Smarandache curves in the Galilean 4-space G4 by Mia Amalia

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Review Paper

Smarandache curves in the Galilean 4-space G4 M. Elzawy a,∗, S. Mosa b a b

Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt Mathematics Department, Faculty of Science, Damanhour University, Damanhour, Egypt

a r t i c l e

i n f o

Article history: Received 13 March 2016 Revised 14 April 2016 Accepted 24 April 2016 Available online xxx

a b s t r a c t In this paper, we study Smarandache curves in the 4-dimensional Galilean space G4 . We obtain FrenetSerret invariants for the Smarandache curve in G4 . The ﬁrst, second and third curvature of Smarandache curve are calculated. These values depending upon the ﬁrst, second and third curvature of the given curve. Examples will be illustrated.

2010 MSC: Primary 53A35 Secondary 51A05

Keywords: Galilean 4-space Smarandache curve Frenent Frame

1. Introduction Galilean space is the space of the Galilean Relativity. For more about Galilean space and pseudo Galilean space may be found in [1–3] The geometry of the Galilean Relativity acts like a bridge from Euclidean geometry to special Relativity. The geometry of curves in Euclidean space have been developed a long time ago [4]. In recent years, mathematicians have begun to investigate curves and surfaces in Galilean space [5]. Galilean space is one of the Cayley-Klein spaces. Smarandache curves have been investigated by some differential geometers such as H.S. Abdelaziz, M. Khalifa and Ahmad T. Ali [6]. In this paper, we study Smarandache curve in 4-dimensional Galilean space G4 and characterize such curves in terms of their curvature functions. 2. Preliminaries The three-dimensional Galilean space G3 , is the Cayley-Klein space equipped with the projective metric of signature (0, 0, +, + ). The absolute of the Galilean geometry is an ordered triple (ω, f, I) where ω is the ideal (absolute) plane, f is a line in ω ∗

Corresponding author. Tel.: +00201024378883. E-mail addresses: mervatelzawy@science.tanta.edu.eg (M. Elzawy), saffamosa@yahoo.com (S. Mosa).

(absolute line) and I is elliptic involution point (0, 0, x2 , x3 ) → (0, 0, x3 , −x2 ). A plane is called Euclidean if it contains f , otherwise it is called isotropic or, i.e. planes x = const. are Euclidean, and so is the plane ω . A vector u = (u1 , u2 , u3 ) is said to be non-isotropic vector if u1 = 0. all unit non-isotropic vectors are of the form u = (1, u2 , u3 ). For isotropic vectors u1 = 0 holds. In the Galilean space G3 there are four classes of lines [7]: 1. The (proper) isotropic lines that don’t belong to the plane ω but meet the absolute line f. 2. The (proper) non-isotropic lines they don’t meet the absolute line f. 3. a proper non-isotropic lines all lines of ω but f. 4. The absolute line f. − → − → Let x = (x1 , x2 , x3 , x4 ) and y = (y1 , y2 , y3 , y4 ) be two vectors in G4 . The Galilean scalar product in G4 can be written as

→ − → − x , y G4 =

x1 y1 if x1 = 0 and y1 = 0 x2 y2 + x3 y3 + x4 y4 if x1 = 0 or y1 = 0

− → The norm of the vector x = (x1 , x2 , x3 , x4 ) is deﬁned by − − → − → → x = x , x G4 . G4

http://dx.doi.org/10.1016/j.joems.2016.04.008 1110-256X/Copyright 2016, 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, S. Mosa, Smarandache curves in the Galilean 4-space G4 , Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.04.008

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The Galilean cross product of the vectors x, y, z on G4 is deﬁned by

→ − → − → − x × y × z

⎧ ⎪ ⎪ 0 ⎪ x ⎪ ⎪ ⎪ y1 ⎪ ⎪ ⎨ 1 z = 1 e ⎪ 1 ⎪ ⎪ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y1 z1

e2 x2 y2 z2 e2 x2 y2 z2

e3 x3 y3 z3 e3 x3 y3 z3

The vectors {t(s), n(s), b1 (s), b2 (s)} are mutually orthogonal vectors

t (s ), t (s ) G4 = n(s ), n(s ) G4 = b1 (s ), b1 (s ) G4 = b 2 ( s ), b 2 ( s ) G4 = 1

e4 x4 y4 z4 e4 x4 y4 z4

if x1 = 0 or y1 = 0 or z1 = 0

and

if x1 = y1 = z1 = 0

t (s ), n(s ) G4 = t (s ), b1 (s ) G4 = t (s ), b2 (s ) G4 = n ( s ) , b 1 ( s ) G4 = n ( s ) , b 2 ( s ) G4 = b 1 ( s ), b 2 ( s ) G4 = 0

where e1 = (1, 0, 0, 0 ), e2 = (0, 1, 0, 0 ), e3 = (0, 0, 1, 0 ), and e4 = ( 0, 0, 0, 1 ) The Galilean G4 studies all properties invariant under motions of objects in space is even more complex. In addition, it stated this geometry can described more precisely as the study of those properties of 4D space with coordinate which are invariant under general Galilean transformation as follows [8].

