Spatial Generalized Octonionic Curves

Abstract

This study investigates curves in a 7-dimensional space, represented by spatial generalized octonion-valued functions of a single variable, where the general octonions include real, split, semi, split semi, quasi, split quasi, and para octonions. We begin by constructing a new frame, referred to as the $G_2$-frame, for spatial generalized octonionic curves, and subsequently derive the corresponding derivative formulas. We also present the connection between the $G_2$-frame and the standard orthonormal basis of spatial generalized octonions. Moreover, we verify that Frenet–Serret formulas hold for spatial generalized octonionic curves. We establish the $G_2$-congruence of two spatial generalized octonionic curves and present the correspondence between the Frenet–Serret frame and the $G_2$-frame. A key advantage of the $G_2$-frame is that the associated frame equations involve lower-order derivatives. This method is both time-efficient and computationally efficient. To demonstrate the theory, we present an example of a unit-speed spatial generalized octonionic curve and compute its $G_2$-frame and invariants using MATLAB.

1. Introduction

After Sir Rowan Hamilton introduced quaternions, his friend Graves came up with a new algebraic system in 1843. He called them octonions [1]. Graves shared his discovery with Hamilton in a letter written in December 1843. Around the same time, Cayley independently discovered the same structure. Later, this structure became known as the Cayley numbers or Cayley algebra, and Cayley published a detailed paper about it [2]. Hamilton later admitted that Graves had both discovered and named the system before Cayley.

Today, the terms “generalized octonions” and “Cayley numbers” are often used to mean the same thing. In physics, octonions have mostly been used in theoretical ideas. For example, in the 1970s, some researchers tried to model quarks using octonionic Hilbert spaces. It is also known that octonions are closely related to the idea that only four normed division algebras exist. This fact connects them to the possible dimensions of spacetime where supersymmetric quantum field theories can exist. Some researchers have even tried to explain the Standard Model of particle physics using octonions. In addition to physics, octonions have appeared in many other fields. These include black hole entropy, quantum information theory, string theory, and even image processing [3,4,5].

The set of generalized octonions includes several types of number systems as special cases. By changing the parameters ${\mu }_1,{\mu }_2,$and${\mu }_3$, different types of octonions can be defined—as long as not all of these parameters are zero.

• If ${\mu }_1={\mu }_2={\mu }_3=-1$ yields the real octonions [6];
• If ${\mu }_1={\mu }_2=-1,{\mu }_3=1$ corresponds to the split octonions [7];
• If ${\mu }_1={\mu }_2=-1,{\mu }_3=0$ defines the semi octonions [8];
• If ${\mu }_1={\mu }_2=1,{\mu }_3=0$ gives the split semi octonions [9];
• If ${\mu }_1=-1,{\mu }_2={\mu }_3=0$ produces the quasi octonions [10];
• If ${\mu }_1=1,{\mu }_2={\mu }_3=0$ leads to the split quasi octonions [11];
• If ${\mu }_1={\mu }_2={\mu }_3=0$ results in the para octonions [12].

Numerous studies in the literature have explored these various types of octonions, including works such as [13,14,15,16,17,18,19,20]. Generalized octonions are also termed three-parameter generalized octonions (3PGO). In [21] a classification is made based on the inner product and vector components, and the polar forms of light-like generalized octonions are derived. Furthermore, matrix representations of generalized octonions are introduced, and several properties of these representations are established. In addition, the powers and roots of the matrix representations are presented.

The frame fields and curvatures of curves in n-dimensional Euclidean and Minkowski spaces are calculated using the Frenet–Serret frame and its associated formulas [22]. Bharathi and Nagaraj [23] utilized spatial quaternions and quaternions to study the differential geometry of curves in 4-dimensional Euclidean space. Then, Çöken and Tuna [24] obtained the characterizations of non-null semi-quaternionic curves in 3- and 4-dimensional Minkowski spaces. Dağdeviren and Yüce  [25] performed analogous calculations for dual quaternionic curves in 3- and 4-dimensional Galilean spaces. In addition, Akbıyık [26] obtained the characterizations of non-null hybrid curves. The characterizations of null hybrid curves were presented by Alo [27]. The frame fields obtained in these studies coincide with the Frenet–Serret frames due to the algebraic and geometric structure of 3- and 4-dimensional spaces. In their work, Bektaş and Yüce [28] investigated the characterizations of spatial octonionic and octonionic curves. However, a significant algebraic difference between quaternions and octonions is that while quaternions form an associative algebra, octonions do not. Moreover, whereas two vectors in the spatial quaternion space possess only one orthogonal vector, in the spatial octonion space, two vectors may have multiple orthogonal vectors. As a result, certain differences emerge in the computation of the frame fields of octonionic curves.

The geometric properties of spatial (pure) octonionic curves in Euclidean spaces were investigated by Ohashi [29,30,31,32]. In these studies, a novel moving frame, called the $G_2$-frame, was introduced for spatial octonionic curves. The corresponding derivative formulas were derived, and their connection to the classical Frenet frame was established. Non-null split octonionic curves were studied by Alo and Akbıyık in [33], where they constructed a $G_2$-frame along non-null spatial split octonionic curves. They derived the corresponding derivative formulas and established the relation between the elements of the $G_2$-frame and those of the classical Frenet–Serret frame. Since the computation of Serret–Frenet formulas in 7- and 8-dimensional spaces involves higher-order derivatives, it can lead to memory-related problems in software environments such as MATLAB. The $G_2$-frame offers lower-order derivative formulations as an alternative to the higher-order classical Frenet–Serret formulas.

This paper is structured as follows. In Section 2, we present the inner product $⟨·,·⟩$ and the vector product ▴ in the Euclidean space ${\mathbb{R}}^7$. Based on these structures, we recall the generalized octonionic product× and the associated inner product $h(·,·)$ in the space of generalized octonions, denoted by $\mathbb{GO}$. We also present several fundamental properties of these operations. In Section 3, we define spatial generalized octonionic curves and construct a moving frame, referred to as the $G_2$-frame, for non-null spatial generalized octonionic curves. Furthermore, we derive the differential equations associated with this frame. In addition to these, we give a relation between the $G_2$-frame and the standard orthonormal basis of spatial generalized octonions. In Section 4, we discuss the $G_2$-invariants and establish the correspondence between the elements of the $G_2$-frame and those of the classical Frenet–Serret frame. Finally, in Section 5, we provide an explicit example of a unit spatial generalized octonionic curve. We compute its $G_2$-frame and associated $G_2$-invariants numerically using MATLAB 2023b. Appendix A includes the MATLAB 2023b codes utilized for performing the computations and illustrative example discussed in the main text.

This work makes two main contributions. First, it extends the concept of the $G_2$-frame to the setting of spatial generalized octonionic curves. Second, it establishes explicit relations between the new invariants and the classical Frenet–Serret frame, offering computational advantages through lower-order derivative formulations.

2. Preliminaries

In this section, we recall some fundamental properties of generalized octonions, following the exposition in [21]. Let $\mathbf{r}$ and $\mathbf{s}$ be two arbitrary 7-tuples in ${\mathbb{R}}^7$, written as follows:

$$\mathbf{r}=(r_1,r_2,r_3,r_4,r_5,r_6,r_7),\mathbf{s}=(s_1,s_2,s_3,s_4,s_5,s_6,s_7).$$

A real-valued function $⟨·,·⟩:{\mathbb{R}}^7\times {\mathbb{R}}^7\to \mathbb{R}$ is defined as follows:

$$\begin{array}{c}\hfill ⟨\mathbf{r},\mathbf{s}⟩=-({\mu }_1{r}_1{s}_1+{\mu }_2{r}_2{s}_2-{\mu }_1{\mu }_2{r}_3{s}_3+{\mu }_3{r}_4{s}_4-{\mu }_1{\mu }_3{r}_5{s}_5-{\mu }_2{\mu }_3{r}_6{s}_6+{\mu }_1{\mu }_2{\mu }_3{r}_7{s}_7),\end{array}$$

where ${\mu }_1,{\mu }_2, \mathrm{and} {\mu }_3$ are real numbers not simultaneously zero. This function defines a non-degenerate inner product on ${\mathbb{R}}^7$. If ${\mu }_1,{\mu }_2,{\mu }_3<0$, then $⟨·,·⟩$ is positive definite. Let $\{e_1,e_2,e_3,e_4,e_5,e_6,e_7\}$ be a standard basis for $({\mathbb{R}}^7,⟨,⟩)$. The vector product is defined according to Table 1.

Table 1. Vector products in $({\mathbb{R}}^7,⟨,⟩)$.

▴	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$	${\mathit{e}}_6$	${\mathit{e}}_7$
$e_1$	0	$e_3$	${\mu }_1{e}_2$	$e_5$	${\mu }_1{e}_4$	$-e_7$	$-{\mu }_1{e}_6$
$e_2$	$-e_3$	0	$-{\mu }_2{e}_1$	$e_6$	$e_7$	${\mu }_2{e}_4$	${\mu }_2{e}_5$
$e_3$	$-{\mu }_1{e}_2$	${\mu }_2{e}_1$	0	$e_7$	${\mu }_1{e}_6$	$-{\mu }_2{e}_5$	$-{\mu }_1{\mu }_2{e}_4$
$e_4$	$-e_5$	$-e_6$	$-e_7$	0	$-{\mu }_3{e}_1$	$-{\mu }_3{e}_2$	$-{\mu }_3{e}_3$
$e_5$	$-{\mu }_1{e}_4$	$-e_7$	$-{\mu }_1{e}_6$	${\mu }_3{e}_1$	0	${\mu }_3{e}_3$	${\mu }_1{\mu }_3{e}_2$
$e_6$	$e_7$	$-{\mu }_2{e}_4$	${\mu }_2{e}_5$	${\mu }_3{e}_2$	$-{\mu }_3{e}_3$	0	$-{\mu }_2{\mu }_3{e}_1$
$e_7$	${\mu }_1{e}_6$	$-{\mu }_2{e}_5$	${\mu }_1{\mu }_2{e}_4$	${\mu }_3{e}_3$	$-{\mu }_1{\mu }_3{e}_2$	${\mu }_2{\mu }_3{e}_1$	0

