International Journal of Analysis and Applications Volume 17, Number 4 (2019), 479-502 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-479 COMPUTING SPECIAL SMARANDACHE CURVES ACCORDING TO DARBOUX FRAME IN EUCLIDEAN 4-SPACE M. KHALIFA SAAD1,2,∗ AND M. A. ABD-RABO3 1Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Madinah, KSA 2Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt 3School of Mathematics and Statistics, Zhengzhou University, 450001 Zhengzhou, China ∗Corresponding author: m khalifag@yahoo.com, mohammed.khalifa@iu.edu.sa Abstract. In this paper, we study some special Smarandache curves and their differential geometric prop- erties according to Darboux frame in Euclidean 4-space E4. Also, we compute some of these curves which lie fully on a hypersurface in E4. Moreover, we defray some computational examples in support our main results. 1. Introduction The geometric modeling of free-form curves and surfaces is of central importance for sophisticated CAD/CAM systems. Among all space curves, Smarandache curves have special emplacement regarding their properties, because of this, they deserve especial attention in Euclidean geometry as well as in other geometries. It is known that Smarandache geometry is a geometry which has at least one Smarandache denied axiom [1]. An axiom is said to be Smarandache denied, if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. Smarandache curves are the objects of Smarandache geometry. By definition, if the position Received 2019-03-27; accepted 2019-04-24; published 2019-07-01. 2010 Mathematics Subject Classification. 53A04, 53A07, 53A35. Key words and phrases. Smarandache curves; hypersurfaces; Darboux frame; Euclidean 4-space. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 479 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-479 Int. J. Anal. Appl. 17 (4) (2019) 480 vector of a curve δ is composed by Frenet frame’s vectors of another curve β, then the curve δ is called a Smarandache curve [2]. In differential geometry, frame fields constitute an important tool while studying curves and surfaces. The most familiar frame fields are Frenet frame along a space curve and Darboux frame along a surface curve [3]. It is analogous to the Frenet frame as applied to surface geometry. The Darboux frame exists at any non- umbilic point of a surface embedded in Euclidean space. In [4] M. Düldü et al. extend the Darboux frame field into Euclidean 4-space E4. In the light of the existing studies in the field of geometry, many interesting results on Smarandache curves have been obtained by many mathematicians, see for example, [2, 3, 5–12]. Turgut and Yilmaz [11] have introduced a particular circumstance of such curves, they entitled it Smarandache TB2 curves in the space E14. They studied special Smarandache curves which are defined by the tangent and second binormal vector fields. In [8], the author has illustrated certain special Smarandache curves in the Euclidean 3-space. Recently, H.S. Abdel-Aziz and M. Khalifa Saad [2, 5] have studied special Smarandache curves of an arbi- trary curve such as TN, TB and TNB with respect to Frenet frame in the three-dimensional Galilean and pseudo-Galilean spaces. Also, in [6, 7] they have investigated Smarandache curves according to Darboux frame in the three-dimensional Minkowski space E31. The main goal of this article is to investigate Smarandache curves in E4 for a given curve with reference to its Darboux frame of first kind. The results presented in this paper generalize and refine some of the existing results in the literature and significant in mathematical modeling and other applications. 2. Basic notions and properties We now review some basic concepts on classical differential geometry of space curves and surfaces in Euclidean 4-space E4 [4, 13–15]. Let {ei | i = 1, 2, 3, 4} be the standard basis of E4, therefore E4 = {X =∑4 i=1 xiei | xi ∈ R}. The scalar product and vector product of vectors X,Y,Z ∈ E 4 are respectively defined by 〈X,Y 〉 = 4∑ i=1 xiyi , X ×Y ×Z = ∣∣∣∣∣∣∣∣∣∣∣∣ e1 e2 e3 e4 x1 x2 x3 x4 y1 y2 y3 y4 z1 z2 z3 z4 ∣∣∣∣∣∣∣∣∣∣∣∣ . Definition 2.1. Let f : D ⊂ E4 → R be a differentiable mapping of an open set D. Given c ∈ R, we recall that the level set c of the f is the set defined as f−1(c) which is the set of solutions in D of the equations f(x,y,z,w) = c. Int. J. Anal. Appl. 17 (4) (2019) 481 Proposition 2.1. Consider that f : U ⊂ E4 → R is a differentiable function and c ∈ f(U) is a regular value of f, then f−1(c) is a regular surface in E4. The implicit surface f is regular if ∇f = (fx,fy,fz,fw) 6= 0. The unit surface normal vector of the implicit surface f is given by N = ∇f‖∇f‖. Definition 2.2. Suppose that f : Rn → R, if all second partial derivatives of f exist and are continuous on the domain of f, then the Hessian matrix Hf is a square n×n matrix, usually defined as Hij = ∂ 2f ∂xi∂xj . 