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 n space were studied by Özdamar and Hacısalihoğlu [4]. The Serret-Frenet formulae for an octonionic curves in R7 and R8 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: e0,e1,e2,e3,e4,e5,e6,e7 where e0 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 A=∑i=07aiei. Hence an octonion can be decomposed in terms of its scalar (SA) and vector (VA) parts as SA=a0 and VA=∑i=17aiei. 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]:
(1)×e0e1e2e3e4e5e6e7e0e0e1e2e3e4e5e6e7e1e1-e0e3-e2e5-e4-e7e6e2e2-e3-e0e1e6e7-e4-e5e3e3e2-e1-e0e7-e6e5-e4e4e4-e5-e6-e7-e0e1e2e3e5e5e4-e7e6-e1-e0-e3e2e6e6e7e4-e5-e2e3-e0-e1e7e7-e6e5e4-e3-e2e1-e0
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 e0 is an operand. The table can be epitomized by the relations: [7], eiej=-δije0+εijkek, where is a completely antisymmetric tensor with value +1 when ijk=123,145,176,246,257,347,365, and e0ei=eie0=ei, with e0 the scalar element, and i,j,k=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
(2)O=A∣A=∑7i=0aiei,e0ei=eie0=ei,e0=+1,eiej=-δije0+εijkek∑i=07,
where ijk=123,145,176,246,257,347,365. δij is Kronecker delta. εijk is completely antisymmetric tensor with value +1. O is spanned by +1 and the +1 imaginary units e1,e2,e3,e4,e5,e6,e7 each with square -1, so that O=R⊕R7 [10]. Then octonions are isomorphic to R8 [11]. Just as the complex numbers and quaternions could be used to describe R2 and R4, the octonions may be used to describe points in R8 using the obvious identification [12]. Let A=∑i=07aiei be an octonion. If a0=0, then A is called a spatial (pure) octonion (or A is called a spatial (pure) octonion whenever A+A¯=0). Before we define the octonionic product, we give information about vector product in R7. 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, SA and a pure part, VA in R7. So we get A=SA+VA. This guides to a vector product on R7 described by VA∧VB=VAVB+VA,VB [13]. Moreover this is given [13] by VA=a1,a2,a3,a4,a5,a6,a7 and VB=b1,b2,b3,b4,b5,b6,b7.

Let ζ=a1,a2,a3,a4,a5,a6,a7∧b1,b2,b3,b4,b5,b6,b7, then,
(3)ζ=a2b3-a3b2+a4b5-a5b4+a7b6-a6b7,a3b1-a1b3+a4b6-a6b4+a5b7-a7b5,a1b2-a2b1+a4b7-a7b4+a6b5-a5b6,a5b1-a1b5+a6b2-a2b6+a7b3-a3b7,a1b4-a4b1+a3b6-a6b3+a7b2-a2b7,a1b7-a7b1+a2b4-a4b2+a5b3-a3b5,a2b5-a5b2+a3b4-a4b3+a6b1-a1b6.

The vector product in R7 has all the properties expect Jacobi identity. So generally for pure octonions VA∧VB∧VC+VB∧VC∧VA+VC∧VA∧VB≠0 or equivalently the triple vector product identity fails VA∧VB∧VC≠VA,VCVB-VA,VBVC. The identities which it does satisfy are as follows.

Theorem 1.

Let VA, VB, and VC be spatial octonions and let θ be the angle between VA and VB. If c is an random real number then the succeeding identicalnesses run for vector products in R7 [13]. Consider the following:

VA∧VB+VC=VA∧VB+VA∧VC;

VA∧VA=0;

VA∧VB=-VB∧VA;

VA,VA∧VB=0;

VA∧VB=VAVBsinθ;

VA∧VB,VC=VB∧VC,VA=VC∧VA,VB;

VA∧VA∧VB=VA,VBVA-VA,VAVB.

