On Weingarten tube surfaces with null curve in Minkowski 3-space

NTMSCI 3, No. 3, 168-174 (2015) 168 New Trends in Mathematical Sciences http://www.ntmsci.com On Weingarten tube surfaces with null curve in Minkowski 3-space Pelin (Pospos) Tekin 1 and F. Nejat Ekmekci 2 1 Aksaray University, Faculty of Sciences and Arts, Department of Mathematics 68100, Aksaray, Turkey. 2 Ankara University, Faculty of Sciences, Department of Mathematics 06100, Ankara,, Turkey. Received: 16 February 2015, Revised: 19 May 2015, Accepted: 31 May 2015 Published online: 30 September 2015 Abstract: In this paper, we study a tube surfaces, consisted spline curve is null. Furthermore, we have investigated Weingarten conditions of this surface using the the mean curvatre H, the Gaussian curvature K and the second Gaussian curvature K II, respectively. Keywords: Tube surfaces, Weingarten conditions, null curve, curvatures. 1 Introduction A surface S in either R 3 or E1 3 is called a Weingarten surface if there exists a non-trivial functional relation Ω(K,H) = 0 with respect to its Gaussian curvature K and its mean curvature H. The existence of a non-trivial functional relation Ω(K,H) = 0 on the surface S parametrized by Φ(u,v) is equivalent to the vanishing of the corresponding Jacobian determinant, namely (K,H) (u,v) = 0 Several geometers have studied on Weingarten surfaces and obtained many interesting results. For study of these surfaces, in 1994 W. Kühnel and in 1999 G. Stamou, investigated ruled Weingarten surface in a Euclidean 3-space E 3. Also, in 1997 C. Baikoussis and Th. Koufogiorgos studied helicoidal (H,K II )-Weingarten surfaces. in 1999 F. Dillen and W. Kühnel, in 2005 F. Dillen and W.Sodsiri gave a classification of ruled Weingarten surface in a Minkowski 3-space E 3 1. In 2009 J. Suk Ro and D. Won Yoon studied tubular Weingarten and linear Weingaren surface in three dimension Euclidean space E 3. In this paper we study a tube surface with null curve in a three dimension Minkowski space. 2 Preliminaries Let R 3 = {(x,y,z) : x,y,z R} be a 3-dimension space, and let X = (x 1,x 2,x 3 ) and Y = (y 1,y 2,y 3 ) be two vectors in R 3. The Lorentz scalar product of X and Y is defined by X,Y = x 1 y 1 + x 2 y 2 x 3 y 3 (2.1) Corresponding author e-mail: pelinpospos@gmail.com

