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International J.Math. Combin. Vol.3 (2008), 51-55 


Smarandache Curves in Minkowski Space-time 


Melih Turgut and Süha Yilmaz 


(Department of Mathematics of Buca Educational Faculty of Dokuz Eylül University, 35160 Buca-Izmir,Turkey. ) 


E-mail: melih.turgut@gmail.com, suha.yilmaz@yahoo.com 


Abstract: A regular curve in Minkowski space-time, whose position vector is composed 
by Frenet frame vectors on another regular curve, is called a Smarandache Curve. In this 
paper, we define a special case of such curves and call it Smarandache TB2 Curves in the 
space Ef. Moreover, we compute formulas of its Frenet apparatus according to base curve 
via the method expressed in [3]. By this way, we obtain an another orthonormal frame of 
Ej. 

Key Words: Minkowski space-time, Smarandache curves, Frenet apparatus of the curves. 


AMS(2000): 53C50, 51B20. 
§1. Introduction 


In the case of a differentiable curve, at each point a tetrad of mutually orthogonal unit vectors 
(called tangent, normal, first binormal and second binormal) was defined and constructed, 
and the rates of change of these vectors along the curve define the curvatures of the curve in 
Minkowski space-time [1]. It is well-known that the set whose elements are frame vectors and 
curvatures of a curve, is called Frenet Apparatus of the curves. 

The corresponding Frenet’s equations for an arbitrary curve in the Minkowski space-time 
E} are given in [2]. A regular curve in Minkowski space-time, whose position vector is composed 
by Frenet frame vectors on another regular curve, is called a Smarandache Curve. We deal with 
a special Smarandache curves which is defined by the tangent and second binormal vector 
fields. We call such curves as Smarandache TB Curves. Additionally, we compute formulas 
of this kind curves by the method expressed in [3]. We hope these results will be helpful to 


mathematicians who are specialized on mathematical modeling. 


§2. Preliminary notes 


To meet the requirements in the next sections, here, the basic elements of the theory of curves 
in the space Ef are briefly presented. A more complete elementary treatment can be found in 
the reference [1]. 

Minkowski space-time E# is an Euclidean space E4 provided with the standard flat metric 
given by 


lReceived August 16, 2008. Accepted September 2, 2008. 


52 Melih Turgut and Suha Yilmaz 


g = —dx? + dx3, + dx? + dai, 


where (x1, £2, £3, 24) is a rectangular coordinate system in Ef. 

Since g is an indefinite metric, recall that a vector v € Ef can have one of the three 
causal characters; it can be space-like if g(v,v) > 0 or v = 0, time-like if g(v,v) < 0 and null 
(light-like) if g(v,v)=0 and v 4 0. Similarly, an arbitrary curve a = a(s) in Ej can be locally 
be space-like, time-like or null (light-like), if all of its velocity vectors a’(s) are respectively 
space-like, time-like or null. Also, recall the norm of a vector v is given by ||vl| = \/|g(v, v). 
Therefore, v is a unit vector if g(v,v) = +1. Next, vectors v, w in E} are said to be orthogonal 





if g(v, w) = 0. The velocity of the curve a(s) is given by ||a’(s)|| . 

Denote by {T(s), N(s), Bi(s), Bo(s)} the moving Frenet frame along the curve a(s) in 
the space E}. Then T, N, B,, B2 are, respectively, the tangent, the principal normal, the first 
binormal and the second binormal vector fields. Space-like or time-like curve a(s) is said to be 





parametrized by arclength function s, if g(a’(s),a’(s)) = +1. 
Let a(s) be a curve in the space-time Ef, parametrized by arclength function s. Then for 
the unit speed space-like curve @ with non-null frame vectors the following Frenet equations 


are given in [2]: 


T 0 k 0 0 T 
N’ -k 0 7 0 N 
m ’ (1) 
By 0 -r o Bı 
B; 0 0 o Bə 


where T, N, Bı and Bz are mutually orthogonal vectors satisfying equations 
g(T,T) = IN, N) = g(Bi, Bi) T 1, g(B2, B2) =-1. 


Here «,7 and o are, respectively, first, second and third curvature of the space-like curve a. 
In the same space, in [3] authors defined a vector product and gave a method to establish the 


Frenet frame for an arbitrary curve by following definition and theorem. 


Definition 2.1 Let a = (a1, a2, a3, a4), b = (bı, b2, b3,b4) and c = (c1, C2, C3, C4) be vectors in 


E$. The vector product in Minkowski space-time E} is defined by the determinant 


—€] €2 €3 ÈA 
a a a a 

a\b\c=— i a ; (2) 
bı bo b3 b4 


Ci C2 C3 C4 


where e€1,€2,e3 and e4 are mutually orthogonal vectors (coordinate direction vectors) satisfying 


equations 


ei Nea \e3 =e€4 , C2 Neg Nea =61 ,€e3Ne4AN€1 =e€2 , €4 ^ei A €2 = —€3. 


Smarandache Curves in Minkowski Space-time 53 


Theorem 2.2 Let a = a(t) be an arbitrary space-like curve in Minkowski space-time Ef with 
above Frenet equations. The Frenet apparatus of a can be written as follows; 


a’ 


T= Tall’ os 


llo“||? a” — g(a’, a”’).a! 














