Spacelike Salkowski and anti-salkowski Curves With a Spacelike Principal Normal in Minkowski 3-Space

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1 Int. J. Open Problem Compt. Math., Vol., No. 3, September 009 ISSN ; Copyright c ICSRS Publication, Spacelike Salkowki and anti-salkowki Curve With a Spacelike Principal Normal in Minkowki 3-Space Ahmad Tawfik Ali Mathematic Department, Faculty of Science, Al-Azhar Univerity, Nar City, 448, Cairo, Egypt. atali7@yahoo.com Abtract A century ago, Salkowki [9] introduced a family of curve with contant curvature but non-contant torion (Salkowki curve) and a family of curve with contant torion but noncontant curvature (anti-salkowki curve). In thi paper, we adapt definition of uch curve to pacelike curve in Minkowki 3-pace. Thereafter, we introduce an explicit parametrization of a pacelike Salkowki curve with a pacelike principal normal vector and a pacelike anti-salkowki curve with a pacelike principal normal vector in Minkowki 3-pace. Moreover, we characterize them a a pace curve with contant curvature or contant torion and whoe normal vector make a contant angle with a fixed traight line. MSC: 53C40, 53C50 Keyword: Salkowki curve; contant curvature and torion; Minkowki 3-pace; pacelike curve. Introduction Salkowki (rep. anti-salkowki) curve in Euclidean pace E 3 are generally known a family of curve with contant curvature (rep. torion) but noncontant torion (rep. curvature) with an explicit parametrization [7, 9]. They were defined in an earlier paper [9] and retrieved, a an example of tangentially cubic curve, in a firt verion of Pottmann and Hofer [8]. Recently, Monterde

2 45 Ahmad T. Ali [7] tudied ome of characterization of thee curve and he prove that the normal vector make a contant angle with a fixed traight line. Analogouly, we define in thi paper, Salkowki curve and anti-salkowki curve in Minkowki pace E 3. Alo, we introduce the explicit parametrization of a pacelike Salkowki curve with a pacelike principal normal vector and pacelike anti-salkowki curve with a pacelike principal normal vector in Minkowki pace E 3 and we tudy ome characterization of thee curve. Preliminarie Firt, we briefly preent theory of the curve in Minkowki 3-pace a follow: The Minkowki three-dimenional pace E 3 i the real vector pace R 3 endowed with the tandard flat Lorentzian metric given by:, = dx + dx + dx 3, where (x, x, x 3 ) i a rectangular coordinate ytem of E 3. If u = (u, u, u 3 ) and v = (v, v, v 3 ) are arbitrary vector in E 3, we define the (Lorentzian) vector product of u and v a the following: u v = i j k u u u 3 v v v 3 An arbitrary vector v E 3 i aid to be a pacelike if v, v > 0 or v = 0, timelike if v, v < 0, and lightlike (or null) if v, v = 0 and v 0. The norm (length) of a vector v i given by v = v, v. An arbitrary regular (mooth) curve α : I R E 3 i locally pacelike if all of it velocity vector α (t) are pacelike for each t I R. If α i pacelike, there exit a change of the parameter t, namely, = (t), uch that α () =. We ay then that α i a unit peed curve [,, 6]. Given a unit peed curve α in Minkowki pace E 3 it i poible to define a Frenet frame {T(), N(), B()} aociated for each point [5, 0]. Here T, N and B are the tangent, principal normal and binormal vector field, repectively. We uppoe that α i a pacelike curve with a pacelike principal normal vector N. Then T () 0 i a pacelike vector independent with T(). We define the curvature of α at a κ() = T (). The principal normal vector N() and the binormal vector B() are defined a N() = T () κ() = α, B() = T() N(), α.

3 Spacelike Salkowki and anti-salkowki Curve with 453 where the vector B() i unitary and timelike. For each, {T, N, B} i an orthonormal bae of E 3 which i called the Frenet trihedron of α. We define the torion of α at a: Then the Frenet formula i where T N B τ() = N (), B(). = 0 κ 0 κ 0 τ 0 τ 0 T N B, () T, T = N, N =, B, B =, T, N = N, B = B, T = 0. 3 Spacelike Salkowki curve with a pacelike principal normal and ome characterization In thi ection, we introduce the explicit parametrization of a pacelike Salkowki curve with a pacelike principal normal vector in Minkowki pace E 3 a the following: Definition 3. For any m R with m >, let u define the pace curve γ m (t) = n 4m with n = ( m. m coh[t] +n n coh[( n)t] coh[( + n)t], n +n coh[t] +n n inh[( n)t] inh[( + n)t], n +n ), coh[nt] m () We will call a pacelike Salkowki curve with a pacelike principal normal vector in Minkowki pace E 3. One can ee a pecial example (m = 5 and t [ 5, 5]) of uch curve in the right figure. The geometric element of thi curve γ m are the following: (): γ m, γ m = coh [nt] m, o γ m = coh[nt] m (): The arc-length parameter i = inh[nt] m. (3): The curvature κ(t) = and the torion τ(t) = tanh[nt].

