September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 International Journal of Geometric Methods in Modern Physics Vol. 14, No. 10 (2017) 1750140 (10 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S0219887817501407 Characterizations of special time-like curves in Lorentzian plane L2 Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. Abdullah Mağden∗,§ , Süha Yılmaz†,¶ and Yasin Ünlütürk‡, ∗Department †Buca of Mathematics, Atatürk University 25400 Erzurum, Turkey Faculty of Education, Dokuz Eylül University 35150 Buca-Izmir, Turkey ‡Department of Mathematics, Kırklareli University 39100 Kırklareli, Turkey §[email protected] ¶[email protected] [email protected] Received 28 March 2017 Accepted 17 May 2017 Published 4 July 2017 In this paper, we ﬁrst obtain the diﬀerential equation characterizing position vector of time-like curve in Lorentzian plane L2 . Then we study the special curves such as Smarandache curves, circular indicatrices, and curves of constant breadth in Lorentzian plane L2 . We give some characterizations of these special curves in L2 . Keywords: Lorentzian plane; time-like curve; circular indicatrices; Smarandache curves; curves of constant breadth. Mathematics Subject Classiﬁcation 2010: 53A35, 53A40, 53B25 1. Introduction There are lots of interesting and important problems in the theory of curves at diﬀerential geometry. One of the interesting problems is the problem of characterization of a regular curve in the theory of curves in the Euclidean and Minkowski spaces, see [1, 2]. Also, there are special curves which are obtained under some deﬁnitions such as Smarandache curves, spherical indicatrices, curves of constant breadth, etc. Smarandache curves are regular curves whose position vectors are obtained by Frénet frame vectors on another regular curve. These curves were examined by Turgut and Yılmaz in [3]. Then many researches occurred about the diﬀerent characterizations of Smarandache curves in Euclidean and Minkowsi spaces, see [4, 5]. Corresponding author. 1750140-1 September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. A. Mağden, S. Yılmaz & Y. Ünlütürk Spherical indicatrix is the locus of a point whose position vector is equal to one of the frame ﬁeld vectors such as the unit tangent T, the principal normal vector N, and the principal binormal vector B at any point of a given curve in the space. If this vector is chosen as the unit tangent vector ﬁeld T of a given curve in the space, then the spherical indicatrix is said to be tangent spherical indicatrix. The principal normal and binormal spherical indicatrices are also obtained by using the position vectors of the principal normal vector N, and the principal binormal vector B, respectively. Spherical indicatrices as special curves have been studied in many works, see [6, 7]. As you know, spherical indicatrix turns into circular indicatrix in the plane. Curves of constant breadth as another special curve, were ﬁrst introduced by Euler [8]. Then in chronological order, it was studied in [9–12]. Our motivation was to see the corresponding results of the special curves mentioned above in Lorentzian plane L2 . As much as we look at the classical diﬀerential geometry literature of the works in Lorentzian plane L2 , the works were rare, see [13–16]. First, we obtain the diﬀerential equation characterizing position vector of time-like curve in Lorentzian plane L2 . Then we study the special curves mentioned above in Lorentzian plane L2 . We give some characterizations of these special curves in L2 . 