The vector product of $\mathbf{r},\mathbf{s}\in {\mathbb{R}}^7,$ can be expressed as follows:

$$\begin{array}{ccc}\hfill \mathbf{r}▴\mathbf{s}& =& \left[\begin{array}{ccccccc}0& {\mu }_2{r}_3& -{\mu }_2{r}_2& {\mu }_3{r}_5& -{\mu }_3{r}_4& {\mu }_2{\mu }_3{r}_7& -{\mu }_2{\mu }_3{r}_6\\ -{\mu }_1{r}_3& 0& {\mu }_1{r}_1& {\mu }_3{r}_6& -{\mu }_1{\mu }_3{r}_7& -{\mu }_3{r}_4& {\mu }_1{\mu }_3{r}_5\\ -r_2& r_1& 0& {\mu }_3{r}_7& -{\mu }_3{r}_6& {\mu }_3{r}_5& -{\mu }_3{r}_4\\ -{\mu }_1{r}_5& -{\mu }_2{r}_6& {\mu }_1{\mu }_2{r}_7& 0& {\mu }_1{r}_1& {\mu }_2{r}_2& -{\mu }_1{\mu }_2{r}_3\\ -r_4& -{\mu }_2{r}_7& {\mu }_2{r}_6& r_1& 0& -{\mu }_2{r}_3& {\mu }_2{r}_2\\ {\mu }_1{r}_7& -r_4& -{\mu }_1{r}_5& r_2& {\mu }_1{r}_3& 0& -{\mu }_1{r}_1\\ r_6& -r_5& -r_4& r_3& r_2& -r_1& 0\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\\ s_4\\ s_5\\ s_6\\ s_7\end{array}\right]\hfill \\ & =& ({\mu }_2{r}_3{s}_2-{\mu }_2{r}_2{s}_3+{\mu }_3{r}_5{s}_4-{\mu }_3{r}_4{s}_5+{\mu }_2{\mu }_3{r}_7{s}_6-{\mu }_2{\mu }_3{r}_6{s}_7)e_1\hfill \\ & +& (-{\mu }_1{r}_3{s}_1+{\mu }_1{r}_1{s}_3+{\mu }_3{r}_6{s}_4-{\mu }_1{\mu }_3{r}_7{s}_5-{\mu }_3{r}_4{s}_6+{\mu }_1{\mu }_3{r}_5{s}_7)e_2\hfill \\ & +& (-r_2{s}_1+r_1{s}_2+{\mu }_3{r}_7{s}_4-{\mu }_3{r}_6{s}_5+{\mu }_3{r}_5{s}_6-{\mu }_3{r}_4{s}_7)e_3\hfill \\ & +& (-{\mu }_1{r}_5{s}_1-{\mu }_2{r}_6{s}_2+{\mu }_1{\mu }_2{r}_7{s}_3+{\mu }_1{r}_1{s}_5+{\mu }_2{r}_2{s}_6-{\mu }_1{\mu }_2{r}_3{s}_7)e_4\hfill \\ & +& (-r_4{s}_1-{\mu }_2{r}_7{s}_2+{\mu }_2{r}_6{s}_3+r_1{s}_4-{\mu }_2{r}_3{s}_6+{\mu }_2{r}_2{s}_7)e_5\hfill \\ & +& ({\mu }_1{r}_7{s}_1-r_4{s}_2-{\mu }_1{r}_5{s}_3+r_2{s}_4+{\mu }_1{r}_3{s}_5-{\mu }_1{r}_1{s}_7)e_6\hfill \\ & +& (r_6{s}_1-r_5{s}_2-r_4{s}_3+r_3{s}_4+r_2{s}_5-r_1{s}_6)e_7.\hfill \end{array}$$

Equivalently, as follows:

$$\begin{array}{ccc}\hfill \mathbf{r}▴\mathbf{s}& =& [{\mu }_2(r_3{s}_2-r_2{s}_3)+{\mu }_3(r_5{s}_4-r_4{s}_5)+{\mu }_2{\mu }_3(r_7{s}_6-r_6{s}_7)]e_1\hfill \\ & +& [{\mu }_1(r_1{s}_3-r_3{s}_1)+{\mu }_3(r_6{s}_4-r_4{s}_6)+{\mu }_1{\mu }_3(r_5{s}_7-r_7{s}_5)]e_2\hfill \\ & +& [(r_1{s}_2-r_2{s}_1)+{\mu }_3(r_5{s}_6-r_6{s}_5+r_7{s}_4-r_4{s}_7)]e_3\hfill \\ & +& [{\mu }_1(r_1{s}_5-r_5{s}_1)+{\mu }_2(r_2{s}_6-r_6{s}_2)+{\mu }_1{\mu }_2(r_7{s}_3-r_3{s}_7)]e_4\hfill \\ & +& [(r_1{s}_4-r_4{s}_1)+{\mu }_2(r_2{s}_7-r_7{s}_2+r_6{s}_3-r_3{s}_6)]e_5\hfill \\ & +& [(r_2{s}_4-r_4{s}_2)+{\mu }_1(r_3{s}_5-r_5{s}_3+r_7{s}_1-r_1{s}_7)]e_6\hfill \\ & +& [(r_2{s}_5-r_5{s}_2)+(r_3{s}_4-r_4{s}_3)+(r_6{s}_1-r_1{s}_6)]e_7.\hfill \end{array}$$

The vector product defined with Table 1 satisfies the following fundamental properties:

i.

$\mathbf{r}▴(\mathbf{s}+\mathbf{k})=\mathbf{r}▴\mathbf{s}+\mathbf{r}▴\mathbf{k}$,

ii.

$\mathbf{r}▴\mathbf{r}=\mathbf{0}$,

iii.

$\mathbf{r}▴\mathbf{s}=-\mathbf{s}▴\mathbf{r}$,

iv.

$\mathbf{r}▴(\mathbf{r}▴\mathbf{s})=⟨\mathbf{r},\mathbf{s}⟩\mathbf{r}-⟨\mathbf{r},\mathbf{r}⟩\mathbf{s}$,

v.

$⟨\mathbf{a},\mathbf{b}▴\mathbf{c}⟩=⟨\mathbf{a}▴\mathbf{b},\mathbf{c}⟩.$

Let $\mathbb{GO}$ denote the space of generalized octonions, equipped with the basis $\{e_i,i=0,1,...,7\}.$ Then, every $R\in \mathbb{GO}$ can be expressed uniquely in the following form:

$$R=r_0+r_1{e}_1+r_2{e}_2+r_3{e}_3+r_4{e}_4+r_5{e}_5+r_6{e}_6+r_7{e}_7.$$

$S_R=r_0$ is the scalar part and $V_R=r_1{e}_1+r_2{e}_2+r_3{e}_3+r_4{e}_4+r_5{e}_5+r_6{e}_6+r_7{e}_7$ the vector part of R. The operations of addition and scalar multiplication in $\mathbb{GO}$ are defined as follows:

$$\begin{array}{cc}\hfill R+S=& (r_0+s_0)e_0+(r_1+s_1)e_1+(r_2+s_2)e_2+(r_3+s_3)e_3\\ & +(r_4+s_4)e_4+(r_5+s_5)e_5+(r_6+s_6)e_6+(r_7+s_7)e_7\end{array}$$

and

$$\lambda R=\lambda r_0{e}_0+\lambda r_1{e}_1+\lambda r_2{e}_2+\lambda r_3{e}_3+\lambda r_4{e}_4+\lambda r_5{e}_5+\lambda r_6{e}_6+\lambda r_7{e}_7.$$

The product of two generalized octonions is given in Table 2.

Table 2. Generalized octonionic product.

×	${\mathit{e}}_0$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$	${\mathit{e}}_6$	${\mathit{e}}_7$
$e_0$	$e_0$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_6$	$e_7$
$e_1$	$e_1$	${\mu }_1$	$e_3$	${\mu }_1{e}_2$	$e_5$	${\mu }_1{e}_4$	$-e_7$	$-{\mu }_1{e}_6$
$e_2$	$e_2$	$-e_3$	${\mu }_2$	$-{\mu }_2{e}_1$	$e_6$	$e_7$	${\mu }_2{e}_4$	${\mu }_2{e}_5$
$e_3$	$e_3$	$-{\mu }_1{e}_2$	${\mu }_2{e}_1$	$-{\mu }_1{\mu }_2$	$e_7$	${\mu }_1{e}_6$	$-{\mu }_2{e}_5$	$-{\mu }_1{\mu }_2{e}_4$
$e_4$	$e_4$	$-e_5$	$-e_6$	$-e_7$	${\mu }_3$	$-{\mu }_3{e}_1$	$-{\mu }_3{e}_2$	$-{\mu }_3{e}_3$
$e_5$	$e_5$	$-{\mu }_1{e}_4$	$-e_7$	$-{\mu }_1{e}_6$	${\mu }_3{e}_1$	$-{\mu }_1{\mu }_3$	${\mu }_3{e}_3$	${\mu }_1{\mu }_3{e}_2$
$e_6$	$e_6$	$e_7$	$-{\mu }_2{e}_4$	${\mu }_2{e}_5$	${\mu }_3{e}_2$	$-{\mu }_3{e}_3$	$-{\mu }_2{\mu }_3$	$-{\mu }_2{\mu }_3{e}_1$
$e_7$	$e_7$	${\mu }_1{e}_6$	$-{\mu }_2{e}_5$	${\mu }_1{\mu }_2{e}_4$	${\mu }_3{e}_3$	$-{\mu }_1{\mu }_3{e}_2$	${\mu }_2{\mu }_3{e}_1$	${\mu }_1{\mu }_2{\mu }_3$