169 P. (Pospos) Tekin and F. Nejat Ekmekci: On Weingarten tube surfaces with null curve in Minkowski 3-space E1 3 = (R3, X,Y ) is called Minkowski 3-space. Since the metric is indefinite, x E1 3 has three causal characters: these are spacelike vector, null (lightlike) vector or timelike vector if x,x > 0 or x = 0, x,x = 0 and x 0, x,x < 0, respectively. For x E1 3, the norm of the vector x is given by x = x,x. Therefore, x is a unit vector if x,x = ±1. Similarly, an arbitrary curve α = α(s) E1 3 can locally be spacelike, timelike or null (lightlike) curve, if all of its velocity vectors α (s) are respectively spacelike, timelike or null (lightlike) i.e., α (s),α (s) > 0, α (s),α (s) < 0, α (s),α (s) = 0. So, α(s) is a unit speed curve if α (s),α (s) = ±1, where s is the arc-length parameter of α. Any two vectors X,Y E1 3 are called orthogonal if X,Y = 0 (O Neill, 1983). The vector product of two vectors X = (x 1,x 2,x 3 ), Y = (y 1,y 2,y 3 ) belong to E1 3, is defined as X Y = (x 3 y 2 x 2 y 3,x 1 y 3 x 3 y 1,x 1 y 2 x 2 y 1 ). We also recall that the pseudosphere of radius 1 and center at the origin is the hyperquadric in E1 3 S1 2(1) = {v E3 1 : v,v = 1} (O Neill, 1983). defined by Let α = α(u) : (a,b) E1 3 be a spacelike unit speed curve with a timelike binormal B, where u is the arc length parameter of α. Considering that T = 1, B = T T and N = T T T T, we obtain the orthonormal frame field {T (u),n(u),b(u)}. Consider M is a timelike tube surface parametrized by Ψ : j R E1 3. Then, the position vector of Ψ can be written in the following form Ψ(u,v) = α(u) + r(n(u)coshv + B(u)sinhv) (2.2) where α,n,b : j E1 3 and r : j R. We call α a base curve and a pair of two vectors N,B a director frame of the timelike tube surface Ψ. We must have Ψ(u,v) α(u) 2 = r 2 The last equation expresses analytically the geometric fact that Ψ(u,v) lies on a Lorentzian sphere S1 2 (u) of radius r centered at α(u) (Abdel-Aziz and Saad, 2011). Theorem 2.1. Let Ψ be a timelike tube surface in Minkowski 3-space E1 3 defined in (2.2) satisfying the Jacobi condition Ω(K,H) = 0 (2.3) for the Gaussian curvature K and the mean curvature H of Ψ. Then, Ψ is a Weingarten surface (Abdel-Aziz and Saad, 2011). Theorem 2.2. Let Ψ be a timelike tube surface with non-degenerate second fundamental form in Minkowski 3-space E 3 1 satisfying the Jacobi equation Ω(K II,K) = 0 (2.4) for the second Gaussian curvature K II and the Gaussian curvature K. Then, the surface Ψ is a Weingarten surface (Abdel- Aziz and Saad, 2011).