4 454 Ahmad T. Ali (4): The Frenet frame i T(t) = ( n coh[t] inh[nt] inh[t] coh[nt], n inh[t] inh[nt] coh[t] coh[nt], n m inh[nt]), ) N(t) = m( n coh[t], inh[t], m, B(t) = ( inh[t] inh[nt] n coh[t] coh[nt], coh[t] inh[nt] n inh[t] coh[nt], n m coh[nt]). (3) From the expreion of the normal vector, ee Equation (3), we can ee that the normal indicatrix, or nortrix, of a Salkowki curve () in Minkowki pace E 3 decribe a parallel of the unit phere. The hyperbolic angle between the normal vector and the vector (0, 0, ) i contant and equal to φ = ±arccoh[n]. Thi fact i reminicent of what happen with another important cla of curve, the general helice in Minkowki pace E 3. Such a condition implie that the tangent indicatrix, or tantrix, decribe a parallel in the unit phere. Lemma 3. Let α : I E 3 be a pacelike curve with a pacelike principal normal vector parameterized by arc-length with κ =. The normal vector make a contant hyperbolic angle, φ, with a fixed traight line in pace if and only if τ() = ±. +tanh [φ] proof: ( ) Let d be the unitary pacelike fixed vector make a contant hyperbolic angle φ with the normal vector N. Therefore N, d = coh[φ]. (4) Differentiating Equation (4) and uing Frenet equation, we get Therefore, If we put B, d = b, we can write T + τb, d = 0. (5) T, d = τ B, d. d = τ b T + coh[φ]n + b B. From the unitary of the vector d we get b = ± inh[φ] τ. Therefore, the vector d can be written a d = ± τ inh[φ] inh[φ] T + coh[φ]n ± B. (6) τ τ

5 Spacelike Salkowki and anti-salkowki Curve with 455 If we differentiate Equation (5) again, we obtain τb + (τ )N, d = 0. (7) Equation (6) and (7) lead to the differential equation Integration the above equation, we get τ ± tanh[φ] + = 0. ( τ ) 3/ τ ± tanh[φ] + + c = 0. (8) τ where c i an integration contant. The integration contant can diappear with a parameter change c. Finally, to olve (8) with τ a unknown we expre the deired reult. ( ) Suppoe that τ = ± and let u conider the pacelike vector +tanh [φ] d = coh[φ] ( T + N + tanh [φ] B ). We will prove that the vector d i a contant vector. Indeed, applying Frenet formula ḋ = coh[φ] ( T + κn T + τb + tanh [φ] B τ + tanh [φ]n ) = 0 Therefore, d i contant and N, d = coh[φ]. Thi conclude the proof of Lemma (3.). Once the intrinic or natural equation of a curve have been determined, the next tep i to integrate Frenet formula with κ = and τ = ± + tanh [φ] = ± ( tanh[φ] tanh[φ] ) = ± tanh [ arcinh[ tanh[φ] ]]. (9) + Theorem 3.3 A pacelike curve ha a pacelike principal normal vector in Minkowki pace E 3 with κ = and uch that their normal vector make a contant angle with a fixed traight line i, up a rigid motion of the pace or up to the antipodal map, p p, pacelike Salkowki curve with a pacelike principal normal vector. Proof: We know from Definition 3. that the arc-length parameter of a Salkowki curve () i = t 0 γ m(u) du = inh[nt]. Therefore, t = m arcinh[m]. In term of the arc-length curvature and torion are then n κ() =, τ() = tanh[arcinh[m]],

6 456 Ahmad T. Ali the ame intrinic equation, with m = coth[φ] and n = m = coh[φ] m (compare with the poitive cae in Equation (9)), a the one hown in Lemma 3.. For the negative cae in Equation (9), let u recall that if a curve α ha torion τ α, then the curve β(t) = α(t) ha a torion τ β (t) = τ α (t), wherea curvature i preerved. Therefore, the fundamental theorem of curve in Minkowki pace tate in our ituation that, up a rigid motion or up to the antipodal map, the curve we are looking for are pacelike Minkowki curve with a pacelike principal normal vector. 4 Spacelike anti-salkowki curve with a pacelike principal normal A an additional material we will how in thi ection how to build, from a curve in Minkowki pace E 3 of contant curvature, another curve of contant torion. Let u recall that a curve α : I E 3, i -regular at a point t 0 if α (t 0 ) 0 and if κ α (t 0 ) 0. Lemma 4. Let α : I E 3 be a regular pacelike curve with a pacelike principal normal vector parameterized by arc-length with curvature κ α, torion τ α and Frenet frame {T α, N α, B α }. Let u β(t) = t 0 T α(u) B α(u) du. If α I atifie τ α ( α ) 0, the curve β i -regular at β and κ β = κ α τ α, τ β =, T β = T α, N β = N α, B β = B α. Proof: In order to obtain the tangent vector of β let u compute From the above equation, we get and T β ( β ) = β( β ) = dβ dt = T α B dt d α(t) dt. β d β d β dt = B α(t) = B α d α d α = τ α d α dt dt, (0) T β ( β ) = T α ( α ). Differentiation the above equation uing Frenet Equation () we obtain Ṫ β ( β ) = dt α d α d α dt dt d β.