2.1. Curves on a hypersurface in E4. Let r : I ⊂ R → M be a regular curve in E4, where ‖r′‖ = ‖dr/du‖ = 1, ∀ u ∈ I , we will omit u for simplification. Then the Frenet frame is defined by   t′ n′ b′1 b′2   =   0 κ1 0 0 −κ1 0 κ2 0 0 −κ2 0 κ3 0 0 κ3 0     t n b1 b2   , ( ′ = d du ) (2.1) where t, n, b1 and b2 are the tangent, principal normal, first binormal and the second binormal vector fields of r, respectively and the functions κi | i = 1, 2, 3, are the curvature functions of r. Theorem 2.1. [2] Let r : I → E4 be a regular curve. Then t = r ′ ‖r′‖ , b2 = r ′ ×r ′′ ×r(3) ‖r′ ×r′′ ×r(3)‖ , b1 = b2 ×r ′ ×r ′′ ‖b2 ×r ′ ×r′′‖ , n = b1 × b2 ×r ′ ‖b1 × b2 ×r ′‖ , (2.2) and κ1 = 〈n,r ′′ 〉 ‖r′‖2 , κ2 = 〈b1,r(3)〉 ‖r′‖3κ1 , κ3 = 〈b2,r(4)〉 ‖r′‖4κ1κ2 . (2.3) Definition 2.3. Let r be a regular curve in E4, which is parameterized by arc length and n-times continuously differentiable. Then r is called a Frenet curve, if at every point the vectors r ′ ,r ′′ , ...,r(n−1) are linearly independent. 2.2. Darboux frame field of first kind in E4. Let M be an oriented hypersurface in E4 and r be a Frenet curve of class Cn(n ≥ 4) with arc-length parameter u lying on M. We denote the unit tangent and unit normal vector fields of the curve by T and N, and P, U ∈ TuM. Definition 2.4. Let r : I → E4 be a regular parameterized curve lying on M in E4, then the frame field {T, P, U, N} along r is called extended Darboux frame field of first kind if {T, N,r′′} is linearly independent, therefore T = r ′ ‖r′‖, N = ∇f ‖∇f‖, P = r′′−(r′′.N)N ‖r′′−(r′′.N)N‖, U = [T, N, E] Int. J. Anal. Appl. 17 (4) (2019) 482 The derivatives of the Darboux frame field of first kind in E4 are given by [16]  T′ P′ U′ N′   =   0 κ1g 0 κn −κ1g 0 κ2g τ1g 0 −κ2g 0 τ2g −κn −τ1g −τ2g 0     T P U N   , (2.4) where κn,κ 1 g,κ 2 g,τ 1 g and τ 2 g are real valued functions denote the normal curvature, geodesic curvatures and the geodesic torsions, respectively. These functions are given by κn = 〈T′, N〉 = 1 ‖∇f‖ 〈r ′′ ,∇f 〉 = −1 ‖∇f‖ (r′Hf (r ′)t), κ1g = 〈T ′, P〉 = ( r ′′ (r ′′ )t − 1 ‖∇f‖2 (r ′ Hf (r ′ )t)2 )1 2 , κ2g = 〈P ′, U〉 = 1 (κ1g) 2∇f ( m1 + 1 ‖∇f‖2 (r ′ Hf (r ′ )t) m2 ) , τ1g = 〈P ′, N〉 = −1 (κ1g) ( 1 ‖∇f‖ (r ′ Hf (r ′′ )t) + 1 ‖∇f‖3 (r ′ Hf (∇f)t)(r ′ Hf (r ′ )t) ) , τ2g = 〈U ′, N〉 = ( −1 (κ1g)‖∇f‖2 ) m3, (2.5) where m1 = ∣∣∣∣∣∣∣∣∣∣∣∣ x ′ y ′ z ′ w ′ x ′′ y ′′ z ′′ w ′′ x(3) y(3) z(3) w(3) fx fy fz fw ∣∣∣∣∣∣∣∣∣∣∣∣ , m2 = ∣∣∣∣∣∣∣∣∣∣∣∣ x ′ y ′ z ′ w ′ x ′′ y ′′ z ′′ w ′′ a b c d fx fy fz fw ∣∣∣∣∣∣∣∣∣∣∣∣ , m3 = ∣∣∣∣∣∣∣∣∣∣∣∣ x ′ y ′ z ′ w ′ fx fy fz fw a b c d p q t v ∣∣∣∣∣∣∣∣∣∣∣∣ , and [a,b,c,d] = (∇f) ′ = r ′ Hf , [p,q,t,v] = (∇f) ′′ = α ′′ Hf + α ′ dHf du ′ , dHf du = [ ∂Hf ∂x (r ′ )t ... ∂Hf ∂w (r ′ )t]. Remark 2.1. [2] Let r : I → E4 be a regular parameterized curve in E4, then • r is called asymptotic curve if and only if kn = 0. • r is called a line of curvature if and only if τ1g = τ2g = 0. 3. First kind Smarandache curves in E4 Consider r = r(u) is a curve lying fully on an oriented hypersurface M in E4. Let {T, P, U, N} be a Darboux frame field of first kind along r(u) and κn,κ 1 g,κ 2 g,τ 1 g ,τ 2 g are real valued functions in arc length parameter u of r. So, we have the following definition. Int. J. Anal. Appl. 17 (4) (2019) 483 Definition 3.1. [11] A regular curve α(s(u)) in E4, whose position vector is obtained by extended Darboux frame vectors of another regular curve r(u) is called Smarandache curve . In the following we continue our studies of special Smarandache curves that we started in [2, 5, 6]. Here we investigate some special Smarandache curves of first kind called TP, TU, PU and PN ( the other special Smarandache curves can be computed in the same manner) and then obtain some of their differential geometric properties which represent the main results. Let r(u) = (x(u),y(u),z(u),w(u)) be a curve of class Cn(n ≥ 4) lying on M. Then, by using proposition 2.1, the unit normal vector field along r is given by N̄ = ∇f ‖∇f‖ . (3.1) 3.1. TP-Smarandache curves. Definition 3.2. Let M be an oriented hypersurface in E4 and the Frent curve r = r(u) lying fully on M with Darboux frame {T, P, U, N} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the TP-Smarandache curve of r is defined as α(s) = 1 √ 2 (T + P). (3.2) Theorem 3.1. