Smarandache curves in Euclidean 4 space E4

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Original Article

Smarandache curves in Euclidean 4- space E4 Mervat Elzawy Mathematics Departement, Faculty of Science, Tanta University, Tanta, Egypt

a r t i c l e

i n f o

Article history: Received 1 January 2017 Revised 15 February 2017 Accepted 5 March 2017 Available online xxx

a b s t r a c t The purpose of this paper is to study Smarandache curves in the 4-dimensional Euclidean space E4 , and to obtain the Frenet–Serret and Bishop invariants for the Smarandache curves in E4 . The first, the second and the third curvatures of Smarandache curves are calculated. These values depending upon the first, the second and the third curvature of the given curve.

2010 MSC: 14H45 14H50 53A04

© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords: Euclidean 4- space Smarandache curves Ferent frame Parallel transport frame

1. Introduction It is well known that, if a curve is differentiable in an open interval, at each point, a set of mutually orthogonal unit vectors can be constructed, and these vectors are called Frenet frame or moving frame vectors. The rates of these frame vectors along the curve define curvatures of the curve. The set, whose elements are frame vectors and curvatures of a curve is called Frenet apparatus of the curves. In recent years, the theory of degenerate submanifolds has been treated by researchers and some classical differential geometry topics have been extended to Minkowski space [1–3] and Galilean space [4]. For instance in [1,2] the authors extended and studied Smarandache curves in Minkowski space-time. A regular curve in Euclidean space E4 , whose position vectors is composed by Frenet frame vectors on another regular curve is called Smarandache curve. Special Smarandache curves in three dimensional Euclidean space studied in [5]. The Bishop frame [6] or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative. We can parallel transport an othonormal frame along a curve simply by parallel transporting each component of the frame in Euclidean 4space. The parallel transport frame is based on the observation that while T(s) for the given curve model is unique, we may chose any convenient arbitrary basis which consists of relatively paral-

E-mail address: mervatelzawy@science.tanta.edu.eg

lel vector fields {M1 (s), M2 (s), M3 (s)} of the frame, such that they are perpendicular to T(s) at each point[7,8]. The parallel transport frame in four dimensional Euclidean space is studied in [9]. Smarandache curves were studied from deferent researchers in three dimensional Euclidean space [10–14]. Smarandache curves in 4-dimensional Galilean space are presented in [15]. In this paper we study Smarandache curves in 4-dimensional space according to the Frenet frame and parallel transport frame.

2. Preliminaries Let α : R → E4 be an arbitrary curve in the Euclidean space E4 . − → − → − → Let a = (a1 , a2 , a3 , a4 ), b = (b1 , b2 , b3 , b4 ) and c = (c1 , c2 , c3 , c4 ) 4 be three vectors in E , equipped with the standard inner product → − → − given by < a , b >= a1 b1 + a2 b2 + a3 b3 + a4 b4 . The norm of a vec − → − → − → tor a ∈ E4 is given by a = < a , a >. The curve α is said to be of a unit speed or parametrized by arc length function s if → − → − − → < α , α >= 1. The vector product of a , b , and c is defined by the determinant

e1 → − − → − → a1 a × b × c = b1 c1

e2 a2 b2 c2

e3 a4 b3 c3

e4 a4 b4 c4

where e1 × e2 × e3 = e4 , e2 × e3 × e4 = e1 , e3 × e4 × e1 = e2 , e4 × e1 × e2 = e3 , e3 × e2 × e1 = e4 .

http://dx.doi.org/10.1016/j.joems.2017.03.003 1110-256X/© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Please cite this article as: M. Elzawy, Smarandache curves in Euclidean 4- space E 4 , Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.03.003

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The vectors t(s), n(s), b1 (s), b2 (s) are the moving Frenet frame along the unit speed curve α . Then the Frenet formulas are given by

