How to determine properties of the special curves using real Octonions in Euclidean 8-Space ¨ ¨ Ozcan BEKTAS¸*, Salim YUCE** *
[email protected], **
[email protected] Department of Mathematics, Faculty of Arts and Sciences, Recep Tayyip Erdogan University, Rize, TURKEY Corresponding Author Department of Mathematics, Faculty of Arts and Sciences, Yildiz Technical University, Istanbul, TURKEY October 5, 2018 Abstract In this study, we approach to the issue of how to determine properties of special octonionic curves (octonionic involute-evolute curves, octonionic Bertrand curves, octonionic Smarandache curves, and octonionic Mannheim curves) by means of real octonions in Euclidean 8-space. Firstly, we give some information about octonion algebras, and octonionic curves in Euclidean 8-space. After that, the chapter of the special octonionic curves are divided into four part. Finally, we obtain some characterizations of the special octonionic curves.
Keywords: Serret-Frenet formula, Spatial real octonionic curve, octonionic involute-evolute curve, octonionic Bertrand curve, octonionic Mannheim curve, octonionic Smarandache curve. M.S.C. 2010: 53A04; 14H50; 11R52.
1
Introduction
Nowadays, the octonion algebras are studied in differential geometry, and clifford algebras.The scientists use especially the octonion algebra in matrix, curve, and surface theories. In the light of these information, we ask a question to ourselves: Can we determine the special octonionic curves, and some characterizations of these curves by using real octonion algebra, and real octonionic curves in Euclidean 8-space? To answer the question, we have to speak of the special curves by using quaternion algebras, and quaternionic curves. Involute-evolute curves were studied by Ozyilmaz, Yilmaz [27], Turgut and Esin, [31]. Bertrand [7] demonstrated that a necessary and sufficient condition for the existence of such a second curve is required; in fact, a linear relationship computed with constant coefficients should exist between the first and second 1
curvatures of the given original curve. The authors studied the Bertrand curves in papers [10, 11, 15]. Liu and Wang [23] obtained the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space E 3 , and Minkowski space E13 . Ali [3] defined the special Smarandache curves in the Euclidean space, and determined Serret-Frenet apparatus of these curves. Quaternions were discovered by Hamilton [21]. Bharathi and Nagaraj [8] computed the Serret-Frenet formulas for quaternionic curves in Euclidean 3space, and Euclidean 4-space. Then, the authors used the quaternionic curve in differential geometry. Soyfidan [30] determined the quaternionic involute-evolute curve, and gave some properties about these curves. Kecilioglu and Ilarslan [22] defined (1, 3) type Bertrand curves for quatenionic curve in Euclidean 4-space. They proved that if bitorsion of a quatenionic curve α does not vanish, then there is no quaternionic curve in Euclidean 4-space which is a Bertrand curve. Gok and Kahraman [19] constructed quaternionic Bertrand curves in Euclidean 4-space. Okuyucu [26] considered a quaternionic Mannheim curve, and gave some characterizations of them in Euclidean 3-space, and Euclidean 4-space. Cetin and Kocayigit [13] introduced the quaternionic Smarandache curves in 3-dimensional Euclidean space. The octonions can be thought of as octal of real numbers. Real, and complex numbers, quaternions, and octonions are the four normed division algebras [12]. The octonions have got a lot of properties in [1, 2, 12, 29]. Octonions are noncommutative, and non-associative algebra in mathematics. The octonion is the one of the highest normed division algebra. Moreover, the octonion analysis is largely determined by Baez [4]. Now, we can answer the questions in the first paragraph. Special curves (involute-evolute, Bertrand, Mannheim, and Smarandache) in Euclidean 8-space can be determined by using real octonion. We will introduce octonionic involuteevolute, octonionic Bertrand, octonionic Mannheim, and octonionic Smarandache curves.
2
Preliminaries
In this part, we denote the spatial octions briefly by OS , and the real octonions with O. Let us firstly refer fundamental notion on the octonions. The octonion 7 P A is written as A = A0 e0 + Ai ei , where terms Ai are the real numbers i=1
coefficients of the real octonions, and ei (i = 1, 2, . . . , 7) are called the unit octonions basis elements, and e0 = +1 is called the scalar element. The rules of the unit octonion basis elements are defined in tabular form, and are serviced
2
as such in Table 1 [17]. Table 1.The × e0 e1 e2 e3 e4 e5 e6 e7
Multiplication Table of the Unit Octonion Basis Elements e0 e1 e2 e3 e4 e5 e6 e7 e0 e1 e2 e3 e4 e5 e6 e7 e1 −e0 e3 −e2 e5 −e4 −e7 e6 e2 −e3 −e0 e1 e6 e7 −e4 −e5 e3 e2 −e1 −e0 e7 −e6 e5 −e4 e4 −e5 −e6 −e7 −e0 e1 e2 e3 e5 e4 −e7 e6 −e1 −e0 −e3 e2 e6 e7 e4 −e5 −e2 e3 −e0 −e1 e7 −e6 e5 e4 −e3 −e2 e1 −e0
The octonion addition, the scalar multiplication, and the octonion multiplication are the operations of the set of octonions. The sum of two elements of O is defined by their sum as elements of R8 . That is, the sum of two octonions are written in the form
A±B
=
7 X
(Ai ± Bi ) ei
i=0
=
(A0 e0 + A1 e1 + A2 e2 + A3 e3 + A4 e4 + A5 e5 + A6 e6 + A7 e7 ) ± (B0 e0 + B1 e1 + B2 e2 + B3 e3 + B4 e4 + B5 e5 + B6 e6 + B7 e7 ) .