Using this table, the generalized octonionic product of $R=r_0+r_1{e}_1+r_2{e}_2+r_3{e}_3+r_4{e}_4+r_5{e}_5+r_6{e}_6+r_7{e}_7$ and $S=s_0+s_1{e}_1+s_2{e}_2+s_3{e}_3+s_4{e}_4+s_5{e}_5+s_6{e}_6+s_7{e}_7$ is defined as follows:

$$\begin{array}{ccc}\hfill R\times S=r_0{s}_0& +& {\mu }_1{r}_1{s}_1+{\mu }_2{r}_2{s}_2-{\mu }_1{\mu }_2{r}_3{s}_3+{\mu }_3{r}_4{s}_4-{\mu }_1{\mu }_3{r}_5{s}_5-{\mu }_2{\mu }_3{r}_6{s}_6+{\mu }_1{\mu }_2{\mu }_3{r}_7{s}_7\hfill \\ & +& (r_0{s}_1+r_1{s}_0-{\mu }_2{r}_2{s}_3+{\mu }_2{r}_3{s}_2-{\mu }_3{r}_4{s}_5+{\mu }_3{r}_5{s}_4-{\mu }_2{\mu }_3{r}_6{s}_7+{\mu }_2{\mu }_3{r}_7{s}_6)e_1\hfill \\ & +& (r_0{s}_2+r_2{s}_0+{\mu }_1{r}_1{s}_3-{\mu }_1{r}_3{s}_1-{\mu }_3{r}_4{s}_6+{\mu }_1{\mu }_3{r}_5{s}_7+{\mu }_3{r}_6{s}_4-{\mu }_1{\mu }_3{r}_7{s}_5)e_2\hfill \\ & +& (r_0{s}_3+r_3{s}_0+r_1{s}_2-r_2{s}_1-{\mu }_3{r}_4{s}_7+{\mu }_3{r}_5{s}_6-{\mu }_3{r}_6{s}_5+{\mu }_3{r}_7{s}_4)e_3\hfill \\ & +& (r_0{s}_4+r_4{s}_0+{\mu }_1{r}_1{s}_5+{\mu }_2{r}_2{s}_6-{\mu }_1{\mu }_2{r}_3{s}_7-{\mu }_1{r}_5{s}_1-{\mu }_2{r}_6{s}_2+{\mu }_1{\mu }_2{r}_7{s}_3)e_4\hfill \\ & +& (r_0{s}_5+r_5{s}_0+r_1{s}_4-r_4{s}_1+{\mu }_2{r}_2{s}_7-{\mu }_2{r}_3{s}_6+{\mu }_2{r}_6{s}_3-{\mu }_2{r}_7{s}_2)e_5\hfill \\ & +& (r_0{s}_6+r_6{s}_0-{\mu }_1{r}_1{s}_7+r_2{s}_4+{\mu }_1{r}_3{s}_5-r_4{s}_2-{\mu }_1{r}_5{s}_3+{\mu }_1{r}_7{s}_1)e_6\hfill \\ & +& (r_0{s}_7+r_7{s}_0-r_1{s}_6+r_2{s}_5+r_3{s}_4-r_4{s}_3-r_5{s}_2+r_6{s}_1)e_7.\hfill \end{array}$$

The product of two generalized octonions R and S can be expressed in the following form:

$$R\times S=S_R{S}_S-⟨V_R,V_S⟩+S_R{V}_S+S_S{V}_R+V_R▴V_S,$$

where $V_R▴V_S$ represents the vector product and $⟨V_R,V_S⟩$ denotes the inner product in $({\mathbb{R}}^7,⟨,⟩)$. Although the multiplication of generalized octonions is neither commutative nor associative, it satisfies the property of alternativity, i.e., the following identities hold

$$\begin{array}{c}\hfill R\times (R\times S)=(R\times R)\times S\\ \hfill (S\times R)\times R=S\times (R\times R).\end{array}$$

For every generalized octonion R, the conjugate of R, denoted by $\overline{R}$ is defined as follows:

$$\overline{R}=r_0{e}_0-r_1{e}_1-r_2{e}_2-r_3{e}_3-r_4{e}_4-r_5{e}_5-r_6{e}_6-r_7{e}_7.$$

The conjugate operator satisfies the following properties:

i.

$\overline{R+S}=\overline{R}+\overline{S}$,

ii.

$\overline{\overline{R}}=R$,

iii.

$\overline{R\times S}=\overline{S}\times \overline{R}$.

The $\mathbb{R}$-valued symmetric bilinear form $h:\mathbb{GO}\times \mathbb{GO}\to \mathbb{R}$ is defined as follows:

$$h(R,S)={\displaystyle \frac{1}{2}}(R\times \overline{S}+S\times \overline{R})$$

or

$$\begin{array}{ccc}\hfill h(R,S)& =& r_0{s}_0-{\mu }_1{r}_1{s}_1-{\mu }_2{r}_2{s}_2+{\mu }_1{\mu }_2{r}_3{s}_3-{\mu }_3{r}_4{s}_4+{\mu }_1{\mu }_3{r}_5{s}_5+{\mu }_2{\mu }_3{r}_6{s}_6\hfill \\ & -& {\mu }_1{\mu }_2{\mu }_3{r}_7{s}_7.\hfill \end{array}$$

The scalar product defined on $\mathbb{GO}$ possesses the following properties for all $R,S,C\in \mathbb{GO}$:

i.

$h(R,S)=S_{R\times \overline{S}}$,

ii.

$h(R,S+C)=h(R,S)+h(R,C),$

iii.

$\alpha h(R,S)=h(\alpha R,S)=h(R,\alpha S).$

A generalized octonion R is classified as space-like, time-like, or null depending on the sign of $h(R,R)$; that is, if $h(R,R)>0$, R is space-like; if $h(R,R)<0$, R is time-like; and if $h(R,R)=0$, R is null. We define ${\theta }_R$ as the sign of R, where ${\theta }_R=1$ for space-like R, and ${\theta }_R=-1$ for time-like R. The norm of R is defined as follows:

$$\begin{array}{ccc}\hfill \left|\right|R\left|\right|& =& \sqrt{{\theta }_{R}h(R,R)}\hfill \\ & =& \sqrt{{\theta }_R(r_0{s}_0-{\mu }_1{r}_1{s}_1-{\mu }_2{r}_2{s}_2+{\mu }_1{\mu }_2{r}_3{s}_3-{\mu }_3{r}_4{s}_4+{\mu }_1{\mu }_3{r}_5{s}_5+{\mu }_2{\mu }_3{r}_6{s}_6-{\mu }_1{\mu }_2{\mu }_3{r}_7{s}_7)}.\hfill \end{array}$$

The norm satisfies the following properties:

i.

$\parallel R\times S\parallel =\parallel R\parallel \parallel S\parallel ,$

ii.

$h(R\times C,S\times C)=\parallel C\parallel h(R,S),$

iii.

$h(R,S\times R)=S_S\parallel R\parallel .$

Any generalized octonion can be written as $R=S_R+V_R$, where $S_R={\displaystyle \frac{1}{2}}(R+\overline{R})$ is the scalar part, and $V_R={\displaystyle \frac{1}{2}}(R-\overline{R})$ is the vector part. The set is as follows:

$${\mathbb{GO}}^p=\{R\in \mathbb{GO}|R+\overline{R}=0\}$$

is referred to as the space of spatial generalized octonions, and its elements are called spatial generalized octonions. For any $R,S\in {\mathbb{GO}}^p$, the product $R\times S$ can be expressed as follows:

$$R\times S=-⟨R,S⟩+R▴S.$$

In particular, if R and S are orthogonal spatial generalized octonions, then the scalar product vanishes, and the expression simplifies to the following:

$$R\times S=R▴S.$$

The inverse of the generalized octonion $R\in \mathbb{GO}$, provided that $R\ne 0$, is defined as follows:

$$R^{-1}={\displaystyle \frac{\overline{R}}{{\parallel R\parallel }^2}},$$

and it satisfies the following properties:

i.

$\parallel R^{-1}{\parallel =\parallel R\parallel }^{-1}$

ii.

$R\times R^{-1}=R^{-1}\times R$

iii.

$(S\times R^{-1})\times R=S$ and $R^{-1}\times (R\times S)=S$.

3. G₂-Frame Fields Along Spatial Generalized Octonionic Curves

In this section we construct the $G_2$-frame and the derivative formulas associated with this frame. We present a relation between the $G_2$ frame and the standard orthonormal basis of spatial generalized octonions. Let $I\subset \mathbb{R}$ and $s\in I$. A smooth map $\beta :I\to {\mathbb{GO}}^p$, given in the form $\beta \left(s\right)={\sum }_{i=1}^7{\beta }_i\left(s\right)e_i,$ is called a spatial generalized octonionic curve.

Definition 1.

Let $\beta :I\to {\mathbb{GO}}^p$ be a spatial generalized octonionic curve. If for every $s\in I$,

$$h({\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right))={\displaystyle \frac{1}{2}}({\beta }^{\prime }\left(s\right)\times \overline{{\beta }^{\prime }\left(s\right)}+{\beta }^{\prime }\left(s\right)\times \overline{{\beta }^{\prime }\left(s\right)}={\beta }^{\prime }\left(s\right)\times \overline{{\beta }^{\prime }\left(s\right)})={\theta }_0,$$

where ${\theta }_0\in \{1,-1\}$, then the curve is said to be a unit-speed spatial generalized octonionic curve. Furthermore, the causal character of the curve is determined by the value of $h({\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right))$ as follows:

• If $h({\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right))=1$, the curve is called space-like,
• If $h({\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right))=-1,$ the curve is called time-like
• If $h({\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right))=0,$ the curve is called null.

The causal character of a curve is determined by its classification as space-like, time-like, or null.

Throughout this study, we consider only frames consisting entirely of non-null vectors.

Theorem 1.

Let $\beta :I\subset \mathbb{R}\to {\mathbb{GO}}^p$ be a smooth, unit-speed, non-null spatial generalized octonionic curve defined by $\beta \left(s\right)={\sum }_{i=1}^7{\beta }_i\left(s\right)e_i$ for $s\in I,$ where the tangent vector is given by $t\left(s\right)={\beta }^{\prime }\left(s\right)$. Then the following properties hold:

(i)

The derivative $t^{\prime }\left(s\right)$ is orthogonal to $t\left(s\right)$, i.e., $h(t^{\prime }\left(s\right),t\left(s\right))=0$,

(ii)

The product $t^{\prime }\left(s\right)\times \overline{t\left(s\right)}$ belongs to the space of spatial generalized octonion ${\mathbb{GO}}^p$.

Proof. 

Let $\beta \left(s\right)={\sum }_{i=1}^7{\beta }_i\left(s\right)e_i$ be a generalized non-null unit-speed octonionic curve. Its tangent vector is given by $t\left(s\right)={\beta }^{\prime }\left(s\right)={\sum }_{i=1}^7{\beta }_i^{\prime }\left(s\right)e_i$ and the conjugate of t is $\overline{t}=-{\sum }_{i=1}^7{\beta }_i^{\prime }\left(s\right)e_i$. It is straightforward to verify that $\overline{t^{\prime }}={\overline{t}}^{\prime }$. Assuming the generalized inner product satisfies $h(t\left(s\right),t\left(s\right))={\theta }_0$, the following is written:

$$t\prime \times \overline{t}+t\times \overline{t}\prime =0.$$

(1)

Since $h(t\left(s\right),t\left(s\right))={\theta }_0,$ using Equation (1) we find $h(t,t\prime )=0$, which confirms the orthogonality and thus establishes the claim in part i.

