D Tangent Surfaces of Timelike Biharmonic D Helices according to Darboux Frame on Non-degenerate Timelike Surfaces in the Lorentzian Heisenberg GroupH

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1 Bol. Soc. Paran. Mat. (3s.) v (2014): c SPM ISSN on line ISSN in press SPM: doi: /bspm.v32i D Tangent Surfaces of Timelike Biharmonic D Helices according to Darboux Frame on Non-degenerate Timelike Surfaces in the Lorentzian Heisenberg GroupH Talat Körpinar and Essin Turhan abstract: In this paper, we study D tangent surfaces of timelike biharmonic D-helices according to Darboux frame on non-degenerate timelike surfaces in the Lorentzian Heisenberg group H. We obtain parametric equation D tangent surfaces of timelike biharmonic D-helices in the Lorentzian Heisenberg group H. Moreover, we illustrate the figure of our main theorem. Key Words: D Biharmonic curve, D tangent surfaces, Darboux frame, Heisenberg group. Contents 1 Introduction 35 2 Preliminaries 36 3 Timelike Biharmonic D-Helices According to Darboux Frame on a Non-Degenerate Timelike Surface in the Lorentzian Heisenberg Group H 37 4 D Tangent Surfaces According to Darboux Frame on a Non-Degenerate Timelike Surface in the Lorentzian Heisenberg Group H Introduction A smooth map φ : N M is said to be biharmonic if it is a critical point of the bienergy functional: E 2 (φ) = where T(φ) := tr φ dφ is the tension field of φ N 1 2 T(φ) 2 dv h, 2000 Mathematics Subject Classification: 31B30, 58E20 35 Typeset by B S P M style. c Soc. Paran. de Mat.

2 36 Talat Körpinar and Essin Turhan The Euler Lagrange equation of the bienergy is given by T 2 (φ) = 0. Here the section T 2 (φ) is defined by T 2 (φ) = φ T(φ)+trR(T(φ),dφ)dφ, and called the bitension field of φ. Non-harmonic biharmonic maps are called proper biharmonic maps. In this paper, we study D tangent surfaces of timelike biharmonic D-helices according to Darboux frame on non-degenerate timelike surfaces in the Lorentzian Heisenberg group H. We obtain parametric equation D tangent surfaces of timelike biharmonic D-helices in the Lorentzian Heisenberg group H. Moreover, we illustrate the figure of our main theorem. 2. Preliminaries Heisenberg group plays an important role in many branches of mathematics such as representation theory, harmonic analysis, PDEs or even quantum mechanics, where it was initially defined as a group of 3 3 matrices 1 x z 0 1 y : x,y,z R with the usual multiplication rule. The left-invariant Lorentz metric on H is ρ = dx 2 +dy 2 +(xdy +dz) 2. The following set of left-invariant vector fields forms an orthonormal basis for the corresponding Lie algebra: { e 1 = z, e 2 = y x z, e 3 = }. (2.1) x The characterising properties of this algebra are the following commutation relations: ρ(e 1,e 1 ) = ρ(e 2,e 2 ) = 1, ρ(e 3,e 3 ) = 1. (2.2)

3 D Tangent Surfaces of Timelike Biharmonic D Helices Timelike Biharmonic D-Helices According to Darboux Frame on a Non-Degenerate Timelike Surface in the Lorentzian Heisenberg Group H LetΠ H be a timelike surface with the unit normal vectornin the Lorentzian Heisenberg group H. If γ is a timelike curve on Π H, then we have the Frenet frame {T,N,B} and Darboux frame {T,n,g} with spacelike vector g = T n along the curve γ. Let {T,N,B} be the Frenet frame fields tangent to the Lorentzian Heisenberg group H along γ defined as follows: T is the unit vector field γ tangent to γ, N is the unit vector field in the direction of T T (normal to γ) and B is chosen so that {T,N,B} is a positively oriented orthonormal basis. Then, we have the following Frenet formulas: T T = κn, T N = κt+τb, (3.1) T B = τn, where κ is the curvature of γ and τ is its torsion and ρ(t,t) = 1, ρ(n,n) = 1, ρ(b,b) = 1, (3.2) ρ(t,n) = ρ(t,b) = ρ(n,b) = 0. Let θ be the angle between N and n. The relationships between {T,N,B} and {T, n, g} are as follows: and T = T, N = cosθn+sinθg, (3.3) B = sinθn cosθg, T = T, g = sinθn cosθb, (3.4) n = cosθn+sinθb. By differentiating (3.4), using (3.1), (3.3) and Frenet formulas we obtain T T = (κcosθ)n+(κsinθ)g, ( T g = ( κsinθ)t+ τ + dθ ) n, (3.5) ds ( T n = ( κcosθ)t τ + dθ ) g. ds