2. Preliminaries Let L2 be the Lorentzian plane with metric g = dx21 − dx22 , (1) 2 where x1 and x2 are rectangular coordinate system. A vector r of L is said to be space-like if g(r, r) > 0, or r = 0, time-like if g(r, r) < 0 and null if g(r, r) = 0 for r = 0 [6]. A curve x is a smooth mapping x : I → L2 , from an open interval I onto L2 . Let s be an arbitrary parameter of α, then we denote the orthogonal coordinate representation of α as α = (α1 (s), α2 (s)) and also the vector dα1 dα2 dα = , =t (2) ds ds ds is called the tangent vector ﬁeld of the curve α = α(s). If tangent vector ﬁeld of α(s) is a space-like, time-like or null then, the curve α(s) is called space-like, time-like or null, respectively [13]. In the rest of the paper, we shall consider time-like curves. While the tangent vector ﬁeld T is time-like vector so N is space-like one. We can have the arc-length parameter s and the Frénet formula as 0 κ T T = · , (3) N κ 0 N 1750140-2 September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 Characterizations of special time-like curves where κ = κ(s) is the curvature of the unit speed curve α = α(s) [15]. We should also indicate that the space-like vector ﬁeld N is the normal vector ﬁeld of the curve α(s), the tangent vector ﬁeld T is a time-like vector, and also T, N = 0. Given φ(s) is the slope angle of the curve, then as in [13] we have dφ = κ(s). ds (4) Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. 3. Position Vector of a Curve in L2 Let α = α(s) be an unit speed time-like curve on the plane L2 . Then we can write the position vector of α(s) with respect to Frénet frame as α = α(s) = λ1 T + λ2 N, (5) where λ1 and λ2 are arbitrary functions of s. Diﬀerentiating (5) and using Frénet equtaions we have a system of ordinary diﬀerential equations as follows: dλ1 ds + λ2 κ − 1 = 0, (6) dλ2 + λ1 κ = 0. ds Using (6)1 in (6)2 we obtain d 1 dλ1 + 1 + λ1 κ = 0. − ds κ ds (7) The diﬀerential equation of second order, according s to λ1 , is a characterization for the curve α = α(s). Using change of variable θ = 0 κ ds in (7), we arrive d2 λ1 dρ , − λ1 = 2 dθ dθ (8) where κ = 1ρ . By the method of variation of parameters and solution of (8), we have θ θ θ θ −θ −θ λ1 = e A − κe dθ + e κe dθ , B+ 0 0 where A, B ∈ R. Rewriting the change of variable, we get θ Rθ Rθ κ ds θ − κ ds λ1 = e 0 κe dθ + e 0 A− B+ 0 θ κe −θ dθ . (9) 0 Denoting diﬀerentiation of Eq. (9) as dλ1 ds = ξ(s), and using (6), we have 1 [ξ(s) − 1]. κ Hence, we give the following theorem: λ2 = (10) Theorem 3.1. Let α = α(s) be an arbitrary unit speed time-like curve in Lorentzian plane L2 . Position vector of the curve α = α(s) with respect to Frénet 1750140-3 September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 A. Mağden, S. Yılmaz & Y. Ünlütürk frame can be composed by the following equation: θ θ 1 θ θ −θ −θ [ξ(s) − 1] N. α = α(s) = e A − κe dθ + e κe dθ T+ B+ κ 0 0 Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. 3.1. Diﬀerential equation of the third-order characterizing Lorentzian plane curves Theorem 3.2. Let α = α(s) be an arbitrary unit speed time-like curve in Lorentzian plane L2 . Position vector and curvature of it is to satisfy the diﬀerential equations of third order d 1 d2 α dα = 0. −κ ds κ ds2 ds Proof. Let α = α(s) be an arbitrary unit speed time-like curve in Lorentzian plane L2 . Then Frénet derivative formula holds (3)1 in (3)2 , we easily have d 1 dT − κT = 0. (11) ds κ ds Let dα ds = t = α̇. So, expression of (11) can be written as follows: dα d 1 d2 α = 0, −κ ds κ ds2 ds (12) formula (12) completes the proof. Here, we try to solve the diﬀerential equation (11) with respect to t. We know that t = (t1 , t2 ) = (α̇1 , α̇2 ), s using the change of variable θ = 0 κ ds in (12) we get d2 t − θ = 0, dθ2 (13) or in parametric form it is d2 t1 d2 t2 − θ = 0, − θ = 0, dθ2 dθ2 as the solution of (14), we obtain t1 = ψ1 eθ + ψ2 e−θ , t2 = ψ3 eθ + ψ4 e−θ , (14) (15) where ψi ∈ R for 1 ≤ i ≤ 4. 4. Special Curves in L2 In this section, we will study some special curves such as Smarandache curves, circular indicatrices, and curves of constant breadth in Lorentzian plane L2 . 