⎡ ⎤

⎡

t 0 ⎢ n ⎥ ⎢−k1 ⎣b ⎦ = ⎣ 0 1 b 2 0

k1 0 −k2 0

⎤⎡ ⎤

0 k2 0 −k3

0 t 0 ⎥⎢ n ⎥ k3 ⎦⎣b1 ⎦ 0 b2

3. Main results 3.1. tb1 Smarandache curves in E4 according to the Frenet frame.

t, n, b1 , and b2 are called, respectively, the tangent, the principal normal, the first binormal and the second binormal vector fields of the curves. The functions k1 (s), k2 (s) and k3 (s) are called respectively, the first, the second and the third curvature of the curve α . The curve is called W − curve if it has constant curvatures k1 , k2 and k3 . Let α = α (t ) be an arbitrary curve in E4 . The Frenet apparatus of the curve α can be calculated by the following equations.

α

t =

α α 2 α − < α , α > α n= α 2 α − < α , α > α b1 = η b2 × t × n t × n × α b2 = η t × n × α 2 α α − < α , α > α

⎡

K1 0 0 0

2

By differentiating β (sβ ) with respect to s we obtain

d β ( sβ ) ds

K2 0 0 0

⎤⎡

⎤

K3 T 0 ⎥⎢M1 ⎥ 0 ⎦⎣M2 ⎦ 0 M3

K3 = k1 (sin φ sin ψ + cos φ sin θ cos ψ ) and

where

k23

− ( θ )2

k3 k21 + k22

, ψ = − ( k2 + k3

ds

1 = √ ((k1 − k2 )n + k3 b2 ) 2

The tangent vector of the curve β is given by

tβ = A1 n + A2 b2

where

dsβ ds

=

(3.1)

(k1 −k2 )2 +k23 √

2

(k −k ) , A1 = 1 2

(k1 −k2 )2 +k23

and A2 =

k3

(k1 −k2 )2 +k23

Again differentiating the tangent vector of β with respect to sβ we can obtain β as follows

β =

√ 2[−k1 (k1 − k2 )t + (k1 k2 − k22 − k23 )b1 ]

(k1 − k2 )2 + k23

A3 =

where

k2 − ( θ )2

3

k21 + k22

(3.2) −k1 (k1 −k2 ) k21 (k1 −k2 )2 +(k1 k2 −k22 −k23 )2

A4 =

and

k1 k2 −k22 −k23 k21 (k1 −k2 )2 +(k1 k2 −k22 −k23 )2

β = A5 n + A6 b2 2(−k21 (k1 −k2 )−k2 (k1 k2 −k22 −k23 )) 3 ((k1 −k2 )2 +k23 ) 2

and A6 =

k3 (k1 k2 −k22 −k23 ) 3

((k1 −k2 )2 +k23 ) 2

The second binormal vector of the curve β is given easily as follows

θ , sin ψ

K12 + K22 + K32 , k2 = −ψ + φ sin θ , k3 =

θ =

dsβ

where A5 =

K2 = k1 (− cos φ sin ψ + sin φ sin θ cos ψ ),

φ cos θ + θ cot ψ = 0

dβ (sβ ) dsβ

nβ = A3 t + A4 b1

K1 = k1 cos θ cos ψ ,

=

The principal normal of the curve β is

where K1 , K2 , and K3 are the principal curvature functions according to parallel transport frame of the curve α . The set {T, M1 , M2 , M3 } is called the parallel transport frame of α [6,14]. The expressions of the principal curvatures are given as follows:

k1 =

2

Proof. Let β = β (sβ ) be tb1 Smarandache curve of the curve α . Then From Definition (2) we have β (sβ ) = √1 (t (s ) + b1 (s ))

where η is taken ± 1 such that determinant of matrix [t,n,b1,b2] is equal to one. The Bishop frame or parallel transport frame is an alternative approach to define a moving frame that is well defined even when the curve has vanishing second derivative [9]. One can express parallel transport of an orthonormal frame along a curve simply by parallel transporting each component of the frame. The tangent vector and the convenient arbitrary basis for the remainder of the frame are used. The parallel transport equations in E4 can be expressed as

⎤

Definition 2. Let α = α (s ) be a unit-speed curve with constant and nonzero curvatures k1 , k2 , k3 and {t, n, b1 , b2 } be moving frame on it, tb1 Smarandache curves are defined by β (sβ ) = √1 (t (s ) +

tus of α ({t, n, b1 , b2 , k1 , k2 , k3 }).

