SMARANDACHE CURVES ACCORDING TO CURVES ON A SPACELIKE SURFACE IN MINKOWSKI 3-SPACE

SMARANDACHE CURVES ACCORDING TO CURVES ON A SPACELIKE SURFACE IN MINKOWSKI 3-SPACE R31 UFUK OZTURK AND ESRA BETUL KOC OZTURK

Abstract. In this paper, we introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space R31 . Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. And, we give an example to illustrate these curves.

1. Introduction In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problem is the characterization of a regular curve. In the solution of the problem, the curvature functions κ and τ of a regular curve have an effective role. It is known that the shape and size of a regular curve can be determined by using its curvatures κ and τ . Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, i.e. a geometry which has at least one Smarandachely denied axiom ([3]). The axiom is said 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.

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 ([7]). Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors ([1, 4, 5, 6, 9, 10]). For instance the special Smarandache curves according to Darboux frame in E3 are characterized in [8].

In this paper, we define Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space R31 . Inspired by above papers we investigate the geodesic curvature and the Sabban Frame’s vectors of Smarandache curves. In section 2, we explain the basic concepts of Minkowski 3-space and give Lorentzian Darboux frame that will be use throughout the paper. Section 3 is devoted to the study of four Smarandache curves, Tη-Smarandache curve, Tξ-Smarandache curve, ηξ-Smarandache curve and Tηξ-Smarandache curve by considering the relationship with invariants kn , kg (s) and τg (s) of curve on spacelike surface in Minkowski 3-space R31 . Also, we give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. Finally, we illustrate these curves with an example.

2. Basic Concepts The Minkowski 3-space metric given by (2.1)

R31

is the Euclidean 3-space R3 provided with the standard flat h·, ·i = −dx21 + dx22 + dx23 ,

2010 Mathematics Subject Classification. 53A04;53A35;53B30;53C22.

Key words and phrases. Smarandache curves, Darboux Frame, Sabban frame, geodesic curvature, Lorentz–Minkowski space.

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U. OZTURK AND E. B. KOC OZTURK

where (x1 , x2 , x3 ) is a rectangular Cartesian coordinate system of R31 . Since h·, ·i 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 hx, xi > 0, timelike if hx, xi < 0 and null 3 (lightlike) p if hx, xi = 0. In particular, the norm (length) of a vector x ∈ R1 is given by kxk = |hx, xi| and two vectors x and y are said to be orthogonal, if hx, yi = 0. For any x = (x1 , x2, x3 ) and y = (y1 , y2 , y3 ) in the space R31 , the pseudo vector product of x and y is defined by (2.2)

x × y = (−x2 y3 + x3 y2 , x3 y1 − x1 y3 , x1 y2 − x2 y1 ).

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 α0 (s) are respectively spacelike, timelike or null (lightlike) for every s ∈ I.([2]). If kα0 (s)k 6= 0 for every s ∈ I, then α is a regular curve in R31 . A spacelike (timelike) regular curve α is parameterized by pseudo-arclength parameter s which is given by α : I ⊂ R −→ R31 , then the tangent vector α0 (s) along α has unit length, that is, hα0 (s), α0 (s)i = 1 (hα0 (s), α0 (s)i = −1) for all s ∈ I, respectively.

Remark 1. Let x = (x1 , x2, x3 ), y = (y1 , y2 , R31 . Then:

x1

(i) hx × y, zi =

y1

z1 (ii) x × (y × z) = − hx, (iii) hx × y, x × yi = − hx,

y3 ) and z = (z1 , z2 , z3 ) be vectors in

x2 x3

y2 y3

z2 z3

zi y + hx, yi z, xi hy, yi + hx, yi2 ,

where × is the pseudo vector product in the space R31 .

Lemma 1. In the Minkowski 3-space R31 , following properties are satisfied ([2]): (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.

Let φ : U ⊂ R2 −→ R31 , φ(U ) = M and γ : I ⊂ R −→ U be a spacelike embedding and a regular curve, respectively. Then we have a curve α on the surface M is defined by α(s) = φ(γ(s)) and since φ is a spacelike embedding, we have a unit timelike normal vector field η along the surface M is defined by η≡

(2.3)

φx × φy . kφx × φy k

Since M is a spacelike surface, we can choose a future directed unit timelike normal vector field η along the surface M . Hence we have a pseudo-orthonormal frame {T, η, ξ} which is called the Lorentzian Darboux frame along the curve α where ξ(s) = T(s) × η(s) is an unit spacelike vector. The corresponding Frenet formulae of α read  0     T 0 kn kg T 0  η  =  kn 0 τg   η  , (2.4) ξ0 −kg τg 0 ξ where kn (s) = − hT0 (s), η(s)i, kg (s) = hT0 (s), ξ(s)i and τg (s) = − hξ 0 (s), η(s)i are the asymptotic curvature, the geodesic curvature and the principal curvature of α on the surface M in R31 , respectively, and s is arclength parameter of α. In particular, the following relations hold (2.5)

T × η = ξ,

η × ξ = −T,

ξ × T = η.