1750140-4 September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 Characterizations of special time-like curves 4.1. Smarandache curves Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. A regular curve in Lorentzian plane L2 whose position vector is composed of Frénet frame vectors on another regular curve is called a Smarandache curve. In this section, we will study TN –Smarandache curve as the only Smarandache curve of Lorentzian plane L2 . Definition 4.1 (TN–Smarandache curves). Let α = α(s) be a unit speed timelike curve in L2 and {T α , N α } be its moving Frénet frame, here T α , N α are the tangent and principal normal vectors of the Smarandache curve of the curve α. The curve α = α(s) is said to be TN –Smarandache curve whose form is −1 (16) β(s∗ ) = √ (T α − N α ). 2 We can investigate Frénet invariants of TN –Smarandache curves according to α = α(s). Diﬀerentiating (16) with respect to s gives us −1 dβ ds∗ = √ (καN α − κα T α ). ∗ ds ds 2 Rearranging of this expression we get −1 ds∗ = √ (καN α − κα T α ), Tβ ds 2 by (18) we have β̇ = (17) (18) ds∗ = κα , (19) ds hence using (18) and (19) we ﬁnd the tangent vector of the curve β as follows: (T α − N α ) √ 2 and diﬀerentiating (20) with respect to s, we have Tβ = dTβ ds∗ (−κα T α + καN α ) √ = . ds∗ ds 2 (20) (21) Substituting (19) in (21), we obtain Tβ = The curvature of the curve β is Tβ = κβ = −(T α − N α ) √ . 2 (T α )2 − (N α )2 = 0. 2 4.2. Circular indicatrices Circular indicatrix is the locus of a point whose position vector is equal to the unit tangent T or the principal normal vector N at any point of a given curve. We will characterize tangent and normal circular indicatrices of time-like curves in Lorentzian plane L2 . 1750140-5 September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 A. Mağden, S. Yılmaz & Y. Ünlütürk 4.2.1. The tangent circular indicatrices of time-like curves in L2 Let ε = ε(s) be a time-like curve in Lorentzian plane L2 . If we translate the ﬁrst vector ﬁeld of Frénet frame to the center of the unit Lorentzian circle S 1 , then we have the tangent circular indicatrix δ = δ(sδ ) in Lorentzian plane L2 . Here, we shall denote diﬀerentiation according to s by a dash and diﬀerentiation according to sδ by a dot. By using the Frénet frame, we obtain the tangent vector of δ = δ(sδ ) as Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. δ = dsδ dδ dsδ = Tδ = κN, dsδ ds ds (22) dsδ = κ(s). ds (23) where Tδ = N and From (22), we have Ṫδ = κN + κ2 T and we also arrive at κδ = Ṫδ = (−(κ )2 + κ4 ). (24) Therefore, we have the principal normal of the curve δ = δ(sδ ) as −κ2 T + κN . Nδ = √ −κ2 + κ4 (25) 4.2.2. The principal normal circular indicatrices of time-like curves in L2 Let ε = ε(s) be a time-like curve in Lorentzian plane L2 . If we translate the second vector ﬁeld of Frénet frame to the center of the unit Lorentzian circle S 1 , then we have the principal normal circular indicatrix φ = φ(sφ ) in Lorentzian plane L2 . We will follow the procedure used to determine the tangent circular indicatrix for the principal normal circular indicatrices of time-like curves in L2 . Let us ﬁrst deﬁne dφ dsφ φ = = κT, dsφ ds where Tφ = N and dsφ ds (26) = κ(s). Diﬀerentiating (26), we obtain Tφ = Ṫφ dsφ = −κN, ds (27) or in other words, Ṫφ = N. 1750140-6 (28) September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 Characterizations of special time-like curves Using (27) and (28) we have the ﬁrst curvature and the principal normal vector of the principal normal circular indicatrix φ = φ(sφ ) as κφ = Ṫφ = 1 and Nφ = N. Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. 