< α (IV ) , b2 > t × n × α α

T 0 ⎢M1 ⎥ ⎢−K1 ⎣M ⎦ = ⎣−K 2 2 M3 −K3

Definition 1. A regular curve in E4 , whose position vector is obtained by Frenet frame vectors on another regular curve, is called Smarandache curve.

Theorem 1. Let α (s) be a unit speed curve with constant non zero curvatures k1 , k2 , k3 and β (sβ ) be tb1 Smarandache curves in E4 defined by the frame vectors of α (s) . Then the Frenet apparatus of β ({tβ , nβ , b1β , b2β , k1β , k2β , k3β } ) can be formed by Frenet appara-

α 4 t × n × α α k2 = α 2 α − < α , α > α

⎡

In this subsection we define tb1 Smarandache curves and obtain their Frenet apparatus.

b1 (s )).

k1 =

k3 =

Note that k1 , k2 , k3 are the principal curvature functions according to Frenet frame and K1 , K2 , K3 are the principal curvature functions according to the parallel transport frame of the curve α .

b2β =

), φ =

(k k − k22 − k23 )t + k1 (k1 − k2 )b1 1 2 (k1 k2 − k22 − k23 )2 + k21 (k1 − k2 )2

(3.3)

The first binormal vector of the curve β is

b1β =

−k3 n + (k1 − k2 )b2

k23 + (k1 − k2 )2

(3.4)

cos θ

Please cite this article as: M. Elzawy, Smarandache curves in Euclidean 4- space E 4 , Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.03.003

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The first, second and third curvature of the of the curve β are

2[(k1 k2 − k22 − k23 )2 + k21 (k1 − k2 )2 ]

k1β =

(3.5)

(k23 + (k1 − k2 )2 )2

√

2k3 [k1 (k1 k2 − k22 − k23 ) + k21 (k1 − k2 )]

k2β =

(k23 + (k1 − k2 )2 ) (k1 k2 − k22 − k23 )2 + k21 (k1 − k2 )2

k3β =

(3.6)

√ 2(−k1 A4 A5 − k2 A3 A5 + k3 A3 A6 ) k23 + (k1 − k2 )2

The third derivative of the curve β reads

β = (λ0 Tα + λ1 M1 α + λ2 M2 α + λ3 M3 α )

√ 2 2K12α + K22α + K32α

√ 2[(−λ1 K1α − λ2 K2α − λ3 K3α )Tα + λ0 K1α M1α + λ0 K2α M2α + λ0 K3α M3α ]

β =

2K12α + K22α + K32α

(3.11) (3.7)

(k1 k2 − k22 − k23 )2 + k21 (k1 − k2 )2

This completes the proof.

3

Tβ × nβ × β = C1 M1α + C2 M2α + C3 M3α where √

√

3.2. TM1 Smarandache curves in E4 according to the parallel transport frame.

2[λ0 λ3 K1α K2α − λ0 λ2 K1α K3α − (λ1 K1α + λ2 K2α + λ3 K3α )(λ3 K2α − λ2 K3α )]

C1 =

C2 = √

2[λ0 λ

2 3 K1α

2[λ0 λ

2 2 K1α

λ20 + λ21 + λ22 + λ23 (2K12α + K22α + K32α )

− λ0 λ1 K1α K3α − (λ1 K1α + λ2 K2α + λ3 K3α )(λ3 K1α − λ1 K3α )]

λ20 + λ21 + λ22 + λ23 (2K12α + K22α + K32α )

− λ0 λ1 K1α K2α − (λ1 K1α + λ2 K2α + λ3 K3α )(λ1 K2α − λ2 K2α )]

In this subsection we define TM1 Smarandache curves and obtain their parallel transport frame and the principal curvatures.