x = (cos β cos α − cos γ sin β sin α )x

The derivatives of the Frenet-Serret equations are deﬁned as in [9].

t ( s ) = k1 ( s )n ( s )

n ( s ) = k2 ( s )b1 ( s )

b 1 (s ) = −k2 (s )n(s ) + k3 (s )b1 (s ) b 2 (s ) = −k3 (s )b1 (s )

3. tb2 Smarandache curves in G4

+ (sin β cos α − cos γ cos β sin α )y + (sin γ sin α )z + (v cos δ1 )t + a

Deﬁnition 1. A curve in G4 , whose position vector is obtained by Frenet frame vectors on another curve, is called Smarandache curve.

y = −(cos β sin α + cos γ sin β cos α )x + (− sin β sin α + cos γ cos β cos α )y

Let us deﬁne special forms of Smarandache curves.

+ (sin γ cos α )z + (v cos δ2 )t + b

Deﬁnition 2. Let α (s) be a unit speed curve in G4 with constant curvatures k1 , k2 and k3 and {t(s), n(s), b1 (s), b2 (s)} be Frenet frame on it. The tb2 Smarandache curves are deﬁned by

z = (sin γ sin β )x − (sin γ cos β )y + (cos γ )z + (v cos δ3 )t + c

t = t + d

with cos2 δ1 + cos2 δ2 + cos2 δ3 = 1. A curve α : I → G4 of C∞ , I ⊂ R in the Galilean G4 is defined by α (s ) = (s, y(s ), z(s ), w(s )) where the curve α is parameterized by the Galilean invariant arc-length. The first Frenet-Serret frame , that is, the tangent vector of α (s) in G4 , is defined by

t (s ) = α (s ) = (1, y (s ), z (s ), w (s ))

(2.1)

The second vector of the Frenet-Serret frame , that is called, the principle normal of α (s) is deﬁned by n(s).

1 1 n (s ) = α (s ) = (0, y (s ), z (s ), w (s )) k1 ( s ) k1 ( s )

b1 ( s ) =

1 0, k2 ( s )

y (s )

k1 ( s )

z (s )

k1 ( s )

w (s )

k1 ( s )

Theorem 1. Let α = α (s ) be a unit speed curve with constant curvatures k1 (s), k2 (s) and k3 (s) and β (sβ (s)) be tb2 Smarandache curve deﬁned by frame vectors of α (s), then

tβ (sβ (s )) = nβ (sβ (s )) =

(2.4)

where k1 (s), k2 (s) and k3 (s) are the ﬁrst, second and third curvature functions of the curve α (s) which are deﬁned by

k1 (s ) = t (s ) = (y (s ))2 + (z (s ))2 + (w (s ))2 G4 k2 (s ) = n (s ) = n , n G4 G 4

k3 (s ) = b 1 (s ), b2 (s ) G4 If the curvature k1 (s), k2 (s) and k3 (s) are constants, then the curve α (s) is called W-curve. The set {t(s), n(s), b1 (s), b2 (s), k1 (s), k2 (s), k3 (s)} is called the Frenet-Serret pparatus of the curve α .

k1 n − k3 b1

k21 + k23

k2 k3 n + k1 k2 b1 − k23 b2

k22 k23 + k21 k22 + k43

b1β (sβ (s )) =

(−k1 k22 n + k3 (k22 + k23 )b1 + k1 k2 k3 b2 )

b2β (sβ (s )) =

(2.3)

Thus the vector b1 (s) is perpendicular to both t(s) and n(s). The second binormal vector of α (s) which is the fourth vector of the Frenet-Serret frame is deﬁned by b2 (s).

b2 ( s ) = t ( s ) × n ( s ) × b1 ( s )

(2.2)

The third vector of the Frenet-Serret frame , that is called, the ﬁrst binormal vector of is deﬁned by