Both kn and kg may be positive or negative. Specifically, kn is positive if α curves towards the normal vector η, and kg is positive if α curves towards the tangent normal vector ξ.

SMARANDACHE CURVES IN MINKOWSKI 3-SPACE R3 1

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Also, the curve α is characterized by kn , kg and τg as the follows:   an asymptotic curve iff kn ≡ 0, a geodesic curve iff kg ≡ 0, (2.6) α is  a principal curve iff τg ≡ 0.

Since α is a unit-speed curve, α ¨ is perpendicular to T, but α ¨ may have components in the normal and tangent normal directions: α ¨ = kn η+kg ξ.

These are related to the total curvature κ of α by the formula κ2 = kαk ¨ 2 = kg2 − kn2 .

(2.7)

From (2.7) we can give the following Proposition.

Proposition 1. Let M be a spacelike surface in R31 . Let α = α(s) be a regular unit speed curves lying fully with the Lorentzian Darboux frame {T, η, ξ} on the surface M in R31 .

There is not a geodesic curve on M .

The pseudosphere with center at the origin and of radius r = 1 in the Minkowski 3-space R31 is a quadric defined by

S12 = ~ x ∈ R31 − x21 + x22 + x23 = 1 .

Let β : I ⊂ R −→ S12 be a curve lying fully in pseudosphere S12 in R31 . Then its position vector β is a spacelike, which means that the tangent vector Tβ = β 0 can be a spacelike, a timelike or a null. Depending on the causal character of Tβ , we distinguish the following three cases, [5].

Case 1. Tβ is a unit spacelike vector Then we have orthonormal Sabban frame {β(s), Tβ (s), ξβ (s)} along the curve α, where ξβ (s) = −β(s) × Tβ (s) is the unit timelike vector. The corresponding Frenet formulae of β, according to the Sabban frame read  0     0 1 0 β β ¯g (s)   Tβ  .  Tβ0  =  −1 0 −k (2.8) ¯g (s) ξβ0 0 −k 0 ξβ ¯g (s) = det(β(s), Tβ (s), Tβ0 (s)) is the geodesic curvature of β and s is the arclength where k parameter of β. In particular, the following relations hold (2.9)

β × Tβ = −ξβ ,

Tβ × ξβ = β,

ξβ × β = Tβ .

Case 2. Tβ is a unit timelike vector Hence we have orthonormal Sabban frame {β(s), Tβ (s), ξβ (s)} along the curve β, where ξβ (s) = β(s) × Tβ (s) is the unit spacelike vector. The corresponding Frenet formulae of β, according to the Sabban frame read    0   β 0 1 0 β 0 ¯g (s)   Tβ  .  Tβ  =  1 0 k (2.10) ¯g (s) ξβ0 0 k 0 ξβ ¯g (s) = det(β(s), Tβ (s), Tβ0 (s)) is the geodesic curvature of β and s is the arclength where k parameter of β. In particular, the following relations hold (2.11)

β × Tβ = ξβ ,

Tβ × ξβ = β,

ξβ × β = −Tβ .

Case 3. Tβ is a null vector It is known that the only null curves lying on pseudosphere S12 are the null straight lines, which are the null geodesics.

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U. OZTURK AND E. B. KOC OZTURK

3. Smarandache curves according to curves on a spacelike surface in Minkowski 3-space R31 In the following section, we define the Smarandache curves according to the Lorentzian Darboux frame in Minkowski 3-space. Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves lying on pseudosphere S12 and give some characterizations on the curves when the curve Îą is an asymptotic curve or a principal curve.

Definition 1. Let Îą = Îą(s) be a spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then TΡ-Smarandache curve of Îą is defined by 1 (3.1) β(s? (s)) = â&#x2C6;&#x161; (aT(s) + bΡ(s)), 2 where a, b â&#x2C6;&#x2C6; R0 and a2 â&#x2C6;&#x2019; b2 = 2.

Definition 2. Let Îą = Îą(s) be a spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then TΞ-Smarandache curve of Îą is defined by 1 (3.2) β(s? (s)) = â&#x2C6;&#x161; (aT(s) + bΞ(s)), 2 where a, b â&#x2C6;&#x2C6; R0 and a2 + b2 = 2.

Definition 3. Let Îą = Îą(s) be a spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then ΡΞ-Smarandache curve of Îą is defined by 1 (3.3) β(s? (s)) = â&#x2C6;&#x161; (aΡ(s) + bΞ(s)), 2 where a, b â&#x2C6;&#x2C6; R0 and b2 â&#x2C6;&#x2019; a2 = 2.

