Characterizations of Type-2 Harmonic Curvatures and General Helices in Euclidean space E⁴ Faik Babadag

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1 Characterization of Type-2 Harmonic Curvature and General Helice in Euclidean pace E⁴ Faik Babadag Department of MATHEMATICS, KIRIKKALE Univerity, KIRIKKALE Abtract In thi tudy, we introduce quaternionic type-2 Harmonic Curvature and General Helice according to Frenet frame in 4-dimenional Euclidean Space E⁴ and invetigate it propertie for two cae. In the firt cae; we ue a contant angle φ between a unit and fixed direction vector field U and the firt relatively Frenet frame vector field V₁ of the curve, that i, H ( V₁, U ) = co φ = cont. where h (V₁, U) i the real quaternion inner product. Since the relatively Frenet frame vector field V₁ of the curve make a contant angle with the unit and fixed direction vector field U, we call thi curve a a General helix in 4-dimenional Euclidean Space E⁴. And then, in the other cae, we define new type-2 harmonic curvature function and we give a vector field D which we call Darboux vector field for General helix. And then we obtain ome characterization for General helix in term of type-2 harmonic curvature function and the Darboux vector field D. Keyword Euclidean pace, General helix, type-2 harmonic curvature, Quaternion algebra, Quaternionic frame. I. INTRODUCTION The curve are a part of our live are the indipenable. For example, heart chet film with X-ray curve, how to act i important to u. Curve give the movement of the particle in Phyic. Helical curve are very important type of curve. Becaue, helice are among the implet object in the art, molecular tructure, nature, etc. For example, the path, arroued by the climbing of bean and the orbit where the progreing of the crew are a helix curve. Alo, in medicine DNA molecule i formed a two intertwined helice and many protein have helical tructure, known a alpha helice. So, uch curve are very important for undertand to nature. Therefore, lot of author intereted in the helice and they publihed many paper in Euclidean 3 and 4 - pace [1-2],]. Helix curve i defined by the property that the tangent vector field make a contant angle with a fixed direction. In 1802, M. A. Lancert firt propoed a theorem and in 1845, B. de Saint Venant firt proved thi theorem: "A neceary and ufficient condition that a curve be a general helix i that the ratio of curvature to torion be contant" [3-5]. In 1987, The Serret-Frenet formulae for quaternionic curve in R³ are introduced by K. Bharathi and M. Nagaraj. Moreover, they obtained the Serret-Frenet formulae for the quaternionic curve in R⁴ by the formulae in R³, [6]. Then, lot of tudie have been publihed by uing thi tudy. One of them i A. C. Çöken and A. Tuna' tudy [7] which they gave Serret-Frenet formula, inclined curve, harmonic curvature and ome characterization for a quaternionic curve in the emi- Euclidean pace E 1 3 and E 2 4. In thi work, we give quaternionic type-2 Harmonic Curvature and General Helice according to Frenet frame in 4- dimenional Euclidean Space E⁴ and invetigate it propertie. We ue a contant angle φ between a unit and fixed direction vector field U and the firt relatively Frenet frame vector field V₁ of the curve, that i, H (V₁, U) = co φ = cont. where h(v₁, U) i the real quaternion inner product. Since the relatively Frenet frame vector field V₁ of the curve make a contant angle with the unit and fixed direction vector field U, we call thi curve a a General helix in 4-dimenional Euclidean Space E⁴. And then, we define new type 2 harmonic curvature function and we give a vector field D which we call Darboux vector field for General helix. And then we obtain ome characterization for General helix in term of type-2 harmonic curvature function and the Darboux vector field D. Page 16