The set of the octonions O is stated by e0 ∈ R, and the seven unit octonion basis elements e1 , e2 , e3 , e4 , e5 , e6 , e7 ; all these elements’ square are −1 [17], hence we can write the set of the octonions O as follows O = R ⊕ R7 [25]. Consequently, we can say that the octonions are isomorphic to R8 [32]. The points in R8 are represented by the octonions [4]. A is called conjugate of the octonion A, and is defined by
A
= A0 e 0 − A1 e 1 − A2 e 2 − A3 e 3 − A4 e 4 − A5 e 5 − A6 e 6 − A7 e 7 7 X Ai e i , = A0 e 0 − i=1
where e0 = e0 , and ej = −ej (j = 1, . . . , 7) [16]. The octonion A has real part, and vectorial part as well. Therefore, the octonion A is separated related to its real (SA ) in R and vectorial (VA ) in R7 parts as follows: SA =
1 A + A = A0 , 2
7
VA =
X 1 A−A = Ai e i . 2 i=1
Thus, an octonion is given by A = SA +VA . The multiplication of two octonions is introduced by A × B = SA SB − hVA , VB i + SA VB + SB VA + VA ∧ VB
3
∀A, B ∈ O. Here, the inner product, and the cross product in R7 are evaluated, respectively [24]. The symmetric non-degenerate reel valued bilinear form g is given by 1 A×B+B×A 2 ∀A, B ∈ O. g is determined with the help of the octonionic multiplication. 7 P g is called the octonionic inner product. Thus, we get g (A, B) = Ai Bi g : O × O → R, g (A, B) =
i=1
[9,14]. If A + A = 0, then the octonion A is called spatial octonion. The 7 P spatial octonion set is represented by OS = { Ai ei ; Ai ∈ R}. Here, e2i = −1, i=1
ei ej = −δij e0 + εijk ek , (i, j, k = 1, 2, ..., 7), (i 6= j 6= k, i 6= 0, j 6= 0, k 6= 0). The spatial octonions are isomorphic to R7 . The vector product of two vectors is only defined in 3- dimensional Euclidean space, R3 , and 7- dimensional Euclidean space, R7 . We express the vector product in R7 . Let A, B be the spatial octonions. The vector product in R7 is defined by A ∧ B = AB + hA, Bi [16,20]. Moreover, this is given by [16, 28] for 7 7 P P A= Ai ei = (Ai ), 1 ≤ i ≤ 7 , and B = Bi ei = (Bi ), 1 ≤ i ≤ 7. The norm i=1
i=1
of the octonion A is denoted by v u 7 p uX kAk = A × A = t A2i . i=0
If kA0 k = 1, then A0 is called unit octonion. Inverse properties: Let A, B be two octonions in O. In this condition, the properties B × A−1 × A = B and A−1 × (A × B) = B are satified [33]. Let A and B be unit octonions. Since the definition of norm of octonion, we have B × A × A = B and A × (A × B) = B. Moufang implies alternative: A Moufang loop is an alternative loop, i.e., it satisfies A × (A × B) = (A × A) × B left alternative identity (A × B) × B = A × (B × B) right alternative identity (A × B) × A = (A × B) × A flexible identity, for all A, B ∈ O [33]. The Serret-Frenet frame and curvatures in R8 : Let Γ : I ⊂ R → R8 be an unit speed space curve in R8 , and {Uj }, 1 ≤ j ≤ 8 be the Serret-Frenet 8- frame related to Γ. The Serret-Frenet formulas for the curve Γ : I ⊂ R → R8 are given as follows: 0
U1 (s) = k1 (s) U2 (s) 0 Um (s) = −km−1 (s) Um−1 (s) + km+1 (s) Um+1 (s) , 2 ≤ m ≤ 7 0 U8 (s) = −k7 (s) U7 (s) . On the other hand, Uj (s) =
Ej (s) kEj (s)k ,
D 0 E kj (s) = Uj (s) , Uj+1 (s) =
0
for 1 ≤ j ≤ 8, where E1 (s) = Γ (s),
4
kEj +1(s)k kEj (s)k
and Ej (s) = Γ(j) (s) −
P
Γ(j) (s) , Ui (s) Uj (s) [18]. These concepts were
i