β (sβ (s )) = √ (t (s ) + b2 (s ))

k1β (sβ (s )) = k2β (sβ (s )) =

k22 + k23

k22 k23 + k21 k22 + k43 2 2 k1 k3 k1 k2 + k43 − k22 k23 t k22 + k23 k21 + k23 k22 k23 + k21 k22 +

(

)

(

√ 2 k22 k23 + k21 k22 + k43

k43 )

k21 + k23 √ 2 k22 + k23

k21 + k23

k3β (sβ (s )) = 0 Proof. Let β = β (sβ (s )) be a tb2 Smarandache curve of the curve α (s). Then

β = β (sβ (s )) = √ (t (s ) + b2 (s )) β ( sβ ) =

dβ (sβ (s )) ds

2 d β ( sβ ) dsβ

dsβ ds

1 = √ (t (s ) + b 2 (s )) 2

1 = √ ( k1 n − k3 b1 ) 2

Please cite this article as: M. Elzawy, S. Mosa, Smarandache curves in the Galilean 4-space G4 , Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.04.008

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tÎ˛ =

k n â&#x2C6;&#x2019; k3 b1

1 Î˛ Âˇ ( sÎ˛ ) =

where

dsÎ˛ ds

Î˛ ( sÎ˛ ) = ÂˇÂˇ

(3.1)

k21 + k23

k21 +k23 â&#x2C6;&#x161;

dtÎ˛

dÎ˛ (sÎ˛ (s )) Âˇ

â&#x2C6;&#x161; 2(k1 n â&#x2C6;&#x2019; k3 b 1 )

nÎ˛ ( sÎ˛ ) =

k21 + k23

â&#x2C6;&#x161; 2 k22 k23 + k21 k22 + k43

(3.2)

k21 + k23

nÎ˛ ( sÎ˛ ) = Âˇ

Î˛ ÂˇÂˇ (sÎ˛ ) k k n + k1 k2 b1 â&#x2C6;&#x2019; k23 b2 = 2 3 Î˛ ÂˇÂˇ (sÎ˛ ) k22 k23 + k21 k22 + k43

â&#x2C6;&#x161; 2(â&#x2C6;&#x2019;k1 k22 n + k3 (k22 + k23 )b1 + k1 k2 k3 b2 )

k21 + k23

k2Î˛ (sÎ˛ ) = nÂˇÎ˛ (sÎ˛ )

(3.3)

k22 k23 + k21 k22 + k43

(3.4)

k21 + k23

b 1Î˛ ( sÎ˛ ) =

nÎ˛ ( sÎ˛ )

k2Î˛ ( sÎ˛ )

(

â&#x2C6;&#x2019;k1 k22 n + k3 k22 + k23

)

+ k23 b1 + k1 k2 k3 b2 k22 k23 + k21 k22 + k43

k22

nÎ˛ ( sÎ˛ ) =

)

(3.5)

= ct where ct is constant

db1Î˛ (sÎ˛ )

(3.7)

Theorem 2. Let Îą = Îą (s ) be a unit speed curve in G4 with constant curvatures k1 (s), k2 (s) and k3 (s). Then the Smarandache nb1 curves of Îą (s) has b2Î˛ (sÎ˛ ) = 0. Proof. Let Î˛ = Î˛ (sÎ˛ (s )) be a nb1 Smarandache curve of Îą (s). Then

k22 n

k22

k2 n â&#x2C6;&#x2019; k2 b1 â&#x2C6;&#x2019; k3 b2

(

)

2k22 + k23 2 + k3 b1 â&#x2C6;&#x2019; k2 k3 b2 2k22 + k23

)

(

)

â&#x2C6;&#x161; 2[k22 n + (k22 + k23 )n â&#x2C6;&#x2019; k2 k3 b2 ]

(4.4)

(2k22 + k23 )

â&#x2C6;&#x2019;k2 n + k2 b1 + k3 b2

2k22 + k23

Ă&#x2014;

[â&#x2C6;&#x2019;k22 n â&#x2C6;&#x2019; (k22 + k23 )b1 + k2 k3 b2 ]

(k22 + k23 ) (2k22 + k23 )

[k2 n â&#x2C6;&#x2019; k2 b1 â&#x2C6;&#x2019; k3 b2 ]

(2k22 + k23 )

= 0t + 0n + 0b1 + 0b2

(4.5)

DeďŹ nition 4. Let Îą = Îą (s ) be a curve in Galilean space G4 and {t(s), n(s), b1 (s), b2 (s)} be its moving Frenet frame . The tnb1 b2 Smarandache curves are deďŹ ned as

Î˛ (sÎ˛ (s )) = â&#x2C6;&#x161; (t (s ) + n(s ) + b1 (s ) + b2 (s ))

(4.6)

Remark 1. The Frenet- Serret invariants of tnb1 b2 Smarandache curves can easily obtained by the apparatus of the curve Îą = Îą (s ). Remark 2. There are another types of Smarandache curves in G4 such as tb1 , tn, nb2 , b1 b2 , tnb1 , tnb2 , tb1 b2 , nb1 b2 .