Applying Equation (1) along with the properties of conjugation in $\mathbb{GO}$, we obtain the following:

$$t\prime \times \overline{t}+\overline{t\prime \times \overline{t}}=0.$$

Therefore, $t\prime \times \overline{t}$ is a spatial generalized octonion. This completes the proof of part ii.    □

Let t be a spatial generalized octonion. Define the unit spatial generalized octonion $n_1$ and the non-negative scalar function ${\kappa }_1$ through the following relation:

$$t\prime ={\theta }_1{\kappa }_1{n}_1,h(n_1,n_1)={\theta }_1,{\theta }_1\in \{1,-1\}$$

(2)

where

$${\kappa }_1=\parallel t\prime \parallel .$$

(3)

denotes the first curvature. Using property (i) of Theorem 1, we compute the following:

$$h(n_1,t)={\displaystyle \frac{{\theta }_1}{{\kappa }_1}}h(t\prime ,t)=0$$

which shows that $n_1$ is orthogonal to t with respect to bilinear form h. Differentiating the orthogonality equation $h(t,n_1)=0$ with respect to the parameter s, we obtain the following equation:

$$h(t\prime ,n_1)+h(t,n_1\prime )=0.$$

(4)

From Equation (2), the following is written:

$$h(t\prime ,n_1)=h({\theta }_1{\kappa }_1{n}_1,n_1)={\kappa }_1.$$

Substituting this expression into Equation (4), we obtain the following:

$$h(n_1\prime ,t)=-{\kappa }_1.$$

(5)

Next, we define the second vector of the orthonormal frame by the following:

$$n_2=t\times n_1.$$

By property ($ii$) of Theorem 1, we conclude that $n_2=t\times n_1$ is a spatial generalized octonion, i.e., $n_2\in {\mathbb{GO}}^p$. Moreover, using the properties of the bilinear form h, we compute $h(n_2,t)=0$ and $h(n_2,n_1)=0$, which implies that the vectors $t,n_1,$ and $n_2$ are mutually orthogonal spatial generalized octonions. Furthermore, the following identities hold:

$$t\times n_2=t\times (t\times n_1)=⟨t,n_1⟩t-⟨t,t⟩n_1=-{\theta }_0{n}_1,$$

$$n_1\times n_2={\theta }_{1}t.$$

Next, the unit spatial generalized octonion $n_3$, which is orthogonal to t, $n_1$, and $n_2$, is defined by the following expression:

$$n_3={\displaystyle \frac{{\theta }_3}{{\kappa }_2}}(n_1^{\prime }-{\theta }_{0}h(n_1^{\prime },t)t-{\theta }_{2}h(n_1^{\prime },n_2)n_2),$$

(6)

where

$${\kappa }_2=\sqrt{{\theta }_3(\parallel n_1^{\prime }{\parallel }^2-{\theta }_{0}h{(n_1^{\prime },t)}^2-{\theta }_{2}h{(n_1^{\prime },n_2)}^2)}$$

and $h(n_3,n_3)={\theta }_3$. This construction ensures that $n_3$ is orthogonal to the previously defined vectors. We then define the following elements of the frame as follows:

$$\begin{array}{ccccc}\hfill n_4=t\times n_3,& \hfill & \hfill n_5=n_1\times n_3,& \hfill & \hfill n_6=n_2\times n_3\end{array}$$

(7)

Using the properties of vector product in the algebra of spatial generalized octonions, the following identities can be established:

$$\begin{array}{ccc}\hfill & t\times n_4=-{\theta }_0{n}_3,\hfill & \hfill n_3\times n_4={\theta }_{3}t,\\ \hfill & n_1\times n_5=-{\theta }_1{n}_3,\hfill & \hfill n_3\times n_5={\theta }_3{n}_1,\\ \hfill & n_5\times n_6=-{\theta }_{5}t,\hfill & \hfill t\times n_6={\theta }_0{n}_5,\\ \hfill & n_1\times n_6=-{\theta }_1{n}_4,\hfill & \hfill n_4\times n_6={\theta }_4{n}_1,\\ \hfill & n_2\times n_6=-{\theta }_2{n}_3,\hfill & \hfill n_3\times n_6={\theta }_3{n}_2,\end{array}$$

where ${\theta }_0=h(t,t)$ and ${\theta }_i=h(n_i,n_i)$ for $i=1,...,6$, and ${\theta }_i\in \{-1,1\}$ for $i=0,...,6$. Moreover, the following relations hold:

$$\begin{array}{ccccccc}\hfill {\theta }_2={\theta }_0{\theta }_1,& \hfill & \hfill {\theta }_4={\theta }_0{\theta }_3,& \hfill & \hfill {\theta }_5={\theta }_1{\theta }_3,& \hfill & \hfill {\theta }_6={\theta }_0{\theta }_5.\end{array}$$

(8)

Thus, we obtain a complete orthonormal set $\{t,n_1,n_2,n_3,n_4,n_5,and n_6\}$ of spatial generalized octonions, for which a multiplication Table 3 can be constructed based on the identities above.

Table 3. Multiplication table of $t,n_1,n_2,n_3,n_4,n_5,n_6$.

×	t	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	${\mathit{n}}_4$	${\mathit{n}}_5$	${\mathit{n}}_6$
t	$-{\theta }_0$	$n_2$	$-{\theta }_0{n}_1$	$n_4$	$-{\theta }_0{n}_3$	$-n_6$	${\theta }_0{n}_5$
$n_1$	$-n_2$	$-{\theta }_1$	${\theta }_{1}t$	$n_5$	$n_6$	$-{\theta }_1{n}_3$	$-{\theta }_1{n}_4$
$n_2$	${\theta }_0{n}_1$	$-{\theta }_{1}t$	$-{\theta }_2$	$n_6$	$n_5$	$-{\theta }_2{n}_4$	$-{\theta }_2{n}_3$
$n_3$	$-n_4$	$-n_5$	$-n_6$	$-{\theta }_3$	${\theta }_{3}t$	${\theta }_3{n}_1$	${\theta }_3{n}_2$
$n_4$	${\theta }_0{n}_3$	$-n_6$	$-n_5$	${\theta }_{3}t$	$-{\theta }_4$	${\theta }_4{n}_2$	${\theta }_4{n}_1$
$n_5$	$n_6$	${\theta }_1{n}_3$	${\theta }_2{n}_4$	$-{\theta }_3{n}_1$	$-{\theta }_4{n}_2$	$-{\theta }_5$	$-{\theta }_{5}t$
$n_6$	$-{\theta }_0{n}_5$	${\theta }_1{n}_4$	${\theta }_2{n}_3$	${\theta }_3{n}_2$	$-{\theta }_4{n}_1$	${\theta }_{5}t$	$-{\theta }_6$

In addition, if we define a new set of basis elements, written as follows:

$$\begin{array}{ccccccc}\hfill E_0=e_0,& \hfill & \hfill E_1=e_1/\sqrt{|{\mu }_1|},& \hfill & \hfill E_2=e_2/\sqrt{|{\mu }_2|},& \hfill & \hfill E_3=e_3/\sqrt{|{\mu }_1{\mu }_2|},\\ \hfill E_4=e_4/\sqrt{|{\mu }_3|}& \hfill & \hfill E_5=e_5/\sqrt{|{\mu }_1{\mu }_3|}& \hfill & \hfill E_6=e_6/\sqrt{|{\mu }_2{\mu }_3|}& \hfill & \hfill E_7=e_7/\sqrt{|{\mu }_1{\mu }_2{\mu }_3|},\end{array}$$

where ${\mu }_i\ne 0,i=1,2,3,$ then $\{E_0,E_1,E_2,E_3,E_4,E_5,E_6,E_7\}$ forms a standard orthonormal basis for the algebra of generalized octonions with respect to the associated quadratic form. The multiplication rules for this orthonormal basis are presented in Table 4.

Table 4. Multiplication table of orthonormal basis $\{E_0,E_1,E_2,E_3,E_4,E_5,E_6,E_7\}$.

×	${\mathit{E}}_0$	${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$	${\mathit{E}}_4$	${\mathit{E}}_5$	${\mathit{E}}_6$	${\mathit{E}}_7$
$E_0$	$E_0$	$E_1$	$E_2$	$E_3$	$E_4$	$E_5$	$E_6$	$E_7$
$E_1$	$E_1$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}$	$E_3$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_2$	$E_5$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_4$	$-E_7$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{e}_6$
$E_2$	$E_2$	$-E_3$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_1$	$E_6$	$E_7$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_4$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$
$E_3$	$E_3$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_2$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_1$	$-{\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}}$	$E_7$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_6$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$	$-{\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}}{E}_4$
$E_4$	$E_4$	$-E_5$	$-E_6$	$-E_7$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_1$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_2$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$
$E_5$	$E_5$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_4$	$-E_7$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_6$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_1$	$-{\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}}$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$	${\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}}{E}_2$
$E_6$	$E_6$	$E_7$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_4$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_2$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$	$-{\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}}$	$-{\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}}{E}_1$
$E_7$	$E_7$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_6$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$	${\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}}{E}_4$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$	$-{\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}}{E}_2$	${\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}}{E}_1$	${\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3}{|{\mu }_1{\mu }_2{\mu }_3|}}$

Furthermore, the multiplication rules for the standard orthonormal basis of spatial generalized octonions are presented in Table 5.

Table 5. Multiplication table of spatial orthonormal basis $\{E_0,E_1,E_2,E_3,E_4,E_5,E_6,E_7\}$.