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame {T, P, U, N} and non-zero constant curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the curvature functions of the TP− Smarandache curve of r satisfy the following equations: κ̄n = 1 λ1‖∇f‖   (κ1g(−κn + τ1g ) + κ2gτ2g )fw − ((κ1g)2+ κn(κn + τ 1 g ))fx − ((κ1g)2 + (κ2g)2+ τ1g (κn + τ 1 g ))fy + (κ 1 gκ 2 g − (κn + τ1g )τ2g )fz   , κ̄1g = 1 λ1λ6   (κ1g(−κn + τ1g ) + κ2gτ2g )λ5 + ((κ1g)2+ κn(κn + τ 1 g ))λ2 + ((κ 1 g) 2 + (κ2g) 2+ τ1g (κn + τ 1 g ))λ3 + (κ 1 gκ 2 g − (κn + τ1g )τ2g )λ4   , κ̄2g = 1 λ26λ11   −(λ2λ7 + λ3λ8 + λ4λ9 + λ5λ10)λ′6+ λ6((κ 1 gλ2 −κ2gλ4)λ8 −λ5(κnλ7 + τ1g λ8+ τ2g λ9) + λ3(−κ1gλ7 + κ2gλ9 + τ1g λ10) + λ7λ′2+ λ8λ ′ 3 + λ9λ ′ 4 + λ10(κnλ2 + τ 2 g λ4 + λ ′ 5))   , Int. J. Anal. Appl. 17 (4) (2019) 484 τ̄1g = 1 λ26‖∇f‖   −(fxλ2 + fyλ3 + fzλ4 + fwλ5)λ′6+ λ6(−κ1gfxλ3 + κ2gfzλ3 −κnfxλ5− τ2g fzλ5 + fxλ ′ 2 + fy(κ 1 gλ2 −κ2gλ4 − τ1g λ5+ λ′3) + fzλ ′ 4 + fw(κnλ2 + τ 1 g λ3 + τ 2 g λ4 + λ ′ 5))   , τ̄2g = 1 λ211‖∇f‖   −(fxλ7 + fyλ8 + fzλ9 + fwλ10)λ′11+ λ11(−κ1gfxλ8 + κ2gfzλ8 −κnfxλ10− τ2g fzλ10 + fxλ ′ 7 + fy(κ 1 gλ7 −κ2gλ9 − τ1g λ10+ λ′8) + fzλ ′ 9 + fw(κnλ7 + τ 1 g λ8 + τ 2 g λ9 + λ ′ 10))   . (3.3) Proof. Because α = α(s) is a TP− Smarandache curve reference to Frenet curve r, then differentiating Eq. (3.2), we get α′ = 1 √ 2 ( −κ1gT + κ 1 gP + κ 2 gU + (τ 1 g + κn)N ) . (3.4) Again, differentiating Eq. (3.4), we obtain α′′ = 1 √ 2   (−(κ1g)2 −κn(κn + τ1g ))T − ((κ1g)2 + (κ2g)2 + τ1g (κn + τ1g ))P +(κ1gκ 2 g − (κn + τ1g )τ2g )U + (−κ1gκn + κ1gτ1g + κ2gτ2g )N   . Also, Eq. (3.4) leads to T̄ = −κ1gT + κ1gP + κ2gU + (τ1g + κn)N λ1 , where λ1 = √ 2(κ1g) 2 + (κ2g) 2 + (τ1g + κn) 2, and we get N̄ = fxT + fyP + fzU + fwN ‖∇f‖ . On the other hand, we have P̄ = λ2T + λ3P + λ4U + λ5N λ6 , where λ2 = 1 √ 2 ( (κ1g) 2 + (κn) 2 + κnτ 1 g ) + σ1 ‖∇f‖ fx , λ3 = 1 √ 2 ( (κ1g) 2 + (κ2g) 2 + κnτ 1 g + (τ 1 g ) 2 ) + σ1 ‖∇f‖ fy , λ4 = 1 √ 2 ( −κ1gκ 2 g + κnτ 2 g + τ 1 g τ 2 g ) + σ1 ‖∇f‖ fz , λ5 = 1 √ 2 ( κ1gκn −κ 1 gτ 1 g −κ 2 gτ 2 g ) + σ1 ‖∇f‖ fw , λ6 = √ (λ2)2 + (λ3)2 + (λ4)2 + (λ5)2, Int. J. Anal. Appl. 17 (4) (2019) 485 σ1 = 〈α′′, N〉 = 1 √ 2‖∇f‖   (−(κ1g)2 −κn(κn + τ1g ))fx − ((κ1g)2 + (κ2g)2+ τ1g (κn + τ 1 g ))fy + (κ 1 gκ 2 g − (κn + τ1g )τ2g )fz+ (−κ1gκn + κ1gτ1g + κ2gτ2g )fw   . Also, we get Ū = λ7T + λ8P + λ9U + λ10N λ11 , where, λ7 =   κ2gfwλ3 −κnfzλ3 − τ1g fzλ3 −κ1gfwλ4+ κnfyλ4 + τ 1 g fyλ4 −κ2gfyλ5 + κ1gfzλ5   , λ8 =   −κ2gfwλ2 + κnfzλ2 + τ1g fzλ2 −κ1gfwλ4− κnfxλ4 − τ1g fxλ4 + κ2gfxλ5 + κ1gfzλ5   , λ9 =   κ1gfwλ2 −κnfyλ2 − τ1g fyλ2 + κ1gfwλ3+ κnfxλ3 + τ 1 g fxλ3 −κ1gFxλ5 −κ1gFyλ5   , λ10 = ( κ2gfyλ2 −κ1gfzλ2 −κ2gfxλ3 −κ1gfzλ3 + κ1gfxλ4 + κ1gfyλ4 ) , λ11 = λ1 λ6 ‖∇f‖. In the light of the above calculations, the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of α are computed as in Eqs. (3.3). � Corollary 3.1. If r is an asymptotic curve. Then, the following equations hold: T̄ = −κ1gT + κ1gP + κ2gU + τ1g N λ1 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = λ13T + λ14P + λ15U + λ16N λ17 , Ū = λ18T + λ19P + λ20U + λ21N λ22 , and then the curvature functions are computed as follows κ̄n = (κ1gτ 1 g + κ 2 gτ 2 g )fw − (κ1g)2fx − ((κ1g)2 + (κ2g)2 + (τ1g )2)fy + (κ1gκ2g − τ1g τ2g )fz λ12‖∇f‖ , κ̄1g = (κ1gτ 1 g + κ 2 gτ 2 g )λ16 − (κ1g)2λ13 − ((κ1g)2 + (κ2g)2 + (τ1g )2)λ14 + (κ1gκ2g − τ1g τ2g )λ15 λ12λ17 , κ̄2g = 1 λ22λ 2 17   λ17(κ 1 gλ13λ19 −κ2gλ15λ19 − τ1g λ16λ19− τ2g λ16λ20 + λ14(−κ1gλ18 + κ2gλ20) + λ18λ′13+ λ19λ ′ 14 + λ20λ ′ 15 + λ21(τ 1 g λ14 + τ 2 g λ15+ λ′16)) − (λ21λ16 + λ13λ18 + λ14λ19 + λ15λ20)λ′17   , Int. J. Anal. Appl. 17 (4) (2019) 486 τ̄1g = 1 λ217‖∇f‖   −fx(κ1gλ14λ17 −λ17λ′13 + λ13λ′17) + fy(λ17(κ1gλ13− κ2gλ15 − τ1g λ16 + λ′14) −λ14λ′17)+ fz(λ17(κ 2 gλ14 − τ2g λ16 + λ′15) −λ15λ′17)+ fw(λ17(τ 1 g λ14 + τ 2 g λ15 + λ ′ 16) −λ16λ′17)   , τ̄2g = 1 λ222‖∇f‖   −(λ21fw + λ18fx + λ19fy + λ20fz)λ′22 + λ22((−τ1g λ21+ κ1gλ18)fy + λ20(τ 2 g fw −κ2gfy) + λ19(τ1g fw −κ1gfx+ κ2gfz) + fwλ ′ 21 + fxλ ′ 18 + fyλ ′ 19 + fz(−τ2g λ21 + λ′20))   , where λ12 = √ 2(κ1g) 2 + (κ2g) 2 + (τ1g ) 2, λ13 = √ 2(κ1g) 2‖∇f‖ + 2σ1fx, λ14 = √ 2((κ1g) 2 + (κ2g) 2 + (τ1g ) 2)‖∇f‖ + 2σ1fy, λ15 = √ 2(−κ1gκ 2 g + τ 1 g τ 2 g )‖∇f‖ + 2σ1fz, λ16 = √ 2(−κ1gτ 1 g −κ 2 gτ 2 g )‖∇f‖ + 2σ1fw, λ17 = √ (λ13)2 + (λ14)2 + (λ15)2 + (λ16)2, λ18 = κ 2 gfwλ14 − τ 1 g fzλ14 −κ 1 gfwλ15 + τ 1 g fyλ15 −κ 2 gfyλ16 + κ 1 gfzλ16, λ19 = −κ2gfwλ13 + τ 1 g fzλ13 −κ 1 gfwλ15 − τ 1 g fxλ15 + κ 2 gfxλ16 + κ 1 gfzλ16, λ20 = k 1 gλ13fw + k 1 gλ14fw + τ 1 g λ14fx −k 1 gλ16fx − τ 1 g λ13fy −k 1 gλ16fy, λ21 = κ 2 gfyλ13 −κ 1 gfzλ13 −κ 2 gfxλ14 −κ 1 gfzλ14 + κ 1 gfxλ15 + κ 1 gfyλ15, λ22 = λ12 λ17 ‖∇f‖, σ1 = 1 √ 2‖∇f‖   −(κ1g)2fx − ((κ1g)2 + (κ2g)2 + (τ1g )2)fy+ (κ1gκ 2 g − τ1g τ2g )fz + (κ1gτ1g + κ2gτ2g )fw   . Corollary 3.2. If r is a line of curvature. Then, the following equations hold: T̄ = −κ1gT + κ1gP + κ2gU + κnN λ1 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = λ24T + λ25P + λ26U + λ27N λ28 , Ū = λ29T + λ30P + λ31U + λ32N λ33 , and then the curvature functions are computed as follows κ̄n = − κ1gκnfw + ((κ 1 g) 2 + κ2n)fx + ((κ 1 g) 2 + (κ2g) 2)fy −κ1gκ2gfz λ23‖∇f‖ , κ̄1g = − κ1gκnλ27 + ((κ 1 g) 2 + κ2n)λ24 + ((κ 1 g) 2 + (κ2g) 2)λ25 −κ1gκ2gλ26 λ23λ28 , Int. J. Anal. Appl. 17 (4) (2019) 487 κ̄2g = 1 λ33λ 2 28   λ28(λ25(−κ1gλ29 + κ2gλ31) + λ29(−κnλ27 + λ′24)+ λ30(κ 1 gλ24 −κ2gλ26 + λ′25) + λ31λ′26 + λ32(κnλ24+ λ′27)) − (λ32λ27 + λ24λ29 + λ25λ30 + λ26λ31)λ′28   , τ̄1g = 1 λ228‖∇f‖   −fx(λ28(κ1gλ25 + κnλ27 −λ′24) + λ24λ′28)+ fy(λ28(κ 1 gλ24 −κ2gλ26 + λ′25) −λ25λ′28)+ fz(λ28(κ 2 gλ25 + λ ′ 26) −λ26λ′28)+ fw(λ28(κnλ24 + λ ′ 27) −λ27λ′28)   , τ̄2g = 1 λ233‖∇f‖   −(λ32fw + λ29fx + λ30fy + λ31fz)λ′33+ λ33(−κnλ32fx −κ1gλ30fx+ λ29(κnfw + κ 1 gfy) + κ 2 gλ30fz + fwλ ′ 32+ fxλ ′ 29 + fy(−κ2gλ31 + λ′30) + fzλ′31)   , where λ23 = √ 2(κ1g) 2 + (κ2g) 2 + (κn)2, λ24 = √ 2((κ1g) 2 + κ2n) 2‖∇f‖ + 2σ1fx, λ25 = √ 2((κ1g) 2 + (κ2g) 2)‖∇f‖ + 2σ1fy, λ26 = − √ 2κ1gκ 2 g‖∇f‖ + 2σ1fz, λ27 = √ 2κ1gκn‖∇f‖ + 2σ1fw, λ28 = √ (λ24)2 + (λ25)2 + (λ26)2 + (λ27)2, λ29 = κ 2 gfwλ25 −κnfzλ25 −κ 1 gfwλ26 + κnfyλ26 −κ 2 gfyλ27 + κ 1 gfzλ27, λ30 = −κ2gfwλ24 + κnfzλ24 −κ 1 gfwλ26 −κnfxλ26 + κ 2 gfxλ27 + κ 1 gfzλ27, λ31 = κ 1 gfwλ24 −κnfyλ24 + κ 1 gfwλ25 + κnfxλ25 −κ 1 gFxλ27 −κ 1 gFyλ27, λ32 = κ 2 gfyλ24 −κ 1 gfzλ24 −κ 2 gfxλ25 −κ 1 gfzλ25 + κ 1 gfxλ26 + κ 1 gfyλ26, λ33 = λ23 λ28 ‖∇f‖, σ1 = 1 √ 2‖∇f‖ (−κ1gκnfw − ((κ 1 g) 2 + κ2n)fx − ((κ 1 g) 2 + (κ2g) 2)fy + κ 1 gκ 2 gfz). 3.2. TU-Smarandache curves. Definition 3.3. Let M be an oriented hypersurface in E4 and the Frenet curve r = r(u) lying fully on M with Darboux frame {T, P, U, N} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the TU-Smarandache curve of r is defined as β(u) = 1 √ 2 (T + U). (3.5) Theorem 3.2. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame {T, P, U, N} and non-zero constant curvatures; κn,κ1g,κ2g,τ1g and τ2g . Then the curvature functions of the Int. J. Anal. Appl. 17 (4) (2019) 488 TU− Smarandache curve of r satisfy the following equations: κ̄n = 1 µ1‖∇f‖   (κ1g −κ2g)τ1g fw − (κ1g(κ1g −κ2g) + κn(κn + τ2g ))fx+ (κ1g −κ2g)κ2gfy − (κn + τ2g )(τ1g fy + τ2g fz)   , κ̄1g = 1 µ1µ6   (κ1g −κ2g)τ1g µ5 − (κ1g(κ1g −κ2g) + κn(κn + τ2g ))µ2+ (κ1g −κ2g)κ2gµ3 − (κn + τ2g )(τ1g fy + τ2g µ4)   , κ̄2g = 1 µ26µ11   −(µ2µ7 + µ3µ8 + µ4µ9 + µ5µ10)µ′6 + µ6((κ1gµ2− κ2gµ4)µ8 −µ5(κnµ7 + τ1g µ8 + τ2g µ9) + µ3(−κ1gµ7+ κ2gµ9 + τ 1 g µ10) + µ7µ ′ 2 + µ8µ ′ 3+ µ9µ ′ 4 + µ10(κnµ2 + τ 2 g µ4 + µ ′ 5))   , τ̄1g = 1 µ26‖∇f‖   −(fxµ2 + fyµ3 + fzµ4 + fwµ5)µ′6 + µ6(fxµ′2 −κ1gfxµ3 + κ2gfzµ3 −κnfxµ5 − τ2g fzµ5+ fy(κ 1 gµ2 −κ2gµ4 − τ1g µ5 + µ′3) + fzµ′4+ fw(κnµ2 + τ 1 g µ3 + τ 2 g µ4 + µ ′ 5))   , τ̄2g = 1 µ211‖∇f‖   −(fxµ7 + fyµ8 + fzµ9 + fwµ10)µ′11 + µ11(fxµ′7 −κ1gfxµ8 + κ2gfzµ8 −κnfxµ10 − τ2g fzµ10+ fy(κ 1 gµ7 −κ2gµ9 − τ1g µ10 + µ′8)+ fzµ ′ 9 + fw(κnµ7 + τ 1 g µ8 + τ 2 g µ9 + µ ′ 10))   . (3.6) Proof. Since β = β(s) is a TU− Smarandache curve reference to Frenet curve r. Then, by differentiating Eq. (3.5), we get β′ = 1 √ 2 ( (κ1g −κ 2 g)P + (κn + τ 2 g )U ) , (3.7) Again, by differentiating Eq. (3.7), we have β′′ = 1 √ 2   (−κ1g(κ1g −κ2g) −κn(κn + τ2g ))T − τ1g (κn + τ2g )P +((κ1g −κ2g)κ2g − (κn + τ2g )τ2g )U + (κ1g −κ2g)τ1g N   . Also, from Eq. (3.7), we obtain T̄ = (κ1g −κ2g)P + (κn + τ2g )U µ1 , where µ1 = √ 2(κ1g −κ2g)2 + (κn + τ2g )2, and we have N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = µ2T + µ3P + µ4U + µ5N µ6 , Int. J. Anal. Appl. 17 (4) (2019) 489 where µ2 = 1 √ 2 ( ((κ1g) 2 −κ1gκ 2 g + κn(κn + τ 2 g )) ) + σ2 ‖∇f‖ fx , µ3 = 1 √ 2 ( τ1g (κn + τ 2 g ) ) + σ2 ‖∇f‖ fy , µ4 = 1 √ 2 ( (κ2g(−κ 1 g + κ 2 g) + τ 2 g (κn + τ 2 g )) ) + σ2 ‖∇f‖ fz , µ5 = 1 √ 2 ( (−κ1g + κ 2 g)τ 1 g ) + σ2 ‖∇f‖ fw , µ6 = √ (µ2)2 + (µ3)2 + (µ4)2 + (µ5)2 σ2 = 〈β′′, N〉 = 1 √ 2‖∇f‖   (−(κ1g)2 + κ1gκ2g −κn(κn + τ2g ))fx + τ1g (κn + τ2g )fy +((κ1g −κ2g)κ2g − τ2g (κn + τ2g ))fz + (κ1g −κ2g)τ1g fw   . Also, we get Ū = µ7T + µ8P + µ3U + µ10N δ , where µ7 = −κnfzµ3 − τ2g fzµ3 −κ 1 gfwµ4 + κ 2 gfwµ4 + κnfyµ4 + τ 2 g fyµ4 + κ 1 gfzµ5 −κ 2 gfzµ5, µ8 = κnfzµ2 + τ 2 g fzµ2 −κnfxµ4 − τ 2 g fxµ4, µ9 = κ 1 gfwµ2 −κ 2 gfwµ2 −κnfyµ2 − τ 2 g fyµ2 + κnfxµ3 + τ 2 g fxµ3 −κ 1 gfxµ5 + κ 2 gfxµ5, µ10 = −κ1gfzµ2 + κ 2 gfzµ2 + κ 1 gfxµ4 −κ 2 gfxµ4, µ11 = µ1 µ6 ‖∇f‖. In the light of the above calculations, the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of β are computed as in Eqs. (3.6). � Corollary 3.3. If r is an asymptotic curve. Then, the following equations hold: T̄ = (κ1g −κ2g)P + τ2g U µ1 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = µ13T + µ14P + µ15U + µ16N µ17 , Ū = µ18T + µ19P + µ20U + µ21N µ22 , Int. J. Anal. Appl. 17 (4) (2019) 