If we take widely information about cross product in R7, we can read the references [14]. Now we can describe octonion product. The octonionic product of two octonions is served as follows [15]:
(4)A×B=SASB-VA,VB+SAVB+SBVA+VA∧VB,∀A,B∈O,
where we have used the dot and cross products in R7. A¯ is called conjugate of A and described as noted below
(5)A¯=SA-VA=a0-∑i=17aiei,
where we have used the conjugates of basis elements as e0¯=e0 and ej¯=-ejj=1,…,7. The inner product of octonions qualifies as follows:
(6)A,B:O×O⟶R,A,B⟶A,B=12A×B¯+B×A¯=∑7i=0aibi.
Hence it is called the octonionic inner product. The norm of an octonion A is defined by
(7)A=A×A¯=∑7i=0ai.
If A=1, then A is called a unit octonion. The only octonion with norm 0 is 0, and every nonzero octonion has a unique inverse; namely [16],
(8)A-1=A¯A2.
For all the normed division algebras, the norm provides the identicalness [16]
(9)A×B=AB.
If we take n=7,8 in the study named “A characterization of inclined curves in Euclidean n space,” we can obtain the following definitions.

Definition 2.

Let γ:I→R7 be a curve in R7 with the arc length parameter s and let u be a unit constant vector of R7. For all s∈I, if
(10)γ′s,u=cosφ=constant,φ≠π2
then the curve is called an inclined curve in R7, where γ′s is the unit tangent vector to the curve γ at its point γs, and φ is a constant angle between the vectors γ′ and u [4].

We can give same definition in R8.

Definition 3.

Let γ:I→R7 be a curve in R7 with an arc length parameter s and let u be an unit constant vector. Let t,n1,n2,n3,n4,n5,n6, 3≤r≤7 be the Frenet 7-frame of γ at its point γs. If the angle, between γ′s and u, is φ=φs we define the function
(11)Hi:I⟶R,3≤i≤r-2
by
(12)ni+1s,u=Hiscosφ
as the harmonic curvature, with order i, of the curve γ at its point γs. We define also H0=0 [4].

We can give same definition in R8.

Now we are going to give some definitions and theorems about octonionic curves in R7 and R8.

Definition 4.

The seven-dimensional Euclidean space R7 is consubstantiated by the space of spatial real octonions OP=γ∈O∣γ+γ-=0 in an obvious manner. Let I=[0,1] be an interval in R and let s∈I be the parameter along the smooth curve
(13)γ:I⊂R⟶OPs⟶γs=∑7i=1γisei.
Then the curve is called spatial octonionic curve or octonionic curve in R7 [5].

Theorem 5.

The seven-dimensional Euclidean space R7 is consubstantiated by the space of spatial real octonions OP=γ∈O∣γ+γ-=0 in an obvious manner. Let I=[0,1] be an interval in R and let s∈I be the parameter along the smooth curve
(14)γ:I⊂R⟶OPs⟶γs=∑7i=1γisei.
Let t,n1,n2,n3,n4,n5,n6 be the Frenet trihedron of the differentiable Euclidean space curve in the Euclidean space R7. Then Frenet equations are
(15)t′s=k1sn1s,n1′s=-k1sts+k2sn2s,n2′s=-k2sn1s+k3sn3s,n3′s=-k3sn2s+k4sn4s,n4′s=-k4sn3s+k5sn5s,n5′s=-k5sn4s+k6sn6s,n6′s=-k6sn5s,
where ki, 1≤i≤6 curvature functions.

We may state Frenet formulae of the Frenet apparatus in the matrix form:
(16)t′n1′n2′n3′n4′n5′n6′=0k100000-k10k200000-k20k300000-k30k400000-k40k500000-k50k600000-k60tn1n2n3n4n5n6.
This is the Serret-Frenet formulae for the spatial octonionic curve γ in R7 [5].

t,n1,n2,n3,n4,n5,n6,k1,k2,k3,k4,k5,k6 is the Frenet apparatus for spatial octonionic curve γ in R7.

Remark 6.

