begin{array}{c}{A}^{\prime }\\ B^{\prime }\\ C^{\prime }\\ D^{\prime }\end{array}\right]=\left[\begin{array}{cccc}0& 0& K_1& 0\\ 0& 0& K_2& K_3\\ -\eta K_2& -\eta K_1& 0& \delta \\ \eta K_3& 0& -\delta & 0\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\\ D\end{array}\right].$$

(42)

If, in addition, the associated curve (γ) is a curve parameterized by a distinguished parameter, then the Frenet equations of type 1 are

$$\left[\begin{array}{c}{A}^{\prime }\\ B^{\prime }\\ C^{\prime }\\ D^{\prime }\end{array}\right]=\left[\begin{array}{cccc}0& 0& K_1& 0\\ 0& 0& K_2& K_3\\ -\eta K_2& -\eta K_1& 0& 0\\ \eta K_3& 0& 0& 0\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\\ D\end{array}\right].$$

(43)

Proof. 

Since $\Gamma$ is parameterized by a distinguished parameter (p), $\varphi =h(A^{\prime },B)=0$. According to Theorem 5, we find that $K_4=\delta$. Then, according to Theorem 2, there exists a curve (${\gamma }_p$) with a distinguished parameter (p) with curvatures of $\kappa$ and $\tau$ and a Frenet frame ($\{a,b,c\}$), that is, $\delta =0$. □

Corollary 2.

There is no Cartan null spatial hybrid curve associated to the Cartan null hybrid curve.

Proof. 

According to Theorem 5, we find that $K_1+\kappa =0$, where $K_1$ is the first curvature of the null spatial hybrid curve and $\kappa$ is the first curvature of its associated curve. But in the Cartan frame, $K_1=1$ and ${\kappa }_1=1$, which is a contradiction. □

In the following example, we construct a Frenet frame for the null hybrid curve ($\Gamma$). We first find the associated null spatial hybrid curve ($\gamma$) corresponding to the $\Gamma$ curve. We construct a Frenet frame for the $\gamma$ curve; then, by using this frame and the associated Frenet formulas, we construct the Frenet frame for the original curve ($\Gamma$). We also present the relationship between the curvatures of the $\Gamma$ curve and its associated null spatial hybrid curve ($\gamma$), as well as a graphic of the $\gamma$ curve and a projection of $\Gamma$ in the space spanned by $\{\mathbf{i},\mathit{ϵ},\mathbf{h}\}$.

Example 1.

Let

$$\Gamma =({\displaystyle \frac{1}{3}}{t}^3,{\displaystyle \frac{1}{2}}{t}^2,{\displaystyle \frac{1}{8}}{t}^4,{\displaystyle \frac{1}{2}}{t}^2)$$

be a null hybrid curve and let ${\Gamma }^{\prime }=A$. Then,

$$A=(t^2,t,{\displaystyle \frac{1}{2}}{t}^3,t),h(A,A)=0.$$

According to $A^{\prime }=(2t,1,{\displaystyle \frac{3}{2}}{t}^2,1)$, we find $h(A^{\prime },A^{\prime })=t^2$, and based on (23), we define

$$C={\displaystyle \frac{1}{t}}(A^{\prime }-\varphi A)={\displaystyle \frac{1}{t}}(2t-\varphi t^2,1-\varphi t,{\displaystyle \frac{3}{2}}{t}^2-{\displaystyle \frac{1}{2}}\varphi t^3,1-\varphi t).$$

To simplify the calculations, we define

$$\widehat{C}={\displaystyle \frac{1}{t}}{A}^{\prime }={\displaystyle \frac{1}{t}}(2t,1,{\displaystyle \frac{3}{2}}{t}^2,1)=(2,{\displaystyle \frac{1}{t}},{\displaystyle \frac{3}{2}}t,{\displaystyle \frac{1}{t}}).$$

Then,

$$C=\widehat{C}+{\displaystyle \frac{\varphi }{t}}A.$$

(44)

We calculate $h(\widehat{C},\widehat{C})=1$ and find that $\eta =h(C,C)=h(\widehat{C}+{\displaystyle \frac{\varphi }{t}}A,\widehat{C}+{\displaystyle \frac{\varphi }{t}}A)=1$, that is, C is a time-like hybrid. By using the hybridian product defined in (7), we obtain

$$A\times \overline{C}=A\times (\overline{\widehat{C}}+{\displaystyle \frac{\varphi }{t}}A)=A\times \overline{\widehat{C}}=t(0,1,-{\displaystyle \frac{1}{2}}{t}^2-t,t+1),$$

(45)

which is a spatial hybrid, and denote it as

$$a=(0,t,-{\displaystyle \frac{1}{2}}{t}^3-t^2,t^2+t).$$

(46)

It can be seen that $h(a,a)=0$, so a is a null spatial hybrid.

