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. )


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 B; 0 0 o

where T, N, 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) bo b3 b4

Ci C2 C3 C4

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


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;


T= Tall’ os

llo“||? a” g(a’, a”’).a!

N= (4) le"? a!” g(a’, a"”’).a!

By =uNATA Bo, (5)


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’


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.


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


[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.