N= (4) 
le"? a!” — g(a’, a"”’).a! 

By =uNATA Bo, (5) 

TANAa” 

By = p, 6 

2 PUT AN Aa] (6) 
ha"? a” = g(a’, a”). 

ge l (7) 


4 
la| 


T N In . 1 
„TAN ^a" lol (8) 


lei? a!” — g(a’, a").a’ 








and 


gal), Bə) 


° = EAN najot 


where u is taken —1 or +1 to make +1 the determinant of |T, N, Bi, B2] matriz. 
§3. Smarandache Curves in Minkowski Space-time 


Definition 3.1 A regular curve in Eł, whose position vector is obtained by Frenet frame vectors 


on another regular curve, is called a Smarandache Curve. 


Remark 3.2 Formulas of all Smarandache curves’ Frenet apparatus can be determined by the 
expressed method. 


Now, let us define a special form of Definition 3.1. 


Definition 3.3 Let € = €(s) be an unit space-like curve with constant and nonzero curvatures 

k,T and o; and {T, N, Bı, B2} be moving frame on it. Smarandache TB curves are defined 
with í 

X = X (sx) = —= (T (s) + Ba(s)). 10 

(x) = Faas CO + Pale) (10) 


Theorem 3.4 Let € = &(s) be an unit speed space-like curve with constant and nonzero cur- 
vatures k, T and o and X = X (sx) be a Smarandache TB curve defined by frame vectors of 
€ =€(s). Then 


54 Melih Turgut and Suha Yilmaz 


(i) The curve X = X(sx) is a space-like curve. 
(it) Frenet apparatus of {Tx,Nx,Bix, Box,kx,Tx,0x} Smarandache TB curve X = 
X(sx) can be formed by Frenet apparatus {T, N, B1, Bo, K, T,0o} of € = €(s). 


Proof Let X = X(sx) be a Smarandache TBə2 curve defined with above statement. Dif- 
ferentiating both sides of (10), we easily have 


dX dsx 1 
— = —— (kN +0 B). 11 
dsx ds K2(s) + 02(s) ( 1) L 
The inner product g(X’, X’) follows that 
GX", X’) = 1, (12) 


where ’ denotes derivative according to s. (12) implies that X = X(sx) is a space-like curve. 


Thus, the tangent vector is obtained as 
1 
Tx = ——= (KN + 0B). 13 
= KETO, i) (13) 


Then considering Theorem 2.1, we calculate following derivatives according to s: 





1 
X" = ——— (-6°T — ToN + KTB, +0°Ba). 14 
Vere : 2) 
mw = 1 3 ool f 3 2 
X = er K? — KT“ )N + (0° — T°0) By + KTO Ba). (15) 
(IV) 1 
O ee (TE + (IN + (...) Ba + (0t — 720) Bo]. (16) 


Vere 
Then, we form 
1 
Vator 


Equation (17) yields the principal normal of X as 


IXI? X" — g(X', X").X' = |-K°T — ToN + «7B, + By]. (17) 





2T N +rKrBı +0B 
Nx = K TO. KT D1 oO 2 (18) 
Van F T202 + eT? + OF 


Thereafter, by means of (17) and its norm, we write first curvature 


[—K4 + 720? + K272 + 0? 
© = ee ee 19 
= k? +0? a 


The vector product Ty A Nx A X” follows that 


Tx \Nx AX” = L [Ko(K? +0°)(T? — o)T + To? (K? +0)N 


, (20) 
A —r?To (k? + 0o)B1 + KT(K? + 0?) (K? + 77) Bo] 


where, A = 7 Shortly, let us denote Ty A Nx A X” with aT + 


lgN + 13B, + l4B2. And therefore, we have the second binormal vector of X = X(sx) as 


LT +bN+13B,4+UB 
Braja + l21V + l3Dı + l4 2 (21) 


V- +++ 


Smarandache Curves in Minkowski Space-time 55 


Thus, we easily have the second and third curvatures as follows: 





(= + +1 4+ (K +6?) 





= _ 3 AN 22 
fi Spo HRT bo (72) 
2/72 2 
pa a*(o* — T°) (23) 
(K2? + 0?) /- +15 +13 +l 
Finally, the vector product Nx A Tx A Box gives us the first binormal vector 
1 [(Kol3 — o?lz — T(K? +07 )la]T — o(r?l4 + ol) N 

Bix = ET , (24) 


+K(K7l4 + ol) Bi + [k?(ol2 — 6713) + Th (K? + 07)] Bo 





1 
(—13 412 +12 412) (62 +0?) (—K 447202 44272402) j 








where L = 





Thus, we compute Frenet apparatus of Smarandache TBə curves. 


Corollary 3.1 Suffice it to say that {Tx, Nx,Bix, B2x} is an orthonormal frame of Et. 


Acknowledgement 


The first author would like to thank TUBITAK-BIDEB for their financial supports during his 
Ph.D. studies. 


References 


[1] B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983. 

[2] J. Walrave, Curves and surfaces in Minkowski space. Dissertation, K. U. Leuven, Fac. of 
Science, Leuven, 1995. 

[3] S. Yilmaz and M. Turgut, On the Differential Geometry of the curves in Minkowski space- 
time I, Int. J. Contemp. Math. Sci. 3(27), 1343-1349, 2008.