NTMSCI 3, No. 3, 168-174 (2015) / www.ntmsci.com 170 3 Tubular weingarten surface with a null curve A parametric equation of a Ψ tube surface with C(t) : (a,b) E1 3 null curve is given by Ψ(t,θ) = C(t) + θn(t) + θ 2 B(t) (3.1) where T, N, B vectors are null, spacelike and null vectors respectively. The orthonormal frame is {T (t), N(t), B(t)} and we have the following conditions T,T = B,B = 0 N,N = T,B = 1 N,B = N,T = 0 κ(t) and τ(t) are curvature and torsion respectively and we can give differential formulas for this system as below T (t) = N N (t) = τt B B (t) = τn therefore, differentiation of the tube surface Ψ can be calculate as follows, Ψ t = (1 + τθ)t (t) + ( τθ 2) N(t) + ( θ)b(t) Ψ θ = N(t) + 2θB(t) Ψ t Ψ θ = ( 2τθ 3 + θ ) T (t) + ( 2θ + 2τθ 2) N(t) + ( 1 τθ)b(t) Ψ t Ψ θ = 8τ 2 θ 4 + 12τθ 3 2τθ 2 + 4θ 2 2θ we find the the unit normal vector field of the tube surface Ψ express by ξ (u) = Ψ t Ψ θ Ψ t Ψ θ = 1 ( 2τθ 3 + θ ) T (t) + ( 2θ + 2τθ 2) N(t) + ( 1 τθ)b(t) λ where λ = 8τ 2 θ 4 + 12τθ 3 2τθ 2 + 4θ 2 2θ. furthermore components of the first fundamental form of Ψ are E = Ψ t,ψ t = 2θ (1 + τθ) + τ 2 θ 4 F = Ψ t,ψ θ = τθ 2 + 2θ G = Ψ θ,ψ θ = 1 Components of the second fundamental form of Ψ are P = Ψ tt,σ = 1 λ [( 1 + 2τθ τ θ 2)( 2θ + 2τθ 2) + ( 1 + τ θ τ 2 θ 2) ( 1 τθ) + ( τθ 2)( 2τθ 3 + θ ) ] Q = Ψ tθ,σ = 1 [ ( τ ( 1 τθ) + ( 2τθ) 2θ + 2τθ 2 ) + ( 2τθ 3 θ )] λ W = Ψ θθ,σ = 1 λ [ 4τθ 3 + 2θ ]

171 P. (Pospos) Tekin and F. Nejat Ekmekci: On Weingarten tube surfaces with null curve in Minkowski 3-space The Gaussian curvature K, second Gaussian curvature K II and the mean curvature H are given by K = PW Q2 EG F 2 = 1 2 [ 8τ3 θ 8 + 8τ 2 τ θ 7 + ( 12τ 2 + 8ττ + 20τ 4) θ 6 + ( 4τ 2 τ + 36τ 3 8τ 2 4ττ ) θ 5 + ( 4τ 4ττ + 16τ 2 14τ 6τ 3 8τ 4) θ 4 + ( 6τ 2 + 2ττ 16τ 3 + 4τ ) θ 3 + ( 6τ 2 + 2τ + τ 4 + 5 ) θ 2 + ( 2τ 3 + 2τ ) θ + τ 2 ]/ ( θ ( 1 + 2τθ 2 + (τ + 2)θ )) and H = EW 2FQ + GP 2(EG F 2 ) = 1 4 [4τ3 θ 6 + ( 4τ 2 + 8τ 3 )θ 4 + (24τ 2 2ττ )θ 3 +( 2τ + 17τ + 4τ 2 3τ 3 )θ 2 + ( 7τ 2 + 6τ + ττ )θ 4τ + 2 + τ ]/(1 + 2τθ 2 + (τ + 2)θ). Also, we get K