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On Smarandache Curves Lying in Lightcone inMinkowski 3-SpaceUfuk Ozturkab, Esra Betul Koc Ozturkab, Kazim İlarslanc & Emilija Neŝovićda Department of Mathematics, Faculty of Sciences, University of Çankiri Karatekin, 18100,Çankiri, Turkeyb School of Mathematics & Statistical Sciences, Arizona State University, Tempe, Arizona85287-1804c Department of Mathematics, Faculty of Sciences and Art, University of Kirikkale, 71450Kirikkale, Turkeyd Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science,34000 Kragujevac, Serbia. E-MAILS: , , ,Published online: 08 Jul 2014.

To cite this article: Ufuk Ozturk, Esra Betul Koc Ozturk, Kazim İlarslan & Emilija Neŝović (2014) On Smarandache Curves Lyingin Lightcone in Minkowski 3-Space, Journal of Dynamical Systems and Geometric Theories, 12:1, 81-91

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ON SMARANDACHE CURVES LYING IN LIGHTCONE IN

MINKOWSKI 3-SPACE

UFUK OZTURK1, ESRA BETUL KOC OZTURK1, KAZIM ILARSLAN2,

EMILIJA NESOVIC3

1DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES, UNIVERSITY OF

CANKIRI KARATEKIN, 18100 CANKIRI, TURKEY

SCHOOL OF MATHEMATICS & STATISTICAL SCIENCES, ARIZONA STATE

UNIVERSITY, TEMPE, ARIZONA 85287-1804

2DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ART, UNIVERSITY

OF KIRIKKALE, 71450 KIRIKKALE, TURKEY

3DEPARTMENT OF MATHEMATICS AND INFORMATICS, UNIVERSITY OF

KRAGUJEVAC, FACULTY OF SCIENCE, 34000 KRAGUJEVAC, SERBIA

E-MAILS: UUZTURK@ASU.EDU, EKOCOZTU@ASU.EDU, KILARSLAN@YAHOO.COM,

NESOVICKG@SBB.RS

Abstract. In this paper we define the spacelike and the null lightcone Smaran-

dache curves of a spacelike lightcone curve α according to the lightcone Frenet

frame of a α in Minkowski 3-space. We obtain the lightcone curvature and the

expressions for the lightcone Frenet frame’s vectors of a spacelike Smarandache

curve of α. We prove that if the null straight line is the lightcone Smarandache

curve of α, then α has non-zero constant lightcone curvature. Finally, we give

some examples of lightcone Smarandache curves.

(Received: 4 April 2014, Accepted: 5 May 2014)

AMS Classification: 53B30, 53A35, 53C22.

Keywords: Smarandache curves; Pseudosphere; Sabban frame; Geodesic curva-

ture; Minkowski space

JOURNAL OF DYNAMICAL SYSTEMS & GEOMETRIC THEORIES

VOL. 12, NUMBER 1 (2014) 81-91.

c©TARU PUBLICATIONS

81

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1. Introduction

It is known that a Smarandache geometry is a geometry which has at least one

Smarandachely denied axiom ([2]). An axiom is said to be Smarandachely 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 vector of a curve β is composed by the Frenet frame’s vectors of another

curve α, then the curve β is called a Smarandache curve ([9]). Special Smarandache

curves in the Euclidean and Minkowski spaces are studied by some authors ([1, 7,

8, 3, 4]). In this paper we define the spacelike and the null lightcone Smarandache

curves of a spacelike lightcone curve α according to the lightcone Frenet frame of

a α in Minkowski 3-space. We obtain the lightcone curvature and the expressions

for the lightcone Frenet frame’s vectors of a spacelike Smarandache curve of α. We

prove that if the null straight line is the lightcone Smarandache curve of α, then

α has non-zero constant lightcone curvature. Finally, we give some examples of

lightcone Smarandache curves in Minkowski 3-space.

2. Basic Concepts

The Minkowski 3-space R31 is the Euclidean 3-space R3 provided with the stan-

dard flat metric given by

(1) 〈· , ·〉 = −dx21 + dx2

2 + dx23,

where (x1, x2, x3) is a rectangular coordinate system of R31. Since g is an indefinite

metric, recall that a non-zero vector ~x ∈ R31 can have one of three Lorentzian

causal characters: it can be spacelike if 〈~x, ~x〉 > 0, timelike if 〈~x, ~x〉 < 0 and null

(lightlike) if 〈~x, ~x〉 = 0. In particular, the norm (length) of a vector ~x ∈ R31 is given

by ‖~x‖ =√|〈~x, ~x〉| and two vectors ~x and ~y are said to be orthogonal, if 〈~x, ~y〉 = 0.