7 Spacelike Salkowki and anti-salkowki Curve with 457 Uing Frenet Equation () and Equation (0), the above equation write From the above equation, we get κ β N β ( β ) = κ α τ α N α ( α ) and So we have κ β = κ α τ α, N β ( β ) = N α ( α ). B β ( β ) = T β ( β ) N β ( β ) = T α ( α ) N α ( α ) = B α ( α ). Differentiating the above equation with repect to β we get τ β =. Let u apply the previou reult to the curve γ m defined in Equation () we have the explicit parametrization of a pacelike anti-salkowki curve a follow: ( β m (t) = n 4m n +n inh[( + n)t] + inh[( n)t] + n inh[t], +n n n +n coh[( + n)t] coh[( n)t] + n coh[t], +n n ), (nt inh[nt]) m () where n = m m. Let u call thee curve by the name pacelike anti-salkowki curve with a pacelike principal normal vector. The preence of the nontrigonometric term nt in the third component of β m make that the change of variable tudied in Section for Salkowki curve doe not work for anti- Salkowki. Moreover, an example (m = 3.5 and t [ 3, 3]) of uch curve can be een in the left figure. Applying Lemma 4. we get the following Propoition 4. The curve β m in Equation () are curve of contant torion equal to and non-contant curvature equal to coth[nt]. Finally, we tate here the following Lemma 4.3 Let α : I E 3 be a regular pacelike curve with a pacelike principal normal vector parameterized by arc-length with curvature κ α, torion τ α and Frenet frame {T α, N α, B α }. Let u conider the curve β(t) = t 0 T α(u) T α(u) du. Then at a parameter α I uch that κ α ( α ) 0, the curve β i -regular at β and κ β =, τ β = τ α κ α, T β = T α, N β = N α, B β = B α.

8 458 Ahmad T. Ali Proof: The proof of thi Lemma i imilar a the proof of Lemma 4.. Theorem 4.4 The pacelike curve with τ = and uch that their principal normal vector make a contant hyperbolic angle with a fixed traight line are the pacelike anti-salkowki curve defined in Equation (). Proof: Let α be a pacelike curve with τ = and let β(t) = t 0 T α(u) T α(u) du. By Lemma 4.3, β i a curve with contant curvature κ =, non-contant torion τ = κ α and with the ame principal normal vector. Therefore, β i a Salkowki curve and α i an anti-salkowki curve in Minkowki 3-pace Figure : Spacelike Salkowki (right) and anti-salkowki (left) curve. 5 Concluion In thi paper, we defined the Salkowki and anti-salkowki curve in Minkowki pace E 3. Next, we introduced an explicit parametrization of a pacelike Salkowki curve with a pacelike principal normal vector and a pacelike anti-salkowki curve with a pacelike principal normal vector in Minkowki 3-pace. Moreover, we characterized them a a pace curve with contant curvature or contant torion and whoe principal normal vector make a contant angle with a fixed traight line.

9 Spacelike Salkowki and anti-salkowki Curve with Open Problem In recent year, by the coming of the theory of relativity, reearcher treated ome of claical differential geometry topic to extend analogou problem to Lorentzian manifold. In a imilar way, we tudy a claical topic in Minkowki pace. In thi work, we have invetigated pacelike Salkowki curve in Minkowki pace. We have given ome explicit characterization of thee curve in term of Frenet equation. Additionally, problem uch a; invetigation of timelike Salkowki curve or extending uch kind curve to Minkowki pace-time or higher dimenional Euclidean pace can be preented a further reearche. Reference [] A.T. Ali and R. Lopez, Slant helice in Minkowki pace E 3. Preprint 008: arxiv: v [math.dg]. [] A. Ferrandez, A. Gimenez and P. Luca, Null helice in Lorentzian pace form. Int. J. Mod. Phy. A. 6 (00) [3] K. Ilarlan and O. Boyacioglu, Poition vector of a pacelike W-curve in Minkowki pace E 3. Bull. Korean Math. Soc. 44(3) (007) [4] K. Ilarlan and O. Boyacioglu, Poition vector of a timelike and a null helix in Minkowki 3-pace. Chao Soliton and Fractal 38 (008) [5] W. Kuhnel, Differential geometry: Curve, Surface, Manifold, Weibaden: Braunchweig, (999). [6] R. Lopez, Differential Geometry of Curve and Surface in Lorentz- Minkowki Space, Preprint 008: arxiv: v [math.dg]. [7] J. Monterde, Salkowki curve reviited: A family of curve with contant curvature and non-contant torion. Computer Aided Geometric Deign. 6 (009) [8] H. Pottmann and J.M. Hofer, A variational approach to pline curve on urface. Computer Aided Geometric Deign. (005) [9] E. Salkowki, Zur tranformation von raumkurven. Mathematiche Annalen. 66(4) (909)

10 460 Ahmad T. Ali [0] J. Walrave, Curve and urface in Minkowki pace. Doctoral Thei, K.U. Leuven, Fac. Sci., Leuven, (995).

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