Computing Special Smarandache Curves According to Darboux Frame in Euclidean 4-Space

Home , Darboux frame

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 [email protected], [email protected]

Abstract. In this paper, we study some special Smarandache curves and their diﬀerential geometric prop-

4 erties according to Darboux frame in Euclidean 4-space E . Also, we compute some of these curves which 4 lie fully on a hypersurface in E . 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 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. Düldüet 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

1 E4. 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

3 frame in the three-dimensional Minkowski space E1. 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

4 4 4 Euclidean 4-space E [4, 13–15]. Let {ei | i = 1, 2, 3, 4} be the standard basis of E , therefore E = {X = P4 4 i=1 xiei | xi ∈ R}. The scalar product and vector product of vectors X,Y,Z ∈ E are respectively deﬁned by 4 X hX,Y i = xiyi , i=1

e e e e 1 2 3 4

x1 x2 x3 x4 X × Y × Z = .

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 −1 4 of f, then f (c) is a regular surface in E . The implicit surface f is regular if ∇f = (fx, fy, fz, fw) 6= 0. ∇f The unit surface normal vector of the implicit surface f is given by N = k∇fk .

Definition 2.2. Suppose that f : Rn → R, if all second partial derivatives of f exist and are continuous on ∂2f the domain of f, then the Hessian matrix Hf is a square n × n matrix, usually defined as Hij = . ∂xi∂xj 2.1. Curves on a hypersurface in E4. Let r : I ⊂ R → M be a regular curve in E4, where kr0k = kdr/duk = 1, ∀ u ∈ I , we will omit u for simplification. Then the Frenet frame is defined by

 0      t 0 κ1 0 0 t        n0   −κ 0 κ 0   n     1 2    0 d   =     , = (2.1)  0      du  b1   0 −κ2 0 κ3   b1        0 b2 0 0 κ3 0 b2

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

0 0 00 0 00 0 r r × r × r(3) b × r × r b × b × r t = , b = , b = 2 , n = 1 2 , (2.2) 0 2 0 00 (3) 1 0 00 0 kr k kr × r × r k kb2 × r × r k kb1 × b2 × r k

and

00 (3) (4) hn, r i hb1, r i hb2, r i κ1 = 0 2 , κ2 = 0 3 , κ3 = 0 4 . (2.3) kr k kr k κ1 kr k κ1κ2

Deﬁnition 2.3. Let r be a regular curve in E4, which is parameterized by arc length and n-times continuously 0 00 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, r00} is linearly independent,

r0 ∇f r00−(r00.N)N therefore T = kr0k , N = k∇fk , P = kr00−(r00.N)Nk , 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]

 0   1    T 0 κg 0 κn T        0   1 2 1     P   −κg 0 κg τg   P    =     , (2.4)  0   2 2     U   0 −κg 0 τg   U        0 1 2 N −κn −τg −τg 0 N

1 2 1 2 where κn, κg, κg, τg and τg are real valued functions denote the normal curvature, geodesic curvatures and the geodesic torsions, respectively. These functions are given by

1 00 −1 κ = hT0, Ni = hr , ∇f i = (r0H (r0)t), n k∇fk k∇fk f

1 2 00 00 1 0 0 κ1 = hT0, Pi = r (r )t − (r H (r )t)2 , g k∇fk2 f 2 0 1 1 0 0 t κg = hP , Ui = 1 2 m1 + 2 (r Hf (r ) ) m2 , (κg) ∇f k∇fk 1 0 −1 1 0 00 t 1 0 t 0 0 t τg = hP , Ni = 1 (r Hf (r ) ) + 3 (r Hf (∇f) )(r Hf (r ) ) , (κg) k∇fk k∇fk 2 0 −1 τg = hU , Ni = 1 2 m3, (2.5) (κg)k∇fk where

0 0 0 0 0 0 0 0 0 0 0 0 x y z w x y z w x y z w

00 00 00 00 00 00 00 00 x y z w x y z w fx fy fz fw m1 = , m2 = , m3 = , (3) (3) (3) (3) x y z w a b c d a b c d

fx fy fz fw fx fy fz fw p q t v and

0 0 0 00 00 0 dH [a, b, c, d] = (∇f) = r H , [p, q, t, v] = (∇f) = α H + α f , f f du

dH ∂H 0 ∂H 0 f = [ f (r )t ... f (r )t]. du ∂x ∂w

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.

1 2 • r is called a line of curvature if and only if τg = τg = 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 1 2 1 2 Darboux frame field of first kind along r(u) and κn, κg, κg, τg , τg 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

∇f N¯ = . (3.1) k∇fk

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 1 2 1 2 with Darboux frame {T, P, U, N} and non-zero curvatures κn, κg, κg, τg and τg . Then the TP-Smarandache curve of r is deﬁned as

1 α(s) = √ (T + P). (3.2) 2

Theorem 3.1. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame 1 2 1 2 {T, P, U, N} and non-zero constant curvatures κn, κg, κg, τg and τg . Then the curvature functions of the TP− Smarandache curve of r satisfy the following equations:

 1 1 2 2 1 2  (κg(−κn + τg ) + κgτg )fw − ((κg) + 1   κ¯ =  1 1 2 2 2  , n  κn(κn + τg ))fx − ((κg) + (κg) +  λ1k∇fk   1 1 1 2 1 2 τg (κn + τg ))fy + (κgκg − (κn + τg )τg )fz

 1 1 2 2 1 2  (κg(−κn + τg ) + κgτg )λ5 + ((κg) + 1   κ¯1 =  1 1 2 2 2  , g  κn(κn + τg ))λ2 + ((κg) + (κg) +  λ1λ6   1 1 1 2 1 2 τg (κn + τg ))λ3 + (κgκg − (κn + τg )τg )λ4

 0  −(λ2λ7 + λ3λ8 + λ4λ9 + λ5λ10)λ6+   1 2 1  λ6((κ λ2 − κ λ4)λ8 − λ5(κnλ7 + τ λ8+  ¯2 1  g g g  κg = 2   , λ6λ11  2 1 2 1 0   τg λ9) + λ3(−κgλ7 + κgλ9 + τg λ10) + λ7λ2+    0 0 2 0 λ8λ3 + λ9λ4 + λ10(κnλ2 + τg λ4 + λ5)) Int. J. Anal. Appl. 17 (4) (2019) 484

 0  −(fxλ2 + fyλ3 + fzλ4 + fwλ5)λ6+   1 2  λ6(−κ fxλ3 + κ fzλ3 − κnfxλ5−  ¯1 1  g g  τg = 2   , λ6k∇fk  2 0 1 2 1   τg fzλ5 + fxλ2 + fy(κgλ2 − κgλ4 − τg λ5+    0 0 1 2 0 λ3) + fzλ4 + fw(κnλ2 + τg λ3 + τg λ4 + λ5))

 0  −(fxλ7 + fyλ8 + fzλ9 + fwλ10)λ11+   1 2  λ11(−κ fxλ8 + κ fzλ8 − κnfxλ10−  ¯2 1  g g  τg = 2   . (3.3) λ11k∇fk  2 0 1 2 1   τg fzλ10 + fxλ7 + fy(κgλ7 − κgλ9 − τg λ10+    0 0 1 2 0 λ8) + fzλ9 + fw(κnλ7 + τg λ8 + τg λ9 + λ10))

Proof. Because α = α(s) is a TP− Smarandache curve reference to Frenet curve r, then diﬀerentiating Eq. (3.2), we get