4.3. Curves of constant breadth Let ϕ = ϕ(s) and ϕ∗ = ϕ∗ (s) be simple closed time-like curve in Lorentzian plane L2 . These curves will be denoted by C and C ∗ . The normal plane at every point p on the curve meets the curve at a single point q other than p. We call the point q as the opposite point of p. We consider curves in the class Γ as in Fujivara [17] having parallel tangents T and T ∗ in opposite directions at the opposite points ϕ and ϕ∗ of the curve. A simple closed curve of constant breadth having parallel tangents in opposite directions at opposite points can be represented with respect to Frénet frame by the following equation: ϕ∗ = ϕ + m1 T + m2 N, (29) ∗ where mi (s), 1 ≤ i ≤ 2 are arbitrary functions of s and ϕ and ϕ are opposite points. The vector d = ϕ∗ − ϕ is called “the distance vector” between the opposite points of C and C ∗ . Diﬀerentiating (29), and considering Frénet derivative equations (3), we have dm1 dϕ∗ ds∗ dm2 = T∗ =T + T + m1 κN + N + m2 κT. ds ds ds ds Since dϕ∗ dϕ = T and = T ∗, ds ds∗ and using Frénet derivative formulas, we get ∗ dm1 dm2 ∗ ds = 1+ + m2 κ T + m1 κ + T N. ds ds ds (30) Since the deﬁnition of curve of constant breadth, there is the relation T ∗ = −T, (31) between the tangent vectors of the curves, and also using (31) in (30), we obtain ds∗ dm1 dm2 = −m2 κ − − 1 and m1 κ + = 0. (32) ds ds ds Let θ be the angle between the tangent vector T at a point α(s) of an oval and a ﬁxed direction, then we have 1 ds =ρ= dθ κ and ds∗ 1 = ρ∗ = ∗ . dθ κ 1750140-7 (33) September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 A. Mağden, S. Yılmaz & Y. Ünlütürk Substituting (33) into (32), Eq. (32) turns into dm1 ∗ −m2 − dθ = ρ + ρ = f (θ), (34) dm2 = −m1 , dθ Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. eliminating m1 in (34) we obtain the linear diﬀerential equation of the second order as d2 m2 − m2 = f (θ), dθ2 where f (θ) = ρ + ρ∗ . By general solution of Eq. (35) we ﬁnd m2 = e θ −θ t f (t)e dt 0 + ε2 − e θ (35) θ f (t)e −t dt 0 + ε1 . 2 Using m1 = − dm dθ in (34) we obtain the value of m1 as θ θ f (t) cos t dt + ε2 − e−θ f (t) sin t dt + ε1 . m1 = −eθ 0 0 Hence using (29) the position vector of the curve ϕ ∗ is given as follows: θ θ ∗ θ −θ ϕ = ϕ + −e f (t) cos t dt + ε2 − e f (t) sin t dt + ε1 T + e −θ 0 θ t f (t)e dt 0 + ε2 − e θ 0 θ f (t)e −t dt 0 + ε1 N. If the distance between the opposite points of C and C ∗ is constant, then we can write that ϕ∗ − ϕ = −m21 + m22 = const. (36) and diﬀerentiating (36) we have dm1 dm2 + m2 =0 (37) dθ dθ and also taking the system (34) and (37) together into consideration, we obtain dm1 m1 + m2 = 0, (38) dθ −m1 so we arrive at dm1 = −m2 . dθ Due to the cases in (39), we will study the following conditions: m1 = 0 or 1750140-8 (39) September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 Characterizations of special time-like curves Case 1: If m1 = 0, then from (34) we ﬁnd that f (θ) = const. and m2 = const. If dm1 = −m2 , dθ m1 = 0 = const. and also Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. then we obtain m2 = 0. If m1 = k1 (k1 ∈ R), then Eq. (29) turns into ϕ∗ = ϕ + k1 T. (40) Case 2: If dm1 = −m2 , dθ then from (34) we have f (θ) = 0 and m1 = − 0 θ m2 dθ. If dm1 = −m2 = 0 = k2 = const., dθ then from (34) we obtain f (θ) = 0 and m1 = 0. Hence, Eq. (29) becomes as follows: ϕ∗ = ϕ + k2 N. (41) References [1] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Monthly 110 (2003) 147–152. [2] S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turkish J. Math. 28(2) (2004) 531–537. [3] M. Turgut and S. Yılmaz, Smarandache curves in Minkowski space-time, Int. J. Math. Combin. 3 (2008) 51–55. [4] A. T. Ali, Special Smarandache curves in the Euclidean space, Math. Combin. Book Ser. 2(2) (2010) 30–36. [5] M. Çetin, Y. Tuncer and M. K. Karacan, Smarandache curves according to Bishop frame in Euclidean 3-space, Gen. Math. Note 20 (2014) 50–56. [6] S. Yılmaz and M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl. 371 (2010) 764–776. 1750140-9 September 4, 2017 14:57 WSPC/S0219-8878 IJGMMP-J043 1750140 Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com by TUFTS UNIVERSITY on 10/28/17. For personal use only. A. Mağden, S. Yılmaz & Y. Ünlütürk [7] S. 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