490 and the curvature functions are obtained as follows: κ̄n = (κ1g −κ2g)τ1g fw + κ1g(−κ1g + κ2g)fx − τ1g τ2g fy − (−κ1gκ2g + (κ2g)2 + (τ2g )2)fz µ12‖∇f‖ , κ̄1g = κ1g(−κ1g + κ2g)µ13 + (κ1g −κ2g)κ2gµ15 − τ2g (τ1g µ14 + τ2g µ15) + (κ1g −κ2g)τ1g µ16 µ12µ17 , κ̄2g = 1 µ217µ22   µ17(κ 1 gµ13µ19 −κ2gµ15µ19 − τ1g µ16µ19 − τ2g µ16µ20+ τ2g µ15µ21 + µ14(−κ1gµ18 + κ2gµ20 + τ1g µ21) + µ18µ′13 + µ19µ′14+ µ20µ ′ 15 + µ21µ ′ 16) − (µ13µ18 + µ14µ19 + µ15µ20 + µ16µ21)µ′17   , τ̄1g = 1 µ217‖∇f‖   −fx(κ1gµ14µ17 −µ17µ′13 + µ13µ′17) + fy(µ17(κ1gµ13− κ2gµ15 − τ1g µ16 + µ′14) −µ14µ′17) + fz(µ17(κ2gµ14 − τ2g µ16+ µ′15) −µ15µ′17) + fw(µ17(τ1g µ14 + τ2g µ15 + µ′16) −µ16µ′17)   , τ̄2g = 1 µ222‖∇f‖   −fx(κ1gµ19µ22 −µ22µ′18 + µ18µ′22) + fy(µ22(κ1gµ18 −κ2gµ20− τ1g µ21 + µ ′ 19) −µ19µ′22) + fz(µ22(κ2gµ19 − τ2g µ21 + µ′20)− µ20µ ′ 22) + fw(µ22(τ 1 g µ19 + τ 2 g µ20 + µ ′ 21) −µ21µ′22)   , where µ12 = √ (κ1g −κ2g)2 + (τ2g )2, µ13 = √ 2((κ1g) 2 −κ1gκ 2 g)‖∇f‖ + 2σ2fx , µ14 = √ 2τ1g τ 2 g‖∇f‖ + 2σ2fy , µ15 = √ 2(κ2g(−κ 1 g + κ 2 g) + (τ 2 g ) 2)‖∇f‖ + 2σ2fz, µ16 = √ 2(−κ1g + κ 2 g)τ 1 g‖∇f‖ + 2σ2fw, µ217 = (µ13) 2 + (µ14) 2 + (µ15) 2 + (µ16) 2, µ18 = −(κ1gµ15fw −κ 2 gµ15fw − τ 2 g µ15fy + τ 2 g µ14fz −κ 1 gµ16fz + κ 2 gµ16fz), µ19 = −(τ2g µ15fx − τ 2 g µ13fz), µ20 = −(−κ1gµ13fw + κ 2 gµ13fw − τ 2 g µ14fx + κ 1 gµ16fx −κ 2 gµ16fx + τ 2 g µ13fy), µ21 = −(−κ1gµ15fx + κ 2 gµ15fx + κ 1 gµ13fz −κ 2 gµ13fz), µ22 = µ12 µ17 ‖∇f‖. Int. J. Anal. Appl. 17 (4) (2019) 491 Corollary 3.4. If r is a line of curvature curve. Then, the following equations hold: T̄ = (κ1g −κ2g)P + κnN µ23 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = µ24T + µ25P + µ26U + µ27N µ28 , Ū = µ29T + µ30P + µ31U + µ32N µ33 , and the curvature functions are obtained as follows: κ̄n = −((κ1g)2 −κ1gκ2g + κ2n)fx + (κ1g −κ2g)κ2gfz µ23‖∇f‖ , κ̄1g = −((κ1g)2 −κ1gκ2g + κ2n)µ24 + (κ1g −κ2g)κ2gµ26 µ23µ28 , κ̄2g = 1 µ217µ22   µ28(−κnµ27µ29 + µ25(−κ1gµ29 + κ2gµ31) + µ29µ′24+ µ30(κ 1 gµ24 −κ2gµ26 + µ′25) + µ31µ′26 + µ32(κnµ24+ µ′27)) − (µ24µ29 + µ25µ30 + µ26µ31 + µ27µ32)µ′28   , τ̄1g = 1 µ228‖∇f‖   −fx(µ28(κ1gµ25 + κnµ27 −µ′24) + µ24µ′28)+ fy(µ28(κ 1 gµ24 −κ2gµ26 + µ′25) −µ25µ′28)+ fz(µ28(κ 2 gµ25 + µ ′ 26) −µ26µ′28)+ fw(µ28(κnµ24 + µ ′ 27) −µ27µ′28)   , τ̄2g = 1 µ233‖∇f‖   −fx(µ33(κ1gµ30 + κnµ32 −µ′29) + µ29µ′33)+ fy(µ33(κ 1 gµ29 −κ2gµ31 + µ′30) −µ30µ′33)+ fz(µ33(κ 2 gµ30 + µ ′ 31) −µ31µ′33)+ fw(µ33(κnµ29 + µ ′ 32) −µ32µ′33)   , where µ23 = √ (κ1g −κ2g)2 + κ2n, µ24 = √ 2((κ 1 g) 2 −κ1gκ 2 g + κ 2 n)‖∇f‖ + 2σ2fx , µ25 = 2σ2fy , µ26 = √ 2κ 2 g(−κ 1 g + κ 2 g)‖∇f‖ + 2σ2fz, µ27 = 2σ2fw, µ 2 28 = (µ24) 2 + (µ25) 2 + (µ26) 2 + (µ27) 2 , µ29 = −(κ1gµ26fw −κ 2 gµ26fw −κnµ26fy + κnµ25fz −κ 1 gµ27fz + κ 2 gµ27fz), µ30 = −(κnµ26fx −κnµ25fz), µ31 = −(−κ1gµ24fw + κ 2 gµ24fw −κnµ25fx + κ 1 gµ27fx −κ 2 gµ27fx + κnµ24fy), µ32 = −(−κ1gµ26fx + κ 2 gµ26fx + κ 1 gµ24fz −κ 2 gµ24fz), µ33 = µ24 µ28 ‖∇f‖. Int. J. Anal. Appl. 17 (4) (2019) 492 3.3. PU-Smarandache curves. Definition 3.4. Let M be an oriented hypersurface in E4and the Frenet curve γ = γ(s) lying fully on M with Darboux frame {T, P, U, N} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the PU-Smarandache curve of γ is defined as γ(u) = 1 √ 2 (P + U). (3.8) Theorem 3.3. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame {T, P, U, N} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the curvature functions of the PU− Smarandache curve of r satisfy the following equations: κ̄n = −1 ν1‖∇f‖   (κ1gκn + κ 2 g(τ 1 g − τ2g ))fw + (−κ1gκ2g+ κn(τ 1 g + τ 2 g ))fx + ((κ 1 g) 2 + (κ2g) 2 + τ1g (τ 1 g + τ2g ))fy + ((κ 2 g) 2 + τ2g (τ 1 g + τ 2 g ))fz   , κ̄1g = 1 ν1ν6   (κ1gκn + κ 2 g(τ 1 g − τ2g ))ν5 + (−κ1gκ2g+ κn(τ 1 g + τ 2 g ))ν2 + ((κ 1 g) 2 + (κ2g) 2 + τ1g (τ 1 g + τ2g ))ν3 + ((κ 2 g) 2 + τ2g (τ 1 g + τ 2 g ))ν4   , κ̄2g = 1 ν26ν11   −(ν2ν7 + ν3ν8 + ν4ν9 + ν5ν10)ν′6 + ν6((κ1gν2 −κ2gν4)ν8 −ν5(κnν7 + τ1g ν8 + τ2g ν9) + ν3(−κ1gν7 + κ2gν9 + τ1g ν10)+ ν7ν ′ 2 + ν8ν ′ 3 + ν9ν ′ 4 + ν10(κnν2 + τ 2 g ν4 + ν ′ 5))   , τ̄1g = 1 ν26‖∇f‖   −(fxν2 + fyν3 + fzν4 + fwν5)ν′6 + ν6(−κ1gfxν3 + κ2gfzν3 −κnfxν5 − τ2g fzν5 + fxν′2 + fy(κ1gν2 −κ2gν4 − τ1g ν5 +ν′3) + fzν ′ 4 + fw(κnν2 + τ 1 g ν3 + τ 2 g ν4 + ν ′ 5))   , τ̄2g = 1 ν211‖∇f‖   −(fxν7 + fyν8 + fzν9 + fwν10)ν′11 + ν11(−κ1gfxν8 + κ2gfzν8 −κnfxν10 − τ2g fzν10 + fxν′7 + fy(κ1gν7 −κ2gν9 − τ1g ν10 +ν′8) + fzν ′ 9 + fw(κnν7 + τ 1 g ν8 + τ 2 g ν9 + ν ′ 10))   . (3.9) Proof. Since γ = γ(s) is a PU− Smarandache curve reference to Frenet curve r. Then, by differentiating Eq. (3.8), we get γ′ = −κ1gT + κ2gP + κ2gU + (τ1g + τ2g )N√ 2 , γ′′ = 1 √ 2 (κ1gκ 2 g −κn(τ 1 g + τ 2 g ))T + (−(κ 1 g) 2 − (κ2g) 2 − τ1g (τ 1 g + τ 2 g ))P +(−(κ2g) 2 − τ2g (τ 1 g + τ 2 g ))U + (−κ 1 gκn −κ 2 gτ 1 g + κ 2 gτ 