Definition 4. Let Îą = Îą(s) be a spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then TΡΞ-Smarandache curve of Îą is defined by 1 (3.4) β(s? (s)) = â&#x2C6;&#x161; (aT(s) + bΡ(s) + cΞ(s)), 3 where a, b â&#x2C6;&#x2C6; R0 and a2 â&#x2C6;&#x2019; b2 + c2 = 3.

Thus, there are two following cases: Case 4. (Îą is an asymptotic curve). Then, we have the following theorems.

Theorem 1. Let Îą = Îą(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then (i) if akg + bĎ&#x201E;g 6= 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΡ-Smarandache curve β is given by     a  â&#x2C6;&#x161; â&#x2C6;&#x161;b 0 T β 2 2    Tβ  =  0 0  Ρ , Ξβ Ξ â&#x2C6;&#x161;b2 â&#x2C6;&#x161;a2 0 ÂŻg of the curve β reads and the geodesic curvature k â&#x2C6;&#x2019; bkg ÂŻg = aĎ&#x201E;gâ&#x2C6;&#x161; , k 2 where = sign (akg + bĎ&#x201E;g ) for all s and dsâ&#x2C6;&#x2014; akg + bĎ&#x201E;g â&#x2C6;&#x161; = . ds 2

SMARANDACHE CURVES IN MINKOWSKI 3-SPACE R3 1

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(ii) if akg + bĎ&#x201E;g = 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΡ-Smarandache curve β is a null geodesic.

Proof. We assume that the curve ι is an asymptotic curve. Differentiating the equation (3.1) with respect to s and using (2.4) we obtain β0

= =

dβ ds 1 â&#x2C6;&#x161; (akg + bĎ&#x201E;g ) Ξ, 2

and

(akg + bĎ&#x201E;g )2 . β0, β0 = 2 Then, there are two following cases: dβ dsâ&#x2C6;&#x2014; (i). If akg + bĎ&#x201E;g 6= 0 for all s, since β 0 = ds â&#x2C6;&#x2014; ds , then the tangent vector Tβ of the curve β is a spacelike vector such that

(3.5)

Tβ = Ξ,

where dsâ&#x2C6;&#x2014; akg + bĎ&#x201E;g â&#x2C6;&#x161; . = ds 2 On the other hand, from the equations (3.1) and (3.5) it can be easily seen that Ξβ (3.6)

= =

â&#x2C6;&#x2019;β Ă&#x2014; Tβ b a â&#x2C6;&#x161; T + â&#x2C6;&#x161; Ρ, 2 2

is a unit timelike vector. ÂŻg of the curve β = β(sâ&#x2C6;&#x2014; ) is given by Consequently, the geodesic curvature k

ÂŻg = det β, Tβ , T0β k aĎ&#x201E;g â&#x2C6;&#x2019; bkg â&#x2C6;&#x161; = 2 From (3.1), (3.5) and (3.6) we obtain the Sabban frame {β, Tβ , Ξβ } of β. (ii). If akg + bĎ&#x201E;g = 0 for all s, then β 0 is null. So, the tangent vector Tβ of the curve β is a null vector. It is known that the only null curves lying on pseudosphere S12 are the null straight lines, which are the null geodesics.

In the theorems which follow, in a similar way as in Theorem 1 we obtain the Sabban ÂŻg of a spacelike Smarandache curve. We frame {β, Tβ , Ξβ } and the geodesic curvature k omit the proofs of Theorems 2, 3 and 4, since they are analogous to the proof of Theorem 1 Theorem 2. Let Îą = Îą(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then (i) if 2kg2 â&#x2C6;&#x2019; (bĎ&#x201E;g )2 6= 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΞ-Smarandache curve β is given by   a â&#x2C6;&#x161; â&#x2C6;&#x161;b 0 2 2     â&#x2C6;&#x2019;bkg bĎ&#x201E;g akg  r  T β r r

  2 2 2 2 â&#x2C6;&#x2019;(bĎ&#x201E; ) 2 â&#x2C6;&#x2019;(bĎ&#x201E; )  Ρ ,  Tβ  =  2kg2 â&#x2C6;&#x2019;(bĎ&#x201E;g ) 2kg 2kg g g   bĎ&#x201E;g â&#x2C6;&#x2019;2kg abĎ&#x201E;g   r Ξ Ξβ r r

2 â&#x2C6;&#x2019;(bĎ&#x201E; )2 2 â&#x2C6;&#x2019;(bĎ&#x201E; )2 2 â&#x2C6;&#x2019;(bĎ&#x201E; )2 2kg 2kg 2kg g g g

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U. OZTURK AND E. B. KOC OZTURK