C3 =

Definition 3. Let α = α (s ) be a unit speed curve in E4 and {Tα , M1α , M2α , M3α } be its moving parallel transport frame. TM1 Smarandache curves is defined by β (sβ ) = √1 (Tα + M1α ).

The second binormal of the curve β is given by the following formula

2

Theorem 2. Let α = α (s ) be a unit speed curve with constant principal curvatures K1α , K2α , K3α and β (sβ ) be TM1 Smarandache curves in E4 defined by the parallel transport frame vectors of α = α (s ). Then the parallel transport frame of β can be formed by the parallel transport frame of α and the principle curvatures of β (K1β , K2β , K3β )can be obtained by the principal curvatures of α . Proof. To investigate the parallel transport frame of TM1 Smarandache curve according to α (s) differentiating β (sβ ) = √1 (Tα + 2

M1α ) with respect to s

Tβ

dsβ

1 = √ (−K1α Tα + K1α M1α + K2α M2α + K3α M3α ) 2

ds

The tangent vector of the curve β can be written as follows

Tβ =

−K1α Tα + K1α M1α + K2α M2α + K3α M3α

where

dsβ

=

√1 2

(3.8)

α

α

dTβ dsβ

where λ0 =

α

, λ1 =

k1β =

α

α

α

α

2 K12α + K22α + K32α

λ20 + λ21 + λ22 + λ23 =

(3.9)

2K12α + K22α + K32α

γ0 Tα + γ1 M1α + γ2 M2α + γ3 M3α

λ0 Tα + λ1 M1α + λ2 M2α + λ3 M3α λ20 + λ21 + λ22 + λ23

(3.13)

where the constants are given by

γ0 =

C1 λ3 K2α − C1 λ2 K3α + C2 λ1 K3α − C2 λ3 K1α + C3 λ2 K1α − C3 λ1 K2α

C12 + C22 + C32

λ20 + λ21 + λ22 + λ23 2K12α + K22α + K32α

C2 λ3 K1α + C2 λ0 K3α − C3 λ2 K1α − C3 λ0 K1α

γ1 =

C12 + C22 + C32

λ20 + λ21 + λ22 + λ23 2K12α + K22α + K32α

C1 λ3 K1α + C1 λ0 K3α − C3 λ1 K1α − C3 λ0 K1α

γ2 =

+ C22 + C32

λ20 + λ21 + λ22 + λ23 2K12α + K22α + K32α

C1 λ2 K1α + C1 λ0 K2α − C2 λ1 K1α − C2 λ0 K1α + C22 + C32

λ20 + λ21 + λ22 + λ23 2K12α + K22α + K32α

λ0 cos θβ cos ψβ = + γ0 cos θβ sin ψβ Tα λ20 + λ21 + λ22 + λ23

λ1 cos θβ cos ψβ + + γ1 cos θβ sin ψβ λ20 + λ21 + λ22 + λ23

λ2 cos θβ cos ψβ C1 sin θβ − M1α + 2 2 2 C1 + C2 + C3 λ20 + λ21 + λ22 + λ23 C2 sin θβ + γ2 cos θβ sin ψβ − M2α C12 + C22 + C32

The principal normal of the curve β is given by the following formula

nβ =

=

M1β

The first curvature of the curve β according to Frenet frame is

√

b1β = b2β × Tβ × nβ

√ − 2K12α

(2K12 + K22 + K32 ) (2K12 + K22 + K32 ) α α α α √α √ α − 2K1α K3α − 2K1α K2α λ2 = , λ = 3 (2K12 + K22 + K32 ) (2K12 + K22 + K32 ) α

