Biharmonic Constant Π 1 − Slope Curves according to Type-2 Bishop Frame in Heisenberg Group Heis 3

: In this paper, we introduce constant Π 1 − slope curves according to type-2 Bishop frame in the Heisenberg group Heis 3 . We characterize the biharmonic constant Π 1 − slope curves in terms of their Bishop curvatures. Finally, we ﬁnd out their explicit parametric equations in the Heisenberg group Heis 3 . Additionally, we illustrate our main theorem.


Introduction
A smooth map φ : N −→ M is said to be biharmonic if it is a critical point of the bienergy functional: where T(φ) := tr∇ φ dφ is the tension field of φ The Euler-Lagrange equation of the bienergy is given by T 2 (φ) = 0. Here the section T 2 (φ) is defined by and called the bitension field of φ. Non-harmonic biharmonic maps are called proper biharmonic maps, [5,6,7].
This study is organised as follows: Firstly, we introduce constant Π 1 − slope curves according to type-2 Bishop frame in the Heisenberg group Heis 3 . We characterize the biharmonic constant Π 1 − slope curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the Heisenberg group Heis 3 . Additionally, we illustrate our main theorem. 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 ∇ e1 e 1 = ∇ e2 e 2 = ∇ e3 e 3 = 0, ∇ e2 e 3 = ∇ e3 e 2 = 1 2 e 1 . Assume that {T, N, B} be the Frenet frame field along γ. Then, the Frenet frame satisfies the following Frenet-Serret equations:  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

Biharmonic Constant
Here, we shall call the set {T, M 1 , M 2 } as Bishop trihedra, k 1 and k 2 as Bishop curvatures and U (s) = arctan k2 k1 , τ (s) = U ′ (s) and κ(s) = k 2 1 + k 2 2 . Bishop curvatures are defined by Let γ be a unit speed regular curve in Heis 3 and (3.1) be its Frenet-Serret frame. Let us express a relatively parallel adapted frame: We shall call this frame as Type-2 Bishop Frame. In order to investigate this new frame's relation with Frenet-Serret frame, first we write The relation matrix between Frenet-Serret and type-2 Bishop frames can be expressed So by (3.5), we may express By this way, we conclude The frame {Π 1 , Π 2 , B} is properly oriented, and τ and A (s) = s 0 κ(s)ds are polar coordinates for the curve γ. We shall call the set {Π 1 , Π 2 , B,ǫ 1 , ǫ 2 } as type-2 Bishop invariants of the curve γ, [19].
Proof: The vector Π 1 is a unit vector, we have the following equation where L 1 , L 2 ∈ R. Applying above equation and (3.9), we get Then, a combination of these equations with the second equation of (5.
where L 3 is constant of integration. Also, where L 4 is constant of integration. Again, by combining (2.2) and (3.8) we have Integrating both sides, we have theorem. Thus, the proof of theorem is completed. ✷ Theorem 3.3. Let γ : I −→ Heis 3 be a unit speed non-geodesic biharmonic constant Π 1 −slope curves according to type-2 Bishop frame in the Heis 3 . Then, the position vector of γ is