¯g of the curve β reads and the geodesic curvature k

a2 b3 Ď&#x201E;g3 â&#x2C6;&#x2019; b4 Ď&#x201E;g3 + 4bĎ&#x201E;g kg2 kg0 + b4 Ď&#x201E;g2 kg â&#x2C6;&#x2019; a 2 b3 Ď&#x201E;g2 kg â&#x2C6;&#x2019; 4bkg3 Ď&#x201E;g0 + 2ab2 + 2ab â&#x2C6;&#x2019; 4a Ď&#x201E;g kg4 â&#x2C6;&#x2019; ab4 + ab3 Ď&#x201E;g3 kg2 + ab4 Ď&#x201E;g5 ÂŻg = k , 2 4kg2 â&#x2C6;&#x2019; 2 (bĎ&#x201E;g )2

where = sign 2kg2 â&#x2C6;&#x2019; (bĎ&#x201E;g )2 for all s and s 2kg2 â&#x2C6;&#x2019; (bĎ&#x201E;g )2 dsâ&#x2C6;&#x2014; = . ds 2 (ii) if akg + bĎ&#x201E;g = 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΡ-Smarandache curve β is a null geodesic.

Theorem 3. Let Îą = Îą(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then (i) if (bkg )2 â&#x2C6;&#x2019; 2Ď&#x201E;g2 6= 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the ΡΞ-Smarandache curve β is given by   a â&#x2C6;&#x161; â&#x2C6;&#x161;b 0 2 2     bĎ&#x201E;g aĎ&#x201E;g  r â&#x2C6;&#x2019;bkg  T β r r

  2 2 2  Ρ , (bkg ) â&#x2C6;&#x2019;2Ď&#x201E;g2 (bkg ) â&#x2C6;&#x2019;2Ď&#x201E;g2 (bkg ) â&#x2C6;&#x2019;2Ď&#x201E;g2  Tβ  =     r  â&#x2C6;&#x2019;2Ď&#x201E;g b2 k g â&#x2C6;&#x2019;abkg Ξβ Ξ r r

2 2 2 2 (bkg ) â&#x2C6;&#x2019;2Ď&#x201E;g2 2 (bkg ) â&#x2C6;&#x2019;2Ď&#x201E;g2 2 (bkg ) â&#x2C6;&#x2019;2Ď&#x201E;g2 ÂŻg of the curve β reads and the geodesic curvature k â&#x2C6;&#x161; 2 3 â&#x2C6;&#x161; 5 3 â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 5 0

2 2 3 4 2bĎ&#x201E;g3 â&#x2C6;&#x2019; 2a 2a b +â&#x2C6;&#x161; 2b kg â&#x2C6;&#x2019; 4 2bĎ&#x201E;g2 kg Ď&#x201E;g0 â&#x2C6;&#x161; b 4+ 5 2b â&#x2C6;&#x161;Ď&#x201E;g kg4 k2 g 3+ â&#x2C6;&#x161; â&#x2C6;&#x2019; 2ab kg + 2 2ab Ď&#x201E;g kg + 4 2a â&#x2C6;&#x2019; 4 2ab2 Ď&#x201E;g4 kg ÂŻg = k , 2 2 b2 kg2 â&#x2C6;&#x2019; 2Ď&#x201E;g2

where = sign (bkg )2 â&#x2C6;&#x2019; 2Ď&#x201E;g2 for all s and s (bkg )2 â&#x2C6;&#x2019; 2Ď&#x201E;g2 dsâ&#x2C6;&#x2014; = . ds 2 (ii) if (bkg )2 â&#x2C6;&#x2019;2Ď&#x201E;g2 = 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΡ-Smarandache curve β is a null geodesic.

Theorem 4. Let Îą = Îą(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then (i) if 3 kg2 â&#x2C6;&#x2019; Ď&#x201E;g2 + (bkg + aĎ&#x201E;g )2 6= 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΡΞ-Smarandache curve β is given by β

=

Tβ

=

Ξβ

=

aT + bΡ + cΞ â&#x2C6;&#x161; , 3 â&#x2C6;&#x2019;ckg T + cĎ&#x201E;g Ρ + (akg + bĎ&#x201E;g ) Ξ q

, 3 kg2 â&#x2C6;&#x2019; Ď&#x201E;g2 + (bkg + aĎ&#x201E;g )2

b (akg + bĎ&#x201E;g ) â&#x2C6;&#x2019; c2 Ď&#x201E;g T + c2 kg + a (akg + bĎ&#x201E;g ) Ρ â&#x2C6;&#x2019; (acĎ&#x201E;g + bckg ) Ξ q ,

9 kg2 â&#x2C6;&#x2019; Ď&#x201E;g2 + 3 (bkg + aĎ&#x201E;g )2

¯g of the curve β reads and the geodesic curvature k

2 2 g f1 â&#x2C6;&#x2019; c kg + a (akg + bĎ&#x201E;g ) f2 â&#x2C6;&#x2019; (acĎ&#x201E;g + bckg ) f3 ÂŻg = b (akg + bĎ&#x201E;g ) â&#x2C6;&#x2019; c Ď&#x201E;â&#x2C6;&#x161; k ,