2 g )N. (3.10) Int. J. Anal. Appl. 17 (4) (2019) 493 Using Eqs. (3.10), we have T̄ = −κ1gT + κ2gP + κ2gU + (τ1g + τ2g )N ν1 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , where ν1 = √ 2(κ2g) 2 + (τ1g + τ 2 g ) 2 + (κ1g) 2. On the other hand, we get P̄ = ν2T + ν3P + ν4U + ν5N ν6 , where ν2 = √ 2(−κ1gκ 2 g + κn(τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fx , ν3 = √ 2((κ1g) 2 + (κ2g) 2 + τ1g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fy , ν4 = √ 2((κ2g) 2 + τ2g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fz , ν5 = √ 2(κ1gκn + κ 2 gτ 1 g −κ 2 gτ 2 g )‖∇f‖ + 2σ3fw , ν26 = ν 2 2 + ν 2 3 + ν 2 4 + ν 2 5 , σ3 = 〈γ′′, N〉 = 1 2‖∇f‖   −(κ1gκn + κ2g(τ1g − τ2g ))fw + (κ1gκ2g− κn(τ 1 g + τ 2 g ))fx − ((κ1g)2 + (κ2g)2 + τ1g (τ1g + τ2g ))fy − ((κ2g)2 + τ2g (τ1g + τ2g ))fz   . Also, we get Ū = ν7T + ν8P + ν9U + ν10N ν11 , where ν7 = κ 2 gfwν3 − τ 1 g fzν3 − τ 2 g fzν3 + κ 2 gfwν4 + τ 1 g fyν4 + τ 2 g fyν4 −κ 2 g(fy + fz)ν5, ν8 = −fw(κ2gν2 + κ 1 gν4) + fz((τ 1 g + τ 2 g )ν2 + κ 1 gν5) + fx(−(τ 1 g + τ 2 g )ν4 + κ 2 gν5), ν9 = fw(−κ2gν2 + κ 1 gν3) −fy((τ 1 g + τ 2 g )ν2 + κ 1 gν5) + fx((τ 1 g + τ 2 g )ν3 + κ 2 gν5), ν10 = κ 2 gfyν2 + κ 2 gfzν2 −κ 2 gfxν3 −κ 1 gfzν3 −κ 2 gfxν4 + κ 1 gfyν4, ν11 = ν1‖∇f‖ν6. In the light of the above calculations, the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of γ are computed as in Eqs. (3.9). � Int. J. Anal. Appl. 17 (4) (2019) 494 Corollary 3.5. If r is an asymptotic curve. Then, the following equations hold: T̄ = −κ1gT −κ2gP + κ2gU + (τ1g + τ2g )N ν12 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = ν13T + ν14P + ν15U + ν16N ν17 , Ū = ν18T + ν19P + ν20U + ν21N ν22 , and the curvature functions are obtained as follows: κ̄n = 1 ν17‖∇f‖   κ2g(−τ1g + τ2g )fw + κ1gκ2gfx − ((κ1g)2 + (κ2g)2+ τ1g (τ 1 g + τ 2 g ))fy − ((κ2g)2 + τ2g (τ1g + τ2g ))fz   , κ̄1g = 1 ν12ν17   κ1gκ2gν13 − ((κ1g)2 + (κ2g)2 + τ1g (τ1g + τ2g ))ν14− ((κ2g) 2 + τ2g (τ 1 g + τ 2 g ))ν15 + κ 2 g(−τ1g + τ2g )ν16   , κ̄2g = 1 ν217ν22   ν17(κ 1 gν13ν19 −κ2gν15ν19 − τ1g ν16ν19 − τ2g ν16ν20+ τ2g ν15ν21 + ν14(−κ1gν18 + κ2gν20 + τ1g ν21) + ν18ν′13 + ν19ν′14+ ν20ν ′ 15 + ν21ν ′ 16) − (ν13ν18 + ν14ν19 + ν15ν20 + ν16ν21)ν′17   , τ̄1g = 1 ν217‖∇f‖   −fx(κ1gν14ν17 −ν17ν′13 + ν13ν′17) + fy(ν17(κ1gν13 −κ2gν15− τ1g ν16 + ν ′ 14) −ν14ν′17) + fz(ν17(κ2gν14 − τ2g ν16 + ν′15)− ν15ν ′ 17) + fw(ν17(τ 1 g ν14 + τ 2 g ν15 + ν ′ 16) −ν16ν′17)   , τ̄2g = 1 ν222‖∇f‖   −fx(κ1gν19ν22 −ν22ν′18 + ν18ν′22) + fy(ν22(κ1gν18− κ2gν20 − τ1g ν21 + ν′19) −ν19ν′22) + fz(ν22(κ2gν19 − τ2g ν21+ ν′20) −ν20ν′22) + fw(ν22(τ1g ν19 + τ2g ν20 + ν′21) −ν21ν′22)   , where ν12 = √ 2(κ2g) 2 + (τ1g + τ 2 g ) 2 + (κ1g) 2, ν13 = √ 2(−κ1gκ 2 g)‖∇f‖ + 2σ3fx, ν14 = √ 2((κ 1 g) 2 + (κ 2 g) 2 + τ 1 g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fy, ν15 = √ 2((κ 2 g) 2 + τ 2 g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fz, ν16 = √ 2(κ 2 gτ 1 g −κ 2 gτ 2 g )‖∇f‖ + 2σ3fw, ν 2 17 = ν 2 12 + ν 2 13 + ν 2 14 + ν 2 15, ν18 = −κ2gν16fy + ν15(κ 2 gfw + (τ 1 g + τ 2 g )fy) −κ 2 gν16fz + ν14(κ 2 gfw − (τ 1 g + τ 2 g )fz), ν19 = −κ2gν13fw + κ 2 gν16fx −ν15(κ 1 gfw + (τ 1 g + τ 2 g )fx) + (τ 1 g + τ 2 g )ν13fz + κ 1 gν16fz, ν20 = −κ2gν13fw + κ 2 gν16fx + ν14(κ 1 gfw + (τ 1 g + τ 2 g )fx) − τ 1 g ν13fy − τ 2 g ν13fy −κ 1 gν16fy, ν21 = κ 2 gν13fy + ν15(−κ 2 gfx + κ 1 gfy) + κ 2 gν13fz −ν14(κ 2 gfx + κ 1 gfz), ν22 = ν12 ν17 ‖∇f‖. Int. J. Anal. Appl. 17 (4) (2019) 495 Corollary 3.6. If r is a line of curvature. Then, the following equations hold: T̄ = − κ1gT + κ 2 gP −κ2gU ν12 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = ν13T + ν14P + ν15U + ν16N ν17 , Ū = ν18T + ν19P + ν20U + ν21N ν22 , and the curvature functions are obtained as follows: κ̄n = −1 ν23‖∇f‖ ( κ1gκnfw −κ 1 gκ 2 gfx + ((κ 1 g) 2 + (κ2g) 2)fy + (κ 2 g) 2fz ) , κ̄1g = −1 ν23ν28 ( −κ1gκ 2 gν24 + ((κ 1 g) 2 + (κ2g) 2)ν25 + (κ 2 g) 2ν26 + κ 1 gκnν27 ) , κ̄2g = 1 ν228ν33   ν28(−κnν27ν29 + ν25(−κ1gν29 + κ2gν31) + ν29ν′24+ ν30(κ 1 gν24 −κ2gν26 + ν′25) + ν31ν′26 + ν32(κnν24 + ν′27))− (ν24ν29 + ν25ν30 + ν26ν31 + ν27ν32)ν ′ 28   , τ̄1g = 1 ν228‖∇f‖   −fx(ν28(κ1gν25 + κnν27 −ν′24) + ν24ν′28) + fy(ν28(κ1gν24− κ2gν26 + ν ′ 25) −ν25ν′28) + fz(ν28(κ2gν25 + ν′26) −ν26ν′28)+ fw(ν28(κnν24 + ν ′ 27) −ν27ν′28)   , τ̄2g = 1 ν233‖∇f‖   −fx(ν33(κ1gν30 + κnν32 −ν′29) + ν29ν33) + fy(ν33(κ1gν29− κ2gν31 + ν ′ 30) −ν30ν′33) + fz(ν33(κ2gν30 + ν′31) −ν31ν′33)+ fw(ν33(κnν29 + ν ′ 32) −ν32ν′33)   , where ν23 = √ 2(κ2g) 2 + (κ1g) 2, ν24 = − √ 2κ1gκ 2 g‖∇f‖ + 2σ3fx, ν25 = √ 2((κ1g) 2 + (κ2g) 2)‖∇f‖ + 2σ3fy, ν26 = √ 2(κ2g) 2‖∇f‖ + 2σ3fz, ν27 = √ 2κ1gκn‖∇f‖ + 2σ3fw, ν 2 28 = ν 2 12 + ν 2 13 + ν 2 14 + ν 2 15, ν29 = κ 2 gν25fw + κ 2 gν26fw −κ 2 gν27fy −κ 2 gν27fz, ν19 = −κ2gν24fw −κ 1 gν26fw + κ 2 gν27fx + κ 1 gν27fz, ν20 = −κ2gν24fw + κ 1 gν25fw + κ 2 gν27fx −κ 1 gν27fy, ν21 = κ 2 gν24fy + ν26(−κ 2 gfx + κ 1 gfy) + κ 2 gν24fz −ν25(κ 2 gfx + κ 1 gfz), ν22 = ν23 ν28 ‖∇f‖. Int. J. Anal. Appl. 17 (4) (2019) 496 3.4. PN-Smarandache curves. Definition 3.5. Let M be an oriented hypersurface in E4and the Frenet curve δ = δ(s) lying fully on M with Darboux frame {T, P, U, N} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the PN-Smarandache curve of δ is defined as δ(u) = 1 √ 2 (P + N). (3.11) Theorem 3.4. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame {T, P, U, N} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . Then the curvature functions of the PN− Smarandache curve of r satisfy the following equations: κ̄n = −1 ξ1‖∇f‖   (κn(κ1g + κn) + (τ1g )2 + τ2g (−κ2g + τ2g ))fw + (−κ1g + κn)τ1g fx +((κ2g) 2 + κ1g(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ2g )fy + τ1g (κ2g + τ2g )fz   , κ̄1g = −1 ξ1ξ6   (κn(κ1g + κn) + (τ1g )2 + τ2g (−κ2g + τ2g ))ξ5 + (−κ1g + κn)τ1g ξ2+ ((κ2g) 2 + κ1g(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ2g )ξ3 + τ1g (κ2g + τ2g )ξ4   , κ̄2g = 1 ξ26ξ11   −(ξ2ξ7 + ξ3ξ8 + ξ4ξ9 + ξ5ξ10)ξ′6 + ξ6((κ1gξ2 −κ2gξ4)ξ8 −ξ5(κnξ7 + τ1g ξ8 + τ2g ξ9) + ξ3(−κ1gξ7 + κ2gξ9 + τ1g ξ10)+ ξ7ξ ′ 2 + ξ8ξ ′ 3 + ξ9ξ ′ 4 + ξ10(κnξ2 + τ 2 g ξ4 + ξ ′ 5))   , τ̄1g = 1 ξ26‖∇f‖   −(fxξ2 + fyξ3 + fzξ4 + fwξ5)ξ′6 + ξ6(−κ1gfxξ3 + κ2gfzξ3 −κnfxξ5 − τ2g fzξ5 + fxξ′2 + fy(κ1gξ2 −κ2gξ4 − τ1g ξ5 +ξ′3) + fzξ ′ 4 + fw(κnξ2 + τ 1 g ξ3 + τ 2 g ξ4 + ξ ′ 5))   , τ̄2g = 1 ξ211‖∇f‖   −(fxξ7 + fyξ8 + fzξ9 + fwξ10)ξ′11 + ξ11(−κ1gfxξ8+ κ2gfzξ8 −κnfxξ10 − τ2g fzξ10 + fxξ′7 + fy(κ1gξ7− κ2gξ9 − τ1g ξ10 + ξ′8) + fzξ′9 + fw(κnξ7 + τ1g ξ8 + τ2g ξ9 + ξ′10))   . (3.12) Proof. Let δ = δ(s) be a PN− Smarandache curve reference to Frenet curve r. Then, by differentiating Eq. (3.11), we obtain δ′ = −(κ1g + κn)T − τ1g P + (κ2g − τ2g )U + τ1g N√ 2 , δ′′ = 1 √ 2 (κ1g −κn)τ 1 g T + (−(κ 2 g) 2 −κ1g(κ 1 g + κn) − (τ 1 g ) 2 + κ2gτ 2 g )P −τ1g (κ 2 g + τ 2 g )U + (−κn(κ 1 g + κn) − (τ 1 g ) 2 + κ2gτ 2 g − (τ 2 g ) 2)N. (3.13) Therefore, from Eqs. (3.13), we get T̄ = −(κ1g + κn)T − τ1g P + (κ2g − τ2g )U + τ1g N ξ1 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , Int. J. Anal. Appl. 17 (4) (2019) 497 where ξ1 = √ 2(τ1g ) 2 + (κ1g + κn) 2 + (κ2g − τ2g )2. On the other hand, we obtain P̄ = ξ2T + ξ3P + ξ4U + ξ5N ξ6 , where ξ2 = √ 2(−κ1g + κn)τ 1 g‖∇f‖ + 2σ4fx , ξ3 = √ 2((κ2g) 2 + κ1g(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ 2 g )‖∇f‖ + 2σ4fy , ξ4 = √ 2τ1g (κ 2 g + τ 2 g )‖∇f‖ + 2σ4fz , ξ5 = √ 2(κn(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ 2 g + (τ 2 g ) 2)‖∇f‖ + 2σ4fw , ξ26 = ξ 2 2 + ξ 2 3 + ξ 2 4 + ξ 2 5 , σ4 = 〈δ′′, N〉 = 1 2‖∇f‖   −(κn(κ1g + κn) + (τ1g )2 + τ2g (−κ2g + τ2g ))fw+ (κ1g −κn)τ1g fx − ((κ2g)2 + κ1g(κ1g + κn)+ (τ1g ) 2 −κ2gτ2g )fy − τ1g (κ2g + τ2g )fz   . Also, we have Ū = ξ7T + ξ8P + ξ9U + ξ10N ξ11 , where, ξ7 = κ 2 gfwξ3 − τ 2 g fwξ3 − τ 1 g fzξ3 + τ 1 g fwξ4 + τ 1 g fyξ4 + (−(κ 2 g − τ 2 g )fy − τ 1 g fz)ξ5, ξ8 = fw((−κ2g + τ 2 g )ξ2 − (κ 1 g + κn)ξ4) + fz(τ 1 g ξ2 + (κ 1 g + κn)ξ5) −fx(τ 1 g ξ4 + (−κ 2 g + τ 2 g )ξ5), ξ9 = fw(−τ1g ξ2 + (κ 1 g + κn)ξ3) + τ 1 g fx(ξ3 + ξ5) −fy(τ 1 g ξ2 + (κ 1 g + κn)ξ5), ξ10 = (κ 2 gfyξ2 − τ 2 g fyξ2 + τ 1 g fzξ2 −κ 2 gfxξ3 + τ 2 g fxξ3 −κ 1 gfzξ3 −κnfzξ3 − τ1g fxξ4 + κ 1 gfyξ4 + κnfyξ4), ξ11 = ξ1‖∇f‖ξ6. In the light of the above calculations , the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of δ are computed as in Eqs. (3.12). � Int. J. Anal. Appl. 17 (4) (2019) 498 Corollary 3.7. If r be asymptotic curve. Then, the following equations hold: T̄ = −κ1gT − τ1g P + (κ2g − τ2g )U + τ1g N ξ12 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = ξ13T + ξ14P + ξ15U + ξ16N ξ17 , Ū = ξ18T + ξ19P + ξ20U + ξ21N ξ22 . Thus, the curvature functions can be computed as follows: κ̄n = −1 ‖∇f‖ξ12   ((τ1g )2 + τ2g (−κ2g + τ2g ))fw −κ1gτ1g fx + ((κ1g)2 + (κ2g)2+ (τ1g ) 2 −κ2gτ2g )fy + τ1g (κ2g + τ2g )fz   , κ̄1g = −1 ξ12ξ17   ((τ1g )2 + τ2g (−κ2g + τ2g ))ξ16 −κ1gτ1g ξ13 + ((κ1g)2+ (κ2g) 2 + (τ1g ) 2 −κ2gτ2g )ξ14 + τ1g (κ2g + τ2g )ξ15   , κ̄2g = 1 ξ217 ξ22   ξ17(κ 1 gξ13ξ19 −κ2gξ15ξ19 − τ1g ξ16ξ19 − τ2g ξ16ξ20+ τ2g ξ15ξ21 + ξ14(−κ1gξ18 + κ2gξ20 + τ1g ξ21) + ξ18ξ′13 + ξ19ξ′14+ ξ20ξ ′ 15 + ξ21ξ ′ 16) − (ξ13ξ18 + ξ14ξ19 + ξ15ξ20 + ξ16ξ21)ξ′17   , τ̄1g = 1 ξ217‖∇f‖   −fx(κ1gξ14ξ17 − ξ17ξ′13 + ξ13ξ′17) + fy(ξ17(κ1gξ13 −κ2gξ15− τ1g ξ16 + ξ ′ 14) − ξ14ξ′17) + fz(ξ17(κ2gξ14 − τ2g ξ16 + ξ′15)− ξ15ξ ′ 17) + fw(ξ17(τ 1 g ξ14 + τ 2 g ξ15 + ξ ′ 16) − ξ16ξ′17)   , τ̄2g = 1 ξ222‖∇f‖   −fx(κ1gξ19ξ22 − ξ22ξ′18 + ξ18ξ′22) + fy(ξ22(κ1gξ18 −κ2gξ20− τ1g ξ21 + ξ ′ 19) − ξ19ξ′22) + fz(ξ22(κ2gξ19 − τ2g ξ21 + ξ′20)− ξ20ξ ′ 22) + fw(ξ22(τ 1 g ξ19 + τ 2 