2 II. PRELIMINARIES Let Q H be the four-dimenional vector pace over a field H whoe characteritic greater than 2. Let e i ( 1 i 4 ) be a bai for the vector pace. Let the rule of multiplication on Q H be defined on e i and extended to the whole of the vector pace ditributivity a follow [6]: A real quaternion i defined by q = ae₁ + be₂ + ce₃ + d ( or S q = d and V q = ae₁ + be₂ + ce₃). Then a quaternion q can now write a q = S q + V q, where S q and V q are the calar part and vectorial part of q, repectively. we define the et of all real quaternion by Q H = q q = ae 1 + be 2 + ce 3 + d ; a, b, c, d R and e 1, e 2, e 3 R 3. Uing thee baic product we can now expand the product of two quaternion to give p q = S p S q + V p, V q + S p V q + S q V p + V p V q for every p, q Q H. where we have ued the dot and cro product in Euclidean pace E⁴. We ee that the quaternionic product contain all the product of Euclidean pace E⁴. There i a unique involutory antiautomorphim of the quaternion algebra, denoted by the ymbol γ and defined a follow: γq = ae₁ be₂ ce₃ for every q = ae₁ + be₂ + ce₃ + d Q H. which i called the "Hamiltonian conjugation". Thi h-inner product of two quaternion i define by h(p, q) = 1 2 [(p γq) + (q γp)] for every p, q Q H where h i the ymmetric, non-degenerate, real valued and bilinear form. The norm of real quaternion q i denoted by q ² = h_{v}(q, q) = (q γq) = a²b² + c² + d² for p, q Q H where if h(p, q) = 0 then p and q are called h-orthogonal. The concept of a patial quaternion will be made ue throughout our work. q i called a patial quaternion whenever q + γq = 0 [6, 7]. III. SERRET-FRENET FORMULAE FOR QUATERNIONIC CURVES EUCLIDEAN SPACE Definition. The four-dimenional Euclidean pace in E⁴ are identified with the pace of unit quaternion. Let 4 α I R Q V α() = α i ()e i, 1 i 4, e₄ = 1 i=1 be a mooth curve in E⁴. Let the parameter be choen uch that the tangent V₁() = α () ha unit magnitude. Let {V₁; V₂; V₃; V₄} be the Frenet apparatu of the differentiable Euclidean pace curve in the Euclidean pace E⁴. Then Frenet formula are given by V₁() = κ()v₂() V₂() = τ()v₃() κ()v₁() V₃() = τ()v₂() + σ()v₄() V₄() = σ()v₃() (1) We may expre Frenet formulae of the Frenet trihedron in the matrix form: V 1 V 2 V 3 V 4 = 0 κ 0 0 κ 0 τ 0 0 τ 0 σ 0 0 σ 0 V 1 V 2 V 3 V 4 (2) Page 17

3 IV. TYPE-2 HARMONIC CURVATURES AND GENERAL HEICES IN EUCLIDEAN SPACE E⁴ Definition. Let α() be a quaternionic curve in E⁴ with arc-length parameter. The type-2 harmonic curvature of α are the function, uch that G i : I R, 1 i 4 G i = 1 0 κ/τ (1/σ)G₃ i = 1 i = 2 i = 3 i = 4 and atifying the condition dg₄ d = σg₃ Theorem. Let α = α() be a quaternionic curve in Euclidean pace E⁴ with arc-length parameter. Let {V₁; V₂; V₃; V₄} and {G₁; G₂; G₃; G₄} denote the Frenet frame and the type-2 harmonic curvature of curve, repectively. Then α i a general helix if and only if the function i contant. G₁² + G₂² = c Proof. Let α = α() be general helix in Euclidean pace E⁴. Let U be the direction with which V₁ make a contant angle φ and, without lo of generality, we uppoe that < U, U >= 1. that i, U = λ₁v₁ + λ₂v₂ + λ₃v₃ + λ₄v₄ (3) λ₁ =< V₁, U >= coφ = contant λ₂ =< V₂, U > λ₃ =< V₃, U > λ 4 =< V 4, U > (4) By taking the derivate of (3) with repect to and uing the Frenet formula, we have λ₁() = κ() < V₂(), U() >= κ()λ₂() = 0. (5) Then λ₂ = 0 and therefore, U = λ₁v₁ + λ₃v₃ + λ₄v₄ The differentiation of (5) give, (κλ₁ τλ₃)v₂ + (λ₃ σλ₄)v₃ + (λ₄ + σλ₃)v₄ = 0 which implie κλ₁ τλ₃ = 0 λ₃ σλ₄ = 0 λ₄ + σλ₃ = 0. (6) Define the function G i = G i () a follow λ i = G i λ₁, 3 i 4. (7) Page 18

4 We point out that λ₁ 0: on the contrary, (6) give λ i = 0, for 3 i 4 and o, U = 0 contradiction. From the equation (1), we get G₃ = (κ/τ) G₄ = (1/σ)G₃. (8) The lat equation of (6) lead to the following condition: G₄ + σg₃ = 0. (9) In particular, and from lat equation of (8) G₃ + σg₄ = 0. (10) If we take derivative of (9) and ue the lat equation, we get the econd order differential equation G₄ + σg₃ + σg₃ = 0 G₄ + σ( (1/σ)G₄) + σ(σg₄) = 0 G₄ ((σ)/σ)g₄ + σ²g₄ = 0. (11) We do change of variable: t = σ d, dt d = σ(). Then the equation (11) become G₄(t) + G₄(t) = 0. The general olution of thi equation i obtained a G₄(t()) = Aco σ d + Bin σ d (12) where A and B are contant. From equation (9), the function G₃ i given by G₄(t ) = (1/σ)G₄ = (1/σ)( (1/σ)Aco σ d + (1/σ)Bin σ d Ain σ d Bco σ d. (13). From equation (12) and (13), we hawe G₃² + G₄² = A² + B² = c. Converly, aume that the condition (2) i atified for a curve α. Define the unit vector U by By taking (8), (9) and (10), a differentiation of U given that U = (V₁ + G₃V₃ + G₄V₄)coφ. du = [(κ G₃τ)V₂ + (G₃ G₄σ)V₃ + (G₄ + G₃σ)V₄]coφ = 0 d which it mean that U i a contant vector. On the other hand, the inner product between the unit tangent vector V₁ with U i < V₁, U >= coφ. Thu α i a general helix curve and the proof i completed. Page 19