The first binormal of the curve β is given by the following formula

The parallel transport frame for the curve β has the form

= λ0 Tα + λ1 M1α + λ2 M2α + λ3 M3α √ − 2(K12α + K22α + K32α )

(3.12)

C12 + C22 + C32

C12

ds Differentiating (3.8) with respect to s

Tβ =

γ3 =

2K12 + K22 + K32 α

C1 M1α + C2 M2α + C3 M3α

b 2β =

C12

2K12α + K22α + K32α

λ20 + λ21 + λ22 + λ23 (2K12α + K22α + K32α )

(3.10)

λ3 cos θβ cos ψβ + γ3 cos θβ sin ψβ λ20 + λ21 + λ22 + λ23 C3 sin θβ − M3α +

C12 + C22 + C32

(3.14)

Please cite this article as: M. Elzawy, Smarandache curves in Euclidean 4- space E 4 , Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.03.003

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M2β =

The second curvature of the curve β according to Frenet frame is given by

Îť0 (− cos φβ sin Ďˆβ + sin φβ sin θβ cos Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł0 (cos φβ cos Ďˆβ + sin φβ sin θβ sin Ďˆβ ) TÎą

Îť1 (− cos φβ sin Ďˆβ + sin φβ sin θβ cos Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł1 (cos φβ cos Ďˆβ + sin φβ sin θβ sin Ďˆβ )

C1 sin φβ cos θβ + M1Îą

k3β =

Îť2 (− cos φβ sin Ďˆβ + sin φβ sin θβ cos Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł2 (cos φβ cos Ďˆβ + sin φβ sin θβ sin Ďˆβ )

C2 sin φβ cos θβ + M2Îą +

K1β =

K2β =

C12 + C22 + C32

(3.15)

Îť20 + Îť21 + Îť22 + Îť23 cos θβ cos Ďˆβ

Îť1 (sin φβ sin Ďˆβ + cos φβ sin θβ sin Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł1 (− sin φβ cos Ďˆβ + cos φβ sin θβ sin Ďˆβ ) C1 cos φβ cos θβ + M1Îą

Îť20 + Îť21 + Îť22 + Îť23 [− cos φβ sin Ďˆβ + sin φβ sin θβ cos Ďˆβ ]

K3β =

Îť20 + Îť21 + Îť22 + Îť23 [sin φβ sin Ďˆβ + cos φβ sin θβ cos Ďˆβ ] (3.21)

C12 + C22 + C32

Îť2 (sin φβ sin Ďˆβ + cos φβ sin θβ sin Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł2 (− sin φβ cos Ďˆβ + cos φβ sin θβ sin Ďˆβ ) C2 cos φβ cos θβ + M2Îą C12 + C22 + C32

Îť3 (sin φβ sin Ďˆβ + cos φβ sin θβ sin Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł3 (− sin φβ cos Ďˆβ + cos φβ sin θβ sin Ďˆβ ) C3 cos φβ cos θβ + M3Îą +

(3.16)

C12 + C22 + C32

Note that

k3β k21 + k22 β

φβ = −

dsβ ,

Ďˆβ = −

⎥

⎣k2β + k3β

β

k23 − θβ β

2

k21 + k22 β

k23 − θβ β

2

cos θβ

dsβ

References

+

(3.19)

I am grateful to the referee for his/her valuable comments and suggestions.

+

(3.18)

Acknowledgement

+ Îł0 (− sin φβ cos Ďˆβ + cos φβ sin θβ sin Ďˆβ ) TÎą

and

C12 + C22 + C32

The proof is complete.