×	${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$	${\mathit{E}}_4$	${\mathit{E}}_5$	${\mathit{E}}_6$	${\mathit{E}}_7$
$E_1$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}$	$E_3$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_2$	$E_5$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_4$	$-E_7$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{e}_6$
$E_2$	$-E_3$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_1$	$E_6$	$E_7$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_4$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$
$E_3$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_2$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_1$	$-{\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}}$	$E_7$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_6$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$	$-{\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}}{E}_4$
$E_4$	$-E_5$	$-E_6$	$-E_7$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_1$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_2$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$
$E_5$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_4$	$-E_7$	$-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_6$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_1$	$-{\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}}$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$	${\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}}{E}_2$
$E_6$	$E_7$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_4$	${\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_2$	$-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$	$-{\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}}$	$-{\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}}{E}_1$
$E_7$	${\displaystyle \frac{{\mu }_1}{|{\mu }_1|}}{E}_6$	$-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}}{E}_5$	${\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}}{E}_4$	${\displaystyle \frac{{\mu }_3}{|{\mu }_3|}}{E}_3$	$-{\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}}{E}_2$	${\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}}{E}_1$	${\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3}{|{\mu }_1{\mu }_2{\mu }_3|}}$

In particular, if we choose the following parameters:

$$\begin{array}{ccccccc}\hfill {\theta }_0=-{\displaystyle \frac{{\mu }_1}{|{\mu }_1|}},& \hfill & \hfill {\theta }_1=-{\displaystyle \frac{{\mu }_2}{|{\mu }_2|}},& \hfill & \hfill {\theta }_2={\displaystyle \frac{{\mu }_1{\mu }_2}{|{\mu }_1{\mu }_2|}},& \hfill & \hfill {\theta }_3=-{\displaystyle \frac{{\mu }_3}{|{\mu }_3|}},\end{array}$$

(9)

$$\begin{array}{ccccc}\hfill {\theta }_4={\displaystyle \frac{{\mu }_1{\mu }_3}{|{\mu }_1{\mu }_3|}},& \hfill & \hfill {\theta }_5={\displaystyle \frac{{\mu }_2{\mu }_3}{|{\mu }_2{\mu }_3|}},& \hfill & \hfill {\theta }_6=-{\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3}{|{\mu }_1{\mu }_2{\mu }_3|}}\end{array}$$

(10)

then, the multiplication Table 3, corresponding to the orthonormal basis $\{t,n_1,n_2,n_3,n_4,n_5,n_6\}$, matches that of Table 5, which corresponds to the basis $\{E_1,E_2,E_3,E_4,E_5,E_6,E_7\}.$

Moreover, if we consider the rescaled $G_2$-frame, written as follows:

$$\begin{array}{ccc}& & \mathfrak{t}=\sqrt{|{\mu }_1|}t,{\mathfrak{n}}_{\mathfrak{1}}=\sqrt{|{\mu }_2|}{n}_1,{\mathfrak{n}}_{\mathfrak{2}}=\sqrt{|{\mu }_1{\mu }_2|}{n}_2,{\mathfrak{n}}_{\mathfrak{3}}=\sqrt{|{\mu }_3|}{n}_3,\hfill \\ & & {\mathfrak{n}}_{\mathfrak{4}}=\sqrt{|{\mu }_1{\mu }_2|}{n}_4,{\mathfrak{n}}_{\mathfrak{5}}=\sqrt{|{\mu }_2{\mu }_3|}{n}_5,{\mathfrak{n}}_{\mathfrak{6}}=\sqrt{|{\mu }_1{\mu }_2{\mu }_3|}{n}_6,\hfill \end{array}$$

then the corresponding multiplication rules can be presented into Table 6.

Table 6. Multiplication table of spatial orthogonal basis $\{\mathfrak{t},{\mathfrak{n}}_{\mathfrak{i}},i=1,...,6\}$.

×	$\mathfrak{t}$	${\mathfrak{n}}_{\mathfrak{1}}$	${\mathfrak{n}}_{\mathfrak{2}}$	${\mathfrak{n}}_{\mathfrak{3}}$	${\mathfrak{n}}_{\mathfrak{4}}$	${\mathfrak{n}}_{\mathfrak{5}}$	${\mathfrak{n}}_{\mathfrak{6}}$
$\mathfrak{t}$	${\mu }_1$	${\mathfrak{n}}_{\mathfrak{2}}$	${\mu }_1{\mathfrak{n}}_{\mathfrak{1}}$	${\mathfrak{n}}_{\mathfrak{4}}$	${\mu }_1{\mathfrak{n}}_{\mathfrak{3}}$	$-{\mathfrak{n}}_{\mathfrak{6}}$	$-{\mu }_1{\mathfrak{n}}_{\mathfrak{5}}$
${\mathfrak{n}}_{\mathfrak{1}}$	$-{\mathfrak{n}}_{\mathfrak{2}}$	${\mu }_2$	$-{\mu }_2\mathfrak{t}$	${\mathfrak{n}}_{\mathfrak{5}}$	${\mathfrak{n}}_{\mathfrak{6}}$	${\mu }_2{\mathfrak{n}}_{\mathfrak{3}}$	${\mu }_2{\mathfrak{n}}_{\mathfrak{4}}$
${\mathfrak{n}}_{\mathfrak{2}}$	$-{\mu }_1{\mathfrak{n}}_{\mathfrak{1}}$	${\mu }_2\mathfrak{t}$	$-{\mu }_1{\mu }_2$	${\mathfrak{n}}_{\mathfrak{6}}$	${\mu }_1{\mathfrak{n}}_{\mathfrak{5}}$	$-{\mu }_2{\mathfrak{n}}_{\mathfrak{4}}$	$-{\mu }_1{\mu }_2{\mathfrak{n}}_{\mathfrak{3}}$
${\mathfrak{n}}_{\mathfrak{3}}$	$-{\mathfrak{n}}_{\mathfrak{4}}$	$-{\mathfrak{n}}_{\mathfrak{5}}$	$-{\mathfrak{n}}_{\mathfrak{6}}$	${\mu }_3$	$-{\mu }_3\mathfrak{t}$	$-{\mu }_3{\mathfrak{n}}_{\mathfrak{1}}$	$-{\mu }_3{\mathfrak{n}}_{\mathfrak{2}}$
${\mathfrak{n}}_{\mathfrak{4}}$	$-{\mu }_1{\mathfrak{n}}_{\mathfrak{3}}$	$-{\mathfrak{n}}_{\mathfrak{6}}$	$-{\mu }_1{\mathfrak{n}}_{\mathfrak{5}}$	${\mu }_3\mathfrak{t}$	$-{\mu }_1{\mu }_3$	${\mu }_3{\mathfrak{n}}_{\mathfrak{2}}$	${\mu }_1{\mu }_3{\mathfrak{n}}_{\mathfrak{1}}$
${\mathfrak{n}}_{\mathfrak{5}}$	${\mathfrak{n}}_{\mathfrak{6}}$	$-{\mu }_2{\mathfrak{n}}_{\mathfrak{3}}$	${\mu }_2{\mathfrak{n}}_{\mathfrak{4}}$	${\mu }_3{\mathfrak{n}}_{\mathfrak{1}}$	$-{\mu }_3{\mathfrak{n}}_{\mathfrak{2}}$	$-{\mu }_2{\mu }_3$	$-{\mu }_2{\mu }_3\mathfrak{t}$
${\mathfrak{n}}_{\mathfrak{6}}$	${\mu }_1{\mathfrak{n}}_{\mathfrak{5}}$	$-{\mu }_2{\mathfrak{n}}_{\mathfrak{4}}$	${\mu }_1{\mu }_2{\mathfrak{n}}_{\mathfrak{3}}$	${\mu }_3{\mathfrak{n}}_{\mathfrak{2}}$	$-{\mu }_1{\mu }_3{\mathfrak{n}}_{\mathfrak{1}}$	${\mu }_2{\mu }_3\mathfrak{t}$	${\mu }_1{\mu }_2{\mu }_3$

Finally, there exists a $G_2$-valued function $g:I\to G_2$, where $G_2$ is the authomorphism group of generalized octonions, written as follows:

$$G_2=\{g\in GL\left(\mathbb{GO}\right)|g(R\times S)=g\left(R\right)\times g\left(S\right),\forall R,S\in \mathbb{GO}\},$$

such that $(t,n_1,n_2,n_3,n_4,n_5,n_6)=(g\left(E_1\right),g\left(E_2\right),g\left(E_3\right),g\left(E_4\right),g\left(E_5\right),g\left(E_6\right),g\left(E_7\right))$. The action of $G_2$ induces a transition from the standard orthonormal basis to a moving frame, which is why it is commonly referred to as the $G_2$-frame.

Theorem 2.

Let $\beta :I\to {\mathbb{GO}}^p$ be a spatial generalized octonionic curve with curvature functions ${\kappa }_1,{\kappa }_2>0$. The associated $G_2-$ frame field $(t,n_1,n_2,n_3,n_4,n_5,n_6)$ along the curve β satisfies the system of differential equations given by the following:

$${\displaystyle \frac{d}{ds}}\left[\begin{array}{c}t\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\end{array}\right]=M({\kappa }_1,{\kappa }_2,{\lambda }_1...,{\lambda }_6)\left[\begin{array}{c}t\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\end{array}\right],$$

(11)

or

$$\left[\begin{array}{c}{t}^{\prime }\\ n_1^{\prime }\\ n_2^{\prime }\\ n_3^{\prime }\\ n_4^{\prime }\\ n_5^{\prime }\\ n_6^{\prime }\end{array}\right]=\left[\begin{array}{ccccccc}0& {\theta }_1{\kappa }_1& 0& 0& 0& 0& 0\\ -{\theta }_0{\kappa }_1& 0& {\theta }_2{\lambda }_1& {\theta }_3{\kappa }_2& 0& 0& 0\\ 0& -{\theta }_1{\lambda }_1& 0& 0& {\theta }_4{\theta }_0{\kappa }_2& 0& 0\\ 0& -{\theta }_1{k}_1& 0& 0& {\theta }_4{\lambda }_2& {\theta }_5{\lambda }_3& {\theta }_6{\lambda }_4\\ 0& 0& -{\theta }_2{\theta }_0{k}_2& -{\theta }_3{\lambda }_2& 0& {\theta }_5{\lambda }_5& -{\theta }_6{\theta }_0{\lambda }_3\\ 0& 0& 0& -{\theta }_3{\lambda }_3& -{\theta }_4{\lambda }_5& 0& {\theta }_6{\lambda }_6\\ 0& 0& 0& -{\theta }_3{\lambda }_4& -{\theta }_4{\theta }_0{\lambda }_3& -{\theta }_5{\lambda }_6& 0\end{array}\right]\left[\begin{array}{c}t\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\end{array}\right],$$

where

$$\begin{array}{ccccc}\hfill & \hfill & \hfill {\kappa }_1=h(t\prime ,n_1),& \hfill & \hfill {\kappa }_2=h(n_1\prime ,n_3),\end{array}$$

(12)

$$\begin{array}{ccccccc}\hfill & {\lambda }_1=h(n_1\prime ,n_2),\hfill & \hfill {\lambda }_2=h(n_3\prime ,n_4),& \hfill & \hfill {\lambda }_3=h(n_3\prime ,n_5),& \hfill & \hfill {\lambda }_4=h(n_3\prime ,n_6)\end{array}$$

(13)

are curvature functions and the following:

$$\begin{array}{ccc}\hfill {\lambda }_5={\theta }_3{\kappa }_1+{\lambda }_4,& \hfill & \hfill {\lambda }_6={\theta }_3{\lambda }_1+{\theta }_1{\lambda }_2.\end{array}$$

(14)

Proof. 