0 1 1 1 2 1 α = √ −κ T + κ P + κ U + (τ + κn)N . (3.4) 2 g g g g Again, diﬀerentiating Eq. (3.4), we obtain   (−(κ1)2 − κ (κ + τ 1))T − ((κ1)2 + (κ2)2 + τ 1(κ + τ 1))P 00 1 g n n g g g g n g α = √   . 2 1 2 1 2 1 1 1 2 2 +(κgκg − (κn + τg )τg )U + (−κgκn + κgτg + κgτg )N

Also, Eq. (3.4) leads to 1 1 2 1 −κ T + κ P + κ U + (τ + κn)N T¯ = g g g g , λ1 where q 1 2 2 2 1 2 λ1 = 2(κg) + (κg) + (τg + κn) ,

and we get f T + f P + f U + f N N¯ = x y z w . k∇fk On the other hand, we have λ T + λ P + λ U + λ N P¯ = 2 3 4 5 , λ6 where

1 1 2 2 1 σ1 λ2 = √ (κ ) + (κn) + κnτ + fx , 2 g g k∇fk

1 1 2 2 2 1 1 2 σ1 λ3 = √ (κ ) + (κ ) + κnτ + (τ ) + fy , 2 g g g g k∇fk

1 1 2 2 1 2 σ1 λ4 = √ −κ κ + κnτ + τ τ + fz , 2 g g g g g k∇fk

1 1 1 1 2 2 σ1 λ5 = √ κ κn − κ τ − κ τ + fw , 2 g g g g g k∇fk

p 2 2 2 2 λ6 = (λ2) + (λ3) + (λ4) + (λ5) , Int. J. Anal. Appl. 17 (4) (2019) 485

 1 2 1 1 2 2 2  (−(κg) − κn(κn + τg ))fx − ((κg) + (κg) + 1   σ = hα00, Ni = √  1 1 1 2 1 2  . 1  τg (κn + τg ))fy + (κgκg − (κn + τg )τg )fz+  2k∇fk   1 1 1 2 2 (−κgκn + κgτg + κgτg )fw

Also, we get λ T + λ P + λ U + λ N U¯ = 7 8 9 10 , λ11 where,

  2 1 1 κgfwλ3 − κnfzλ3 − τg fzλ3 − κgfwλ4+ λ7 =   , 1 2 1 κnfyλ4 + τg fyλ4 − κgfyλ5 + κgfzλ5   2 1 1 −κgfwλ2 + κnfzλ2 + τg fzλ2 − κgfwλ4− λ8 =   , 1 2 1 κnfxλ4 − τg fxλ4 + κgfxλ5 + κgfzλ5   1 1 1 κgfwλ2 − κnfyλ2 − τg fyλ2 + κgfwλ3+ λ9 =   , 1 1 1 κnfxλ3 + τg fxλ3 − κgFxλ5 − κgFyλ5 2 1 2 1 1 1 λ10 = κgfyλ2 − κgfzλ2 − κgfxλ3 − κgfzλ3 + κgfxλ4 + κgfyλ4 ,

λ11 = λ1 λ6 k∇fk.

¯1 ¯2 ¯1 ¯2 In the light of the above calculations, the curvature functionsκ ¯n, κg, κg, τg and τg of α are computed as in

Eqs. (3.3).

Corollary 3.1. If r is an asymptotic curve. Then, the following equations hold:

−κ1T + κ1P + κ2U + τ 1N f T + f P + f U + f N T¯ = g g g g , N¯ = x y z w , λ1 k∇fk λ T + λ P + λ U + λ N λ T + λ P + λ U + λ N P¯ = 13 14 15 16 , U¯ = 18 19 20 21 , λ17 λ22 and then the curvature functions are computed as follows

1 1 2 2 1 2 1 2 2 2 1 2 1 2 1 2 (κgτg + κgτg )fw − (κg) fx − ((κg) + (κg) + (τg ) )fy + (κgκg − τg τg )fz κ¯n = , λ12k∇fk (κ1τ 1 + κ2τ 2)λ − (κ1)2λ − ((κ1)2 + (κ2)2 + (τ 1)2)λ + (κ1κ2 − τ 1τ 2)λ ¯1 g g g g 16 g 13 g g g 14 g g g g 15 κg = , λ12λ17

 1 2 1  λ17(κgλ13λ19 − κgλ15λ19 − τg λ16λ19−   2 1 2 0  τ λ16λ20 + λ14(−κ λ18 + κ λ20) + λ18λ +  ¯2 1  g g g 13  κg = 2   , λ22λ17  0 0 1 2   λ19λ14 + λ20λ15 + λ21(τg λ14 + τg λ15+    0 0 λ16)) − (λ21λ16 + λ13λ18 + λ14λ19 + λ15λ20)λ17 Int. J. Anal. Appl. 17 (4) (2019) 486

 1 0 0 1  −fx(κgλ14λ17 − λ17λ13 + λ13λ17) + fy(λ17(κgλ13−   2 1 0 0  κ λ15 − τ λ16 + λ ) − λ14λ )+  ¯1 1  g g 14 17  τg = 2   , λ17k∇fk  2 2 0 0   fz(λ17(κgλ14 − τg λ16 + λ15) − λ15λ17)+    1 2 0 0 fw(λ17(τg λ14 + τg λ15 + λ16) − λ16λ17)

 0 1  −(λ21fw + λ18fx + λ19fy + λ20fz)λ22 + λ22((−τg λ21+ 1   τ¯2 =  1 2 2 1 1  , g 2  κgλ18)fy + λ20(τg fw − κgfy) + λ19(τg fw − κgfx+  λ22k∇fk   2 0 0 0 2 0 κgfz) + fwλ21 + fxλ18 + fyλ19 + fz(−τg λ21 + λ20)) where

q 1 2 2 2 1 2 λ12 = 2(κg) + (κg) + (τg ) , √ 1 2 λ13 = 2(κg) k∇fk + 2σ1fx, √ 1 2 2 2 1 2 λ14 = 2((κg) + (κg) + (τg ) )k∇fk + 2σ1fy, √ 1 2 1 2 λ15 = 2(−κgκg + τg τg )k∇fk + 2σ1fz, √ 1 1 2 2 λ16 = 2(−κgτg − κgτg )k∇fk + 2σ1fw,

p 2 2 2 2 λ17 = (λ13) + (λ14) + (λ15) + (λ16) ,

2 1 1 1 2 1 λ18 = κgfwλ14 − τg fzλ14 − κgfwλ15 + τg fyλ15 − κgfyλ16 + κgfzλ16,

2 1 1 1 2 1 λ19 = −κgfwλ13 + τg fzλ13 − κgfwλ15 − τg fxλ15 + κgfxλ16 + κgfzλ16,

1 1 1 1 1 1 λ20 = kgλ13fw + kgλ14fw + τg λ14fx − kgλ16fx − τg λ13fy − kgλ16fy,

2 1 2 1 1 1 λ21 = κgfyλ13 − κgfzλ13 − κgfxλ14 − κgfzλ14 + κgfxλ15 + κgfyλ15,

λ22 = λ12 λ17 k∇fk,   1 2 1 2 2 2 1 2 1 −(κg) fx − ((κg) + (κg) + (τg ) )fy+ σ1 = √   . 2k∇fk 1 2 1 2 1 1 2 2 (κgκg − τg τg )fz + (κgτg + κgτg )fw

Corollary 3.2. If r is a line of curvature. Then, the following equations hold:

1 1 2 −κ T + κ P + κ U + κnN f T + f P + f U + f N T¯ = g g g , N¯ = x y z w , λ1 k∇fk λ T + λ P + λ U + λ N λ T + λ P + λ U + λ N P¯ = 24 25 26 27 , U¯ = 29 30 31 32 , λ28 λ33

and then the curvature functions are computed as follows

1 1 2 2 1 2 2 2 1 2 κgκnfw + ((κg) + κn)fx + ((κg) + (κg) )fy − κgκgfz κ¯n = − , λ23k∇fk κ1κ λ + ((κ1)2 + κ2 )λ + ((κ1)2 + (κ2)2)λ − κ1κ2λ ¯1 g n 27 g n 24 g g 25 g g 26 κg = − , λ23λ28 Int. J. Anal. Appl. 17 (4) (2019) 487