What has been achieved in this theorem is reputable in local differential geometry. We have done this for two especial goals:

to designate the demonstration for the Serret-Frenet formulae and Frenet apparatus of the curve γ in R7. We will roll the outcomes of this theorem comprehensively in the next theorem;

to indicate how octonions are to be used in designating curvature numbers of curves in general.

Definition 7.

The eight-dimensional Euclidean space R8 is assimilated into the space of real octonion. Let I=[0,1] be an interval in R and let s∈I be the parameter along the smooth curve
(17)β:I⊂R⟶Os⟶βs=∑7i=0γisei.
Then the curve is called octonionic curve [5].

Theorem 8.

The eight-dimensional Euclidean space R8 is assimilated into the space of real octonion. Let
(18)β:I⊂R⟶Os⟶βs=∑7i=0γisei
be a smooth curve in R8 described over I. Let the parameter s be selected that T=β′s=∑i=07γi′sei has unit magnitude. Let T,N1,N2,N3,N4,N5,N6,N7 be the Frenet elements of β. Then the Frenet equations are
(19)T′s=KsN1s,N1′s=-KsTs+k1sN2s,N2′s=-k1sN1s+k2-KsN3s,N3′s=-k2-KsN2s+k3sN4s,N4′s=-k3sN3s+k4-KsN5s,N5′s=-k4-KsN4s+k5sN6s,N6′s=-k5sN5s+k6+KsN7s,N7′s=-k6+KsN6s,
where N1=t×T, N2=n1×T, N3=n2×T, N4=n3×T, N5=n4×T, N6=n5×T and N7=n6×T. K=T′s.

We may express Frenet formulae of the Frenet apparatus in the matrix form:(20)T′N1′N2′N3′N4′N5′N6′N7′=0K000000-K0k1000000-k10k2-K000000-k2-K0k3000000-k30k4-K000000-k4-K0k5000000-k50k6+K000000-k6+K0TN1N2N3N4N5N6N7.

This is the Serret-Frenet formulae for octonionic curve β in R8 [5].

3. Octonionic Inclined Curves and Harmonic CurvaturesDefinition 9.

Let γI be spatial octonionic curve with an arc length parameter s and let U be an unit and constant spatial octonion. For all s∈I, let γ′s,U be a constant defined by
(21)γ′s,U=cosφ=constant,φ≠π2.
Then γI is called spatial octonionic inclined curve.

Definition 10.

γI octonionic curve is given by arc length parameter s. Let t,n1,n2,n3,n4,n5,n6 be the Frenet trihedron in the point γs of the curve γ and let U be unit and constant spatial octonion such that angle φs is between γ′s and U,
(22)Hi:I⟶R,1≤i≤5
be a function defined by
(23)ni+1s,U=Hicosφ=constant,φ≠π2.
Then functions Hi are called ith Harmonic curvature in the point γs of the γ spatial octonionic curve with respect to u.

Definition 11.

β:I⊂R→O octonionic curve is given by arc length parameter s such that U is a unit and constant spatial octonion for every s∈I,
(24)β′s,U=cosφ=constant,φ≠π2.

Then curve β is called octonionic inclined curve in O.

Definition 12.

β:I⊂R→O octonionic curve is given by arc length parameter s. Let T,N1,N2,N3,N4,N5,N6,N7 be the Frenet apparatus and let U be unit and constant such that angle φs is between T′s and U. Let
(25)Hi:I⟶R,1≤i≤6
be a function defined by
(26)Ni+1s,U=Hicosφ=constant,φ≠π2.
Then functions Hi are called ith Harmonic curvature in the point βs of the β octonionic curve with respect to U.

Theorem 13.

Let γ:I→R7 be spatial octonionic inclined curve given by arc length parameter s. Curvatures in the point γs of curve γ are kis, ζi=1/kis and His, 1≤i≤6 are harmonic curvatures; they are
(27)H1=k1k2,H2=H1′k3,Hj=Hj-1′+Hj-2′kiζj+1,2≤j≤5.

Proof.