θβ =

The third curvature of the curve β according to parallel transport frame reads

C12 + C22 + C32

−(C1 K1Îą + C2 K2Îą + C3 K3Îą )(Îť1 K1Îą + Îť2 K2Îą + Îť3 K3Îą )

(3.20)

Îť3 (− cos φβ sin Ďˆβ + sin φβ sin θβ cos Ďˆβ ) Îť20 + Îť21 + Îť22 + Îť23 + Îł3 (cos φβ cos Ďˆβ + sin φβ sin θβ sin Ďˆβ )

C3 sin φβ cos θβ + M3Îą Îť0 (sin φβ sin Ďˆβ + cos φβ sin θβ sin Ďˆβ ) = Îť20 + Îť21 + Îť22 + Îť23

(3.17)

The second curvature of the curve β according to parallel transport frame reads

+

M3β

Îť20 + Îť21 + Îť22 + Îť23

The ďŹ rst curvature of the curve β according to parallel transport frame reads

C12 + C22 + C32

C12 + C22 + C32

The third curvature of the curve β according to Frenet frame is given by

+

k2β =

β

⎤ ⎌dsβ ,

[1] M. Turgut, Smarandache breadth pseudo null curves in Minkowski space-time, Int. J.Math. Combin. Vol. 1 (2009) 46–49. [2] M. Turgut, S. Yilmaz, Smarandache curves in Minkowski space-time, Int. J.Math. Combin. Vol. 3 (2008) 51–55. Ë™ ˆ cd ´ , On Smarandache curves [3] U. Ozturkab, E.B.K. Ozturkab, K. Ilarslanc , E. Nesovi lying in lightcone in Minkowski 3-space, J. Dyn. Syst. Geom. Theor. Vol. 12 (Number 1) (2014) 81–91. [4] A. Magden, S. YÄąlmaz, On the curves of constant breadth in four dimensional Galilean space, Int. Math. Forum Vol. 9 (no. 25) (2014) 1229–1236. [5] A.T. Ali, Special Smarandache curves in the Euclidean space, Int. J.Math. Combin. Vol. 2 (2010) 30–36. [6] R. Bishop, There is more than one way to frame a curve, in: The American Mathematical Monthly March, 1975, pp. 246–252. [7] S. Buyukkutuk, G. Ozturk, Constant ratio curves according to parallel transport frame in Euclidean 4-space E4 , New Trends Math. Sci. NTMSCI 3 (No. 4) (2015) 171–178. [8] T. KrpÄąnar, E. Turhan, Biharmonic curves according to parallel transport frame in E4 , Bol. Soc. Paran. Mat. (3s.) v. 31 (2) (2013) 213–217. [9] F. celik, Z. Bozkurt, I. Gok, F.N. Ekmekci, Y. Yayl, Parallel transport frame in 4-dimensional euclidean space E4 , Caspian J. Math. Sci. (CJMS) 3 (1) (2014) 91–103. [10] A.T. Yakut, M. SavaG, T. Tamirci, The Smarandache curves on S21 and its duality on H20 , J. Appl. Math. Volume (2014) 12. Hindawi Publishing Corporation, Article ID 193586. [11] M. Cetin, H. Kocayig˘ it, On the quaternionic Smarandache curves in Euclidean 3-space, Int. J. Contemp. Math. Sciences Vol. 8 (no. 3) (2013) 139–150. [12] M. Cetin, Y. Tuncer, M.K. Karacan, Smarandache curves according to Bishop frame in Euclidean 3-space, Gen. Math. Notes Vol. 20 (No. 2) (February 2014) 50–66. [13] S. SENYURT, A. CALISKAN, N C- Smarandache curves of Mannheim curve couple according to Frenet frame, Int. J.Math. Combin. Vol.1 (2015) 1–13. [14] S. Senyurt, A. Calskan, An application according to spatial quaternionic Smarandache curve, Appl. Math. Sci. Vol. 9 (no. 5) (2015) 219–228. [15] M. Elzawy, S. Mosa, Smarandache curves in the Galilean 4-space G4 , J. Egypt. Math. Soc. 25 (2017) 53–56.

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