 1 2 0  λ28(λ25(−κgλ29 + κgλ31) + λ29(−κnλ27 + λ24)+ 1   κ¯2 =  1 2 0 0  , g 2  λ30(κgλ24 − κgλ26 + λ25) + λ31λ26 + λ32(κnλ24+  λ33λ28   0 0 λ27)) − (λ32λ27 + λ24λ29 + λ25λ30 + λ26λ31)λ28

 1 0 0  −fx(λ28(κgλ25 + κnλ27 − λ24) + λ24λ28)+   1 2 0 0  fy(λ28(κ λ24 − κ λ26 + λ ) − λ25λ )+  ¯1 1  g g 25 28  τg = 2   , λ28k∇fk  2 0 0   fz(λ28(κgλ25 + λ26) − λ26λ28)+    0 0 fw(λ28(κnλ24 + λ27) − λ27λ28)

 0  −(λ32fw + λ29fx + λ30fy + λ31fz)λ33+   1  λ33(−κnλ32fx − κ λ30fx+  ¯2 1  g  τg = 2   , λ33k∇fk  1 2 0   λ29(κnfw + κgfy) + κgλ30fz + fwλ32+    0 2 0 0 fxλ29 + fy(−κgλ31 + λ30) + fzλ31) where

q √ 1 2 2 2 2 1 2 2 2 λ23 = 2(κg) + (κg) + (κn) , λ24 = 2((κg) + κn) k∇fk + 2σ1fx, √ √ 1 2 2 2 1 2 λ25 = 2((κg) + (κg) )k∇fk + 2σ1fy, λ26 = − 2κgκgk∇fk + 2σ1fz, √ 1 p 2 2 2 2 λ27 = 2κgκnk∇fk + 2σ1fw, λ28 = (λ24) + (λ25) + (λ26) + (λ27) ,

2 1 2 1 λ29 = κgfwλ25 − κnfzλ25 − κgfwλ26 + κnfyλ26 − κgfyλ27 + κgfzλ27,

2 1 2 1 λ30 = −κgfwλ24 + κnfzλ24 − κgfwλ26 − κnfxλ26 + κgfxλ27 + κgfzλ27,

1 1 1 1 λ31 = κgfwλ24 − κnfyλ24 + κgfwλ25 + κnfxλ25 − κgFxλ27 − κgFyλ27,

2 1 2 1 1 1 λ32 = κgfyλ24 − κgfzλ24 − κgfxλ25 − κgfzλ25 + κgfxλ26 + κgfyλ26,

λ33 = λ23 λ28 k∇fk,

1 1 1 2 2 1 2 2 2 1 2 σ1 = √ (−κ κnfw − ((κ ) + κ )fx − ((κ ) + (κ ) )fy + κ κ fz). 2k∇fk g g n g g g g

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 1 2 1 2 with Darboux frame {T, P, U, N} and non-zero curvatures κn, κg, κg, τg and τg . Then the TU-Smarandache curve of r is deﬁned as 1 β(u) = √ (T + U). (3.5) 2

Theorem 3.2. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame 1 2 1 2 {T, P, U, N} and non-zero constant curvatures; κn, κg, κg, τg and τg . Then the curvature functions of the Int. J. Anal. Appl. 17 (4) (2019) 488

TU− Smarandache curve of r satisfy the following equations:   1 2 1 1 1 2 2 1 (κg − κg)τg fw − (κg(κg − κg) + κn(κn + τg ))fx+ κ¯n =   , µ1k∇fk 1 2 2 2 1 2 (κg − κg)κgfy − (κn + τg )(τg fy + τg fz)   1 2 1 1 1 2 2 (κ − κ )τ µ5 − (κ (κ − κ ) + κn(κn + τ ))µ2+ ¯1 1 g g g g g g g κg =   , µ1µ6 1 2 2 2 1 2 (κg − κg)κgµ3 − (κn + τg )(τg fy + τg µ4)

 0 1  −(µ2µ7 + µ3µ8 + µ4µ9 + µ5µ10)µ6 + µ6((κgµ2−   2 1 2 1  κ µ4)µ8 − µ5(κnµ7 + τ µ8 + τ µ9) + µ3(−κ µ7+  ¯2 1  g g g g  κg = 2   , µ6µ11  2 1 0 0   κgµ9 + τg µ10) + µ7µ2 + µ8µ3+    0 2 0 µ9µ4 + µ10(κnµ2 + τg µ4 + µ5))

 0 0  −(fxµ2 + fyµ3 + fzµ4 + fwµ5)µ6 + µ6(fxµ2   1 2 2  −κ fxµ3 + κ fzµ3 − κnfxµ5 − τ fzµ5+  ¯1 1  g g g  τg = 2   , µ6k∇fk  1 2 1 0 0   fy(κgµ2 − κgµ4 − τg µ5 + µ3) + fzµ4+    1 2 0 fw(κnµ2 + τg µ3 + τg µ4 + µ5))

 0 0  −(fxµ7 + fyµ8 + fzµ9 + fwµ10)µ11 + µ11(fxµ7   1 2 2  −κ fxµ8 + κ fzµ8 − κnfxµ10 − τ fzµ10+  ¯2 1  g g g  τg = 2   . (3.6) µ11k∇fk  1 2 1 0   fy(κgµ7 − κgµ9 − τg µ10 + µ8)+    0 1 2 0 fzµ9 + fw(κnµ7 + τg µ8 + τg µ9 + µ10))

Proof. Since β = β(s) is a TU− Smarandache curve reference to Frenet curve r. Then, by diﬀerentiating Eq. (3.5), we get

0 1 1 2 2 β = √ (κ − κ )P + (κn + τ )U , (3.7) 2 g g g Again, by diﬀerentiating Eq. (3.7), we have   (−κ1(κ1 − κ2) − κ (κ + τ 2))T − τ 1(κ + τ 2)P 00 1 g g g n n g g n g β = √   . 2 1 2 2 2 2 1 2 1 +((κg − κg)κg − (κn + τg )τg )U + (κg − κg)τg N

Also, from Eq. (3.7), we obtain 1 2 2 (κ − κ )P + (κn + τ )U T¯ = g g g , µ1 where q 1 2 2 2 2 µ1 = 2(κg − κg) + (κn + τg ) ,

and we have f T + f P + f U + f N N¯ = x y z w , k∇fk µ T + µ P + µ U + µ N P¯ = 2 3 4 5 , µ6 Int. J. Anal. Appl. 17 (4) (2019) 489 where

1 1 2 1 2 2 σ2 µ2 = √ ((κ ) − κ κ + κn(κn + τ )) + fx , 2 g g g g k∇fk

1 1 2 σ2 µ3 = √ τ (κn + τ ) + fy , 2 g g k∇fk

1 2 1 2 2 2 σ2 µ4 = √ (κ (−κ + κ ) + τ (κn + τ )) + fz , 2 g g g g g k∇fk

1 1 2 1 σ2 µ5 = √ (−κ + κ )τ + fw , 2 g g g k∇fk

p 2 2 2 2 µ6 = (µ2) + (µ3) + (µ4) + (µ5)

  (−(κ1)2 + κ1κ2 − κ (κ + τ 2))f + τ 1(κ + τ 2)f 00 1 g g g n n g x g n g y σ2 = hβ , Ni = √   . 2k∇fk 1 2 2 2 2 1 2 1 +((κg − κg)κg − τg (κn + τg ))fz + (κg − κg)τg fw

Also, we get

µ T + µ P + µ U + µ N U¯ = 7 8 3 10 , δ where

2 1 2 2 1 2 µ7 = −κnfzµ3 − τg fzµ3 − κgfwµ4 + κgfwµ4 + κnfyµ4 + τg fyµ4 + κgfzµ5 − κgfzµ5,

2 2 µ8 = κnfzµ2 + τg fzµ2 − κnfxµ4 − τg fxµ4,

1 2 2 2 1 2 µ9 = κgfwµ2 − κgfwµ2 − κnfyµ2 − τg fyµ2 + κnfxµ3 + τg fxµ3 − κgfxµ5 + κgfxµ5,

1 2 1 2 µ10 = −κgfzµ2 + κgfzµ2 + κgfxµ4 − κgfxµ4,

µ11 = µ1 µ6 k∇fk.