Next, recall that an arbitrary curve α = α(s) in E31, can locally be spacelike, timelike

or null (lightlike), if all of its velocity vectors α′(s) are respectively spacelike, timelike

or null (lightlike) for all s ∈ I([6]). A spacelike or a timelike curve α is parameterized

by arclength parameter s if 〈α′(s), α′(s)〉 = 1 or 〈α′(s), α′(s)〉 = −1 respectively.

For any two vectors ~x = (x1, x2, x3) and ~y = (y1, y2, y3) in the space R31, the

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ON SMARANDACHE CURVES ... — JDSGT VOL. 12, NUMBER 1 (2014) 83

pseudo vector product of ~x and ~y is defined by

~x× ~y =

∣∣∣∣∣∣∣∣−~e1 ~e2 ~e3

x1 x2 x3

y1 y2 y3

∣∣∣∣∣∣∣∣= (−x2y3 + x3y2, x3y1 − x1y3, x1y2 − x2y1).(2)

Lemma 2.1. Let ~x, ~y and ~z be the vectors in R31. Then:

(i) 〈~x× ~y, ~z〉 = det(~x, ~y, ~z),

(ii) ~x× (~y × ~z) = −〈~x, ~z〉 ~y + 〈~x, ~y〉~z,(iii) 〈~x× ~y, ~x× ~y〉 = −〈~x, ~x〉 〈~y, ~y〉 + 〈~x, ~y〉2,

where × is the pseudo vector product in R31.

Lemma 2.2. In the Minkowski 3-space R31, following properties are satisfied ([6]):

(i) two timelike vectors are never orthogonal;

(ii) two null vectors are orthogonal if and only if they are linearly dependent;

(iii) timelike vector is never orthogonal to a null vector.

The lightcone in the Minkowski 3-space R31 is a quadric defined by

C ={~x ∈ R3

1

∣∣ 〈~x, ~x〉 = 0}.

Let α : I ⊂ R −→ C be a curve lying fully in the lightcone C in R31. Then α is

a spacelike curve or a null straight line.

Assume that α is a spacelike lightcone curve. Then there exists the lightcone

Frenet frame {α, T, ξ} along the curve α, such that the following conditions hold:

(3) 〈α, α〉 = 〈ξ, ξ〉 = 〈α, T 〉 = 〈T, ξ〉 = 0, 〈α, ξ〉 = 〈T, T 〉 = 1.

It follows that

(4) α× T = α, T × ξ = ξ, ξ × α = T.

The Frenet formulae of α, according to the lightcone Frenet frame read [5]

(5)

α′

T ′

ξ′

=

0 1 0

k 0 −1

0 −k 0

α

T

ξ

.

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84 UFUK OZTURK, ESRA BETUL KOC OZTURK, KAZIM ILARSLAN, EMILIJA NESOVIC

where k(s) = 〈T ′(s), ξ(s)〉 6= 0 is the lightcone curvature of α and s is the arclength

parameter of α. If α is a null straight line lying in lightcone, then it hasn’t got

lightcone Frenet frame, so we exclude this case in this paper.

3. Lightcone Smarandache Curves in Minkowski 3-space

In this section, we consider the spacelike lightcone curve α and define its light-

cone Smarandache curve according to the lightcone Frenet frame {α, T, ξ} of α in

Minkowski 3-space.

Definition 3.1. Let α : I ⊂ R 7→ C be a unit speed spacelike curve lying fully in

lightcone C with the lightcone Frenet frame {α, T, ξ}. The lightcone αTξ-Smarandache

curve β of α is defined by

(6) β(s) = aα(s) + bT (s) + cξ(s),

where s is arclength parameter of α, a, b, c ∈ R0 and b2 + 2ac = 0.

Note that the lightcone αTξ-Smarandache curve β of α can be a spacelike or a

null curve.

In the sequel, we obtain the lightcone curvature and the expressions for the

ligthcone Frenet frame’s vectors of a spacelike lightcone αTξ-Smarandache curve β

of α. We also prove that if β is a null lightcone αTξ-Smarandache curve of α, then

α has non-zero constant lightcone curvature.

Finally, we give some examples of lightcone Smarandache curves in Minkowski

3-space.