II = 1 4 [(64τ3 (τ ) 2 + 64τ 4 τ )θ 13 + ( 32τ 4 τ 64(τ ) 3 τ 2 160τ 3 τ τ )θ 12 +( 280τ 5 64τ τ 2 τ + τ τ 3 τ 32τ 3 τ 208τ 5 τ + 240τ 6 + 96(τ ) 2 τ τ 2 +128τ 3 (τ ) 2 256τ 4 (τ ) 2 48τ 3 τ 64(τ ) 3 τ + 128τ 4 τ 192(τ ) 2 τ 2 )θ 11 +(96(τ ) 3 τ 320τ 4 τ + 48τ 3 τ 96(τ ) 3 τ 2 + 16τ 4 τ 240τ 5 τ + 80τ 5 τ +288τ 4 τ τ + 160τ τ 2 τ + 96(τ ) 2 ττ + 64τ τ τ 2 + 64(τ ) 2 τ 2 80τ 3 τ +96(τ ) 3 τ 3 + 160τ 3 τ + 160τ 5 + 744τ 4 τ 320τ 3 (τ ) 2 48τ 3 τ τ )θ 10 +(564τ 4 + 192τ 5 + 128τ 3 τ + 208τ(τ ) 2 376(τ ) 2 τ 2 544τ 4 τ + 1852τ 6 +64(τ ) 3 192τ 7 + 240τ 5 (τ ) 2 352τ 4 (τ ) 2 288τ 3 (τ ) 2 + 128(τ ) 3 τ 2 96(τ ) 3 τ + 512τ 3 τ 288τ τ 2 τ 32τ τ 3 τ 96(τ ) 2 τ τ 2 48(τ ) 2 ττ 16τ τ τ 2 + 16τ ττ + 32τ ττ + 288τ 4 τ + 224τ 4 τ 16τ 3τ 160τ 5 τ 200τ 3 τ + 40τ 2 τ + 120τ 6 τ + 48τ 2 τ )θ 9 + (216τ 4 + 60τ 2 τ + 3196τ 5 1432τ 3 τ 288τ(τ ) 2 600(τ ) 2 τ 2 + 488τ 4 τ 296τ 6 16(τ ) 3 + 120τ 6 τ 1072τ 5 τ + 512τ 4 (τ ) 2 752τ 3 (τ ) 2 48(τ ) 3 τ 3 + 16(τ ) 3 τ 2 + 48(τ ) 3 τ 3 +16(τ ) 3 τ 2 + 48τ 3 τ 32τ 3 τ τ + 208τ τ 2 τ 240τ 4 τ τ 96(τ ) 2 ττ 64τ τ τ 2 256τ ττ 32τ ττ 480τ 4 τ 24τ 4 τ + 184τ 3 τ + 296τ 5 τ +504τ 3 τ 72τ 5 τ 8τ 2 τ 88τ 2 τ 48(τ ) 2 τ )θ 8 + ( 370τ 3 + 1876τ 4 184τ 2 τ 180τ 5 + 1360τ 3 τ 96(τ ) 2 48τ(τ ) 2 180(τ ) 2 τ 2 2808τ 4 τ +1272τ 6 + 32(τ ) 3 578τ 7 96τ 5 (τ ) 2 + 320τ 5 τ + 128τ 4 (τ ) 2 + 768τ 3 (τ ) 2 64(τ ) 3 τ 2 + 36τ 8 488τ 3 τ τ + 48τ τ 2 τ + 8τ τ 3 τ + 24(τ ) 2 τ τ 2 +48(τ ) 2 ττ + 16τ τ τ 2 + 152τ ττ 32τ ττ + 208τ 4 τ 208τ 4 τ 32τ 3 τ + 4τ 5 τ 684τ 3 τ + 352τ 2 τ 48τ 6 τ + 56τ 2 τ + 44ττ +8τ τ 56ττ 16τ τ )θ 7 + (236ττ + 528τ 3 + 396τ 4 + 1970τ 2 τ +4188τ 5 2376τ 3 τ + 96τ 2 + 532τ(τ ) 2 + 716(τ ) 2 τ 2 + 472τ 4 τ 1468τ 6 8(τ ) 3 + 84τ 7 58τ 6 τ + 624τ 5 τ 224τ 4 (τ ) 2 + 324τ 3 (τ ) 2 + 24(τ ) 3 τ 2 +258τ 6 + 8(τ ) 3 5τ 7 + 4τ 6 τ 92τ 5 τ + 28τ 4 τ 74τ 3 (τ ) 2 50τ + 48τ τ 2 τ 6τ 4 τ τ + 8τ ττ 8τ ττ + 56τ 3 τ + 16τ 5 τ 2τ 3 τ 2τ 5 τ + 114τ 2 τ +12τ 2 τ 12(τ ) 2 + 48ττ 54τ τ 10ττ )θ 4 + ( 31ττ 46τ + 416τ 2 226τ 3 234τ 858τ 4 + 212τ 2 τ + 158τ 5 80τ 3 τ 97(τ ) 2 + 10τ(τ ) 2 28(τ ) 2 τ 2 226τ 4 τ + 22τ 6 6τ 7 + 7τ 5 τ + τ 4 (τ ) 2 + 20τ 3 (τ ) 2 10τ