2 3 3 kg2 â&#x2C6;&#x2019; Ď&#x201E;g2 + (bkg + aĎ&#x201E;g )2

SMARANDACHE CURVES IN MINKOWSKI 3-SPACE R3 1

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where = sign 3 kg2 â&#x2C6;&#x2019; Ď&#x201E;g2 + (bkg + aĎ&#x201E;g )2 for all s and s (bkg )2 â&#x2C6;&#x2019; 2Ď&#x201E;g2 dsâ&#x2C6;&#x2014; = , ds 2 and

a2 c â&#x2C6;&#x2019; 3c Ď&#x201E;g2 + abcĎ&#x201E;g kg kg0 + 3c â&#x2C6;&#x2019; a2 c Ď&#x201E;g kg â&#x2C6;&#x2019; abckg2 Ď&#x201E;g0

+ 3a + ab2 kg4 + a3 + 2ab2 â&#x2C6;&#x2019; 3a Ď&#x201E;g2 kg2 + b3 + 2a2 b + 3b Ď&#x201E;g kg3

+ a2 b â&#x2C6;&#x2019; 3b Ď&#x201E;g3 kg

f2 = abcĎ&#x201E;g2 + b2 c + 3c Ď&#x201E;g kg kg0 â&#x2C6;&#x2019; abcĎ&#x201E;g kg + 3c + b2 c kg2 Ď&#x201E;g0

+ 3b â&#x2C6;&#x2019; a2 b Ď&#x201E;g4 â&#x2C6;&#x2019; b3 + 2a2 b + 3b Ď&#x201E;g2 kg2 â&#x2C6;&#x2019; ab2 + 3a Ď&#x201E;g kg3

+ 3a â&#x2C6;&#x2019; 2ab2 â&#x2C6;&#x2019; a3 Ď&#x201E;g3 kg

f3 = â&#x2C6;&#x2019;a3 + 3a + ab2 Ď&#x201E;g2 + b3 + 3b â&#x2C6;&#x2019; a2 b Ď&#x201E;g kg kg0

+ â&#x2C6;&#x2019;b3 â&#x2C6;&#x2019; 3b + a2 b kg2 + a3 â&#x2C6;&#x2019; ab2 â&#x2C6;&#x2019; 3a Ď&#x201E;g kg Ď&#x201E;g0

+ 3c + b2 c kg4 + 3c â&#x2C6;&#x2019; a2 c Ď&#x201E;g4 + 2abcĎ&#x201E;g kg3 â&#x2C6;&#x2019; 2abcĎ&#x201E;g3 kg

+ a2 c â&#x2C6;&#x2019; 6c â&#x2C6;&#x2019; b2 c Ď&#x201E;g2 kg2

(ii) if 3 kg2 â&#x2C6;&#x2019; Ď&#x201E;g2 + (bkg + aĎ&#x201E;g )2 = 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the TΡ-Smarandache curve β is a null geodesic. f1

=

Case 5. (Îą is a principal curve). Then, we have the following theorems.

Theorem 5. Let Îą = Îą(s) be a principal spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then the TΡ-Smarandache curve β is spacelike and the Sabban frame {β, Tβ , Ξβ } is given by   a â&#x2C6;&#x161; â&#x2C6;&#x161;b 0     2 2 β akg  q bkn  T akn q q   Tβ  =   (akg )2 â&#x2C6;&#x2019;2kn2 (akg )2 â&#x2C6;&#x2019;2kn2 (akg )2 â&#x2C6;&#x2019;2kn2   Ρ  ,   2 abk a k â&#x2C6;&#x2019;2k g g n Ξ Ξβ q q q 2 2 2 2 2 2 2(akg ) â&#x2C6;&#x2019;4kn 2(akg ) â&#x2C6;&#x2019;4kn 2(akg ) â&#x2C6;&#x2019;4kn ÂŻg of the curve β reads and the geodesic curvature k 2 g g1 â&#x2C6;&#x2019; a kg g2 â&#x2C6;&#x2019;2kn g3 ÂŻg = abkâ&#x2C6;&#x161; k , 2 2 a2 kg2 â&#x2C6;&#x2019; 2kn2

where dsâ&#x2C6;&#x2014; = ds

s

(akg )2 â&#x2C6;&#x2019; 2kn2 , 2

and g1

=

ba2 kg kn kg0 â&#x2C6;&#x2019; ba2 kg2 kn0 + a3 kg4 â&#x2C6;&#x2019; a3 + 2a kg2 kn2 + 2akn4 ,

g2

=

â&#x2C6;&#x2019;a3 kg2 kn0 + a3 kg kn kg0 â&#x2C6;&#x2019; ba2 kg2 kn2 + 2bkn4 ,

g3

=

2akn2 kg0 â&#x2C6;&#x2019; 2akg kn kn0 â&#x2C6;&#x2019; ba2 kg3 kn + 2bkg kn3 .

