Biharmonic B − Slant Helices According To Bishop Frame In The ̃ SL 2 ( R )

In this paper, we study biharmonic B−slant helices in the S̃L2 (R). We characterize the biharmonic B−slant helices in terms of their curvature and torsion. Finally, we find out their explicit parametric equations.


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 is the energy density of f at the point x ∈ M .Critical points of the energy functional are called harmonic maps.
The first variational formula of the energy gives the following characterization of harmonic maps: the map f is harmonic if and only if its tension field τ (f ) vanishes identically, where the tension field is given by τ (f ) = trace∇df. (1.2) As suggested by Eells and Sampson in [6], we can define the bienergy of a map f by 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 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 B−slant helices in the SL 2 (R).Secondly, we characterize the biharmonic B−slant helices in terms of their curvature and torsion.Finally, we find out their explicit parametric equations.

Riemannian Structure of SL 2 (R)
We identify SL 2 (R) with The following set of left-invariant vector fields forms an orthonormal basis for The characterising properties of g SL2(R) defined by The Riemannian connection ∇ of the metric g SL2(R) is given by which is known as Koszul's formula.
Using the Koszul's formula, we obtain Moreover we put where the indices i, j, k and l take the values 1, 2 and 3 (2.3) Assume that {T, N, B} be the Frenet frame field along γ.Then, the Frenet frame satisfies the following Frenet-Serret equations: where κ is the curvature of γ and τ its torsion and The Bishop frame or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative.The Bishop frame is expressed as where Here, we shall call the set {T, M 1 , M 2 } as Bishop trihedra, k 1 and k 2 as Bishop curvatures and

Bishop curvatures are defined by
The relation matrix may be expressed as

Talat Körpınar and Essin Turhan
On the other hand, using above equation we have With respect to the orthonormal basis {e 1 , e 2 , e 3 } we can write ) Theorem 3.1.γ : I −→ SL 2 (R) is a biharmonic curve according to Bishop frame if and only if Definition 3.2.A regular curve γ : I −→ SL 2 (R) is called a slant helix provided the unit vector M 1 of the curve γ has constant angle W with some fixed unit vector u, that is The condition is not altered by reparametrization, so without loss of generality we may assume that slant helices have unit speed.The slant helices can be identified by a simple condition on natural curvatures.
To separate a slant helix according to Bishop frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the curve defined above as B-slant helix.
We shall also use the following lemma.
Lemma 3.3.Let γ : I −→ SL 2 (R) be a unit speed biharmonic curve.Then γ is a biharmonic B-slant helix if and only if Theorem 3.4.Let γ : I −→ SL 2 (R) be a unit speed non-geodesic biharmonic B−slant helix.Then, the parametric equations of γ are (3.9) Proof: We suppose that γ is a unit speed non-geodesic biharmonic B−slant helix.Since γ is biharmonic B−slant helix without loss of generality, we take where W is constant angle.On the other hand, the vector M 1 is a unit vector, we have the following equation where Q 1 , Q 2 are constants of integration.On the other hand, using Bishop formulas (3.3) and (2.1), we have Using above equation and (3.11), we get Using (2.1) in (3.13), we obtain By direct calculations we have (3.9), which proves our assertion.2 In the light of above theorem, we express the following result without proof: Corollary 3.5.Let γ : I −→ SL 2 (R) be a unit speed non-geodesic biharmonic B−slant helix.Then, the parametric equations of γ in terms of Bishop curvatures are where We can use Mathematica in above theorem, yields