¯1 ¯2 ¯1 ¯2 In the light of the above calculations, the curvature functionsκ ¯n, κg, κg, τg and τg of β are computed as in Eqs. (3.6).

Corollary 3.3. If r is an asymptotic curve. Then, the following equations hold:

(κ1 − κ2)P + τ 2U T¯ = g g g , µ1 f T + f P + f U + f N N¯ = x y z w , k∇fk µ T + µ P + µ U + µ N P¯ = 13 14 15 16 , µ17 µ T + µ P + µ U + µ N U¯ = 18 19 20 21 , µ22 Int. J. Anal. Appl. 17 (4) (2019) 490

and the curvature functions are obtained as follows:

1 2 1 1 1 2 1 2 1 2 2 2 2 2 (κg − κg)τg fw + κg(−κg + κg)fx − τg τg fy − (−κgκg + (κg) + (τg ) )fz κ¯n = , µ12k∇fk κ1(−κ1 + κ2)µ + (κ1 − κ2)κ2µ − τ 2(τ 1µ + τ 2µ ) + (κ1 − κ2)τ 1µ ¯1 g g g 13 g g g 15 g g 14 g 15 g g g 16 κg = , µ12µ17

 1 2 1 2  µ17(κgµ13µ19 − κgµ15µ19 − τg µ16µ19 − τg µ16µ20+ 1   κ¯2 =  2 1 2 1 0 0  , g 2  τg µ15µ21 + µ14(−κgµ18 + κgµ20 + τg µ21) + µ18µ13 + µ19µ14+  µ17µ22   0 0 0 µ20µ15 + µ21µ16) − (µ13µ18 + µ14µ19 + µ15µ20 + µ16µ21)µ17

 1 0 0 1  −fx(κgµ14µ17 − µ17µ13 + µ13µ17) + fy(µ17(κgµ13− 1   τ¯1 =  2 1 0 0 2 2  , g 2  κgµ15 − τg µ16 + µ14) − µ14µ17) + fz(µ17(κgµ14 − τg µ16+  µ17k∇fk   0 0 1 2 0 0 µ15) − µ15µ17) + fw(µ17(τg µ14 + τg µ15 + µ16) − µ16µ17)

 1 0 0 1 2  −fx(κgµ19µ22 − µ22µ18 + µ18µ22) + fy(µ22(κgµ18 − κgµ20− 1   τ¯2 =  1 0 0 2 2 0  , g 2  τg µ21 + µ19) − µ19µ22) + fz(µ22(κgµ19 − τg µ21 + µ20)−  µ22k∇fk   0 1 2 0 0 µ20µ22) + fw(µ22(τg µ19 + τg µ20 + µ21) − µ21µ22)

where

q 1 2 2 2 2 µ12 = (κg − κg) + (τg ) , √ 1 2 1 2 µ13 = 2((κg) − κgκg)k∇fk + 2σ2fx , √ 1 2 µ14 = 2τg τg k∇fk + 2σ2fy , √ 2 1 2 2 2 µ15 = 2(κg(−κg + κg) + (τg ) )k∇fk + 2σ2fz, √ 1 2 1 µ16 = 2(−κg + κg)τg k∇fk + 2σ2fw,

2 2 2 2 2 µ17 = (µ13) + (µ14) + (µ15) + (µ16) ,

1 2 2 2 1 2 µ18 = −(κgµ15fw − κgµ15fw − τg µ15fy + τg µ14fz − κgµ16fz + κgµ16fz),

2 2 µ19 = −(τg µ15fx − τg µ13fz),

1 2 2 1 2 2 µ20 = −(−κgµ13fw + κgµ13fw − τg µ14fx + κgµ16fx − κgµ16fx + τg µ13fy),

1 2 1 2 µ21 = −(−κgµ15fx + κgµ15fx + κgµ13fz − κgµ13fz),

µ22 = µ12 µ17 k∇fk. Int. J. Anal. Appl. 17 (4) (2019) 491

Corollary 3.4. If r is a line of curvature curve. Then, the following equations hold:

(κ1 − κ2)P + κ N T¯ = g g n , µ23 f T + f P + f U + f N N¯ = x y z w , k∇fk µ T + µ P + µ U + µ N P¯ = 24 25 26 27 , µ28 µ T + µ P + µ U + µ N U¯ = 29 30 31 32 , µ33

and the curvature functions are obtained as follows:

1 2 1 2 2 1 2 2 −((κg) − κgκg + κn)fx + (κg − κg)κgfz κ¯n = , µ23k∇fk −((κ1)2 − κ1κ2 + κ2 )µ + (κ1 − κ2)κ2µ ¯1 g g g n 24 g g g 26 κg = , µ23µ28

 1 2 0  µ28(−κnµ27µ29 + µ25(−κgµ29 + κgµ31) + µ29µ24+ 1   κ¯2 =  1 2 0 0  , g 2  µ30(κgµ24 − κgµ26 + µ25) + µ31µ26 + µ32(κnµ24+  µ17µ22   0 0 µ27)) − (µ24µ29 + µ25µ30 + µ26µ31 + µ27µ32)µ28

 1 0 0  −fx(µ28(κgµ25 + κnµ27 − µ24) + µ24µ28)+   1 2 0 0  fy(µ28(κ µ24 − κ µ26 + µ ) − µ25µ )+  ¯1 1  g g 25 28  τg = 2   , µ28k∇fk  2 0 0   fz(µ28(κgµ25 + µ26) − µ26µ28)+    0 0 fw(µ28(κnµ24 + µ27) − µ27µ28)

 1 0 0  −fx(µ33(κgµ30 + κnµ32 − µ29) + µ29µ33)+   1 2 0 0  fy(µ33(κ µ29 − κ µ31 + µ ) − µ30µ )+  ¯2 1  g g 30 33  τg = 2   , µ33k∇fk  2 0 0   fz(µ33(κgµ30 + µ31) − µ31µ33)+    0 0 fw(µ33(κnµ29 + µ32) − µ32µ33) where q √ 1 2 2 2 1 2 1 2 2 µ23 = (κg − κg) + κn, µ24 = 2((κg) − κgκg + κn)k∇fk + 2σ2fx ,

√ 2 1 2 µ25 = 2σ2fy , µ26 = 2κg(−κg + κg)k∇fk + 2σ2fz,

2 2 2 2 2 µ27 = 2σ2fw, µ28 = (µ24) + (µ25) + (µ26) + (µ27) ,

1 2 1 2 µ29 = −(κgµ26fw − κgµ26fw − κnµ26fy + κnµ25fz − κgµ27fz + κgµ27fz),

µ30 = −(κnµ26fx − κnµ25fz),

1 2 1 2 µ31 = −(−κgµ24fw + κgµ24fw − κnµ25fx + κgµ27fx − κgµ27fx + κnµ24fy),

1 2 1 2 µ32 = −(−κgµ26fx + κgµ26fx + κgµ24fz − κgµ24fz),

µ33 = µ24 µ28 k∇fk. 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 1 2 1 2 with Darboux frame {T, P, U, N} and non-zero curvatures κn, κg, κg, τg and τg . Then the PU-Smarandache curve of γ is deﬁned as