Next, we calculate the null spatial hybrid as $b={\displaystyle \frac{1}{h(a,v)}}\{v-{\displaystyle \frac{h(v,v)}{2h(a,v)}}a\}$, where v is a space-like spatial hybrid. If we choose $v=(0,0,0,1)$, by using $h(v,v)=-1$ and $h(a,v)=-t(1+t)$, we find

$$b=-{\displaystyle \frac{1}{2t{(1+t)}^2}}(0,-1,{\displaystyle \frac{1}{2}}{t}^2+t,1+t).$$

(47)

It can be seen that $h(b,b)=0$ and $h(a,b)=1$. Next, we define the space-like spatial hybrid (c) as

$$c=b\times a={\displaystyle \frac{1}{1+t}}(0,1,1+{\displaystyle \frac{1}{2}}{t}^2+t,0),$$

(48)

and find that $h(c,c)=-1$. The $\gamma \left(t\right)$ curve with the Frenet frame expressed as $F=\{a,b,c\}$, containing two null and one space-like spatial hybrids, is an associated null spatial hybrid curve of Γ.

Now, by using (7), we calculate

$$\begin{array}{ccc}\hfill c\times \widehat{C}& =& {\displaystyle \frac{1}{t(1+t)}}(2t^2+t,2t+1,t^3+{\displaystyle \frac{3}{2}}{t}^2+t,1-t^2+t),\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill c\times A& =& (t^2,t,{\displaystyle \frac{1}{2}}{t}^3,t).\hfill \end{array}$$

(50)

It can be shown that $c\times \widehat{C}$ is a space-like hybrid and $c\times A$ is a null hybrid. Hybrid D can be calculated by (27) as $D=-c\times C=-c\times \widehat{C}-{\displaystyle \frac{\varphi }{t}}c\times A$. Then, substituting (49) and (50), we find

$$D=-{\displaystyle \frac{1}{(1+t)}}(2t+1+\varphi t^2,2+{\displaystyle \frac{1}{t}}+\varphi t,t^2+{\displaystyle \frac{3}{2}}t+1+{\displaystyle \frac{1}{2}}\varphi t^3,{\displaystyle \frac{1}{t}}-t+1+\varphi t).$$

(51)

It can be seen that D is a unit space-like hybrid. Next, we calculate

$$\begin{array}{ccc}\hfill b\times \widehat{C}& =& -{\displaystyle \frac{1}{2t^2{(1+t)}^2}}(2+2t-t^2,-3t-2,{\displaystyle \frac{5}{2}}{t}^3+3t^2-2t-2,4t^2+3t),\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill b\times A& =& -{\displaystyle \frac{1}{(1+t)}}(1,-1,{\displaystyle \frac{1}{2}}{t}^2-1,t).\hfill \end{array}$$

(53)

It can be shown that $b\times \widehat{C}$ and $b\times A$ are null hybrids. Hybrid B can be calculated by (26) as $B=b\times C=b\times \widehat{C}+{\displaystyle \frac{\varphi }{t}}b\times A$. Then, substituting (52) and (53), we find

$$\begin{array}{cc}\hfill B=-{\displaystyle \frac{1}{t(1+t)}}(& {\displaystyle \frac{2+2t-t^2}{2t(1+t)}}+\varphi ,-{\displaystyle \frac{3t+2}{2t(1+t)}}-\varphi ,\hfill \\ \hfill & {\displaystyle \frac{\frac{5}{2}{t}^3+3t^2-2t-2}{2t(1+t)}}+\varphi ({\displaystyle \frac{1}{2}}{t}^2-1),{\displaystyle \frac{4t^2+3t}{2t(1+t)}}+\varphi t).\hfill \end{array}$$

(54)

It can be shown that B is a null hybrid.

The set expressed as $\mathfrak{F}=\{A,B,C,D\}$, containing two null hybrids, one time-like hybrid, and one space-like hybrid, is a Frenet frame for the null hybrid curve (Γ). The curvature and torsion of γ are calculated as

$$\begin{array}{c}\hfill \kappa =-h(a^{\prime },c)=-t,\\ \hfill \tau =-h(b^{\prime },c)=h(c^{\prime },b)=-{\displaystyle \frac{1}{2t}},\end{array}$$

and $\delta =h(a^{\prime },b)={\displaystyle \frac{2t^2+3t+1}{t{(1+t)}^2}}$. The curvatures of Γ are

$$\begin{array}{ccc}\hfill K_1& =& h(A^{\prime },C)=t,\hfill \\ \hfill K_2& =& -h(C^{\prime },B)=-h{(\widehat{C}+{\displaystyle \frac{\varphi }{t}}A)}^{\prime },B)\hfill \\ & & ={\displaystyle \frac{1}{2t^4{(1+t)}^2}}(7t^3+10t^2+4t)+{\displaystyle \frac{\varphi }{t(1+t)}}({\displaystyle \frac{2}{t^2}}+1+{\displaystyle \frac{1}{t}})+{\left({\displaystyle \frac{\varphi }{t}}\right)}^{\prime }+{\displaystyle \frac{{\varphi }^2}{t}},\hfill \\ \hfill K_3& =& K_2-\tau ,\hfill \\ \hfill K_4& =& \delta -\varphi .\hfill \end{array}$$

