New characterization of b-m 2 developable surfaces

. In this paper, b- m 2 developable surfaces of biharmonic b-slant helices in the special three-dimensional −φ Ricci symmetric para-Sasakian manifold P is studied. Explicit parametric equations of b- m 2 developable surfaces of biharmonic b-slant helices in the special three-dimensional −φ Ricci symmetric para-Sasakian manifold P are characterized. de múltiplo P bi-energia; curva bi-harmônica; múltiplo para-Sasakiano; superfície desenvolvimentável; esquema de Bishop


Introduction
In differential geometry (CADDEO; MONTALDO, 2001), (DIMITRIC, 1992), (LOUBEAU;ONICIUC, 2007), (O'NEILL, 1983) that under the assumption of sufficient differentiability, a developable surface is either a plane, conical surface, cylindrical surface or tangent surface of a curve or a composition of these types. Thus a developable surface is a ruled surface, where all points of the same generator line share a common tangent plane. The rulings are principal curvature lines with vanishing normal curvature and the Gaussian curvature vanishes at all surface points. Therefore developable surfaces are also called single-curved surfaces, as opposed to double-curved surfaces (CARMO, 1976).
On the other hand, a smooth map M N → : φ is said to be biharmonic if it is a critical point of the bienergy functional: and called the bitension field of φ (ARSLAN et al., 2005), (EELLS;LEMAIRE, 1978), (EELLS; SAMPSON, 1964). Non-harmonic biharmonic maps are called proper biharmonic maps. New methods for constructing a canal surface surrounding a biharmonic curve in the Lorentzian Heisenberg group Heis 3 were given, (KORPINAR; 2012), (TURHAN;. Also, in (KORPINAR; TURHAN, 2010), (TURHAN; KORPINAR, 2010) 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.
In this paper, we study b-m 2 developable surfaces of biharmonic b-slant helices in the special three-dimensional − φ Ricci symmetric para-Sasakian manifold P. We characterize the biharmonic curves in terms of their curvature and torsion in the special three-dimensional − φ Ricci symmetric para-Sasakian manifold P. Finally, we find out explicit parametric equations of b-m 2 developable surfaces of biharmonic b-slant helices in the special three-dimensional − φ Ricci symmetric para-Sasakian manifold P. Additionally, we illustrate our results in Figure 1, 2 and 3.

Preliminaries
An n-dimensional differentiable manifold M is said to admit an almost para-contact Riemannian , where φ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form and g is a Riemannian metric on M such that for any vector fields X, Y on M (BLAIR, 1975).
then M is called a para-Sasakian manifold or, briefly a P-Sasakian manifold.
for all vector fields X, Y, Z, W orthogonal to ξ . This notion was introduced by (SATO, 1976) for a Sasakian manifold.
Definition 2: A para-Sasakian manifold M is said where: for all vector fields X, Y, Z, W on M (BLAIR, 1976).
Definition 3: A para-Sasakian manifold M is said where: for all vector fields X and Y on M and ) , , (TAKASHI, 1977).
If X, Y are orthogonal to ξ , then the manifold is said to be locally φ -Ricci symmetric, (SATO, 1976).
The three-dimensional manifold where: (x, y, z) are the standard coordinates in R 3 . We choose the vector fields are linearly independent at each point of P. Let η be the 1-form defined by Let φ be the (1,1) tensor field defined by Then using the linearity of and g we have defines an almost para-contact metric structure on P.
Let ∇ be the Levi-Civita connection with respect to g. Then, we have Biharmonic b-slant helices in the special threedimensional − φ ricci symmetric para-sasakian manifold P Let us consider biharmonicity of curves in the special three-dimensional φ -Ricci Symmetric para-Sasakian manifold P. Let {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, (BISHOP, 1975) 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 Bishop 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.
Now, we illustrate our result in Figure 1.
Theorem 2. Let R be b-m 2 developable of a unit speed non-geodesic biharmonic b-slant helix in P. Then, the vector equation of b-m 2 developable is given by Proof. By the Bishop (1975) Consequently, the equation of R can be found from (14), (17). This concludes the proof of Theorem.
We can prove the following interesting main result.
Theorem 3. Let R be b-m 2 developable of a unit speed non-geodesic biharmonic b-slant helix in P. Then the parametric equations of b-m 2 developable are given by ( ) Proof. It is obvious from Theorem 15 Hence the proof is completed.
Finally, the obtained parametric equations are illustrated in Figure 2 and 3:

Conclusion
In the last decade there has been a growing interest in the theory of biharmonic maps which can be divided in two main research directions. On the one side, constructing the examples and classification results have become important from the differential geometric aspect. The other side is the analytic aspect from the point of view of partial differential equations.
In this paper, b-m 2 developable surfaces of biharmonic b-slant helices in the special three-dimensional − φ Ricci symmetric para-Sasakian manifold P is studied. Explicit parametric equations of b-m 2 developable surfaces of biharmonic b-slant helices in the special three-dimensional − φ Ricci symmetric para-Sasakian manifold P are characterized.