On the Octonionic Inclined Curves in the 8-Dimensional Euclidean Space

Abstract

We describe octonionic inclined curves and harmonic curvatures for the octonionic curves. We give characterizations for an octonionic curve to be an octonionic inclined curve. And finally, we obtain some characterisations for the octonionic inclined curves in terms of the harmonic curvatures.

1. Introduction

Hamilton [1] revealed quaternion in 1843 as an explanation of group construction and also performed to mechanics in three-dimensional space. For quaternions, same features are provided as complex numbers with the discrimination that the commutative rule is not effective in their case. The octonions [2, 3] form the widest normed algebra after the algebra of real numbers, complex numbers, and quaternions. The octonions are also known as Cayley Graves numbers and also have an algebraic structure defined on the eight-dimensional real vector space in such a way that two octonions can be added, multiplied, and divided with the fact that multiplication is neither commutative nor associative. Inclined curves in Euclidean space were studied by Özdamar and Hacısalihoğlu [4]. The Serret-Frenet formulae for an octonionic curves in and are given by Bektaş and Yüce [5]. But, to our knowledge, there has been no study on the octonionic inclined curves in the eight-dimensional Euclidean space. Such a study is the object of this paper. Our main aim in the present work is to study the differential geometry of a smooth curve in the eight-dimensional Euclidean space.

2. Preliminaries

The octonions can be thought of as octal of real numbers. Octonion is a real linear combination of the unit octonions: where is the scalar or real element; it may be assimilated with the real number 1. That is, every real octonion (in this study we use octonion instead of real octonion, since two concepts are the same) A can be expressed in the manner . Hence an octonion can be decomposed in terms of its scalar () and vector () parts as and . Addition and extraction of octonions are made by adding and quarrying corresponding terms and thereby their factors, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions. The product of each term can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one [3, 6]: Most of diagonal elements of the table are antisymmetric, making it almost a skew symmetric matrix except for the elements on the main diagonal, the row, and the column for which is an operand. The table can be epitomized by the relations: [7], , where is a completely antisymmetric tensor with value when , and , with the scalar element, and .

The overhead declaration though is not unique but is only one of 480 possible declarations for octonion multiplication. The others can be acquired by permuting the nonscalar elements, so can be noted to have different bases. Alternately they can be acquired by fixing the product rule for a few terms and deducing the rest from the other properties of the octonions. The 480 different algebras are isomorphic, so are in practice identical, and there is rarely a need to consider which particular multiplication rule is used [8, 9]. We denote the set of octonion where . is Kronecker delta. is completely antisymmetric tensor with value . is spanned by and the imaginary units each with square , so that [10]. Then octonions are isomorphic to [11]. Just as the complex numbers and quaternions could be used to describe and , the octonions may be used to describe points in using the obvious identification [12]. Let be an octonion. If , then is called a spatial (pure) octonion (or is called a spatial (pure) octonion whenever ). Before we define the octonionic product, we give information about vector product in . Seven-dimensional Euclidean space and three-dimensional Euclidean space are the only Euclidean spaces to have a vector product. We know that we can express an octonion as the sum of a real part, and a pure part, in . So we get . This guides to a vector product on described by [13]. Moreover this is given [13] by and .

Let , then,

The vector product in has all the properties expect Jacobi identity. So generally for pure octonions or equivalently the triple vector product identity fails . The identities which it does satisfy are as follows.

Theorem 1. Let , , and be spatial octonions and let be the angle between and . If is an random real number then the succeeding identicalnesses run for vector products in [13]. Consider the following: (i);(ii);(iii); (iv);(v);(vi);(vii).

If we take widely information about cross product in , we can read the references [14]. Now we can describe octonion product. The octonionic product of two octonions is served as follows [15]: where we have used the dot and cross products in . is called conjugate of and described as noted below where we have used the conjugates of basis elements as and . The inner product of octonions qualifies as follows: Hence it is called the octonionic inner product. The norm of an octonion is defined by If , then is called a unit octonion. The only octonion with norm is , and every nonzero octonion has a unique inverse; namely [16], For all the normed division algebras, the norm provides the identicalness [16] If we take in the study named “A characterization of inclined curves in Euclidean space,” we can obtain the following definitions.

Definition 2. Let be a curve in with the arc length parameter and let be a unit constant vector of . For all , if then the curve is called an inclined curve in , where is the unit tangent vector to the curve at its point , and is a constant angle between the vectors and [4].

We can give same definition in .

Definition 3. Let be a curve in with an arc length parameter and let be an unit constant vector. Let , be the Frenet -frame of at its point . If the angle, between and , is we define the function by as the harmonic curvature, with order , of the curve at its point . We define also [4].

We can give same definition in .

Now we are going to give some definitions and theorems about octonionic curves in and .

Definition 4. The seven-dimensional Euclidean space is consubstantiated by the space of spatial real octonions in an obvious manner. Let be an interval in and let be the parameter along the smooth curve Then the curve is called spatial octonionic curve or octonionic curve in [5].

Theorem 5. The seven-dimensional Euclidean space is consubstantiated by the space of spatial real octonions in an obvious manner. Let be an interval in and let be the parameter along the smooth curve Let be the Frenet trihedron of the differentiable Euclidean space curve in the Euclidean space . Then Frenet equations are where , curvature functions.
We may state Frenet formulae of the Frenet apparatus in the matrix form: This is the Serret-Frenet formulae for the spatial octonionic curve in [5].
is the Frenet apparatus for spatial octonionic curve in .