1 γ(u) = √ (P + U). (3.8) 2

Theorem 3.3. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame 1 2 1 2 {T, P, U, N} and non-zero curvatures κn, κg, κg, τg and τg . Then the curvature functions of the PU− Smarandache curve of r satisfy the following equations:

 1 2 1 2 1 2  (κgκn + κg(τg − τg ))fw + (−κgκg+ −1   κ¯ =  1 2 1 2 2 2 1 1  , n  κn(τg + τg ))fx + ((κg) + (κg) + τg (τg +  ν1k∇fk   2 2 2 2 1 2 τg ))fy + ((κg) + τg (τg + τg ))fz

 1 2 1 2 1 2  (κgκn + κg(τg − τg ))ν5 + (−κgκg+ 1   κ¯1 =  1 2 1 2 2 2 1 1  , g  κn(τg + τg ))ν2 + ((κg) + (κg) + τg (τg +  ν1ν6   2 2 2 2 1 2 τg ))ν3 + ((κg) + τg (τg + τg ))ν4

 0 1 2  −(ν2ν7 + ν3ν8 + ν4ν9 + ν5ν10)ν6 + ν6((κgν2 − κgν4)ν8 1   κ¯2 =  1 2 1 2 1  , g 2  −ν5(κnν7 + τg ν8 + τg ν9) + ν3(−κgν7 + κgν9 + τg ν10)+  ν6 ν11   0 0 0 2 0 ν7ν2 + ν8ν3 + ν9ν4 + ν10(κnν2 + τg ν4 + ν5))

 0 1 2  −(fxν2 + fyν3 + fzν4 + fwν5)ν6 + ν6(−κgfxν3 + κgfzν3 1   τ¯1 =  2 0 1 2 1  , g 2  −κnfxν5 − τg fzν5 + fxν2 + fy(κgν2 − κgν4 − τg ν5  ν6 k∇fk   0 0 1 2 0 +ν3) + fzν4 + fw(κnν2 + τg ν3 + τg ν4 + ν5))

 0 1 2  −(fxν7 + fyν8 + fzν9 + fwν10)ν11 + ν11(−κgfxν8 + κgfzν8 1   τ¯2 =  2 0 1 2 1  . (3.9) g 2  −κnfxν10 − τg fzν10 + fxν7 + fy(κgν7 − κgν9 − τg ν10  ν11k∇fk   0 0 1 2 0 +ν8) + fzν9 + fw(κnν7 + τg ν8 + τg ν9 + ν10))

Proof. Since γ = γ(s) is a PU− Smarandache curve reference to Frenet curve r. Then, by diﬀerentiating Eq. (3.8), we get

−κ1T + κ2P + κ2U + (τ 1 + τ 2)N γ0 = g g √g g g , 2

00 1 1 2 1 2 1 2 2 2 1 1 2 γ = √ (κ κ − κn(τ + τ ))T + (−(κ ) − (κ ) − τ (τ + τ ))P 2 g g g g g g g g g 2 2 2 1 2 1 2 1 2 2 +(−(κg) − τg (τg + τg ))U + (−κgκn − κgτg + κgτg )N. (3.10) Int. J. Anal. Appl. 17 (4) (2019) 493

Using Eqs. (3.10), we have

−κ1T + κ2P + κ2U + (τ 1 + τ 2)N T¯ = g g g g g , ν1 f T + f P + f U + f N N¯ = x y z w , k∇fk

where q 2 2 1 2 2 1 2 ν1 = 2(κg) + (τg + τg ) + (κg) .

On the other hand, we get ν T + ν P + ν U + ν N P¯ = 2 3 4 5 , ν6

where

√ 1 2 1 2 ν2 = 2(−κgκg + κn(τg + τg ))k∇fk + 2σ3fx , √ 1 2 2 2 1 1 2 ν3 = 2((κg) + (κg) + τg (τg + τg ))k∇fk + 2σ3fy , √ 2 2 2 1 2 ν4 = 2((κg) + τg (τg + τg ))k∇fk + 2σ3fz , √ 1 2 1 2 2 ν5 = 2(κgκn + κgτg − κgτg )k∇fk + 2σ3fw ,

2 2 2 2 2 ν6 = ν2 + ν3 + ν4 + ν5 ,

 1 2 1 2 1 2  −(κgκn + κg(τg − τg ))fw + (κgκg− 1   σ = hγ00, Ni =  1 2 1 2 2 2 1 1  . 3  κn(τg + τg ))fx − ((κg) + (κg) + τg (τg +  2k∇fk   2 2 2 2 1 2 τg ))fy − ((κg) + τg (τg + τg ))fz

Also, we get ν T + ν P + ν U + ν N U¯ = 7 8 9 10 , ν11

where

2 1 2 2 1 2 2 ν7 = κgfwν3 − τg fzν3 − τg fzν3 + κgfwν4 + τg fyν4 + τg fyν4 − κg(fy + fz)ν5,

2 1 1 2 1 1 2 2 ν8 = −fw(κgν2 + κgν4) + fz((τg + τg )ν2 + κgν5) + fx(−(τg + τg )ν4 + κgν5),

2 1 1 2 1 1 2 2 ν9 = fw(−κgν2 + κgν3) − fy((τg + τg )ν2 + κgν5) + fx((τg + τg )ν3 + κgν5),

2 2 2 1 2 1 ν10 = κgfyν2 + κgfzν2 − κgfxν3 − κgfzν3 − κgfxν4 + κgfyν4,

ν11 = ν1k∇fkν6.

¯1 ¯2 ¯1 ¯2 In the light of the above calculations, the curvature functionsκ ¯n, κg, κg, τg and τg 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:

−κ1T − κ2P + κ2U + (τ 1 + τ 2)N T¯ = g g g g g , ν12 f T + f P + f U + f N N¯ = x y z w , k∇fk ν T + ν P + ν U + ν N P¯ = 13 14 15 16 , ν17 ν T + ν P + ν U + ν N U¯ = 18 19 20 21 , ν22

and the curvature functions are obtained as follows:   2 1 2 1 2 1 2 2 2 1 κg(−τg + τg )fw + κgκgfx − ((κg) + (κg) + κ¯n =   , ν17k∇fk 1 1 2 2 2 2 1 2 τg (τg + τg ))fy − ((κg) + τg (τg + τg ))fz   1 2 1 2 2 2 1 1 2 κ κ ν13 − ((κ ) + (κ ) + τ (τ + τ ))ν14− ¯1 1 g g g g g g g κg =   , ν12ν17 2 2 2 1 2 2 1 2 ((κg) + τg (τg + τg ))ν15 + κg(−τg + τg )ν16

 1 2 1 2  ν17(κgν13ν19 − κgν15ν19 − τg ν16ν19 − τg ν16ν20+ 1   κ¯2 =  2 1 2 1 0 0  , g 2  τg ν15ν21 + ν14(−κgν18 + κgν20 + τg ν21) + ν18ν13 + ν19ν14+  ν17ν22   0 0 0 ν20ν15 + ν21ν16) − (ν13ν18 + ν14ν19 + ν15ν20 + ν16ν21)ν17

 1 0 0 1 2  −fx(κgν14ν17 − ν17ν13 + ν13ν17) + fy(ν17(κgν13 − κgν15− 1   τ¯1 =  1 0 0 2 2 0  , g 2  τg ν16 + ν14) − ν14ν17) + fz(ν17(κgν14 − τg ν16 + ν15)−  ν17k∇fk   0 1 2 0 0 ν15ν17) + fw(ν17(τg ν14 + τg ν15 + ν16) − ν16ν17)