g ξ20 + ξ ′ 21) − ξ21ξ′22)   , where ξ12 = √ 2(τ1g ) 2 + (κ1g) 2 + (κ2g − τ2g )2, ξ13 = √ 2(−κ1g)τ 1 g‖∇f‖ + 2σ4fx , ξ14 = √ 2((κ 2 g) 2 + κ 1 g(κ 1 g) + (τ 1 g ) 2 −κ2gτ 2 g )‖∇f‖ + 2σ4fy , ξ15 = √ 2τ 1 g (κ 2 g + τ 2 g )‖∇f‖ + 2σ4fz , ξ16 = √ 2((τ 1 g ) 2 −κ2gτ 2 g + (τ 2 g ) 2 )‖∇f‖ + 2σ4fw , ξ217 = ξ 2 2 + ξ 2 3 + ξ 2 4 + ξ 2 5 , ξ18 = κ 2 gfwξ14 − τ 2 g fwξ14 − τ 1 g fzξ14 + τ 1 g fwξ15 + τ 1 g fyξ15 + (−(κ 2 g − τ 2 g )fy − τ 1 g fz)ξ16, ξ19 = fw((−κ2g + τ 2 g )ξ14 − (κ 1 g)ξ15) + fz(τ 1 g ξ14 + (κ 1 g)ξ16) −fx(τ 1 g ξ15 + (−κ 2 g + τ 2 g )ξ16), ξ20 = fw(−τ1g ξ14 + (κ 1 g)ξ14) + τ 1 g fx(ξ14 + ξ16) −fy(τ 1 g ξ14 + (κ 1 g)ξ16), ξ21 = κ 2 gfyξ13 − τ 2 g fyξ13 + τ 1 g fzξ13 −κ 2 gfxξ14 + τ 2 g fxξ14 −κ 1 gfzξ14 − τ 1 g fxξ15 + κ 1 gfyξ15, ξ22 = ξ12‖∇f‖ξ17. Int. J. Anal. Appl. 17 (4) (2019) 499 Corollary 3.8. If r be line of curvature. Then, the following equations hold: T̄ = −(κ1g + κn)T + k2gU ξ23 , N̄ = fxT + fyP + fzU + fwN ‖∇f‖ , P̄ = ξ24T + ξ25P + ξ26U + ξ27N ξ28 , Ū = ξ29T + ξ30P + ξ31U + ξ32N ξ33 . Thus, the curvature functions can be computed as follows: κ̄n = − κn(κ 1 g + κn)fw + ((κ 2 g) 2 + κ1g(κ 1 g + κn))fy ‖∇f‖ξ23 , κ̄1g = − κn(κ 1 g + κn)ξ25 + ((κ 2 g) 2 + κ1g(κ 1 g + κn))ξ27 ξ23ξ28 , κ̄2g = 1 ξ228 ξ33   ξ28(−κnξ27ξ29 + ξ25(−κ1gξ29 + κ2gξ31) + ξ29ξ′24+ ξ30(κ 1 gξ24 −κ2gξ26 + ξ′25) + ξ31ξ′26 + ξ32(κnξ24+ ξ′27)) − (ξ24ξ29 + ξ25ξ30 + ξ26ξ31 + ξ27ξ32)ξ′28   , τ̄1g = 1 ξ228‖∇f‖   −fx(ξ28(κ1gξ25 + κnξ27 − ξ′24) + ξ24ξ′28) + fy(ξ28(κ1gξ24− κ2gξ26 + ξ ′ 25) − ξ25ξ′28) + fz(ξ28(κ2gξ25 + ξ′26) − ξ26ξ′28)+ fw(ξ − 28(κnξ24 + ξ′27) − ξ27ξ′28)   , τ̄2g = 1 ξ233‖∇f‖   −fx(ξ33(κ1gξ30 + κnξ32 − ξ′29) + ξ29ξ′33) + fy(ξ33(κ1gξ29− κ2gξ31 + ξ ′ 30) − ξ30ξ′33) + fz(ξ33(κ2gξ30 + ξ′31) − ξ31ξ′33)+ fw(ξ − 33(κnξ29 + ξ′32) − ξ32ξ′33)   , where ξ23 = √ (κ1g + κn) 2 − (κ2g)2, ξ24 = 2σ4fx , ξ25 = √ 2((κ2g) 2 + κ1g(κ 1 g + κn))‖∇f‖ + 2σ4fy , ξ26 = 2σ4fz , ξ27 = √ 2κn(κ 1 g + κn)‖∇f‖ + 2σ4fw , ξ 2 28 = ξ 2 24 + ξ 2 25 + ξ 2 26 + ξ 2 27 , ξ29 = (kg2ξ25fw −κ2gξ27fy), ξ30 = −κ2gξ24fw −κ 1 gξ26fw −κnξ26[u]fw + κ 2 gξ27fx + (κ 1 g + κn)ξ27fz, ξ31 = (κ 1 g + κn)ξ25fw −κ 1 gξ27fy −κnξ27fy, ξ32 = κ 2 gξ24fy + κ 1 gξ26fy + κnξ26fy − ξ25(κ 2 gfx + (κ 1 g + κn)fz), ξ33 = ξ23‖∇f‖ξ28. Int. J. Anal. Appl. 17 (4) (2019) 500 4. Examples Example 4.1. Consider the curve r(u) given by r(u) = (cos(u), sin(u), cos(2u), sin(2u)) . By using the definition 2.6, we can calculate the Darboux frame {T, P, U, N} as follows: T = ( − sin(u) 5 , cos(u) 5 ,− 2 5 sin(2u), 2 5 cos(2u) ) , N = ( cos(u) √ 2 , sin(u) √ 2 , cos(2u) √ 2 , sin(2u) √ 2 ) , P = ( cos(u) √ 2 , sin(u) √ 2 ,− cos(2u) √ 2 ,− sin(2u) √ 2 ) , U = ( 2 sin(u) √ 5 ,− 2 cos(u) √ 5 ,− 1 √ 5 sin(2u), 1 √ 5 cos(2u) ) . The curvature functions of the curve r can be computed as follows: κn = − √ 5 2 ,κ1g = 3 √ 10 , κ2g = −2 √ 2 5 ,τ1g = τ 2 g = 0. Therefore, we can obtain TU−Smarndache curve as φ = 1 √ 2 (T + U) = ( sin(u) √ 10 , −cos(u) √ 10 , −3 sin(2u) √ 10 , 3 cos(2u) √ 10 ) . By using the definition 2.6, we can obtain T̄ = 1 √ 37 (cos(u), sin(u),−6 cos(2u),−6 sin(2u)) , N̄ = 1 2 √ 5 (sin(u), cos(u), 3 sin(2u), 3 cos(2u)) , P̄ = 1 √ 16930 (17 sin(u),−17 cos(u), 129 sin(2u),−129 cos(2u)) , Ū = 1 √ 8465 (−54 cos(u),−54 sin(u),−9 cos(2u),−9 sin(2u)) . The curvature functions of the Smarandache curve φ can be computed as follows: κ̄n = −1 2 √ 37 5 , κ̄1g = 1531 √ 626410 , κ̄2g = −324 √ 2 1693 , τ̄1g = τ̄ 2 g = 0. Example 4.2. Consider the unit speed curve given by r(u) = ( cos(u) 2 , sin(u) 2 , u 2 , u √ 2 ) . By using the definition 2.6 we can calculate the Darboux frame {T, P, U, N} as follow: T = ( −sin(u) 2 , cos(u) 2 , 1 2 , 1 √ 2 ) , N = ( cos(u) 2 , sin(u) 2 , 1 2 , 1 √ 2 ) , Int. J. Anal. Appl. 17 (4) (2019) 501 P = ( − √ 3 2 cos(u), − √ 3 2 sin(u), 1 2 √ 3 , −1 √ 6 ) , U = ( − √ 2 3 sin(u), √ 2 3 cos(u), −1 √ 6 ,− 1 2 √ 3 ) , The curvature functions of the curve r can be computed as follows κn = − 1 4 , κ1g = √ 3 4 , κ2g = − 1 √ 2 ,τ1g = 0, τ 2 g = − 1 √ 6 . Therefore we can obtain TN−Smarndache curve as β = ( cos(u) − sin(u) 2 √ 2 , cos(u) + sin(u) 2 √ 2 , 1 √ 2 , 0 ) . By using the definition 2.6, we can obtain T̄ = ( − cos(u) + sin(u) 2 √ 2 , cos(u) − sin(u) 2 √ 2 , 0, 0 ) , N̄ = ( cos(u) − sin(u) 2 √ 2 , cos(u) + sin(u) 2 √ 2 , 1 2 , 1 2 ) , P̄ = ( −3(cos(u) − sin(u)) √ 22 , 3(cos(u) + sin(u)) √ 22 , 1 √ 11 , 1 √ 11 ) , Ū = ( 0, 0, 1 √ 11 , 1 √ 11 ) . The curvature functions of the Smarandache curve β can be computed as κ̄n = −1 2 , κ̄1g = 3 √ 11 , κ̄2g = 0, τ̄ 1 g = τ̄ 2 g = 0. 5. Conclusion In the four dimensional Euclidean space E4, some special Smarandache curves lying on a hypersurface are investigated. 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