Let φ be an angle between the unit and constant spatial octonion U and ts. Such that t,n1,n2,n3,n4,n5,n6 Frenet apparatus in the point γs we obtain that
(28)ts,U=cosφ.
Here, differentiating with respect to s, we find that t′s,U=0. By the aid of (15), we obtain that n1s,U=0. If derivative of this function with respect to s is taken, we find that n1′s,U=0. Here, using (15),
(29)-k1sts+k2sn2s,U=0
is obtained. Thus, if (21) and (23) are used,
(30)-k1s+k2sH1cosφ=0,cosφ≠π2
is found. Thus,
(31)H1=k1sk2s.

By the aid of (15), we obtain that n2s,U=H1cosφ. If derivative of this function with respect to s is taken, and (15), (21), and (23) are used, we find that H2=H1′/k3. For the higher harmonic curvatures let us differentiate (23), with respect to s for j; then nj+1′s,U=Hj-1′cosφ. By the aid of (15), -kjsnis+kj+1sni+2s,U=Hj-1′cosφ we get Hj=Hj-1′+Hj-2′kiζj+1, 2≤j≤5.

Theorem 14.

Let γ:I→R7 be a spatial octonionic inclined curve. Such that γs=∑i=17γisei,
(32)βs=∑7i=0γisei,
obtained from γ, octonionic curve, is an octonionic inclined curve.

Proof.

Let β:I→O be an octonionic curve given by arc length parameter s.

Let T,N1,N2,N3,N4,N5,N6,N7 be the Frenet apparatus and let U be unit and constant spatial octonion. If we use Definition 11, we get the following statement:
(33)β′s,u=Ts,U,Ts=STs+V→Ts,U=SU+V→U,
where
(34)Ts=STs+V→Ts.

We notice that
(35)Ts,U=12Ts×U¯+U×Ts¯=12STs+V→Ts×U¯+U×STs-V→Ts.

Since U is spatial octonion, then U=V→U, U¯=-U. Here, we can account for the product of octonion
(36)Ts,U=12STs·0-V→Ts,-U+STs·-U+0·V→Ts+V→Ts∧-U+0·STs-U,V→Ts+STs·U+0·V→Ts+U∧-V→Ts=12V→Ts,U-STsU-V→Ts∧U+U,V→Ts+STsU-U∧V→Ts=V→Ts,U
and so β′s,U=cosφ is obtained. Then, β curve is octonionic inclined curve.

Theorem 15.

Let β:I→O be an octonionic inclined curve given by arc length parameter s. Such that Kis are curvatures in the point βs,δis=1/Kis, 1≤i≤7, are curvature radii and His, 1≤i≤6, are harmonic curvatures, they are
(37)H1=1.curvature2.curvature,H2=H1′k2-Ks,H3=H2′+k2-KH1k3,H4=H3′+k3H2k4-Ks,H5=H4′+k4-KH3k5,H6=H5′+k5H4k6+Ks,
where K1s=K, K2s=k1, K3s=k2-K, K4s=k3, K5s=k4-K, K6s=k5, K7s=k6+K.

Proof.