4 38 Talat Körpinar and Essin Turhan If we represent κcosθ, κsinθ and τ + dθ ds with the symbols κ n, κ g, and τ g respectively, then the equations in (3.5) can be written as T T = κ g g+κ n n, T g = κ g T+τ g n, (3.6) T n = κ n T τ g g. With respect to the orthonormal basis {e 1,e 2,e 3 }, we can write T = T 1 e 1 +T 2 e 2 +T 3 e 3, g = g 1 e 1 +g 2 e 2 +g 3 e 3. (3.7) n = n 1 e 1 +n 2 e 2 +n 3 e 3, To separate a curve according to Darboux frame from that of Frenet- Serret frame, in the rest of the paper, we shall use notation for the curve as D-curve. First of all we recall the following well known result (cf. [8]). Theorem 3.1. Let γ : I Π H be a non-geodesic unit speed timelike curve on timelike surface Π in the Lorentzian Heisenberg group H. γ is a unit speed timelike biharmonic curve on Π if and only if κ 2 n +κ2 g = constant 0, (3.8) κ n κ3 n +κ gτ g κ 2 g κ n +κ g τ g +κ g τ g τ2 g κ n = κ n (1 4g 2 1 ) 4κ gn 1 g 1, κ g κ 3 g 2κ nτ g κ 2 nκ g κ n τ g κ g τ 2 g = 4κ n n 1 g 1 +κ g (1 4n 2 1). Theorem 3.2. Let γ : I Π H be a non-geodesic unit speed timelike biharmonic D- helix on timelike surface Π in the Lorentzian Heisenberg group H. Then parametric equations of timelike biharmonic D- helix are x(s) = coshℵ R (cosh[rs]sinh[ ]+cosh[ ]sinh[rs])+ 1, y(s) = coshℵ R (cosh[ ]cosh[rs]+sinh[ ]sinh[rs])+ 2, (3.9) z(s) = sinhℵs+ ( +Rs) 2R 2 cosh 2 ℵ 1 R cosh2 ℵcosh[ ]cosh[rs] 1 R cosh2 ℵsinh[ ]sinh[rs] 1 4R 2 cosh2 ℵsinh2[Rs+ ]+ 3, where, 1, 2, 3, are constants of integration and R =sechℵ κ 2 n +κ2 g 2sinhℵ. Using Mathematica in above system we have Fig.1:

5 D Tangent Surfaces of Timelike Biharmonic D Helices 39 Fig D Tangent Surfaces According to Darboux Frame on a Non-Degenerate Timelike Surface in the Lorentzian Heisenberg Group H A ruled surface is formed by a continuous family of straight line segments. The D tangent surface of γ is a ruled surface O(s,u) = γ(s)+ut(s). (4.1) Theorem 4.1. Let γ : I O H be a non-geodesic unit speed timelike biharmonic D- helix on timelike surface O in the Lorentzian Heisenberg group H. Then,

