Biharmonic S -Curves According to Sabban Frame in Heisenberg Group

: In this paper, we study biharmonic curves accordig to Sabban frame in the Heisenberg group Heis 3 . We characterize the biharmonic curves in terms of their geodesic curvature and we prove that all of biharmonic curves are helices in the Heisenberg group Heis 3 . Finally, we ﬁnd out their explicit parametric equations according to Sabban Frame.


Introduction
Harmonic maps f : (M, g) −→ (N, h) between manifolds are the critical points of the energy where v g is the volume form on (M, g) and As suggested by Eells and Sampson in [6], we can define the bienergy of a map f by 206 Talat Körpinar and Essin Turhan and say that is biharmonic if it is a critical point of the bienergy. Jiang derived the first and the second variation formula for the bienergy in [7,8], showing that the Euler-Lagrange equation associated to E 2 is where J f is the Jacobi operator of f . The equation τ 2 (f ) = 0 is called the biharmonic equation. Since J f is linear, any harmonic map is biharmonic. Therefore, we are interested in proper biharmonic maps, that is non-harmonic biharmonic maps. This study is organised as follows: Firstly, we study biharmonic curves accordig to Sabban frame in the Heisenberg group Heis 3 . Secondly, we characterize the biharmonic curves in terms of their geodesic curvature and we prove that all of biharmonic curves are helices in the Heisenberg group Heis 3 . Finally, we find out their explicit parametric equations according to Sabban Frame.

The Heisenberg Group Heis 3
Heisenberg group Heis 3 can be seen as the space R 3 endowed with the following multipilcation: Heis 3 is a three-dimensional, connected, simply connected and 2-step nilpotent Lie group. The Riemannian metric g is given by The Lie algebra of Heis 3 has an orthonormal basis for which we have the Lie products We obtain Biharmonic S-Curves

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The components {R ijkl } of R relative to {e 1 , e 2 , e 3 } are defined by The non vanishing components of the above tensor fields are

Biharmonic S-Curves According To Sabban Frame In The Heisenberg Group Heis 3
Let γ : I −→ Heis 3 be a non geodesic curve on the Heisenberg group Heis 3 parametrized by arc length. Let {T, N, B} be the Frenet frame fields tangent to the Heisenberg group Heis 3 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: where κ is the curvature of γ and τ is its torsion, Now we give a new frame different from Frenet frame. Let α : I −→ S 2 Heis 3 be unit speed spherical curve. We denote σ as the arc-length parameter of α . Let us denote t (σ) = α ′ (σ) , and we call t (σ) a unit tangent vector of α. We now set a vector s (σ) = α (σ) × t (σ) along α. This frame is called the Sabban frame of α on the Heisenberg group Heis 3 . Then we have the following spherical Frenet-Serret formulae of α :

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Talat Körpinar and Essin Turhan where κ g is the geodesic curvature of the curve α on the S 2 Heis 3 and With respect to the orthonormal basis {e 1 , e 2 , e 3 }, we can write To separate a biharmonic curve according to Sabban frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the curve defined above as biharmonic S-curve.
Proof: Since α is biharmonic, α is a S−helix. So, without loss of generality, we take the axis of α is parallel to the vector e 3 . Then, g (t, e 3 ) = t 3 = cos ϕ, (3.6) where ϕ is constant angle. So, substituting the components t 1 , t 2 and t 3 in the equation (3.3), we have the following equation t = sin ϕ sin µe 1 + sin ϕ cos µe 2 + cos ϕe 3 . (3.7) The covariant derivative of the vector field t is: where M 1 , M 2 are constants of integration.
Integrating both sides, we have (3.9). This proves our assertion. Thus, the proof of theorem is completed. 2 We can use Mathematica in above theorem, yields