New type surfaces in terms of B-Smarandache Curves in Sol

In this work, new type ruled surfaces in terms of BSmarandache TM1 curves of biharmonic Bslant helices in the SOL are studied. We characterize the BSmarandache TM1 curves in terms of their Bishop curvatures. Additionally, we express some interesting relations.


Introduction
Ruled surfaces have been popular in architecture.Structural elegance, these and many other contributions are in contrast to recent free-form architecture.Applied mathematics and in particular geometry have initiated the implementation of comprehensive frameworks for modeling and mastering the complexity of today's architectural needs shapes in an optimal sense by ruled surfaces, (CARMO, 1976;LIUA et al., 2007).
A smooth map M N  :  is said to be biharmonic if it is a critical point of the bienergy functional (CADDEO et al., 2004): is the tension field of .
New methods for constructing a canal surface surrounding a biharmonic curve in the Lorentzian Heisenberg group Heis³ were given, (KORPINAR; TURHAN, 2010;2011;2012).Also, in (TURHAN; KORPINAR, 2010a; 2010b) they characterized biharmonic curves in terms of their curvature and torsion.Also, by using timelike biharmonic curves, they give explicit parametrizations of canal surfaces in the Lorentzian Heisenberg group Heis³.
This study is organised as follows: Firstly, we study B-Smarandache TM 1 curves of biharmonic B-slant helices in the SOL 3 .Additionally, we characterize the B-Smarandache TM 1 curves in terms of their Bishop curvatures.Finally, we express some interesting relations.

Riemannian Structure of Sol Space SOL 3
Sol space, one of Thurston's eight 3-dimensional geometries, can be viewed as  The isometry group of SOL 3 has dimension 3. The connected component of the identity is generated by the following three families of isometries: be the Frenet frame field along  .Then, the Frenet frame satisfies the following Frenet--Serret equations: where  is the curvature of  and  its torsion and 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 } { The relation matrix may be expressed as On the other hand, using above equation we have 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 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 B -slant helix.Then, the parametric equations of  are Then, the position vector of  is where are constants of integration.
To separate a Smarandache    Finally, the obtained parametric equations are illustrated in Figure 3 and 4: We can use Mathematica in Corollary 4.3, yields

Conclusion
Ruled surfaces is that they can be generated by straight lines.A practical application of ruled surfaces is that they are used in civil engineering.Since building materials such as wood are straight, they can be thought of as straight lines.The result is that if engineers are planning to construct something with curvature, they can use a ruled surface since all the lines are straight.
In this paper, new type ruled surfaces in terms of For the covariant derivatives of the Levi-Civita connection of the left-invariant metric 3 SOL g , defined above the following is true:

2 
With respect to the orthonormal basis Bishop frame if and only if speed non-geodesic biharmonic  B slant helix.

1
TM curve 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 Smarandache 1

SOL
Care constants of integration.Proof.Assume that  is a non geodesic biharmonic B-slant helix according to Bishop frame.For non-constant u , we obtain and (4.4) in (4.1) we have (4.2), which completes the proof.In terms of Eqs.(2.1) and (4.2), we may give: Then, the parametric equations of new type ruled surface of  B Smarandache 1 TM curves of biharmonic  B slant helix are given by integration.Proof.Substituting (2.1) to (4.2), we have (4.5) as desired.