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

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.


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 Hacısalihoglu [4].The Serret-Frenet formulae for an octonionic curves in R 7 and R 8 are given by Bektas ¸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.

Preliminaries
The octonions can be thought of as octal of real numbers.Octonion is a real linear combination of the unit octonions: {e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } where e 0 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  = ∑ 7 =0     .Hence an octonion can be decomposed in terms of its scalar (  ) and vector (V  ) parts as   =  0 and V  = ∑ 7 =1     .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 2 Mathematical Problems in Engineering which e 0 is an operand.The table can be epitomized by the relations: [7],     = −   0 +     , where is a completely antisymmetric tensor with value +1 when  = 123, 145, 176, 246, 257, 347, 365, and  0   =    0 =   , with  0 the scalar element, and , ,  = 1, ..., 7.
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  = 123, 145, 176, 246, 257, 347, 365.  is Kronecker delta.  is completely antisymmetric tensor with value +1.O is spanned by +1 and the +1 imaginary units  1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 each with square −1, so that O = R ⊕ R 7 [10].Then octonions are isomorphic to R 8 [11].Just as the complex numbers and quaternions could be used to describe R 2 and R 4 , the octonions may be used to describe points in R 8 using the obvious identification [12].Let  = ∑ 7 =0     be an octonion.If  0 = 0, then  is called a spatial (pure) octonion (or  is called a spatial (pure) octonion whenever  +  = 0).Before we define the octonionic product, we give information about vector product in R 7 .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 R 7 .So we get  =   + V  .This guides to a vector product on R 7 described by [13].Moreover this is given [13] by The vector product in R 7 has all the properties expect Jacobi identity.So generally for pure octonions The identities which it does satisfy are as follows.
Theorem 1.Let V  , V  , and V  be spatial octonions and let  be the angle between V  and V  .If  is an random real number then the succeeding identicalnesses run for vector products in R 7 [13].Consider the following: If we take widely information about cross product in R 7 , 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  7 . is called conjugate of  and described as noted below where we have used the conjugates of basis elements as  0 =  0 and   = −  ( = 1, . . ., 7).The inner product of octonions qualifies as follows: ⟨, ⟩ :  ×  → R, Hence it is called the octonionic inner product.The norm of an octonion  is defined by If ‖‖ = 1, then  is called a unit octonion.The only octonion with norm 0 is 0, 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  = 7, 8 in the study named "A characterization of inclined curves in Euclidean  space, " we can obtain the following definitions.
Definition 2. Let  :  → R 7 be a curve in R 7 with the arc length parameter  and let  be a unit constant vector of R 7 .For all  ∈ , if then the curve is called an inclined curve in R 7 , 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 R 8 .
Definition 3. Let  :  → R 7 be a curve in R 7 with an arc length parameter  and let  be an unit constant vector.Let {t, n 1 , n 2 , n 3 , n 4 , n 5 , n 6 }, 3 ≤  ≤ 7 be the Frenet 7-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  0 = 0 [4].
We can give same definition in R 8 .Now we are going to give some definitions and theorems about octonionic curves in R 7 and R 8 .Definition 4. The seven-dimensional Euclidean space R 7 is consubstantiated by the space of spatial real octonions   = { ∈  |  +  = 0} in an obvious manner.Let  = [0, 1] be an interval in R and let  ∈  be the parameter along the smooth curve Then the curve is called spatial octonionic curve or octonionic curve in R 7 [5].
where   , 1 ≤  ≤ 6 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 R 7 [5].{t, n 1 , n 2 , n 3 , n 4 , n 5 , n 6 ,  1 ,  2 ,  3 ,  4 ,  5 ,  6 } is the Frenet apparatus for spatial octonionic curve  in R 7 .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 R 7 .
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 R 8 is assimilated into the space of real octonion.Let  = [0, 1] be an interval in R and let  ∈  be the parameter along the smooth curve Then the curve is called octonionic curve [5]. where We may express Frenet formulae of the Frenet apparatus in the matrix form: This is the Serret-Frenet formulae for octonionic curve  in R 8 [5].

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 {t, n 1 , n 2 , n 3 , n 4 , n 5 , n 6 } 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 .
Proof.Let  :  →  be an octonionic curve given by arc length parameter .