From Equation (6), we find the following:

$$n_1^{\prime }={\theta }_3{\kappa }_2{n}_3+{\theta }_{0}h(n_1^{\prime },t)t+{\theta }_{2}h(n_1^{\prime },n_2)n_2.$$

Substituting Equation (5) and $h(n_1^{\prime },n_2)={\lambda }_1$ we find the following:

$$n_1^{\prime }=-{\theta }_0{\kappa }_{1}t+{\theta }_2{\lambda }_1{n}_2+{\theta }_3{\kappa }_2{n}_3.$$

(15)

Now, for $i=2,3,4,5,6$ we write the following:

$$n_i^{\prime }={\theta }_{0}h(n_i^{\prime },t)t+\sum _{j=1}^6{\theta }_{j}h(n_i^{\prime },n_j)n_j.$$

From, the following:

$$\begin{array}{ccccc}\hfill h(n_2^{\prime },t)=-h(n_2,t^{\prime })=0,& \hfill & \hfill h(n_2^{\prime },n_1)=-h(n_2,n_1^{\prime })=-{\lambda }_1,& \hfill & \hfill h(n_2^{\prime },n_2)=0,\\ \hfill h(n_2^{\prime },n_3)=h({(t\times n_1)}^{\prime },n_3)=0,& \hfill & \hfill h(n_2^{\prime },n_4)=h({(t\times n_1)}^{\prime },n_4)={\theta }_0{\kappa }_2,\\ \hfill h(n_2^{\prime },n_5)=h({(t\times n_1)}^{\prime },n_5)=0,& \hfill & \hfill h(n_2^{\prime },n_6)=h({(t\times n_1)}^{\prime },n_6)=0\end{array}$$

we find the following:

$$n_2^{\prime }=-{\theta }_1{\lambda }_1{n}_1+{\theta }_4{\theta }_0{\kappa }_2{n}_4.$$

(16)

We define the following:

$$\begin{array}{ccccc}\hfill h(n_3^{\prime },n_4)={\lambda }_2,& \hfill & \hfill h(n_3^{\prime },n_5)={\lambda }_3,& \hfill & \hfill h(n_3^{\prime },n_6)={\lambda }_4,\end{array}$$

(17)

then by using the following:

$$\begin{array}{ccccccc}\hfill h(n_3^{\prime },t)=0,& \hfill & \hfill h(n_3^{\prime },n_1)=-{\kappa }_2,& \hfill & \hfill h(n_3^{\prime },n_2)=0,& \hfill & \hfill h(n_3^{\prime },n_3)=0\end{array}$$

(18)

we find the following:

$$n_3^{\prime }=-{\theta }_1{\kappa }_2{n}_1+{\theta }_4{\lambda }_2{n}_4+{\theta }_5{\lambda }_3{n}_5+{\theta }_6{\lambda }_4{n}_6.$$

(19)

From the following:

$$\begin{array}{ccccc}\hfill h(n_4^{\prime },t)=0,& \hfill & \hfill h(n_4^{\prime },n_1)=0,& \hfill & \hfill h(n_4^{\prime },n_2)=-{\theta }_0{\kappa }_2,\end{array}$$

(20)

$$\begin{array}{ccccc}\hfill h(n_4^{\prime },n_3)=-{\lambda }_2,& \hfill & \hfill h(n_4^{\prime },n_5)={\theta }_3{\kappa }_1-{\lambda }_4,& \hfill & \hfill h(n_4^{\prime },6)=-{\theta }_0{\lambda }_3,\end{array}$$

(21)

defining ${\theta }_3{\kappa }_1+{\lambda }_4={\lambda }_5$, we obtain the following:

$$n_4^{\prime }=-{\theta }_2{\theta }_0{\kappa }_2{n}_2-{\theta }_3{\lambda }_2{n}_3+{\theta }_5{\lambda }_5{n}_5-{\theta }_6{\theta }_0{\lambda }_3{n}_6.$$

(22)

From the following:

$$\begin{array}{ccccccc}\hfill h(n_5^{\prime },t)=0,& \hfill & \hfill h(n_5^{\prime },n_1)=0,& \hfill & \hfill h(n_5^{\prime },n_2)=0,& \hfill & \hfill h(n_5^{\prime },n_3)=-{\lambda }_3,\end{array}$$

(23)

$$\begin{array}{ccccc}\hfill h(n_5^{\prime },n_4)=-{\lambda }_5,& \hfill & \hfill h(n_5^{\prime },n_5)=0,& \hfill & \hfill h(n_5^{\prime },n_6)={\theta }_3{\lambda }_1+{\theta }_1{\lambda }_2,\end{array}$$

(24)

defining ${\theta }_3{\lambda }_1+{\theta }_1{\lambda }_2={\lambda }_6$ we obtain the following:

$$n_5^{\prime }=-{\theta }_3{\lambda }_3{n}_3-{\theta }_4{\lambda }_5{n}_4+{\theta }_6{\lambda }_6{n}_6.$$

(25)

From the following:

$$\begin{array}{ccccccc}\hfill h(n_6^{\prime },t)=0,& \hfill & \hfill h(n_6^{\prime },n_1)=0,& \hfill & \hfill h(n_6^{\prime },n_2)=0,& \hfill & \end{array}$$

(26)

$$\begin{array}{cccccc}\hfill h(n_6^{\prime },n_3)=-{\lambda }_4,& \hfill & \hfill h(n_6^{\prime },n_4)={\theta }_0{\lambda }_3,& \hfill & \hfill h(n_6^{\prime },n_5)=-{\lambda }_6,& \hfill \end{array}$$

(27)

we obtain the following:

$$n_6^{\prime }=-{\theta }_3{\lambda }_4{n}_3+{\theta }_4{\theta }_0{\lambda }_3{n}_4-{\theta }_5{\lambda }_6{n}_5$$

(28)

   □

Remark 1.

Let β be a unit-speed spatial generalized octonionic curve, and let the following:

$$\{V_0,V_1,V_2,V_3,V_4,V_5,V_6\}$$

denote its Frenet–Serret frame, where $V_0={\beta }^{\prime }$ and the following:

$$V_1={\displaystyle \frac{{\eta }_1}{k_1}}{V}_0^{\prime },$$

with $k_1=\parallel V_0^{\prime }\parallel >0$. For $i=2,...,6,$ the frame vectors and curvatures are defined by the following:

$$V_i={\displaystyle \frac{{\eta }_i}{k_i}}(V_{i-1}^{\prime }+{\eta }_{i-2}{k}_{i-1}{V}_{i-2}),$$

and

$$\begin{array}{ccc}\hfill k_i=\parallel V_{i-1}^{\prime }+{\eta }_{i-2}{k}_{i-1}{V}_{i-2}\parallel ,& \hfill & \hfill {\eta }_i=h(V_i,V_i).\end{array}$$

Then, the Frenet–Serret formulas in ${\mathbb{R}}_4^7$ takes the following form:

$$\begin{array}{ccc}\hfill V_0^{\prime }& =& {\eta }_1{k}_1{V}_1\hfill \\ \hfill V_i^{\prime }& =& -{\eta }_{i-1}{k}_i{V}_{i-1}+{\eta }_{i+1}{k}_{i+1}{V}_{i+1},i=1,2,...,5\hfill \\ \hfill V_6^{\prime }& =& -{\eta }_5{k}_6{V}_5,\hfill \end{array}$$

or in matrix form, written as follows:

$$\left[\begin{array}{c}{V}_0^{\prime }\\ V_1^{\prime }\\ V_2^{\prime }\\ V_3^{\prime }\\ V_4^{\prime }\\ V_5^{\prime }\\ V_6^{\prime }\end{array}\right]=\left[\begin{array}{ccccccc}0& {\eta }_1{k}_1& 0& 0& 0& 0& 0\\ -{\eta }_0{k}_1& 0& {\eta }_2{k}_2& 0& 0& 0& 0\\ 0& -{\eta }_1{k}_2& 0& {\eta }_3{k}_3& 0& 0& 0\\ 0& 0& -{\eta }_2{k}_3& 0& {\eta }_4{k}_4& 0& 0\\ 0& 0& 0& -{\eta }_3{k}_4& 0& {\eta }_5{k}_5& 0\\ 0& 0& 0& 0& -{\eta }_4{k}_5& 0& {\eta }_6{k}_6\\ 0& 0& 0& 0& 0& -{\eta }_5{k}_6& 0\end{array}\right]\left[\begin{array}{c}{V}_0\\ V_1\\ V_2\\ V_3\\ V_4\\ V_5\\ V_6\end{array}\right].$$

(29)

Equation (29) is referred to as the Frenet–Serret formulas for a unit-speed spatial generalized octonionic curve. It is important to note that ${\eta }_0={\theta }_0,{\eta }_1={\theta }_1$ and $k_1={\kappa }_1$.

4. G₂-Congruence and Relation Between G₂ and Frenet–Serret Frames

In this section, we investigate the $G_2$-congruence of curves. We present a relation between $G_2$-frame vectors and Frenet–Serret frame vectors. Also, we calculate the $G_2$-curvatures with respect to the Frenet–Serret curvatures.

Definition 2.