Theorem 3.1. Let α : I ⊂ R 7→ C be a unit speed spacelike curve lying fully in

lightcone C with the lightcone Frenet frame {α, T, ξ} and the lightcone curvature

k(s) 6= 0. If β : I ⊂ R 7→ C is a spacelike lightcone αTξ-Smarandache curve of α,

then its frame {β, Tβ , ξβ} is given byβ

Tβ

ξβ

=

a b c

bk(s)|a+ck(s)|

a−ck(s)|a+ck(s)| − b

|a+ck(s)|ck2(s)

(a+ck(s))2 − bk(s)(a+ck(s))2

a(a+ck(s))2

α

T

ξ

,and the corresponding lightcone curvature kβ reads

kβ(s) =1

ε0(a+ ck(s))3[aε1(s)− bk(s)ε2(s) + ck2(s)ε3(s)]

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ON SMARANDACHE CURVES ... — JDSGT VOL. 12, NUMBER 1 (2014) 85

where

ε0 = sgn{a+ ck(s)} = ±1,

ε1(s) =abk′2 − c2k2(s))

ε0(a+ ck(s))2,

ε2(s) =2bk(s)(a+ ck(s))− 2ack′(s)

ε0(a+ ck(s))2,

ε3(s) =bck′2 + c2k2(s)

ε0(a+ ck(s))2,

a, b, c ∈ R0, b2 + 2ac = 0 and a+ ck(s) 6= 0 for all s.

Proof. Assume that β is a spacelike lightcone αTξ-Smarandache curve of α. Then

β has parameter equation

(7) β(s) = aα(s) + bT (s) + cξ(s),

where a, b, c ∈ R0, b2 + 2ac = 0 and s is the arclength parameter of α. Denote by

s∗ the arclength parameter of β. Differentiating the equation (7) with respect to s

and using (5) we obtain

(8) β′(s) =dβ

ds∗ds∗

ds= bk(s)α(s) + (a− ck(s))T (s)− bξ(s).

Since Tβ = dβ/ds? and

(9)ds∗

ds= ||β′(s)|| =

√(a− ck(s))

2 − 2b2k(s) = |a+ ck(s)|,

relation (8) becomes

(10) Tβ(s) =1

|a+ ck(s)|(bk(s)α(s) + (a− ck(s))T (s)− bξ(s))

where |a+ ck(s)| 6= 0 for all s.

Next we will determine the vector ξβ(s) of the lightcone Frenet frame {β, Tβ , ξβ},such that it satisfies the following conditions:

(11) 〈ξβ , ξβ〉 = 0, 〈ξβ , Tβ〉 = 0, 〈β, ξβ〉 = 1.

With respect to the lightcone Frenet frame {α, T, ξ}, the vector ξβ can be decom-

posed as

(12) ξβ(s) = m(s)α(s) + n(s)T (s) + p(s)ξ(s),

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where m(s), n(s) and p(s) are some scalar functions in arclength parameter s of α.

Relations (3), (7), (10), (11) and (12) imply the system of equations

n2(s) + 2m(s)p(s) = 0,

ap(s) + bn(s) + cm(s) = 1,

bp(s)k(s) + (a− ck(s))n(s)− bm(s) = 0.

A straightforward calculation shows that the solution of the previous system of

equations is given by

m(s) =ck2(s)

(a+ ck(s))2n(s) = − bk(s)

(a+ ck(s))2, p(s) =

a

(a+ ck(s))2.

Substituting this in (12), we obtain

(13) ξβ(s) =ck2(s)

(a+ ck(s))2α(s)− bk(s)

(a+ ck(s))2T (s) +

a

(a+ ck(s))2ξ(s).

Differentiating the equation (10) with respect to s and using (9), it follows that

(14)dTβds∗

ds∗

ds= T ′β |a+ ck(s)| = ε1(s)α(s) + ε2(s)T (s) + ε3(s)ξ(s),

where

ε1(s) =abk′2 − c2k2(s))

ε0(a+ ck(s))2,

ε2(s) =2bk(s)(a+ ck(s))− 2ack′(s)

ε0(a+ ck(s))2,

ε3(s) =bck′2 + c2k2(s)

ε0(a+ ck(s))2,

Finally, from (13) and (14) we obtain that the lightcone curvature kβ of β is given

by

(15) kβ(s) = 〈T ′β(s), ξβ(s)〉 =1

ε0(a+ ck(s))3(aε1(s)− bk(s)ε2(s) + ck2(s)ε3(s)),

where ε0 = sgn{a+ ck(s)} = ±1. �

Theorem 3.2. Let α : I ⊂ R 7→ C be a unit speed spacelike curve lying fully in

lightcone C with the lightcone Frenet frame {α, T, ξ} and the lightcone curvature

k(s) 6= 0. If γ is a null lightcone αTξ-Smarandache curve of α, then α has non-zero

constant lightcone curvature given by

k(s) =b2

2c2,

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ON SMARANDACHE CURVES ... — JDSGT VOL. 12, NUMBER 1 (2014) 87

where a, b, c ∈ R0 and b2 + 2ac = 0.