 1 0 0 1  −fx(κgν19ν22 − ν22ν18 + ν18ν22) + fy(ν22(κgν18− 1   τ¯2 =  2 1 0 0 2 2  , g 2  κgν20 − τg ν21 + ν19) − ν19ν22) + fz(ν22(κgν19 − τg ν21+  ν22k∇fk   0 0 1 2 0 0 ν20) − ν20ν22) + fw(ν22(τg ν19 + τg ν20 + ν21) − ν21ν22) where

q √ 2 2 1 2 2 1 2 1 2 ν12 = 2(κg) + (τg + τg ) + (κg) , ν13 = 2(−κgκg)k∇fk + 2σ3fx,

√ 1 2 2 2 1 1 2 ν14 = 2((κg) + (κg) + τg (τg + τg ))k∇fk + 2σ3fy,

√ 2 2 2 1 2 ν15 = 2((κg) + τg (τg + τg ))k∇fk + 2σ3fz,

√ 2 1 2 2 2 2 2 2 2 ν16 = 2(κgτg − κgτg )k∇fk + 2σ3fw, ν17 = ν12 + ν13 + ν14 + ν15,

2 2 1 2 2 2 1 2 ν18 = −κgν16fy + ν15(κgfw + (τg + τg )fy) − κgν16fz + ν14(κgfw − (τg + τg )fz),

2 2 1 1 2 1 2 1 ν19 = −κgν13fw + κgν16fx − ν15(κgfw + (τg + τg )fx) + (τg + τg )ν13fz + κgν16fz,

2 2 1 1 2 1 2 1 ν20 = −κgν13fw + κgν16fx + ν14(κgfw + (τg + τg )fx) − τg ν13fy − τg ν13fy − κgν16fy,

2 2 1 2 2 1 ν21 = κgν13fy + ν15(−κgfx + κgfy) + κgν13fz − ν14(κgfx + κgfz),

ν22 = ν12 ν17 k∇fk. Int. J. Anal. Appl. 17 (4) (2019) 495

Corollary 3.6. If r is a line of curvature. Then, the following equations hold:

κ1T + κ2P − κ2U T¯ = − g g g , ν12 f T + f P + f U + f N N¯ = x y z w , k∇fk ν T + ν P + ν U + ν N P¯ = 13 14 15 16 , ν17 ν T + ν P + ν U + ν N U¯ = 18 19 20 21 , ν22

and the curvature functions are obtained as follows:

−1 1 1 2 1 2 2 2 2 2 κ¯n = κgκnfw − κgκgfx + ((κg) + (κg) )fy + (κg) fz , ν23k∇fk ¯1 −1 1 2 1 2 2 2 2 2 1 κg = − κgκgν24 + ((κg) + (κg) )ν25 + (κg) ν26 + κgκnν27 , ν23ν28

 1 2 0  ν28(−κnν27ν29 + ν25(−κgν29 + κgν31) + ν29ν24+ 1   κ¯2 =  1 2 0 0 0  , g 2  ν30(κgν24 − κgν26 + ν25) + ν31ν26 + ν32(κnν24 + ν27))−  ν28ν33   0 (ν24ν29 + ν25ν30 + ν26ν31 + ν27ν32)ν28

 1 0 0 1  −fx(ν28(κgν25 + κnν27 − ν24) + ν24ν28) + fy(ν28(κgν24− 1   τ¯1 =  2 0 0 2 0 0  , g 2  κgν26 + ν25) − ν25ν28) + fz(ν28(κgν25 + ν26) − ν26ν28)+  ν28k∇fk   0 0 fw(ν28(κnν24 + ν27) − ν27ν28)

 1 0 1  −fx(ν33(κgν30 + κnν32 − ν29) + ν29ν33) + fy(ν33(κgν29− 1   τ¯2 =  2 0 0 2 0 0  , g 2  κgν31 + ν30) − ν30ν33) + fz(ν33(κgν30 + ν31) − ν31ν33)+  ν33k∇fk   0 0 fw(ν33(κnν29 + ν32) − ν32ν33) where

q √ 2 2 1 2 1 2 ν23 = 2(κg) + (κg) , ν24 = − 2κgκgk∇fk + 2σ3fx, √ √ 1 2 2 2 2 2 ν25 = 2((κg) + (κg) )k∇fk + 2σ3fy, ν26 = 2(κg) k∇fk + 2σ3fz, √ 1 2 2 2 2 2 ν27 = 2κgκnk∇fk + 2σ3fw, ν28 = ν12 + ν13 + ν14 + ν15,

2 2 2 2 ν29 = κgν25fw + κgν26fw − κgν27fy − κgν27fz,

2 1 2 1 ν19 = −κgν24fw − κgν26fw + κgν27fx + κgν27fz,

2 1 2 1 ν20 = −κgν24fw + κgν25fw + κgν27fx − κgν27fy,

2 2 1 2 2 1 ν21 = κgν24fy + ν26(−κgfx + κgfy) + κgν24fz − ν25(κgfx + κgfz),

ν22 = ν23 ν28 k∇fk. 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 1 2 1 2 with Darboux frame {T, P, U, N} and non-zero curvatures κn, κg, κg, τg and τg . Then the PN-Smarandache curve of δ is deﬁned as 1 δ(u) = √ (P + N). (3.11) 2

Theorem 3.4. Let r = r(u) be a Frenet curve lying on a hypersurface M in E4 with Darboux frame 1 2 1 2 {T, P, U, N} and non-zero curvatures κn, κg, κg, τg and τg . Then the curvature functions of the PN− Smarandache curve of r satisfy the following equations:   1 1 2 2 2 2 1 1 −1 (κn(κg + κn) + (τg ) + τg (−κg + τg ))fw + (−κg + κn)τg fx κ¯n =   , ξ1k∇fk 2 2 1 1 1 2 2 2 1 2 2 +((κg) + κg(κg + κn) + (τg ) − κgτg )fy + τg (κg + τg )fz   1 1 2 2 2 2 1 1 (κn(κ + κn) + (τ ) + τ (−κ + τ ))ξ5 + (−κ + κn)τ ξ2+ ¯1 −1 g g g g g g g κg =   , ξ1ξ6 2 2 1 1 1 2 2 2 1 2 2 ((κg) + κg(κg + κn) + (τg ) − κgτg )ξ3 + τg (κg + τg )ξ4

 0 1 2  −(ξ2ξ7 + ξ3ξ8 + ξ4ξ9 + ξ5ξ10)ξ6 + ξ6((κgξ2 − κgξ4)ξ8 1   κ¯2 =  1 2 1 2 1  , g 2  −ξ5(κnξ7 + τg ξ8 + τg ξ9) + ξ3(−κgξ7 + κgξ9 + τg ξ10)+  ξ6 ξ11   0 0 0 2 0 ξ7ξ2 + ξ8ξ3 + ξ9ξ4 + ξ10(κnξ2 + τg ξ4 + ξ5))

 0 1 2  −(fxξ2 + fyξ3 + fzξ4 + fwξ5)ξ6 + ξ6(−κgfxξ3 + κgfzξ3 1   τ¯1 =  2 0 1 2 1  , g 2  −κnfxξ5 − τg fzξ5 + fxξ2 + fy(κgξ2 − κgξ4 − τg ξ5  ξ6 k∇fk   0 0 1 2 0 +ξ3) + fzξ4 + fw(κnξ2 + τg ξ3 + τg ξ4 + ξ5))