Let β and $\tilde{\beta }$ be two unit-speed spatial generalized octonionic curves sharing the same causal character. These curves are said to be $G_2$-congruent if there exists an isometry

$$\begin{array}{c}\hfill F=T_p\circ g,\end{array}$$

where $T_p$ denotes translation by the following:

$$\begin{array}{c}\hfill p=(p_1,p_2,p_3,p_4,p_5,p_6,p_7)\in {\mathbb{GO}}^p,\end{array}$$

and $g\in G_2$, together with some parameter shift $s_0\in \mathbb{R}$, such that we obtain the following:

$$\begin{array}{c}\hfill g\left(\beta (s+s_0)\right)+p=\tilde{\beta }\left(s\right)\end{array}$$

holds for all $s\in I$. Equivalently, this relation can be expressed as follows:

$$\tilde{\beta }\left(s\right)=g\left(\beta (s+s_0)\right)+p=\left[\begin{array}{cccccccc}0& E_1& E_2& E_3& E_4& E_5& E_6& E_7\end{array}\right]\left[\begin{array}{cc}1& 0_{1\times 7}\\ p& C\end{array}\right]\left[\frac{1}{\widehat{\beta }\left(s\right)}\right],$$

where

$$C\in G_2\subset M_{7\times 7}\left(\mathbb{R}\right),\beta \left(s\right)=\left[\begin{array}{ccccccc}{E}_1& E_2& E_3& E_4& E_5& E_6& E_7\end{array}\right]\widehat{\beta }\left(s\right)$$

and $\widehat{\beta }:I\to M_{7\times 1}\left(\mathbb{R}\right)$ is the coordinate representation of β.

Theorem 3.

Let β be a unit-speed curve in the spatial generalized octonionic space ${\mathbb{GO}}^p$, and let $\tilde{\beta }$ be a curve $G_2$-congruent to β. Then the functions ${\kappa }_1,{\kappa }_2,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ introduced in Theorem 2, associated with the curve β, remain invariant under the natural action of $G_2$.

Proof. 

Let $t,n_1,n_2,n_3,n_4,n_5,n_6$ and $\tilde{t},{\tilde{n}}_1,{\tilde{n}}_2,{\tilde{n}}_3,{\tilde{n}}_4,{\tilde{n}}_5,{\tilde{n}}_6$ denote the $G_2$-frame fields associated with the curves $\beta$ and $\tilde{\beta }$, respectively. Let $F=T_{a}g$, where $T_a$ is a translation and $g\in G_2$ such that the following is written:

$$F\left(\beta \left(s\right)\right)=\tilde{\beta }\left(s\right)$$

for all s in the domain of the curves, be an isometry. Then, the corresponding frame fields are related by the following:

$$\begin{array}{c}\hfill \left[\begin{array}{ccccccc}\tilde{t}& {\tilde{n}}_1& {\tilde{n}}_2& {\tilde{n}}_3& {\tilde{n}}_4& {\tilde{n}}_5& {\tilde{n}}_6\end{array}\right]=\left[\begin{array}{ccccccc}t& n_1& n_2& n_3& n_4& n_5& n_6)\end{array}\right]C\\ \hfill =\left[\begin{array}{ccccccc}g\left(t\right)& g\left(n_1\right)& g\left(n_2\right)& g\left(n_3\right)& g\left(n_4\right)& g\left(n_5\right)& g\left(n_6\right)\end{array}\right].\end{array}$$

Since g preserve the inner product h, the following is calcualted:

$$\widetilde{{\kappa }_1}=h(\widetilde{t^{\prime }},{\tilde{n}}_1)=h(g\left(t^{\prime }\right),g\left(n_1\right))=h(t^{\prime },n_1)={\kappa }_1.$$

By analogous computations, one obtains the invariance of the remaining invariants:

$$\begin{array}{ccccc}\hfill \widetilde{{\kappa }_2}={\kappa }_2,& \hspace{1em}\hfill & \hfill \widetilde{{\lambda }_i}={\lambda }_i,& \hspace{1em}\hfill & \hfill i=1,2,3,4.\end{array}$$

   □

Two curves are called parallel when their congruence is established exclusively through translation, with no rotational component involved. Consequently, the two curves differ by a constant vector and their coordinate functions coincide up to this translation.

Lemma 1.

Let $\beta ,\tilde{\beta }:I\in \mathbb{R}\to {\mathbb{GO}}^p$ be two spatial generalized octonionic curves. Then β and ${\beta }^{\prime }$ are said to be parallel if their tangent vectors ${\beta }^{\prime }\left(s\right)$ and ${\tilde{\beta }}^{\prime }\left(s\right)$ are linearly dependent for all $s\in I$. Moreover, if ${\beta }^{\prime }\left(s_0\right)=\widetilde{{\beta }^{\prime }}\left(s_0\right)$ at some point $s_0\in I$, then $\beta =\tilde{\beta }$ for all $s\in S$; that is, the two curves coincide.

Proof. 

The result follows directly from the definitions of parallel curves and the velocity vector.    □

Theorem 4.

Let $\beta ,\tilde{\beta }:I\subset \mathbb{R}\to {\mathbb{GO}}^p$ be two unit-speed spatial generalized octonionic curves parametrized by the same arc-length. If their $G_2$-invariants coincide, i.e., the following:

$${\tilde{\kappa }}_i={\kappa }_i(i=1,2),{\tilde{\lambda }}_i={\lambda }_i(i=1,2,3,4)$$

for all $s\in I$, then β and $\tilde{\beta }$ are $G_2$-congruent.

Proof. 

Let $\{t,n_1,n_2,n_3,n_4,n_5,n_6\}$ and $\{\tilde{t},{\tilde{n}}_1,{\tilde{n}}_2,{\tilde{n}}_3,{\tilde{n}}_4,{\tilde{n}}_5,{\tilde{n}}_6\}$ denote the $G_2$-frame fields along the unit-speed spatial generalized octonionic curves $\beta$ and $\tilde{\beta }$. Define the $G_2\times {\mathbb{GO}}^p$-valued functions $\mathbf{L}$ and $\tilde{\mathbf{L}}$ by the following:

$$\begin{array}{ccc}\hfill \mathbf{L}\left(s\right)& =& \left[\begin{array}{cccccccc}\beta & t& n_1& n_2& n_3& n_4& n_5& n_6\end{array}\right]\hfill \\ & =& \left[\begin{array}{cccccccc}0& E_1& E_2& E_3& E_4& E_5& E_6& E_7\end{array}\right]\left[\begin{array}{cc}1& 0_{1\times 7}\\ \widehat{\beta }& C\end{array}\right]\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \tilde{\mathbf{L}}\left(s\right)& =& \left[\begin{array}{cccccccc}\tilde{\beta }& \tilde{t}& \widetilde{n_1}& \widetilde{n_2}& \widetilde{n_3}& \widetilde{n_4}& \widetilde{n_5}& \widetilde{n_6}\end{array}\right]\hfill \\ & =& \left[\begin{array}{cccccccc}0& E_1& E_2& E_3& E_4& E_5& E_6& E_7\end{array}\right]\left[\begin{array}{cc}1& 0_{1\times 7}\\ \widehat{\tilde{\beta }}& \tilde{C}\end{array}\right].\hfill \end{array}$$

Then there exists a $G_2\times {\mathbb{GO}}^p$- valued function $\mathfrak{v}$ such that we obtain the following:

$$\tilde{\mathbf{L}}\left(s\right)=\mathfrak{v}\left(s\right)\mathbf{L}\left(s\right)$$

(30)

for all $s\in I$. Since the $G_2$-invariants of the two curves coincide, i.e., $\widetilde{{\kappa }_i}={\kappa }_i$ for $i=1,2$, and $\widetilde{{\lambda }_i}={\lambda }_i$ for $i=1,2,3,4$, the following is calculated from Theorem 2:

$$\begin{array}{ccc}\hfill \mathfrak{v}\prime \left(s\right)& =& {\left(\tilde{\mathbf{L}}\left(s\right){\mathbf{L}}^{-1}\left(s\right)\right)}^{\prime }\hfill \\ & =& {\tilde{\mathbf{L}}}^{\prime }\left(s\right)(M({\kappa }_1,...,{\lambda }_6)-M({\tilde{\kappa }}_1,...,{\tilde{\lambda }}_6)){\mathbf{L}}^{-1}\left(s\right)M^{-1}({\tilde{\kappa }}_1,...,{\tilde{\lambda }}_6)=0.\hfill \end{array}$$

Hence, ${\mathfrak{v}}^{\prime }\left(s\right)=0$, implying that $\mathfrak{v}$ is constant. Therefore, $\tilde{\beta }\left(s\right)=g\left(\beta \left(s\right)\right)+a$ for some $g\in G_2$ and translation vector $a\in {\mathbb{GO}}^p$, which shows that $\beta$ and $\tilde{\beta }$ are $G_2-$congruent.    □

Theorem 5.

The correspondence between the $G_2-$ frame $\{t,n_1,n_2,n_3,n_4,n_5,n_6\}$ and the Frenet–Serret frame $\{V_0,V_1,V_2,V_3,V_4,V_5,V_6\}$ is established through the following relations:

$$\begin{array}{cc}\hfill & t=V_0,\hfill \\ \hfill & n_1=V_1,\hfill \\ \hfill & n_2=V_0\times V_1,\hfill \\ \hfill & n_3={\displaystyle \frac{1}{{\kappa }_2}}[{\theta }_3{\eta }_2{k}_2{V}_2-{\theta }_3{\theta }_2{\lambda }_1{V}_0\times V_1],\hfill \\ \hfill & n_4={\displaystyle \frac{{\theta }_3{\eta }_2{k}_2}{{\kappa }_2}}{V}_0\times V_2,\hfill \\ \hfill & n_5={\displaystyle \frac{{\theta }_3{\eta }_2{k}_2}{{\kappa }_2}}{V}_1\times V_2,\hfill \\ \hfill & n_6=-{\displaystyle \frac{{\theta }_3{\eta }_2{k}_2}{{\kappa }_2}}{V}_0\times (V_1\times V_2).\hfill \end{array}$$

Proof. 

It is evident that the following are calculated:

$$\begin{array}{ccc}\hfill t& =& {\beta }^{\prime }=V_0,\hfill \\ \hfill n_1& =& {\displaystyle \frac{{\theta }_1}{{\kappa }_1}}{t}^{\prime }={\displaystyle \frac{{\theta }_1}{{\kappa }_1}}{V}_0^{\prime }=V_1,\hfill \\ \hfill n_2& =& t\times n_1=V_0\times V_1.\hfill \end{array}$$

From (6), we find the following:

$$\begin{array}{cc}\hfill n_3& ={\displaystyle \frac{{\theta }_3}{{\kappa }_2}}(n_1^{\prime }-{\theta }_{0}h(n_1^{\prime },t)t-{\theta }_{2}h(n_1^{\prime },n_2)n_2)\hfill \\ \hfill & ={\displaystyle \frac{{\theta }_3}{{\kappa }_2}}(V_1^{\prime }+{\theta }_0{k}_1{V}_0-{\theta }_2{\lambda }_1{V}_0\times V_1)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\kappa }_2}}[{\theta }_3{\eta }_2{k}_2{V}_2-{\theta }_3{\theta }_2{\lambda }_1{V}_0\times V_1].\hfill \end{array}$$

Thus, by proceeding with analogous computations, the desired result follows directly, completing the proof.