5 Lemma. Let α() be a unit peed general helix in Euclidean pace E⁴. Let {V₁; V₂; V₃; V₄} and {G₁; G₂; G₃; G₄} denote the Frenet frame and the type-2 harmonic curvature of curve, repectively. Then, the following equation hold: < V₁, U >= G₁ < V₁, U > < V 2, U >= G 2 < V 1, U > < V 3, U >= G 3 < V 1, U > < V 4, U >= G 4 < V 1, U > (14) where U i an axi of the general helix α. By uing the above Lemma, we have the following corollary: Corollary. If U i an axi of the general helix α, then we can write U = (G₁V₁ + G₃V₃ + G₄V₄)coφ. Definition. Let α() be a non-degenerate unit peed curve in Euclidean pace E⁴. Let {V₁; V₂; V₃; V₄} and {G₁; G₂; G₃; G₄} denote the Frenet frame and the type-2 harmonic curvature of curve, repectively, the vector i called the type-2 Darboux vector of the curve α.alo, i an axi of the general helix α. D = G₁V₁ + G₂V₂ + G₃V₃ + G₄V₄ (15) D = G₁V₁ + G₃V₃ + G₄V₄ Lemma. Let α() be a non-degenerate unit peed curve in Euclidean pace E⁴. Let {V₁; V₂; V₃; V₄} and {G₁; G₂; G₃; G₄} denote the Frenet frame and the type-2 harmonic curvature of curve, repectively, then α i a general helix if and only if D i contant vector. Proof. Let α() be a general helix in Euclidean pace E⁴. From corollary 1, By differentiating the D with repect to D = G₁V₁ + G₃V₃ + G₄V₄. D = (κ G₃τ)V₂ + (G₃ G₄σ)V₃ + (G₄ + G₃σ)V₄. By uing equation (8),(9) and (10), we get D = 0. Therefore, D i contant vector. Converaly, if D i contant vector. Then we can ee that Thu we get < D, V₁ >=< V₁ + G₃V₃ + G₄V₄, V₁ > < D, V₁ >=< V₁, V₁ >= 1. coφ = < D, V₁ > D V₁ = 1 D where φ i contant angle between D and V₁. In thi cae, we can define a unique axi of the general helix uch that : U = Dco. From thi equation < U, V₁ >=< D, V₁ > coφ = coφ=contant. Therefore, U i a contant. So, thi complete the proof. Page 20

6 Lemma. In three- dimenional Euclidean pace, from equation (15), we can write the axi of a non-degenerate curve a: D = G₁t + G₂n + G₃b = G₁t + G₃b = t + (κ/τ)b where {t; n; b} and {κ; τ} are the Frenet apparatu and with non-zero curvature in the Euclidean pace E³,repectively. If we take derivative of D along the curve, we get D = t + ((κ/τ))b + (κ/τ)b = κn + ((κ/τ))b + (κ/τ)( τn) = ((κ/τ))b. Thu, from the equation, if the curve i a general helix, then from Lemma 2, we have D = 0 ((κ/τ)) = 0, (κ/τ) i contant, therefore the curve i a general helix. Lemma. There are no general helice with non-zer contant curvature (i.e., W-curve) in Euclidean pace E⁴. Proof. In four-dimenional Euclidean pace, from equation (15), we get D = G₁V₁ + G₂V₂ + G₃V₃ + G₄V₄ D = V₁ + (κ/τ)v₃ + (1/σ)(κ/τ)V₄ where κ, τ and σ are curvature of the curve. If all the curvature of the curve are non-zero contant, then the curve i a Wcurve, then we get If we take derivative of equation (16), we obtain D = V₁ + (κ/τ)v₃. (16) D = κv₁ + (κ/τ)( τv₂ + σv₄) = (κ/τ)σv₄. So, we can eaily ee that D i not equal to zero, then D i not contant vector. In thi cae, according to Lemma 2 the curve i not general helix. Lemma. There are no general helice with non-zero contant curvature ration (i.e., CCr-Curve) in E⁴. Proof. The proof of thi Lemma i the ame a the proof of Lemma 1. REFERENCES [1] M. Barro, General helice and a theorem of Lancert. Proc. AMS, 1997, 125, [2] B. Bükcü, M. K. Karacan, The Slant Helice According to Bihop Frame, International Journal of Computational and Mathematical Science 3:2, [3] D. J. Struik, Lecture on Claical Differential Geometry, Dover, New-York, [4] Hamilton, W.R., Element of Quaternion, I, II and III, chelea, New YORK, [5] K. Bharathi, M. Nagaraj, Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 16, [6] A. C. Çöken, A. Tuna, On the quaternionic inclined curve in the emi-euclidean pace E 2 4 : Applied Mathematic and Computation 155, 2004, [7] J.P. Ward, Quaternion and Cayley Number, Kluwer Academic Publiher, Boton/London,1997. Page 21