B-tubular surfaces in Lorentzian Heisenberg Group H 3

. In this paper, B - tubular surfaces in terms of biharmonic spacelike new type B - slant helices according to Bishop frame in the Lorentzian Heisenberg group H 3 are studied. The Necessary and sufficient conditions for new type B - slant helices to be biharmonic are obtained. B - tubular surfaces in the cLorentzian Heisenberg group H 3 are characterized. Additionally, main results in Figures 1, 2, 3 and 4 are illustrated.

We remind that, if  is a space curve, a tubular surface associated to this curve is a surface swept by a family of spheres of constant radius (which will be the radius of the tube), having the center on the given curve. Alternatively, as we shall see in the next section, for them we can construct quite easily a parameterization using the Frenet frame associate to the curve. The tubular surfaces are used quite often in computer graphics, but we think they deserve more attention for several reasons. For instance, there is the problem of representing the curves themselves. Usually, the space curves are represented by using solids rather then tubes. There are, today, several very good computer algebra system (such as Maple, or Mathematica) which allow the vizualisation of curves and surfaces, in different kind of representations, (KORPINAR; TURHAN, 2012), (POTTMANN;PETERNELL, 1998).
The aim of this paper is to study tubular surfaces surrounding biharmonic spacelike Bslant helices according to Bishop The identity of the group is (0,0,0) and the inverse of 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: is a positively oriented orthonormal basis. Then, we have the following Frenet formulas, (TURHAN; KORPINAR, 2011b): In the rest of the paper, we suppose everywhere 0   and 0   . 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, (BISHOP, 1975). The Bishop frame is expressed as Using Equation(2.2) and Jacobi operator, we obtain above system. This completes the proof.
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 spacelike new type 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 spacelike new type B-slant helix.

B-tubular surfaces in terms of spacelike biharmonic new type B-Slant helices according to Bishop frame in the Heisenberg group H 3
The envelope of a 1-parameter family of the spheres in the Lorentzian Heisenberg group 3 H is called a tubular surface in the Lorentzian Heisenberg group 3 H . The curve formed by the centers of the spheres is called center curve of the tubular surface. The radius of the tubular surface is the function r such that r is the constant radius of the sphere. To separate a tubular surface according to Bishop frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the surface defined above as B -tubular surface.
where 1 C is a constant of integration. From Equation(4.6), we get On the other hand, using definition of tubular surface, we have   (4.14) The solution of Equation (4.14) can be written in the following form Using Equation (2.1) and Equation (4.1), we have Equation (4.17). Thus proof is complete.
The obtained parametric equations for Equation (4.17) is illustrated in Figures 1, 2 and 3 with helping the programme of Mathematica as follow:

Conclusion
A tubular surface associated to a curve is a surface swept by a family of spheres of constant radius (which will be the radius of the tube), having the center on the given curve. Alternatively, as we shall see in the next section, for them we can construct quite easily a parameterization using the Frenet frame associate to the curve.
In this work, B -tubular surfaces in the Lorentzian Heisenberg group 3 H are characterized.