NTMSCI 3, No. 3, 168-174 (2015) / www.ntmsci.com 172 12τ 3 τ τ 12τ ττ + 14τ 4 τ 6τ 4 τ τ 5 τ 12τ 3 τ + 8τ 2 τ + 12τ 2 τ +17ττ 4ττ )θ 3 + (18τ(τ ) 2 4ττ 2τ 4 τ 6τ 3 τ + 4(τ ) 2 τ 2 10τ 6 44τ 2 5τ 6τ 3 τ + 4(τ ) 2 τ 2 10τ 6 44τ 2 5τ 252τ 3 + 3τ 4 τ + 4τ 3 τ 48(τ ) 3 τ + 48τ 3 τ τ 216τ τ 2 τ + 60τ 4 τ τ + 24τ 2 ττ + 16τ τ τ 2 +112τ ττ + 32τ ττ + 68τ 4 τ + 12τ 4 τ 192τ 3 τ 128τ 5 τ + 60τ 3 τ +20τ 5 τ 504τ 2 τ + 20τ 2 τ + 48(τ ) 2 τ + 64ττ + 108τ τ + 36ττ )θ 6 +(640ττ + 187τ 2 + 264τ 3 + 4851τ 4 1150τ 2 τ 1396τ 5 + 384τ 3 τ +134(τ ) 2 + 216τ(τ ) 2 28(τ ) 2 τ 2 + 1590τ 4 τ 291τ 6 32(τ ) 3 + 120τ 7 +12τ 5 (τ ) 2 136τ 5 τ 34τ 4 (τ ) 2 160τ 3 (τ ) 2 + 32(τ ) 3 τ 3τ 8 30τ 20τ + 120τ 3 τ τ + 56τ τ 2 τ 12(τ ) 2 ττ 4τ τ τ 2 56τ ττ + 8τ ττ 90τ 4 τ + 60τ 4 τ + 24τ 3 τ + 2τ 5 τ + 156τ 3 τ + 78τ 2 τ + 6τ 6 τ 56τ 2 τ 206ττ 4τ τ + 28ττ + 16τ τ )θ 5 + ( 768ττ 114τ 2 + 2239τ 3 107τ 672τ 4 + 16τ 2 τ 962τ 5 + 1154τ 3 τ 20(τ ) 2 338τ(τ ) 2 28(τ ) 2 137τ 4 τ +18τ 4 + 2τ 3 (τ ) 2 142τ 3 τ + 4τ 5 τ + 54ττ 11τ 2 τ + 68τ + 48τ 5 17τ 2 τ 6τ τ 2 τ )θ 2 + ( τ 3 τ + (τ ) 2 8τ 2 τ + 10τ 4 τ 4τ 5 + 8 + 6τ 4τ 3 + 28τ 4 2τ 2 τ ττ + 3(τ ) 2 4τ 2 + 28τ 4 2τ 2 τ ττ + 3(τ ) 2 4τ 2 )θ + 6τ 3 τ +4τ 3 2ττ + 4τ)] /[ ( 8τ 3 )θ 8 + (9τ 2 τ )θ 7 + (20τ 4 + 8ττ + 12τ 2 )θ 6 + ( 4τ 2 τ +36τ 3 8τ 2 4ττ )θ 5 ( 4ττ + 16τ 2 8τ 4 14τ 4τ 6τ 3 )θ 4 + (4τ 6τ 2 16τ 3 + 2ττ )θ 3 + (5 6τ 2 + 2τ + τ 4 )θ 2 + (2τ 3 + 2τ)θ + τ 2 ) 2 ]. We have investigated the Weingarten condition for the Gaussian curvature, Mean curvature and the second Gaussian curvature by taking τ = 1, under this condition we can give the following theorems. Theorem 3.1. Let Ψ be a tube surface in Minkowski 3-space E1 3 defined in (3.1) satisfying the Jacobi condition Ω(K,H) = 0 for the Gaussian curvature K and the mean curvature H of Ψ. Then, Ψ is a Weingarten surface. Proof. Let Ψ be a tube surface in E1 3 parametrized by (3.1) and satisfying Jacobi condition. Then, we have, differentiating K and H with respect to t, K H u v K H v u = 0 (3.2) K t = 1 2 [ 16τ3 τ θ 9 + (8τ 3 τ + 8τ 2 (τ ) 2 24τ 2 τ )θ 8 +(16τ 2 τ + 60τ 4 τ + 16τ(τ ) 2 + 12τ 2 τ )θ 7 +(152τ 3 τ 8τ 2 τ 4τ 2 (τ ) 2 + 8(τ ) 2 4τ 3 τ 4τ 2 τ + 8ττ + 24ττ )θ 6 +(124τ 2 τ 16ττ 8τ 2 τ 8ττ 24τ 4 τ 12τ 3 τ 8τ(τ ) 2 )θ 5 +(32ττ 4ττ 14τ 4τ 64τ 3 τ 24τ 2 τ 4(τ ) 2 + 2τ 2 τ )θ 4 +( 12ττ + 3τ 4 τ + 4ττ τ 54τ 2 τ )θ 3 + (8τ 3 τ 12ττ + 2τ )θ 2 (2τ + 7τ 2 τ )θ + 2ττ ]/((τθ + 1)(1 + 2τθ 2 + (τ + 2)θ)θ),

