One parameter family of S-tangent surfaces

In this paper, we study  S tangent surfaces according to Sabban frame in the Lorentzian Heisenberg group H . We obtained differential equations in terms of their geodesic curvatures in the Lorentzian Heisenberg group H . Finally, we found explicit parametric equations of one parameter family of  S tangent surfaces according to Sabban Frame.


Introduction
Construction of fluid flows constitutes an active research field with a high industrial impact.Corresponding real-world measurements in concrete scenarios complement numerical results from direct simulations of the Navier-Stokes equation, particularly in the case of turbulent flows, and for the understanding of the complex spatiotemporal evolution of instationary flow phenomena.More and more advanced imaging devices (lasers, highspeed cameras, control logic, etc.) are currently developed that allow to record fully timeresolved image sequences of fluid flows at high resolutions.As a consequence, there is a need for advanced algorithms for the analysis of such data, to provide the basis for a subsequent pattern analysis, and with abundant applications across various areas, (O'NEIL, 1983;KWON et al., 2005).
Paper, sheet metal, and many other materials are approximately unstretchable.The surfaces obtained by bending these materials can be flattened onto a plane without stretching or tearing.More precisely, there exists a transformation that maps the surface onto the plane, after which the length of any curve drawn on the surface remains the same.Such surfaces, when sufficiently regular, are well known to mathematicians as developable surfaces.While developable surfaces have been widely used in engineering, design and manufacture, they have been less popular in computer graphics, despite the fact that their isometric properties make them ideal primitives for texture mapping, some kinds of surface modelling, and computer animation (CARMO, 1976;O'NEIL, 1983;KWON et al., 2005).
The Euler-Lagrange equation of the bienergy is given by 0 and called the bitension field of  .Non-harmonic biharmonic maps are called proper biharmonic maps.This study is organised as follows: Firstly, we study  S tangent surfaces according to Sabban frame in the Lorentzian Heisenberg group H . Secondly, we obtain differential equations about in terms of their geodesic curvatures.Finally, we find explicit parametric equations of one parameter family of  S tangent surfaces according to Sabban Frame and we illustrate our results in Figures 1,2,3.
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: For the covariant derivatives of the Levi-Civita connection of the left-invariant metric g , defined above the following is true: We adopt the following notation and sign convention for Riemannian curvature operator: The Riemannian curvature tensor is given by and l take the values 1, 2 and 3. T is the unit vector field '  tangent to  , N is the unit vector field in the direction of is a positively oriented orthonormal basis.Then, we have the following Frenet formulas: where  is the curvature of  and  is its torsion, Now we give a new frame different from Frenet frame, (BISHOP, 1975;BABAARSLAN;YAYLI, 2011).Let  We now set a  This frame is called the Sabban frame of  on   .To separate a biharmonic curve according to Sabban frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the curve defined above as biharmonic S -curve. is a timelike biharmonic S -curve if and only if Using Equation (2.1) and Sabban formulas Equation (3.2), we have Equation (3.4).
All of timelike biharmonic S -curves are helices.

 S Tangent Surfaces Of Timelike Biharmonic S -Curves According To Sabban Frame In The H
To separate a tangent surface according to Sabban frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for this surface as  S tangent surface.The purpose of this section is to study  S tangent surfaces of timelike biharmonic S -curve in the Lorentzian Heisenberg group Heis .

3
The  S tangent surface of  is a ruled surface A surface evolution and its flow t We can define the following one-parameter family of developable ruled surface Hence, we have the following theorem.
Let S R be one-parameter family of the  S tangent surface of a unit speed non-geodesic timelike biharmonic S -curve .Then are smooth functions of time and Assume that  is a unit speed non-geodesic timelike biharmonic S -curve.
From definition of S -helix, we obviously obtain Using the formula of the Sabban, we write a relation: The components of the first fundamental form are frame fields tangent to the Lorentzian Heisenberg group H along  defined as follows: We denote  as the arc-length parameter of  .Let us denote    , the following spherical Frenet-Serret formulae of  :