We can calculate the Frenet elements for $t=1$ and $\varphi =0$ as follows:

$$\begin{array}{ccc}\hfill & A=(1,1,{\displaystyle \frac{1}{2}},1),\hfill & \hfill C=(2,1,{\displaystyle \frac{3}{2}},1),\\ \hfill & A\times \overline{C}=a=(0,1,-{\displaystyle \frac{3}{2}},2),\hfill & \hfill b=-{\displaystyle \frac{1}{8}}(0,-1,{\displaystyle \frac{3}{2}},2),& \hfill & \hfill c=-a\wedge b=(0,{\displaystyle \frac{1}{2}},{\displaystyle \frac{5}{4}},0),\\ \hfill & B=-{\displaystyle \frac{1}{8}}(3,-5,{\displaystyle \frac{3}{2}},7),\hfill & \hfill D=-{\displaystyle \frac{1}{2}}(3,3,{\displaystyle \frac{7}{2}},1).\end{array}$$

The curvatures are calculated as

$$\begin{array}{ccccccc}\hfill K_1=-\kappa =1& ,\hfill & \hfill K_2={\displaystyle \frac{21}{8}}& ,\hfill & \hfill K_3={\displaystyle \frac{25}{8}}& ,\hfill & \hfill K_4=\delta ={\displaystyle \frac{3}{2}}.\end{array}$$

Note that the γ curve is not unique, as there is a family of curves in ${\mathbb{R}}_2^3$ that have the same tangent vector a. Let $\gamma \left(s\right)=({\displaystyle \frac{1}{2}}{t}^2,-{\displaystyle \frac{1}{8}}{t}^4-{\displaystyle \frac{1}{3}}{t}^3,{\displaystyle \frac{1}{3}}{t}^3+{\displaystyle \frac{1}{2}}{t}^2)$ be one of the associated curves of the null hybrid curve (Γ). We present a graphic of this curve and the projection of the Γ curve in the space spanned by $\{\mathbf{i},\mathit{ϵ},\mathbf{h}\}$ in Figure 1.

6. Null Hybrid Bertrand Curves

Definition 1.

Let $\Gamma \left(s\right)$ and $\tilde{\Gamma }\left(\tilde{s}\right)$ be two null hybrid curves such that $\tilde{\Gamma }\left(\tilde{s}\right)=\tilde{\Gamma }\left(\phi \left(s\right)\right)$, where $\phi :I\to \tilde{I}$ is a $C^{\infty }$ map. If, for all $s\in I\subseteq \mathbb{R}$, the principle normals of these curves are identical, then $\Gamma \left(\tilde{s}\right)$ is called the Bertrand curve of $\Gamma \left(s\right)$. This property suggests that Bertrand curves have a certain symmetry.

Lemma 2.

Let $\Gamma \left(p\right)$ be a null hybrid curve, and let $\tilde{\Gamma }\left(\tilde{p}\right)$ be the Bertrand curve of $\Gamma \left(p\right)$. Let $\{\tilde{A},\tilde{B},\tilde{C},\tilde{D};\widetilde{K_1},\widetilde{K_2},\widetilde{K_3}\}$ and $\{A,B,C,D;K_1,K_2,K_3\}$ be the Frenet frames of $\tilde{\Gamma }\left(\tilde{p}\right)$ and $\Gamma \left(p\right)$, respectively. Then, the distance between corresponding points of Bertrand null hybrid mates is constant.

Proof. 

Since $\tilde{\Gamma }\left(\tilde{p}\right)$ is the Bertrand curve of $\Gamma \left(p\right)$,

$$\tilde{\Gamma }\left(\tilde{p}\right)=\Gamma \left(p\right)+\lambda \left(p\right)C\left(p\right),$$

(55)

where $\lambda \left(p\right)$ is a differentiable function. If we differentiate (55), by using (43), we find

$$\tilde{A}\left(\tilde{p}\right){\displaystyle \frac{d\tilde{p}}{dp}}=(1+\lambda K_2)A+{\lambda }^{\prime }C-\lambda K_{1}B.$$

(56)

Since $\tilde{C}$ and C are linearly dependent, $\tilde{C}=\chi \left(p\right)C$, where $\chi \left(p\right)$ is a differentiable function. If we multiply this equation by $\tilde{C}$, we find ${\lambda }^{\prime }=0$ or $\lambda =constant$. □

Theorem 6.