Proof. We assume that the curve ι is a principal curve. Differentiating the equation (3.1) with respect to s and using (2.4) we obtain β0

= =

dβ ds 1 â&#x2C6;&#x161; (bkn T + akn Ρ + akg Ξ), 2

8

U. OZTURK AND E. B. KOC OZTURK

and

(akg )2 − 2kn2 β0, β0 = , 2

where from (2.7) (akg )2 − 2kn2 > 0 for all s. Since β 0 = the curve β is a spacelike vector such that

dβ ds∗ , ds∗ ds

the tangent vector Tβ of

1 Tβ = q (bkn T + akn η + akg ξ), (akg )2 − 2kn2

(3.7) where

ds∗ = ds

s

(akg )2 − 2kn2 .

2

On the other hand, from the equations (3.1) and (3.7) it can be easily seen that ξβ (3.8)

= =

−β × Tβ 1 q (abkg T + a2 kg η−2kn ξ), 2 2 2 (akg ) − 4kn

is a unit timelike vector. ¯g of the curve β = β(s∗ ) is given by Consequently, the geodesic curvature k

¯g = det β, Tβ , T0β k =

abkg g1 − a2 kg g2 −2kn g3 √ 2 2 a2 kg2 − 2kn2

where g1

=

ba2 kg kn kg0 − ba2 kg2 kn0 + a3 kg4 − a3 + 2a kg2 kn2 + 2akn4

g2

=

−a3 kg2 kn0 + a3 kg kn kg0 − ba2 kg2 kn2 + 2bkn4

g3

=

2akn2 kg0 − 2akg kn kn0 − ba2 kg3 kn + 2bkg kn3

From (3.1), (3.7) and (3.8) we obtain the Sabban frame {β, Tβ , ξβ } of β.

In the theorems which follow, in a similar way as in Theorem 5 we obtain the Sabban ¯g of a spacelike Smarandache curve. We frame {β, Tβ , ξβ } and the geodesic curvature k omit the proofs of Theorems 6 and 8, since they are analogous to the proof of Theorem 5 Theorem 6. Let α = α(s) be a principal spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, η, ξ}. Then the Tξ-Smarandache curve β is spacelike and the Sabban frame {β, Tβ , ξβ } is given by   a √ √b     0 2 2 β akg akn  √ −bkg  T √ √  η ,  Tβ  =  2 −(ak )2 2 −(ak )2 2 −(ak )2 2kg 2kg 2kg n n n   2 ξβ ξ √ −abkn √ 2kg √ −a kn 2 −2(ak )2 4kg n

2 −2(ak )2 4kg n

2 −2(ak )2 4kg n

¯g of the curve β reads and the geodesic curvature k 2 ¯g = − abk√n h1 + 2kg h2 +a k n h3 , k 2 2 2 2kg − (akn )2

where ds∗ = ds

s

2kg2 − (akn )2 , 2

SMARANDACHE CURVES IN MINKOWSKI 3-SPACE R3 1

9

and h1

=

â&#x2C6;&#x2019;ba2 kn2 kg0 + ba2 kg kn kn0 + 2akg4 â&#x2C6;&#x2019; a3 kg2 kn2 â&#x2C6;&#x2019; 2akg2 kn2 + a3 kn4 ,

h2

=

2akg kn kg0 â&#x2C6;&#x2019; 2akg2 kn0 â&#x2C6;&#x2019; 2bkg3 kn + ba2 kg kn3 ,

h3

=

a3 kn2 kg0 â&#x2C6;&#x2019; a3 kg kn kn0 â&#x2C6;&#x2019; 2bkg4 + ba2 kg2 kn2 .

Theorem 7. Let Îą = Îą(s) be a principal spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, Ρ, Ξ}. Then (i) if akn â&#x2C6;&#x2019; bkg 6= 0 for all s, the ΡΞ-Smarandache curve β is spacelike and the Sabban frame {β, Tβ , Ξβ } is given by      a â&#x2C6;&#x161; â&#x2C6;&#x161;b 0 β T 2 2   Tβ  =  0 0  Ρ ,  a b â&#x2C6;&#x161; â&#x2C6;&#x161; Ξβ Ξ 0 â&#x2C6;&#x2019; 2 2 ÂŻg of the curve β reads and the geodesic curvature k + bkn ÂŻg = â&#x2C6;&#x2019; akgâ&#x2C6;&#x161; k , 2 where = sign (akn â&#x2C6;&#x2019; bkg ) for all s and dsâ&#x2C6;&#x2014; akn â&#x2C6;&#x2019; bkg â&#x2C6;&#x161; = . ds 2 (ii) if akn â&#x2C6;&#x2019; bkg = 0 for all s, then the Sabban frame {β, Tβ , Ξβ } of the ΡΞ-Smarandache curve β is a null geodesic.