   □

Theorem 6.

The $G_2$-invariants associated with the unit-speed curve β in ${\mathbb{GO}}^p$ are determined by the following expressions:

$$\begin{array}{ccc}& & {\kappa }_1=k_1,\hfill \\ & & {\kappa }_2=\sqrt{{\theta }_2({\eta }_2{k}_2^2-{\theta }_5{\lambda }_1^2)},\hfill \\ & & {\lambda }_1={\eta }_2{k}_{2}h(V_2,V_0\times V_1),\hfill \\ & & {\lambda }_2={\theta }_3{\eta }_1{\displaystyle \frac{k_2}{{\kappa }_2}}{\lambda }_1+{\theta }_3{\eta }_2{\eta }_3{\displaystyle \frac{k_2{k}_3}{{\kappa }_2}}h(V_3,V_0\times V_2)\hfill \\ & & {\lambda }_3={\eta }_3{\displaystyle \frac{k_2^2{k}_3}{{\kappa }_2^2}}h(V_3,V_1\times V_2)\hfill \\ & & {\lambda }_4=-{\eta }_3{\displaystyle \frac{k_2^2{k}_3}{{\kappa }_2^2}}h(V_3,V_0\times (V_1\times V_2)).\hfill \end{array}$$

${\lambda }_5$ and ${\lambda }_6$ can be calculated using (14).

Proof. 

Applying Theorem 2 and the equations in Theorem 5 we obtain the following:

$${\lambda }_1=h(n_1^{\prime },n_2)=h(V_1^{\prime },V_0\times V_1)={\eta }_2{k}_{2}h(V_2,V_0\times V_1)$$

and

$$\begin{array}{cc}\hfill {\kappa }_2=& \sqrt{{\theta }_3(h(n_1^{\prime },n_1^{\prime })-{\theta }_{0}h{(n_1^{\prime },t)}^2-{\theta }_{2}h{(n_1^{\prime },n_2)}^2)}\hfill \\ \hfill =& \sqrt{{\theta }_3(h(V_1^{\prime },V_1^{\prime })-{\theta }_0{k}_1^2-{\theta }_2{\lambda }_2)}\hfill \\ \hfill =& \sqrt{{\theta }_3({\eta }_0{k}_1^2+{\eta }_2{k}_2^2-{\theta }_0{\kappa }_1^2-{\theta }_2{\lambda }_1^2)}\hfill \\ \hfill =& \sqrt{{\theta }_3({\eta }_2{k}_2^2-{\theta }_2{\lambda }_1^2)}.\hfill \end{array}$$

Differentiating $n_3$ we find the following:

$$\begin{array}{cc}\hfill n_3^{\prime }=& {\theta }_2{\eta }_2{\left({\displaystyle \frac{k_2}{{\kappa }_2}}\right)}^{\prime }{V}_2-{\theta }_1{\theta }_2{\eta }_2{\displaystyle \frac{k_2^2}{{\kappa }_2}}{V}_1+{\theta }_2{\eta }_2{\eta }_3{\displaystyle \frac{k_2{k}_3}{{\kappa }_2}}{V}_3\hfill \\ \hfill & -{\theta }_3{\theta }_2{\left({\displaystyle \frac{{\lambda }_1}{{\kappa }_2}}\right)}^{\prime }{V}_0\times V_1-{\theta }_2{\theta }_3{\eta }_2{\displaystyle \frac{{\lambda }_1}{{\kappa }_2}}{k}_2{V}_0\times V_2,\hfill \end{array}$$

(31)

and using Equation (31) we obtain the following:

$$\begin{array}{cc}\hfill {\lambda }_2=& h(n_3^{\prime },n_4)={\theta }_3{\eta }_1{\displaystyle \frac{k_2}{{\kappa }_2}}{\lambda }_1+{\theta }_3{\eta }_2{\eta }_3{\displaystyle \frac{k_2{k}_3}{{\kappa }_2}}h(V_3,V_0\times V_2).\hfill \end{array}$$

Similarly, we obtain the following:

$$\begin{array}{cc}\hfill {\lambda }_3=& h(n_3^{\prime },n_5)={\eta }_3{\displaystyle \frac{k_2^2{k}_3}{{\kappa }_2^2}}h(V_3,V_1\times V_2)\hfill \end{array}$$

and the following:

$$\begin{array}{c}\hfill {\lambda }_4=h(n_3^{\prime },n_6)={\eta }_3{\displaystyle \frac{k_2^2{k}_3}{{\kappa }_2^2}}h(V_3,V_0\times (V_1\times V_2)).\end{array}$$

   □

5. An Application

In this section, we provide an example of a unit spatial generalized octonionic curve. We construct its $G_2$-frame and derive associated $G_2$-invariants numerically using MATLAB.

Example 1.

Consider the space ${\mathbb{R}}^7$ together with the inner product, calculated as follows:

$$\begin{array}{c}\hfill {⟨\mathbf{r},\mathbf{s}⟩}_{2,3,5}=-(2r_1{s}_1+3r_2{s}_2-6r_3{s}_3+5r_4{s}_4-10r_5{s}_5-15r_6{s}_6+30r_7{s}_7),\end{array}$$

The space $({\mathbb{R}}^7,{⟨\mathbf{r},\mathbf{s}⟩}_{2,3,5})$ is identified with the spatial generalized octonions with ${\mu }_1=2,{\mu }_2=3,{\mu }_3=5$. In this space consider the unit-speed curve given by the following:

$$\begin{array}{cc}\hfill \beta \left(s\right)={\displaystyle \frac{1}{\sqrt{8}}}[& {\displaystyle \frac{1}{\sqrt{2}}}sin\left(s\right)e_1+{\displaystyle \frac{1}{\sqrt{3}}}cos\left(s\right)e_2+{\displaystyle \frac{1}{\sqrt{6}}}sin\left(3s\right)e_3+{\displaystyle \frac{1}{\sqrt{5}}}sin\left(2s\right)e_4\hfill \\ \hfill & +{\displaystyle \frac{1}{\sqrt{10}}}cos\left(3s\right)e_5+{\displaystyle \frac{2s}{\sqrt{15}}}{e}_6+{\displaystyle \frac{1}{\sqrt{30}}}cos\left(2s\right)e_7]\hfill \end{array}$$

Using the MATLAB codes provided below, one can compute the $G_2$-frame $(t,n_1,n_2,n_3,n_4,n_5,n_6)$ associated with the curve β at the parameter value $s=0$:

$$\begin{array}{cc}\hfill t\left(0\right)=& 0.25e_1+0e_2+0.433e_3+0.3162e_4+0e_5+0.1826e_6+0e_7,\hfill \\ \hfill n_1\left(0\right)=& 0e_1-0.07217e_2+0e_3+0e_4-0.3558e_5+0e_6-0.09129e_7,\hfill \\ \hfill n_2\left(0\right)=& 0.7187e_1+0e_2+0.4511e_3+0.09882e_4+0e_5-0.2396e_6+0e_7,\hfill \\ \hfill n_3\left(0\right)=& -1.013e_1+0e_2-0.4559e_3-0.2596e_4+0e_5+0.09762e_6+0e_7,\hfill \\ \hfill n_4\left(0\right)=& 0e_1+0.2581e_2+0e_3+0e_4-0.121e_5+0e_6-0.1776e_7,\hfill \\ \hfill n_5\left(0\right)=& 0.2295e_1+0e_2-0.1282e_3-0.4923e_4+0e_5-0.1207e_6+0e_7,\hfill \\ \hfill n_6\left(0\right)=& 0e_1+0.5215e_2+0e_3+0e_4+0.1091e_5+0e_6+0.1006e_7.\hfill \end{array}$$

and

$$\begin{array}{cccccc}\hfill & {\kappa }_1=2.828,\hfill & \hfill & {\kappa }_2=3.009,\hfill & \hfill & \\ \hfill & {\lambda }_1=-3.381,\hfill & \hfill & {\lambda }_2=-2.721,\hfill & \hfill & {\lambda }_3=0.0,\hfill \\ \hfill & {\lambda }_4=-0.7577,\hfill & \hfill & {\lambda }_5=-3.586,\hfill & \hfill & {\lambda }_6=0.6598,\hfill \end{array}$$

where

$$\begin{array}{cccccc}\hfill & {\theta }_0=1.0,\hfill & \hfill & {\theta }_1=1.0,\hfill & \hfill & {\theta }_2=1.0,\hfill \\ \hfill & {\theta }_3=-1.0,\hfill & \hfill & {\theta }_4=-1.0,\hfill & \hfill & {\theta }_5=-1.0,\hfill \\ \hfill & {\theta }_6=-1.0.\hfill \end{array}$$

6. Conclusions

The spatial generalized octonionic-valued functions of a single real variable determine curves in the inner space $({\mathbb{R}}^7,⟨,⟩)$. In this work, we construct a $G_2$-frame and derive the derivative formulas associated with this frame for seven-dimensional curves in $({\mathbb{R}}^7,⟨,⟩)$ using the vector product defined in $({\mathbb{R}}^7,⟨,⟩)$ and the structure of spatial generalized octonions. It is shown that the Frenet–Serret formulas also hold for such curves. Furthermore, we prove the $G_2$-congruence theorem and present the relation between the Frenet–Serret frame and the $G_2$-frame. Lastly, we present an illustrative example with Matlab codes. The resulting frame equations involve lower-order derivatives, which simplifies the overall computational framework. When calculating Serret–Frenet-type derivative formulas for high-dimensional curves using software such as MATLAB, memory limitations and performance issues occur due to the complexity of the symbolic and numerical operations involved. This study proposes an alternative framework that helps reduce these computational difficulties.

In addition to all these results, this study also constitutes a generalization of the works conducted on spatial real octonionic curves and spatial split octonionic curves presented in [31,33].