173 P. (Pospos) Tekin and F. Nejat Ekmekci: On Weingarten tube surfaces with null curve in Minkowski 3-space and H t = 1 4 [8τ3 τ θ 7 + 12τ 2 τ θ 6 + ( 4τ 2 τ + 16τ 3 τ )θ 5 + ( 8ττ + 48τ 2 τ 2τ 2 τ )θ 4 (48ττ 6τ 3 τ 4ττ + 4τ 2 τ )θ 3 + (τ 2 τ 16τ 2 τ + 17τ + 8ττ 2τ )θ 2 (2ττ 14ττ + 4τ )θ + τ 4τ ]/[(τθ + 1)(1 + 2τθ 2 + (τ + 2)θ], differentiating K and H with respect to θ, K θ = 1 2 [ 80τ4 θ 10 + ( 96τ 3 48τ 4 + 64τ 3 τ )θ 9 + (16τ 3 + 120τ 5 + 128τ 2 τ + 40τ 3 τ )θ 8 +(96τ 2 16τ 3 τ + 64τ 2 τ + 16τ 3 + 80τ 5 + 304τ 4 + 64ττ )θ 7 +(224τ 3 + 8ττ + 196τ 4 16τ 2 44τ 2 τ 16τ 5 12τ 3 τ )θ 6 +(4τ 2 24τ 2 τ 44τ 4 56τ + 152τ 3 16τ 5 40ττ 16τ )θ 5 +( 40τ 4 44τ 12τ + 2τ 2 τ 44τ 3 + 40τ 2 2τ 5 12ττ )θ 4 +( 20τ 2 + 8τ + 4ττ 32τ 3 8τ 4 )θ 3 + ( 10τ 3 τ 4 4τ + 2τ + 5 8τ 2 )θ 2 ( 2τ 3 4τ 2 )θ τ 2 ]/(θ 2 (1 + 2τθ 2 + (τ + 2)θ) 2 ), and H θ = 1 4 [32τ4 θ 7 + (40τ 3 + 20τ 4 )θ 6 + (32τ 4 + 8τ 3 )θ 5 ( 4τ 2 τ + 24τ 4 + 84τ 3 24τ 2 )θ 4 ( 4τ 2 τ + 80τ 2 + 80τ 3 8ττ )θ 3 + (34τ 8ττ 2τ 4τ 6τ 3 4ττ )θ 4 + 12τ 2τ 3τ 2 ]/[1 + 2τθ 2 + (τ + 2)θ] 2. From the above equations (3.2) is satisfied, and then, the surface Ψ is a Weingarten surface. Theorem 3.2. Let Ψ be a tube surface with non-degenerate second fundamental form in Minkowski 3-space E 3 1 satisfying the Jacobi condition Ω(K II,K) = 0 (3.3) for the second Gaussian curvature K II and the Gaussian curvature K. Then, the surface Ψ is a Weingarten surface. Proof. For the proof, we can use the similar way with the theorem 3.1. When we take the partial derivative of the Gaussian curvature K and the second Gaussian curvature K II, we can find the following equation Then, Ψ is a Weingarten tube surface. K t K II θ K K II = 0. (3.4) θ t References [1] Abdel-Aziz, H.S., Saad, M.K., Weingarten Timelike Tube Surfaces Around a Spacelike Curve, Int. Journal of Math. Analysis, Vol.5, no.25, 1225-1236, 2011. [2] Balgetir, H., Bektaş, M., Inoguchi, J., Null Bertrand curves in Minkowski 3- space and their Characterizations, Note di Matematica, 23, n.1, 7-13, 2004. [3] Dillen, F.and Kühnel W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math., 98, 307 320, 1999. [4] Doğan, F. Genelleştirilmiş Kanal Yüzeyleri, Doktora Tezi, Ankara, 2012. [5] Duggal, K. L., and Bejancu, A., Lightlike submanifolds of semi-riemannian manifolds and applications, Kluwer Academic, Dordrecht and Boston, Mass., 1996.

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