Pedal Curves of Tangent Surfaces of Biharmonic B-General Helices according to Bishop Frame in Heisenberg Group Heis

In this paper, we study pedal curves of tangent surfaces of biharmonic B-general helices according to Bishop frame in the Heisenberg group Heis. We give necessary and sufficient conditions for B-general helices to be biharmonic according to Bishop frame. We characterize this pedal curves in the Heisenberg group Heis. Additionally, we illustrate our main theorem.


Introduction
Developable surfaces, which can be developed onto a plane without stretching and tearing, form a subset of ruled surfaces, which can be generated by sweeping a line through space. There are three types of developable surfaces: cones, cylinders (including planes) and tangent surfaces formed by the tangents of a space curve, which is called the cuspidal edge of this surface, [3].
In this paper, we study pedal curves of tangent surfaces of biharmonic Bgeneral helices according to Bishop frame in the Heisenberg group Heis 3 . We give necessary and sufficient conditions for B-general helices to be biharmonic according to Bishop frame. We characterize this pedal curves in the Heisenberg group Heis 3 . Additionally, we illustrate our main theorem.

The Heisenberg Group Heis 3
Heisenberg group Heis 3 can be seen as the space R 3 endowed with the following multipilcation:

Talat Körpinar and Essin Turhan
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 ∇ e2 e 3 = ∇ e3 e 2 = 1 2 e 1 .

Biharmonic B-General Helices with Bishop 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.
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 Here, we shall call the set {T, M 1 , M 2 } as Bishop trihedra, k 1 and k 2 as Bishop curvatures.

233
With respect to the orthonormal basis {e 1 , e 2 , e 3 } we can write To separate a general 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-general helix.

Pedal Curves In The Heisenberg Group Heis 3
The purpose of this section is to study pedal curves of tangent developable of biharmonic B-general helices with Bishop frame in the Heisenberg group Heis 3 .
The tangent surface of γ B is a ruled surface Let R be a developable ruled surface given by equation (4.1) in Heis 3 . Since the tangent plane is constant along rulings of R, it is clear that the pedal of R is a curve. Thus, for the pedal of R, we can writē where Π(s) is the distance between the points γ (s) andγ (s).
Using above equation, we have (4.2), the theorem is proved. ✷