Proof. We assume that the curve ι is a principal curve. Differentiating the equation (3.3) with respect to s and using (2.4) we obtain β0

= =

dβ dsâ&#x2C6;&#x2014; dsâ&#x2C6;&#x2014; ds 1 â&#x2C6;&#x161; (akn â&#x2C6;&#x2019; bkg ) T, 2

and

(akn â&#x2C6;&#x2019; bkg )2 β0, β0 = .

2 Then, there are two following cases: dβ dsâ&#x2C6;&#x2014; (i). If akn â&#x2C6;&#x2019; bkg 6= 0 for all s, since β 0 = ds â&#x2C6;&#x2014; ds , then we obtain the unit tangent vector Tβ of the curve β is a spacelike vector such that

(3.9)

Tβ = T,

where

dsâ&#x2C6;&#x2014; akn â&#x2C6;&#x2019; bkg â&#x2C6;&#x161; = , ds 2

and = sign (akn â&#x2C6;&#x2019; bkg ) .

On the other hand, from the equations (3.3) and (3.9) it can be easily seen that Ξβ (3.10)

= =

â&#x2C6;&#x2019;β Ă&#x2014; Tβ b a â&#x2C6;&#x2019; â&#x2C6;&#x161; Ρ + â&#x2C6;&#x161; Ξ, 2 2

is a unit timelike vector. ÂŻg of the curve β = β(sâ&#x2C6;&#x2014; ) is given by Consequently, the geodesic curvature k

ÂŻg = det β, Tβ , T0β k akg + bkn â&#x2C6;&#x161; = â&#x2C6;&#x2019; 2

10

U. OZTURK AND E. B. KOC OZTURK

From (3.3), (3.9) and (3.10) we obtain the Sabban frame {β, Tβ , ξβ } of β. (ii). If akn − bkg = 0 for all s, then β 0 is null. So, the tangent vector Tβ of the curve β is a null vector. It is known that the only null curves lying on pseudosphere S12 are the null straight lines, which are the null geodesics.

Theorem 8. Let α = α(s) be a principal spacelike curve lying fully on the spacelike surface M in R31 with the moving Lorentzian Darboux frame {T, η, ξ}. Then the Tηξ-Smarandache curve β is spacelike and the Sabban frame {β, Tβ , ξβ } is given by β

=

Tβ

=

ξβ

=

1 √ (aT + bη + cξ), 3 (bkn − ckg ) T + akn η + akg ξ q ,

3 kn2 − kg2 + (bkg − ckn )2

(abkg − ackn ) T + 3 + b2 kg − bckn η− 3 − c2 kn + bckg ξ q ,

9 kn2 − kg2 + 3 (bkg − ckn )2

¯g of the curve β reads and the geodesic curvature k

2 ¯g = − (abkg − ackn ) l1 − 3 +√b kg − bckn l 2 − k 2 2 a2 kg2 − 2kn2

3 − c2 kn + bckg l3

,

where q

∗

ds = ds

3 kn2 − kg2 + (bkg − ckn )2 √ , 3

and l1

=

l2

=

l3

=

3abkn2 − 3ackg kn kg0 + 3ackg2 − 3abkg kn kn0 + (bkn − ckg ) (bkg − ckn )3

+3bckg4 − 3bckn4 + 3 b2 + c2 kg kn3 − 3 b2 + c2 kg3 kn

3b2 + 3c2 + 9 kn2 − 6bckg kn kg0 − 3b2 + 3c2 + 9 kg kn − 6bckg2 kn0

− ac3 + 3ac kn4 + 3ab + 3abc2 kg kn3 + ab3 − 3ab kg3 kn + 3ac − 3ab2 c kg2 kn2

3b2 + 3c2 − 9 kg kn − 6bckn2 kg0 + 9 − 3b2 − 3c2 kg2 + 6bckg kn kn0

+ ab3 − 3ab kg4 − 3ac + ac3 kg kn3 + 3ac − 3ab2 c kg3 kn + 3ab + 3abc2 kg2 kn2

Example 1. Let us define a spacelike ruled surface (see Figure 1) in the Minkowski 3-space such as φ : U ⊂ R2 −→ R31 (s, u) −→ φ(s, u) = α (s) + ue (s) and φ(s, u) = (−u sinh s, s, −u cosh s) where u ∈ (−1, 1).