â&#x2C6;&#x161;

Îą = Îą (s ) = (s, sin s, 2 cos s, sin s )

tÎ˛

(4.3)

Example 1. Let us consider the following curve I â&#x160;&#x201A; R â&#x160;&#x201A; G4

Î˛ = Î˛ (sÎ˛ (s )) = â&#x2C6;&#x161; (n(s ) + b1 (s ))

k2Î˛ ( sÎ˛ )

and the proof is complete.

DeďŹ nition 3. Let Îą (s) be a unit speed curve in G4 with constant curvatures k1 (s), k2 (s), k3 (s) and {t(s), n(s), b1 (s), b2 (s)} be Frenet frame on it. The nb1 Smarandache curves in G4 are deďŹ ned by Î˛ (sÎ˛ (s )) = â&#x2C6;&#x161;1 (n(s ) + b1 (s ))

dsÎ˛

Ă&#x2014;

4. nb1 Smarandache curves in G4

nÂˇÎ˛ (sÎ˛ )

(3.6)

k3Î˛ (sÎ˛ ) = bÂˇ1Î˛ (sÎ˛ ), b2Î˛ (sÎ˛ ) G4 = 0

d Î˛ ( sÎ˛ )

(4.2)

(2k22 + k23 ) â&#x2C6;&#x161; 2 (k22 + k23 )

b 1Î˛ ( sÎ˛ ) =

So we can deduce that

Î˛ ( sÎ˛ ) =

(4.1)

b 2Î˛ ( sÎ˛ ) = tÎ˛ ( sÎ˛ ) Ă&#x2014; n Î˛ ( sÎ˛ ) Ă&#x2014; b 1Î˛ ( sÎ˛ )

(k31 k22 k3 + k1 k53 â&#x2C6;&#x2019; k1 k22 k33 )t k22 + k23 k21 + k23 (k22 k23 + k21 k22 + k43 )

and the proof is complete.

[â&#x2C6;&#x2019;k2 n + k2 b 1 + k3 b 2 ]

(2k22 + k23 ) â&#x2C6;&#x161; 2 (k22 + k23 ) k2Î˛ (sÎ˛ (s )) = (2k22 + k23 )

b 2Î˛ ( sÎ˛ ) = tÎ˛ ( sÎ˛ ) Ă&#x2014; n Î˛ ( sÎ˛ ) Ă&#x2014; b 1Î˛ ( sÎ˛ )

2k22 + k23

â&#x2C6;&#x161; 2 (k22 + k23 )[k2 n â&#x2C6;&#x2019; k2 b1 â&#x2C6;&#x2019; k3 b2 ]

Âˇ

dsÎ˛

2k22 + k23

Î˛ ÂˇÂˇ (sÎ˛ ) [â&#x2C6;&#x2019;k22 n â&#x2C6;&#x2019; (k22 + k23 )b1 + k2 k3 b2 ] = Î˛ ÂˇÂˇ (sÎ˛ ) (k22 + k23 ) (2k22 + k23 )

db1Î˛ (sÎ˛ )

Hence we can ďŹ nd Âˇ

From the second Frenet Serret equations we have

â&#x2C6;&#x161; 2 k22 + k23

nÂˇÎ˛ , nÂˇÎ˛ G4 =

â&#x2C6;&#x2019;k2 n + k2 b1 + k3 b2

Âˇ nÎ˛ (sÎ˛ ) =

Now

nÎ˛ ( sÎ˛ ) =

= dsÎ˛ k21 + k23 â&#x2C6;&#x161; 2(k2 k3 n + k1 k2 b1 â&#x2C6;&#x2019; k23 b2 )

k1Î˛ (sÎ˛ ) = tÎ˛Âˇ (sÎ˛ ) =

Î˛ Âˇ ( sÎ˛ ) =

tÎ˛ =

1 = â&#x2C6;&#x161; (n (s ) + b 1 (s ) 2

(4.7)

Differentiating (4.7) we have

â&#x2C6;&#x161;

