Characterization of Spacelike Biharmonic Curves with Timelike Binormal According to Flat Metric in Lorentzian Heisenberg Group Heis

In this paper, we study spacelike biharmonic curves with timelike binormal according to flat metric in the Lorentzian Heisenberg group Heis. We characterize spacelike biharmonic curves with timelike binormal in terms of their curvature and torsion. Additionally, we determine the parametric representation of the spacelike biharmonic curves with timelike binormal according to flat metric from this characterization.


Introduction
The theory of biharmonic maps is an old and rich subject, initially studied due to its implications in the theory of elasticity and uid mechanics.G.B. Airy and J.C. Maxwell were the first to study and express plane elastic problems in terms of the biharmonic equation.Later on, the theory evolved with the study of polyharmonic functions developed by E. Almansi, T. Levi-Civita, M. Nicolaescu.
Let f : (M, g) → (N, h) be a smooth function between two Riemannian manifolds.The bienergy E 2 (f ) of f over compact domain Ω ⊂ M is defined by where τ (f ) = trace g ∇df is the tension field of f and dv g is the volume form of M .Using the first variational formula one sees that f is a biharmonic function if and only if its bitension field vanishes identically, i.e. 102 Talat Körpinar and Essin Turhan where is the Laplacian on sections of the pull-back bundle f −1 (T N )and R N is the curvature operator of (N, h) defined by In this paper, we study spacelike biharmonic curves with timelike binormal according to flat metric in the Lorentzian Heisenberg group Heis 3 .We characterize spacelike biharmonic curves with timelike binormal in terms of their curvature and torsion.

The Lorentzian Heisenberg Group Heis 3
The Heisenberg group Heis 3 is a Lie group which is diffeomorphic to R 3 and the group operation is defined as The identity of the group is (0, 0, 0) and the inverse of (x, y, z) is given by (−x, −y, −z).The left-invariant Lorentz metric on Heis 3 is The following set of left-invariant vector fields forms an orthonormal basis for the corresponding Lie algebra: The characterising properties of this algebra are the following commutation relations: (2.2) Proposition 2.1 For the covariant derivatives of the Levi-Civita connection of the left-invariant metric g, defined above the following is true: where the (i, j)-element in the table above equals ∇ ei e j for our basis Then, the Lorentz metric g is flat.
3. Spacelike Biharmonic Curves with Timelike Binormal According to Flat Metric in the Lorentzian Heisenberg Group Heis 3 An arbitrary curve γ : I −→ Heis 3 is spacelike, timelike or null, if all of its velocity vectors γ ′ (s) are, respectively, spacelike, timelike or null, for each s ∈ I ⊂ R. Let γ : I −→ Heis 3 be a unit speed spacelike curve with timelike binormal and {t, n, b} are Frenet vector fields, then Frenet formulas are as follows where κ 1 , κ 2 are curvature function and torsion function, respectively and With respect to the orthonormal basis {e 1 , e 2 , e 3 } we can write Theorem 3.1 If γ : I −→ Heis 3 is a unit speed spacelike biharmonic curve with timelike binormal according to flat metric, then Proof: Using Equation (3.1), we have On the other hand, from Equation (2.4) we get 3), we obtain This completes the proof. 2 Corollary 3.2 If γ : I −→ Heis 3 is a unit speed spacelike biharmonic curve with timelike binormal, then γ is a helix.
Theorem 3.3 Let γ : I −→ Heis 3 is a unit speed spacelike biharmonic curve with timelike binormal according to flat metric.Then the parametric equations of γ are where C 1 , C 2 , C 3 are constants of integration.
Proof: Assume that γ is a unit speed spacelike biharmonic curve with timelike binormal according to flat metric in the Lorentzian Heisenberg group Heis 3 .Since γ is spacelike biharmonic , γ is a helix.So, without loss of generality, we take the axis of γ is parallel to the spacelike vector e 1 .Then, where ϕ is constant angle.Direct computations show that t = cosh ϕe 1 + sinh ϕ sinh ℘e 2 + sinh ϕ cosh ℘e 3 . (3.7) Using above equation and Frenet equations, we obtain where C is a constant of integration.If we take integrate above system we have Equation (3.5).The proof is completed. 2 Using Mathematica in above Theorem, we have following figure.