 0 1  −(fxξ7 + fyξ8 + fzξ9 + fwξ10)ξ11 + ξ11(−κgfxξ8+ 1   τ¯2 =  2 2 0 1  . (3.12) g 2  κgfzξ8 − κnfxξ10 − τg fzξ10 + fxξ7 + fy(κgξ7−  ξ11k∇fk   2 1 0 0 1 2 0 κgξ9 − τg ξ10 + ξ8) + fzξ9 + fw(κnξ7 + τg ξ8 + τg ξ9 + ξ10))

Proof. Let δ = δ(s) be a PN− Smarandache curve reference to Frenet curve r. Then, by diﬀerentiating Eq. (3.11), we obtain

1 1 2 2 1 −(κ + κn)T − τ P + (κ − τ )U + τ N δ0 = g g √ g g g , 2

00 1 1 1 2 2 1 1 1 2 2 2 δ = √ (κ − κn)τ T + (−(κ ) − κ (κ + κn) − (τ ) + κ τ )P 2 g g g g g g g g 1 2 2 1 1 2 2 2 2 2 −τg (κg + τg )U + (−κn(κg + κn) − (τg ) + κgτg − (τg ) )N. (3.13)

Therefore, from Eqs. (3.13), we get

1 1 2 2 1 −(κ + κn)T − τ P + (κ − τ )U + τ N T¯ = g g g g g , ξ1 f T + f P + f U + f N N¯ = x y z w , k∇fk Int. J. Anal. Appl. 17 (4) (2019) 497

where

q 1 2 1 2 2 2 2 ξ1 = 2(τg ) + (κg + κn) + (κg − τg ) .

On the other hand, we obtain

ξ T + ξ P + ξ U + ξ N P¯ = 2 3 4 5 , ξ6 where

√ 1 1 ξ2 = 2(−κg + κn)τg k∇fk + 2σ4fx , √ 2 2 1 1 1 2 2 2 ξ3 = 2((κg) + κg(κg + κn) + (τg ) − κgτg )k∇fk + 2σ4fy , √ 1 2 2 ξ4 = 2τg (κg + τg )k∇fk + 2σ4fz , √ 1 1 2 2 2 2 2 ξ5 = 2(κn(κg + κn) + (τg ) − κgτg + (τg ) )k∇fk + 2σ4fw ,

2 2 2 2 2 ξ6 = ξ2 + ξ3 + ξ4 + ξ5 ,

 1 1 2 2 2 2  −(κn(κg + κn) + (τg ) + τg (−κg + τg ))fw+ 1   σ = hδ00, Ni =  1 1 2 2 1 1  . 4  (κg − κn)τg fx − ((κg) + κg(κg + κn)+  2k∇fk   1 2 2 2 1 2 2 (τg ) − κgτg )fy − τg (κg + τg )fz

Also, we have

ξ T + ξ P + ξ U + ξ N U¯ = 7 8 9 10 , ξ11

where,

2 2 1 1 1 2 2 1 ξ7 = κgfwξ3 − τg fwξ3 − τg fzξ3 + τg fwξ4 + τg fyξ4 + (−(κg − τg )fy − τg fz)ξ5,

2 2 1 1 1 1 2 2 ξ8 = fw((−κg + τg )ξ2 − (κg + κn)ξ4) + fz(τg ξ2 + (κg + κn)ξ5) − fx(τg ξ4 + (−κg + τg )ξ5),

1 1 1 1 1 ξ9 = fw(−τg ξ2 + (κg + κn)ξ3) + τg fx(ξ3 + ξ5) − fy(τg ξ2 + (κg + κn)ξ5),

2 2 1 2 2 1 ξ10 = (κgfyξ2 − τg fyξ2 + τg fzξ2 − κgfxξ3 + τg fxξ3 − κgfzξ3 − κnfzξ3 −

1 1 τg fxξ4 + κgfyξ4 + κnfyξ4),

ξ11 = ξ1k∇fkξ6.

¯1 ¯2 ¯1 ¯2 In the light of the above calculations , the curvature functionsκ ¯n, κg, κg, τg and τg 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:

−κ1T − τ 1P + (κ2 − τ 2)U + τ 1N T¯ = g g g g g , ξ12 f T + f P + f U + f N N¯ = x y z w , k∇fk ξ T + ξ P + ξ U + ξ N P¯ = 13 14 15 16 , ξ17 ξ T + ξ P + ξ U + ξ N U¯ = 18 19 20 21 . ξ22

Thus, the curvature functions can be computed as follows:   1 2 2 2 2 1 1 1 2 2 2 −1 ((τg ) + τg (−κg + τg ))fw − κgτg fx + ((κg) + (κg) + κ¯n =   , k∇fkξ12 1 2 2 2 1 2 2 (τg ) − κgτg )fy + τg (κg + τg )fz   1 2 2 2 2 1 1 1 2 ((τ ) + τ (−κ + τ ))ξ16 − κ τ ξ13 + ((κ ) + ¯1 −1 g g g g g g g κg =   , ξ12ξ17 2 2 1 2 2 2 1 2 2 (κg) + (τg ) − κgτg )ξ14 + τg (κg + τg )ξ15

 1 2 1 2  ξ17(κgξ13ξ19 − κgξ15ξ19 − τg ξ16ξ19 − τg ξ16ξ20+ 1   κ¯2 =  2 1 2 1 0 0  , g 2  τg ξ15ξ21 + ξ14(−κgξ18 + κgξ20 + τg ξ21) + ξ18ξ13 + ξ19ξ14+  ξ17 ξ22   0 0 0 ξ20ξ15 + ξ21ξ16) − (ξ13ξ18 + ξ14ξ19 + ξ15ξ20 + ξ16ξ21)ξ17

 1 0 0 1 2  −fx(κgξ14ξ17 − ξ17ξ13 + ξ13ξ17) + fy(ξ17(κgξ13 − κgξ15− 1   τ¯1 =  1 0 0 2 2 0  , g 2  τg ξ16 + ξ14) − ξ14ξ17) + fz(ξ17(κgξ14 − τg ξ16 + ξ15)−  ξ17k∇fk   0 1 2 0 0 ξ15ξ17) + fw(ξ17(τg ξ14 + τg ξ15 + ξ16) − ξ16ξ17)

 1 0 0 1 2  −fx(κgξ19ξ22 − ξ22ξ18 + ξ18ξ22) + fy(ξ22(κgξ18 − κgξ20− 1   τ¯2 =  1 0 0 2 2 0  , g 2  τg ξ21 + ξ19) − ξ19ξ22) + fz(ξ22(κgξ19 − τg ξ21 + ξ20)−  ξ22k∇fk   0 1 2 0 0 ξ20ξ22) + fw(ξ22(τg ξ19 + τg ξ20 + ξ21) − ξ21ξ22) where

q √ 1 2 1 2 2 2 2 1 1 ξ12 = 2(τg ) + (κg) + (κg − τg ) , ξ13 = 2(−κg)τg k∇fk + 2σ4fx ,

√ 2 2 1 1 1 2 2 2 ξ14 = 2((κg) + κg(κg) + (τg ) − κgτg )k∇fk + 2σ4fy ,

√ 1 2 2 ξ15 = 2τg (κg + τg )k∇fk + 2σ4fz ,

√ 1 2 2 2 2 2 2 2 2 2 2 ξ16 = 2((τg ) − κgτg + (τg ) )k∇fk + 2σ4fw , ξ17 = ξ2 + ξ3 + ξ4 + ξ5 ,

2 2 1 1 1 2 2 1 ξ18 = κgfwξ14 − τg fwξ14 − τg fzξ14 + τg fwξ15 + τg fyξ15 + (−(κg − τg )fy − τg fz)ξ16,