1 = â&#x2C6;&#x161; (k2 b1 + (â&#x2C6;&#x2019;k2 n + k3 b2 )) 2 1 2 = â&#x2C6;&#x161; 2k2 + k23 2 d Î˛ ( sÎ˛ ) â&#x2C6;&#x2019;k2 n + k2 b1 + k3 b2 = = Î˛ Âˇ ( sÎ˛ ) = dsÎ˛ 2k2 + k2 2

Îą (s ) = (1, cos s, â&#x2C6;&#x2019;

2 sin s, cos s )

(4.8)

The Galilean inner product follows that Îą , Îą G4 = 1 . So the curve is parameterized by arc length and the tangent vector is (4.8). In order to calculate the ďŹ rst curvature let us express

â&#x2C6;&#x161; t (s ) = (0, â&#x2C6;&#x2019; sin s, â&#x2C6;&#x2019; 2 cos s, â&#x2C6;&#x2019; sin s )

Taking the norm of both sides, we have k1 (s ) =

â&#x2C6;&#x161; 2.

Please cite this article as: M. Elzawy, S. Mosa, Smarandache curves in the Galilean 4-space G4 , Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.04.008

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The principal normal n(s) becomes

√ 1 n(s ) = √ (0, − sin s, − 2 cos s, − sin s ) 2

(4.9)

One more differentiating of (4.9), we have

√ 1 n (s ) = √ (0, − cos s, 2 sin s, − cos s ) 2 By using n (s) we have the second curvature k2 (s ) = 1 and the ﬁrst binormal vector

√ 1 b1 (s ) = √ (0, − cos s, 2 sin s, − cos s ) 2

(4.10)

(4.11)

We can obtain easily the third curvature of the curve k3 (s ) = 0. The tb2 Smarandache curve of the curve α (s) is the curve β (sβ (s))

β (sβ (s )) = √

1, cos s −

1 √ 1 , − 2 sin s, cos s + √ 2 2

(4.16)

b 2β = ( 0 , 0 , 0 , 0 )

(4.17)

k3β = 0

(4.18)

In the same way we can obtain the nb1 Smarandache curve and its Frenet-Serret apparatus by using (4.1),(4.2).(4.3),(4.4),(4.5) of Theorem 2. Acknowledgement

The second binormal vector b2 (s ) = t (s ) × n(s )× b1 (s)

1 b2 (s ) = √ (0, −1, 0, 1 ) 2

√ 1 b1β = √ (0, sin s, 2 cos s, sin s ) 2

By using Theorem (1) above we can obtain easily the FrenetSerret apparatus of the curve β (sβ (s)).

√ 1 tβ = √ (0, − sin s, − 2 cos s, − sin s ) 2

(4.12)

√ 1 nβ = √ (0, − cos s, 2 sin s, − cos s ) 2

(4.13)

k1β = 1

(4.14)

k2β = 1

(4.15)

The authors are grateful to the referee for his/her valuable comments and suggestions. References [1] A. Ogrenmis, M. Ergut, M. Bekatas, On the helices in the Galilean space G3 , Iranian J. Sci. Technol. Trans. A 31 (A2) (2007) 177–181. Printed in The Islamic Republic of Iran. [2] M. Dede, Tubuler surfaces in Galilean space, Math. Commun. 18 (2013) 209–217. [3] M. Dede, C. Ekici, A. Coken, On the parallel surfaces in Galilean space, Hacettepe J. Math. Stat. 42 (6) (2013) 605–615. [4] R. S.Millman, G. D.Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977. [5] Z. Erjavec, B. Divjak, The equiform differential of curves in the pseudo-Galilean space, Math. Commun. 13 (2008) 321–332. [6] A. Ali, Special ,Smarandache curves in the Euclidean space, Int. J.Math. Combin. 2 (2010) 30–36. [7] H. Oztekin, S. Tatlipinar, Determination of the position vectors of curves from intrinsic equations in G3 , Walailak J.Sci. Tech. 11 (12) (2014) 1011–1018. [8] M. Bektas, M. Ergut, A.O. Ogrenmus, Special curves of 4D Galilean space, Int. J. Math. Eng. Sci. 2 (3) (2013). [9] M. E.Aydin, M. Ergut, The equiform differential geometry of curves in 4-dimensional Galilean space G4 , Stud. Univ. Babes-Bolyai Math. 58 (3) (2013) 399–406.

Please cite this article as: M. Elzawy, S. Mosa, Smarandache curves in the Galilean 4-space G4 , Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.04.008