6 40 Talat Körpinar and Essin Turhan equation of D tangent surface of timelike biharmonic D- helix is O(s,u) = [sinhℵs+ ( +Rs) 2R 2 cosh 2 ℵ 1 R cosh2 ℵcosh[ ]cosh[rs] 1 R cosh2 ℵsinh[ ]sinh[rs] 1 4R 2 cosh2 ℵsinh2[Rs+ ] + 3 +[ coshℵ R (cosh[rs]sinh[ ]+cosh[ ]sinh[rs])+ 1] (4.2) [ coshℵ R (cosh[ ]cosh[rs]+sinh[ ]sinh[rs])+ 2]+usinhP]e 1 +[ coshℵ R (cosh[ ]cosh[rs]+sinh[ ]sinh[rs]) +ucoshpsinh[ms+n]+ 2 ]e 2 +[ coshℵ R (cosh[rs]sinh[ ]+cosh[ ]sinh[rs]) +ucoshpcosh[ms+n]+ 1 ]e 3, where, 1, 2, 3, are constants of integration and R =sechℵ κ 2 n +κ 2 g 2sinhℵ. Proof: From the assumption we get T = sinhpe 1 +coshpsinh[ms+n]e 2 +coshpcosh[ms+n]e 3. (4.3) Substituting Eq.(3.9) and Eq.(4.3) into Eq.(4.1), we obtain Eq.(4.2). This completes the proof. Theorem 4.2. Let γ : I O H be a non-geodesic unit speed timelike biharmonic D- helix on timelike surface O in the Lorentzian Heisenberg group H. Then, equation of D tangent surface of timelike biharmonic D- helix are x O = [ coshℵ R (cosh[rs]sinh[ ]+cosh[ ]sinh[rs])+ucoshpcosh[ms+n]+ 1] y O = [ coshℵ R (cosh[ ]cosh[rs]+sinh[ ]sinh[rs])+ucoshpsinh[ms+n]+ 2] z O = [sinhℵs+ ( +Rs) 2R 2 cosh 2 ℵ 1 R cosh2 ℵcosh[ ]cosh[rs] 1 R cosh2 ℵsinh[ ]sinh[rs] 1 4R 2 cosh2 ℵsinh2[Rs+ ] + 3 +[ coshℵ R (cosh[rs]sinh[ ]+cosh[ ]sinh[rs])+ 1] [ coshℵ R (cosh[ ]cosh[rs]+sinh[ ]sinh[rs])+ 2]+usinhP] [ coshℵ R (cosh[ ]cosh[rs]+sinh[ ]sinh[rs])+ucoshpsinh[ms+n]+ 2] [ coshℵ R (cosh[rs]sinh[ ]+cosh[ ]sinh[rs])+ucoshpcosh[ms+n]+ 1],

7 D Tangent Surfaces of Timelike Biharmonic D Helices 41 where, 1, 2, 3, are constants of integration and R =sechℵ κ 2 n +κ 2 g 2sinhℵ. Proof: From Theorem 4.1, we easily have above system, which completes the proof. In the light of Theorem 4.2, we give the following figures for the D tangent surface of timelike biharmonic D- helix. Fig. 2 Fig. 3

8 42 Talat Körpinar and Essin Turhan References 1. J. F. Burke: Bertrand Curves Associated with a Pair of Curves, Mathematics Magazine, 34 (1) (1960), R. Caddeo, S. Montaldo, P. Piu: Biharmonic curves on a surface, Rend. Mat. Appl. 21 (2001), J. Eells, J.H. Sampson: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), A.A. Ergin: Timelike Darboux curves on a timelike surface M M1 3, Hadronic Journal 24 (6) (2001), N. Ekmekçi and K. Ilarslan: On Bertrand curves and their characterization, Diff.Geom. Dyn. Syst., 3 (2) (2001), S. Flöry, H. Pottmann, Ruled surfaces for rationalization and design in architecture. In: Proceedings of the conference of the association for computer aided design in architecture (ACADIA); S.Izumiya, D.Pei, T.Sano: The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space, Glasgow Math. J. 42 (2000), T. Körpınar and E. Turhan, On Characterization Timelike Biharmonic D-Helices According to Darboux Frame On Non-Degenerate Timelike Surfaces In The Lorentzian Heisenberg Group, Annals of Fuzzy Mathematics and Informatics, 4 (2) (2012), T. Körpınar and E. Turhan: On Spacelike Biharmonic Slant Helices According to Bishop Frame in the Lorentzian Group of Rigid Motions E(1,1), Bol. Soc. Paran. Mat. 30 (2) (2012), J. Milnor, Curvatures of Left-Invariant Metrics on Lie Groups, Advances in Mathematics 21 (1976), A.W. Nutbourne, R.R. Martin, Differential Geometry Applied to the Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, B. O Neill, Semi-Riemannian Geometry, Academic Press, New York (1983). Talat Körpinar and Essin Turhan Fırat University, Department of Mathematics, 23119, Elazığ, Turkey address: talatkorpinar@gmail.com address: essin.turhan@gmail.com