Remark 6. What has been achieved in this theorem is reputable in local differential geometry. We have done this for two especial goals:(1)to designate the demonstration for the Serret-Frenet formulae and Frenet apparatus of the curve in . We will roll the outcomes of this theorem comprehensively in the next theorem;(2)to indicate how octonions are to be used in designating curvature numbers of curves in general.

Definition 7. The eight-dimensional Euclidean space is assimilated into the space of real octonion. Let be an interval in and let be the parameter along the smooth curve Then the curve is called octonionic curve [5].

Theorem 8. The eight-dimensional Euclidean space is assimilated into the space of real octonion. Let be a smooth curve in described over . Let the parameter be selected that has unit magnitude. Let be the Frenet elements of . Then the Frenet equations are where , , , , , and . .
We may express Frenet formulae of the Frenet apparatus in the matrix form:
This is the Serret-Frenet formulae for octonionic curve in [5].

3. Octonionic Inclined Curves and Harmonic Curvatures

Definition 9. Let be spatial octonionic curve with an arc length parameter and let be an unit and constant spatial octonion. For all , let be a constant defined by Then is called spatial octonionic inclined curve.

Definition 10. octonionic curve is given by arc length parameter . Let be the Frenet trihedron in the point of the curve and let be unit and constant spatial octonion such that angle is between and , be a function defined by Then functions are called th Harmonic curvature in the point of the spatial octonionic curve with respect to .

Definition 11. octonionic curve is given by arc length parameter such that is a unit and constant spatial octonion for every ,
Then curve is called octonionic inclined curve in .

Definition 12. octonionic curve is given by arc length parameter . Let be the Frenet apparatus and let be unit and constant such that angle is between and . Let be a function defined by Then functions are called th Harmonic curvature in the point of the octonionic curve with respect to .

Theorem 13. Let be spatial octonionic inclined curve given by arc length parameter . Curvatures in the point of curve are , and , are harmonic curvatures; they are

Proof. Let be an angle between the unit and constant spatial octonion and . Such that Frenet apparatus in the point we obtain that Here, differentiating with respect to , we find that . By the aid of (15), we obtain that . If derivative of this function with respect to is taken, we find that . Here, using (15), is obtained. Thus, if (21) and (23) are used, is found. Thus,
By the aid of (15), we obtain that . If derivative of this function with respect to is taken, and (15), (21), and (23) are used, we find that . For the higher harmonic curvatures let us differentiate (23), with respect to for ; then . By the aid of (15), we get , .

Theorem 14. Let be a spatial octonionic inclined curve. Such that , obtained from , octonionic curve, is an octonionic inclined curve.

Proof. Let be an octonionic curve given by arc length parameter .
Let be the Frenet apparatus and let be unit and constant spatial octonion. If we use Definition 11, we get the following statement: where
We notice that
Since is spatial octonion, then , . Here, we can account for the product of octonion and so is obtained. Then, curve is octonionic inclined curve.

Theorem 15. Let be an octonionic inclined curve given by arc length parameter . Such that are curvatures in the point , , are curvature radii and , , are harmonic curvatures, they are where , , , , , , .

Proof. curve is given by regular octonionic. is an unit and a constant spatial octonion, and such that is Frenet apparatus in the point , is written. If derivative with respect to of this equation is taken, we obtain that . Here, using (19), is found. Because of , we write as . Thus, is obtained. Here, using (19), is found. In addition, from (26) for we obtain that By taking (24) and (40) into consideration, is found. On the other hand, if derivative of (40) with respect to is taken, is found. Here, using (19), is found. In addition, from (26) for we obtain that By taking (24) and (44) into consideration, is obtained. Thus, If derivative of (44) with respect to is taken, is found. Here, using (19), is found. In addition, from (26) for we obtain that By taking (40) and (49) into consideration, is obtained. Thus, Similarly, If derivative of (49) and following equations with respect to is taken, we get

Theorem 16. is a spatial octonionic curve given by arc length parameter . And let , be harmonic curvatures in the point . is octonionic inclined curve if and only if is constant.

Proof. Let be a spatial octonionic curve given by arc length parameter . Then, there is a unit and constant spatial octonion. Therefore, is constant for spatial octonionic inclined curve with respect to arc length parameter such that is basis of spatial octonion in the point ; spatial octonion is obtained. Since is a unit, Here, using (55), if we use Definition 10 in the last equation we can write From octonionic product, we have where
In contrast, suppose that is constant for spatial octonionic curve. It is study to show that . Therefore, there is angle so that . Thus, we define spatial octonion, where Here, we demonstrate that is a constant. Thus, if derivative of (61) with respect to is taken, is found. On the other hand, is obtained. Here, using (15), is obtained. Similarly, Finally, we get Thus, is a constant. On the other hand, is obtained. Thus, is found. Therefore, is an inclined curve.

Theorem 17. is an octonionic curve given by arc length parameter . And let be harmonic curvatures in the point . is an octonionic inclined curve if and only if is constant.

Proof. The result is straightforward.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.