β:I→O curve is given by regular octonionic. U is an unit and a constant spatial octonion, and such that T,N1,N2,N3,N4,N5,N6,N7 is Frenet apparatus in the point βs,
(38)Ts,U=cosφ=constant
is written. If derivative with respect to s of this equation is taken, we obtain that T′s,U=0. Here, using (19), KsN1s,U=0 is found. Because of KsN1s≠0, we write as N1s,U=0. Thus, N1′s,U=0 is obtained. Here, using (19),
(39)-KsTs,U+k1sN2s,U=0
is found. In addition, from (26) for i=1 we obtain that
(40)N2s,U=H1scosφ.
By taking (24) and (40) into consideration,
(41)-Ks+k1sH1scosφ=0,cosφ≠0H1s=Ksk1s=1.curvature2.curvature
is found. On the other hand, if derivative of (40) with respect to s is taken,
(42)N2′s,U=H1′scosφ
is found. Here, using (19),
(43)-k1sN1s,U+k2-KsN3s,U=H1′scosφ
is found. In addition, from (26) for i=2 we obtain that
(44)N3s,U=H2scosφ.
By taking (24) and (44) into consideration,
(45)k2-KsH2scosφ=H1′scosφ
is obtained. Thus,
(46)H1s=H1′sk2-Ks.
If derivative of (44) with respect to s is taken,
(47)N3′s,u=H2′scosφ
is found. Here, using (19),
(48)-k2-KsN2s,U+k3sN4s,U=H2′scosφ
is found. In addition, from (26) for i=3 we obtain that
(49)N4s,U=H3scosφ.
By taking (40) and (49) into consideration,
(50)-k2-KsH1scosφ+k3sH3scosφ=H2′scosφ
is obtained. Thus,
(51)H3s=H2′s+k2-KsH1sk3s.
Similarly, If derivative of (49) and following equations with respect to s is taken,
(52)N5s,U=H4scosφ,N6s,U=H5scosφ,
we get
(53)H4s=H3′s+k3sH2sk4-Ks,H5s=H4′s+k4-KsH3k5s,H6=H5′s+k5sH4sk6+Ks.

Theorem 16.

γ is a spatial octonionic curve given by arc length parameter s. And let Hi, 1≤i≤5 be harmonic curvatures in the point γs. γ is octonionic inclined curve if and only if ∑i=15Hi2 is constant.

Proof.

⇒ Let γ be a spatial octonionic curve given by arc length parameter s. Then, there is a U unit and constant spatial octonion. Therefore,
(54)γ′s,U=cosφ
is constant for γ spatial octonionic inclined curve with respect to arc length parameter s such that t,n1,n2,n3,n4,n5,n6 is basis of spatial octonion in the point γs; spatial octonion U(55)U=ts,Uts+∑6i=1nis,Unis
is obtained. Since U is a unit,
(56)U2=U×U¯=1.
Here, using (55),
(57)U2=ts,Uts+∑6i=1nis,Unis×ts,Uts+∑6i=1nis,Unis¯,
if we use Definition 10 in the last equation we can write
(58)1=cosφt+∑5i=0Hisni+1scosφ×cosφt+∑5i=0Hisni+1scosφ¯.
From octonionic product, we have
(59)1=cos2φ+∑5i=1Hi2scos2φ,
where
(60)∑5i=1Hi2s=tan2φ=constant.

⇐ In contrast, suppose that ∑i=15Hi2s is constant for γ spatial octonionic curve. It is study to show that γ′s,U=cosφ. Therefore, there is φ angle so that tan2φ=a. Thus, we define U spatial octonion, where
(61)U=cosφt+∑6i=2Hi-1sniscosφ.
Here, we demonstrate that u is a constant. Thus, if derivative of (61) with respect to s is taken,
(62)1cosφdUds=t′+∑6i=2Hi-1′snis+∑6i=2Hi-1sni′s,1cosφdUds=t′+H1′n2+H2′n3+H3′n4+H4′n5+H5′n6+H1n2′+H2n3′+H3n4′+H4n5′+H5n6′
is found. On the other hand,
(63)n3s,U=H2cosφ⟹n3′s,U=H2′cosφ
is obtained. Here, using (15),
(64)H2′=-k3H1+k4H3
is obtained. Similarly,
(65)H3′=-k4H2+k5H4,H4′=-k5H3+k6H5,H5′=-k6H4.
Finally, we get
(66)1cosφdUds=0.
Thus, u is a constant. On the other hand,
(67)U2=U×U¯,(68)U2=cosφt+∑6i=2Hi-1sniscosφ×cosφt+∑6i=2Hi-1sniscosφ¯=cos2φ+cos2φ∑5i=1Hi2s=1
is obtained. Thus,
(69)ts,U=12t×U¯+U×t¯=cosφ
is found. Therefore, γ is an inclined curve.

Theorem 17.

β is an octonionic curve given by arc length parameter s. And let Hi,1≤i≤6 be harmonic curvatures in the point βs. β is an octonionic inclined curve if and only if ∑i=16Hi2 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.

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