Then we get the Lorentzian Darboux frame {T, η, ξ} along the curve α as follows T(s)

=

η(s)

≡

ξ(s)

=

(0, 1, 0) , 1 √ (cosh s, −u, sinh s) , 1 − u2 1 √ (− sinh s, 0, − cosh s) 1 − u2

where ξ(s) is a spacelike vectors and η(s) is a unit timelike vector.

SMARANDACHE CURVES IN MINKOWSKI 3-SPACE R3 1

11

Figure 1. The spacelike surface φ(s, u)

Moreover, the geodesic curvature kg (s), the asymptotic curvature kn (s) and the principal curvature τg (s) of the curve α have the form kg (s)

=

kn (s)

=

τg (s)

=

0

T (s), ξ(s) = 0,

− T0 (s), η(s) = 0,

1 − ξ 0 (s), η(s) = − .

1 − u2

√ Taking a = 3, b = 1 and using (3.1), we obtain that the Tη-Smarandache curve β of the curve α is given by (see Figure 2a) β(s? (s)) =

cosh s √ u sinh s √ , 3− √ , √ 1 − u2 1 − u2 1 − u2

Taking a = b = 1 and using (3.2), we obtain that the Tξ-Smarandache curve β of the curve α is given by (see Figure 2b) β(s? (s)) =

cosh s sinh s , 1, − √ . −√ 1 − u2 1 − u2

√ Taking a = 3, b = 1 and using (3.3), we obtain that the ηξ-Smarandache curve β of the curve α is given by (see Figure 3a)

√ √ √ 1 sinh s − 3 cosh s, 3u, cosh s − 3 sinh s β(s? (s)) = − √ 1 − u2 √ Taking a = 3, b = 1, c = 1 and using (3.4), we obtain that the Tηξ-Smarandache curve β of the curve α is given by (see Figure 3b) β(s? (s)) = √

p 1 cosh s − sinh s, 3 − 3u2 − u, sinh s − cosh s .

1 − u2

12

U. OZTURK AND E. B. KOC OZTURK

(a) The Tη-Smarandache (b) The Tξ-Smarandache curve β on curve β on S12 for u = √1 S12 for u = √1 3

3

Figure 2

(a) The ηξ-Smarandache curve β on S12 (b) The Tηξ-Smarandache curve β on for u = √1 S12 for u = √1 3

3

Figure 3

SMARANDACHE CURVES IN MINKOWSKI 3-SPACE R3 1

Figure 4. The Smarandache curves on S12 for u =

13

√1 3

4. Conflict of Interests The author(s) declare(s) that there is no conflict of interests regarding the publication of this article.

References [1] Ali, A. T.,Special Smarandache Curves in the Euclidean Space, Int. J. Math. Combin., 2, 30–36, (2010).

[2] O’Neill, B., Semi-Riemannian Geometry with applications to relativity, Academic Press, New York, 1983.

[3] Ashbacher, C., Smarandache Geometries, Smarandache Notions J., 8, 212–215, (1997).

[4] Koc Ozturk, E. B., Ozturk, U., Ilarslan, K. and Neˇ sovi´ c, E., On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space, International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 658670, 7 pages, 2013. doi:10.1155/2013/658670 [5] Koc Ozturk, E. B., Ozturk, U., Ilarslan, K. and Neˇ sovi´ c, E., On Pseudospherical Smarandache Curves in Minkowski 3-Space, Journal of Applied Mathematics, vol. 2014, Article ID 404521, 14 pages, 2014. doi:10.1155/2014/404521 [6] Ta¸sk¨ opr¨ u, K. and Tosun, M., Smarandache Curves on S 2 , Bol. Soc. Parana. Mat. (3), 32(1), 51–59, (2014).

[7] Turgut, M. and Yılmaz, S., Smarandache Curves in Minkowski space-time, Int. J. Math. Comb., 3, 51–55, (2008).

[8] Bektas, O. and Yuce, S., Special Smarandache Curves According to Darboux Frame in E 3 , Rom. J. Math. Comput. Sci., 3(1), 48–59, (2013).

[9] Korpinar, T. and Turhan, E., A new approach on Smarandache TN-curves in terms of spacelike biharmonic curves with a timelike binormal in the Lorentzian Heisenberg group Heis3 , Journal of Vectorial Relativity, 6, 8–15, (2011).

[10] Korpinar, T. and Turhan, E., Characterization of Smarandache M1 M2 -curves of spacelike biharmonic B-slant helices according to Bishop frame in E(1, 1), Adv. Model. Optim., 14(2), 327–333, (2012).

Department of Mathematics, Faculty of Sciences, University of C ¸ ankırı Karatekin, 18100 C ¸ ankırı, Turkey E-mail address: ozturkufuk06@gmail.com, uuzturk@asu.edu Department of Mathematics, Faculty of Sciences, University of C ¸ ankırı Karatekin, 18100 C ¸ ankırı, Turkey E-mail address: e.betul.e@gmail.com, ekocoztu@asu.edu