2 2 1 1 1 1 2 2 ξ19 = fw((−κg + τg )ξ14 − (κg)ξ15) + fz(τg ξ14 + (κg)ξ16) − fx(τg ξ15 + (−κg + τg )ξ16),

1 1 1 1 1 ξ20 = fw(−τg ξ14 + (κg)ξ14) + τg fx(ξ14 + ξ16) − fy(τg ξ14 + (κg)ξ16),

2 2 1 2 2 1 1 1 ξ21 = κgfyξ13 − τg fyξ13 + τg fzξ13 − κgfxξ14 + τg fxξ14 − κgfzξ14 − τg fxξ15 + κgfyξ15,

ξ22 = ξ12k∇fkξ17. Int. J. Anal. Appl. 17 (4) (2019) 499

Corollary 3.8. If r be line of curvature. Then, the following equations hold:

1 2 −(κ + κn)T + k U T¯ = g g , ξ23 f T + f P + f U + f N N¯ = x y z w , k∇fk ξ T + ξ P + ξ U + ξ N P¯ = 24 25 26 27 , ξ28 ξ T + ξ P + ξ U + ξ N U¯ = 29 30 31 32 . ξ33

Thus, the curvature functions can be computed as follows:

1 2 2 1 1 κn(κg + κn)fw + ((κg) + κg(κg + κn))fy κ¯n = − , k∇fkξ23 κ (κ1 + κ )ξ + ((κ2)2 + κ1(κ1 + κ ))ξ ¯1 n g n 25 g g g n 27 κg = − , ξ23ξ28

 1 2 0  ξ28(−κnξ27ξ29 + ξ25(−κgξ29 + κgξ31) + ξ29ξ24+ 1   κ¯2 =  1 2 0 0  , g 2  ξ30(κgξ24 − κgξ26 + ξ25) + ξ31ξ26 + ξ32(κnξ24+  ξ28 ξ33   0 0 ξ27)) − (ξ24ξ29 + ξ25ξ30 + ξ26ξ31 + ξ27ξ32)ξ28

 1 0 0 1  −fx(ξ28(κgξ25 + κnξ27 − ξ24) + ξ24ξ28) + fy(ξ28(κgξ24− 1   τ¯1 =  2 0 0 2 0 0  , g 2  κgξ26 + ξ25) − ξ25ξ28) + fz(ξ28(κgξ25 + ξ26) − ξ26ξ28)+  ξ28k∇fk   0 0 fw(ξ − 28(κnξ24 + ξ27) − ξ27ξ28)

 1 0 0 1  −fx(ξ33(κgξ30 + κnξ32 − ξ29) + ξ29ξ33) + fy(ξ33(κgξ29− 1   τ¯2 =  2 0 0 2 0 0  , g 2  κgξ31 + ξ30) − ξ30ξ33) + fz(ξ33(κgξ30 + ξ31) − ξ31ξ33)+  ξ33k∇fk   0 0 fw(ξ − 33(κnξ29 + ξ32) − ξ32ξ33) where

q 1 2 2 2 ξ23 = (κg + κn) − (κg) , ξ24 = 2σ4fx , √ 2 2 1 1 ξ25 = 2((κg) + κg(κg + κn))k∇fk + 2σ4fy , ξ26 = 2σ4fz , √ 1 2 2 2 2 2 ξ27 = 2κn(κg + κn)k∇fk + 2σ4fw , ξ28 = ξ24 + ξ25 + ξ26 + ξ27 ,

2 ξ29 = (kg2ξ25fw − κgξ27fy),

2 1 2 1 ξ30 = −κgξ24fw − κgξ26fw − κnξ26[u]fw + κgξ27fx + (κg + κn)ξ27fz,

1 1 ξ31 = (κg + κn)ξ25fw − κgξ27fy − κnξ27fy,

2 1 2 1 ξ32 = κgξ24fy + κgξ26fy + κnξ26fy − ξ25(κgfx + (κg + κn)fz),

ξ33 = ξ23k∇fkξ28. 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: sin(u) cos(u) 2 2 T = − , , − sin(2u), cos(2u) , 5 5 5 5 cos(u) sin(u) cos(2u) sin(2u) N = √ , √ , √ , √ , 2 2 2 2

cos(u) sin(u) cos(2u) sin(2u) P = √ , √ , − √ , − √ , 2 2 2 2 2 sin(u) 2 cos(u) 1 1 U = √ , − √ , −√ sin(2u), √ cos(2u) . 5 5 5 5 The curvature functions of the curve r can be computed as follows: r r 5 1 3 2 2 1 2 κn = − , κ = √ , κ = −2 , τ = τ = 0. 2 g 10 g 5 g g Therefore, we can obtain TU−Smarndache curve as 1 sin(u) − cos(u) −3 sin(2u) 3 cos(2u) φ = √ (T + U) = √ , √ , √ , √ . 2 10 10 10 10 By using the deﬁnition 2.6, we can obtain 1 T¯ = √ (cos(u), sin(u), −6 cos(2u), −6 sin(2u)) , 37 1 N¯ = √ (sin(u), cos(u), 3 sin(2u), 3 cos(2u)) , 2 5 1 P¯ = √ (17 sin(u), −17 cos(u), 129 sin(2u), −129 cos(2u)) , 16930 1 U¯ = √ (−54 cos(u), −54 sin(u), −9 cos(2u), −9 sin(2u)) . 8465 The curvature functions of the Smarandache curve φ can be computed as follows: r √ −1 37 ¯1 1531 ¯2 −324 2 ¯ 1 ¯2 κ¯n = , κ = √ , κ = , τ = τ = 0. 2 5 g 626410 g 1693 g g

Example 4.2. Consider the unit speed curve given by cos(u) sin(u) u u r(u) = , , , √ . 2 2 2 2 By using the deﬁnition 2.6 we can calculate the Darboux frame {T, P, U, N} as follow: − sin(u) cos(u) 1 1 T = , , , √ , 2 2 2 2 cos(u) sin(u) 1 1 N = , , , √ , 2 2 2 2 Int. J. Anal. Appl. 17 (4) (2019) 501

√ √ ! − 3 − 3 1 −1 P = cos(u), sin(u), √ , √ , 2 2 2 3 6 ! r2 r2 −1 1 U = − sin(u), cos(u), √ , − √ , 3 3 6 2 3 The curvature functions of the curve r can be computed as follows √ 1 1 3 2 1 1 2 1 κn = − , κ = , κ = −√ , τ = 0, τ = −√ . 4 g 4 g 2 g g 6 Therefore we can obtain TN−Smarndache curve as cos(u) − sin(u) cos(u) + sin(u) 1 β = √ , √ , √ , 0 . 2 2 2 2 2 By using the deﬁnition 2.6, we can obtain cos(u) + sin(u) cos(u) − sin(u) T¯ = − √ , √ , 0, 0 , 2 2 2 2 cos(u) − sin(u) cos(u) + sin(u) 1 1 N¯ = √ , √ , , , 2 2 2 2 2 2 −3(cos(u) − sin(u)) 3(cos(u) + sin(u)) 1 1 P¯ = √ , √ , √ , √ , 22 22 11 11 1 1 U¯ = 0, 0, √ , √ . 11 11 The curvature functions of the Smarandache curve β can be computed as

−1 ¯1 3 ¯2 ¯1 ¯2 κ¯n = , κ = √ , κ = 0, τ = τ = 0. 2 g 11 g g g

5. Conclusion

In the four dimensional Euclidean space E4, some special Smarandache curves lying on a hypersurface are investigated. Also, the diﬀerential geometric properties of these curves are obtained. Furthermore, some computational examples in support of our main results are given.

Acknowledgment

This research was supported by Islamic University of Madinah. We would like to thank our colleagues from Deanship of Scientiﬁc Research who provided insight and expertise that greatly assisted the research.

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