Proof. Assume that γ is a null lightcone αTξ-Smarandache curve of α. Then γ is

given by

γ(s) = aα(s) + bT (s) + cξ(s),

where s is arclength parameter of α, a, b, c ∈ R0 and b2 + 2ac = 0. Differentiating

the previous equation with respect to s and using (5), we obtain

Tγ(s) = γ′(s) = bk(s)α(s) + (a− ck(s))T (s)− bξ(s).

The condition 〈Tγ(s), Tγ(s)〉 = 0 implies

(16) (a− ck(s))2 − 2b2k(s) = 0.

By assumption there holds b2 + 2ac = 0 and thus b4 − 4a2c2 = 0. By using the last

equation and (16), we find

k(s) =b2 ±

√b4 − 4a2c2

2c2=

b2

2c2,

which proves the Theorem. �

4. Examples

Example 4.1. Let α be a unit speed spacelike curve lying in lightcone in Minkowski

3-space with parameter equation (see Figure 1)

(17) α(s) = (2 cosh(s

2), 2 sinh(

s

2), 2).

The lightcone Frenet frame {α, T, ξ} along the curve α is given by

α(s) = (2 cosh(s

2), 2 sinh(

s

2), 2),

T (s) = α′(s) = (sinh(s

2), cosh(

s

2), 0),

ξ(s) = (−cosh( s2 )

4,−

sinh( s2 )

4,

1

4).

In particular, the lightcone curvature k(s) of the curve α is given by

(18) k(s) = 〈T ′(s), ξ(s)〉 =1

8.

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88 UFUK OZTURK, ESRA BETUL KOC OZTURK, KAZIM ILARSLAN, EMILIJA NESOVIC

Figure 1. The curve α

Taking a = 1, b =√

2 and using the relation b2 +2ac = 0 we find c = −1. It follows

that the lightcone αTξ-Smarandache curve β of α is given by (see Figure 2)

β(s) = α(s) +√

2T (s)− ξ(s)

= (e

s2 (9 + 4

√2) + e−

s2 (9− 4

√2)

8,e

s2 (9 + 4

√2)− e− s

2 (9− 4√

2)

8,

7

4)

It can be easily checked that 〈β′(s), β′(s)〉 > 0, which means that β is a spacelike

curve. According to Theorem 3.1, its lightcone Frenet frame {β, Tβ , ξβ} is given byβ

Tβ

ξβ

=

1

√2 −1

√2

797 − 8

√2

7

− 149 − 8

√2

496449

α

T

ξ

.By using relation (15), we obtain that the lightcone curvature kβ of β reads

kβ(s) =8

49.

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ON SMARANDACHE CURVES ... — JDSGT VOL. 12, NUMBER 1 (2014) 89

Figure 2. The lightcone αTξ-Smarandache curve β and the curve α

Example 4.2. Let us consider a unit speed spacelike curve α lying on lightcone in

the Minkowski 3-space with parameter equation (17). Then its lightcone curvature is

given by (18). Denote by β a null lightcone αTξ-Smarandache curve of α. According

to Theorem 3.2, the curve α has non-zero constant lightcone curvature given by

k(s) =b2

2c2, b, c ∈ R0.

The last relation together with (18) implies b = ± c2 . By Definition 3.1 there holds

b2 + 2ac = 0 and therefore a = − c8 . Choosing c = 1 and b = c

2 , we find a = − 18 and

b = 12 . Consequently, the null lightcone αTξ-Smarandache curve β of α is given by

(see Figure 3)

β(s) = −1

8α(s) +

1

2T (s) + ξ(s) =

1

2e−s(−1, 1, 0),

which means that β is a null straight line.

References

[1] Ahmad T. Ali, Special Smarandache Curves in the Euclidean Space, International Journal of

Mathematical Combinatorics, Vol. 2, 30–36 (2010).

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Figure 3. The null lightcone αTξ -Smarandache curve β and the

curve α

[2] Charles Ashbacher, Smarandache Geometries, Smarandache Notions Journal, Vol. 8, Number

1-3, 212–215 (1997).

[3] Talat Korpinar and Essin Turhan, A new approach on Smarandache TN-curves in terms of

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[6] Barrett O’neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press,

New Tork, 1983.

[7] Esra Betul Koc Ozturk, Ufuk Ozturk, Kazım Ilarslan and Emilija Nesovic, On pseudohyper-

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