The necessary and sufficient conditions for a null hybrid curve, parameterized by a distinguished parameter (p) with the Frennet frame ($\{A,B,C,D;K_1,K_2,K_3\}$), to be a null hybrid Bertrand curve are: $K_2\ne 0$, $K_3=0$, and φ being a linear map.

Proof. 

Let $\tilde{\Gamma }$ and $\Gamma$ be Bertrand mates. According to (56), we obtain

$$\tilde{A}\left(\tilde{p}\right){\displaystyle \frac{d\tilde{p}}{dp}}=(1-\eta \lambda K_2)A-\eta \lambda K_{1}B.$$

From

$$0=h(\tilde{A}\left(\tilde{p}\right){\displaystyle \frac{d\tilde{p}}{dp}},\tilde{A}\left(\tilde{p}\right){\displaystyle \frac{d\tilde{p}}{dp}})=-(1-\eta \lambda K_2)\lambda K_1,$$

we find $K_2=\eta {\displaystyle \frac{1}{\lambda }}$. Then, (56) becomes

$$\tilde{A}\left(\tilde{p}\right){\displaystyle \frac{d\tilde{p}}{dp}}=-\eta \lambda K_{1}B.$$

(57)

By differentiating (57) and using (43), we find

$$\widetilde{K_1}\tilde{C}{\displaystyle \frac{d\tilde{p}}{dp}}=\eta \lambda K_1{\left({\displaystyle \frac{d\tilde{p}}{dp}}\right)}^{-1}{\displaystyle \frac{d^2\tilde{p}}{d{p}^2}}B-\eta \lambda K_1{K}_{2}C-\eta \lambda K_1{K}_{3}D.$$

(58)

Since the principal normals coincide, we find $K_3=0$ and ${\displaystyle \frac{d^2\tilde{p}}{d{p}^2}}=0$.

Conversely, let $\Gamma$ be a null hybrid curve with $K_2=constant$ and $K_3=0$, and let $\phi \left(p\right)=\tilde{p}$ be a linear transformation. Consider the following curve:

$$\tilde{\Gamma }=\Gamma +{\displaystyle \frac{1}{K_2}}C.$$

(59)

By differentiating with respect to p and by using (43), we obtain

$$\tilde{A}{\displaystyle \frac{d\tilde{p}}{dp}}=-\eta K_{1}B,$$

(60)

so $\tilde{\Gamma }$ is a null hybrid curve. Then, by differentiating (60) and using the hypothesis of the theorem, we find that $\tilde{C}$ and C are linearly dependent. □

Corollary 3.

If $\tilde{\Gamma }$ is a null hybrid Bertrand curve of Γ, then Γ is a null spatial hybrid curve.

Proof. 

According to Theorem 6, we find that $K_3=K_2-\eta \tau =0$ or $K_2=\eta \tau$, and based on $K_1=-\kappa$, we find that $\Gamma$ is a null spatial hybrid curve. □

7. Discussion

In this study, we focus on the geometry of Minkowski space with a hybrid structure. We derive the Frenet frame and Frenet formulas for null curves in ${\mathbb{R}}_2^4$ by using the hybrid numbers. We begin by recalling basic notions about hybrid numbers and null curves, then introduce null spatial hybrid curves. We obtain a Frenet frame and the Cartan frame for a null spatial hybrid curve and derive the Frenet equations (Theorem 3) and the Cartan equations (Theorem 4) regarding this frame. Next, we introduce a null hybrid curve and define its associated null spatial hybrid curve. We use Frenet equations of the associated curve to obtain a general Frenet equations of type 1 for a given null hybrid curve (Theorem 5). In spite of this, we deduce that the Cartan frame of a null hybrid curve cannot be obtained from the Cartan frame of its associated null spatial hybrid curve. We also provide an example of a null hybrid curve. We obtain a Frenet frame and derive Frenet equations for its associated null spatial hybrid curve, then extend them to the Frenet equations for the given null hybrid curve, (Example 1). Moreover, we introduce the Bertrand curve of a null hybrid curve, and we conclude that the only Bertrand null hybrid curves are null spatial hybrid curves (Theorem 6).

In this study, we examine null hybrid curves by using general Frenet equations of type 1, which contains two null hybrids, one space-like hybrid, and one time-like hybrid. Future research will investigate null hybrid curves by using general Frenet equations of type 2, which contain four null hybrids, as well as their applications in physics. Moreover, this method can be used to investigate other special null hybrid curves in ${\mathbb{R}}_2^4$.