On Inextensible Flows Of Tangent Developable of Biharmonic B − Slant Helices according to Bishop Frames in the Special 3-Dimensional

In this paper, we study inextensible flows of tangent developable surfaces of biharmonic B-slant helices in the special three-dimensional Kenmotsu manifold K with η-parallel ricci tensor. We express some interesting relations about inextensible flows of this surfaces.


Introduction
Geometric flows have been extensively used in mesh processing.In particular, surface flows based on functional minimization (i.e., evolving a surface so as to progressively decrease an energy functional) is a common methodology in geometry processing with applications spanning surface diffusion.
On the other hand, a smooth map φ : N −→ M is said to be biharmonic if it is a critical point of the bienergy functional: where T(φ) := tr∇ φ dφ is the tension field of φ The Euler-Lagrange equation of the bienergy is given by T 2 (φ) = 0.Here the section T 2 (φ) is defined by and called the bitension field of φ.Non-harmonic biharmonic maps are called proper biharmonic maps.
In this paper, we study inextensible flows of tangent developable surfaces of biharmonic B−slant helices in the special three-dimensional Kenmotsu manifold K with η-parallel ricci tensor.We express some interesting relations about inextensible flows of this surfaces.

Preliminaries
Let M 2n+1 (φ, ξ, η, g) be an almost contact Riemannian manifold with 1-form η, the associated vector field ξ, (1, 1)-tensor field φ and the associated Riemannian metric g.It is well known that [2] for any vector fields X, Y on M .
3. Biharmonic Curves in the Special Three-Dimensional Kenmotsu Manifold K Let {t, n, b} be the Frenet frame field along γ.Then, the Frenet frame satisfies the following Frenet-Serret equations: where κ = |T(γ)| = |∇ t t| is the curvature of γ and τ its torsion and In the rest of the paper, we suppose everywhere κ = 0 and τ = 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.The Bishop frame is expressed as where Here, we shall call the set {t, m 1 , m 1 } as Bishop trihedra, k 1 and k 2 as Bishop curvatures, τ (s) = ζ ′ (s) and κ(s Bishop curvatures are defined by ) The relation matrix may be expressed as On the other hand, using above equation we have With respect to the orthonormal basis {e 1 , e 2 , e 3 } we can write Lemma 3.1.γ : I −→ K is a biharmonic curve with Bishop frame if and only if A regular curve γ : I −→ K is called a slant helix provided the unit vector m 1 of the curve γ has constant angle θ with unit vector field u along γ such that ∇ t u = 0, that is g (m 1 (s) , u) = cos θ for all s ∈ I. (3.9) 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.Theorem 3.3.(see [10]) Let γ : I −→ K be a unit speed curve with non-zero Bishop curvatures.Then γ is a B−slant helix if and only if Proof: Using Theorem 3.3.we have above system.2 Theorem 3.5.(see [10]) Let γ : I −→ K be a unit speed non-geodesic biharmonic B-slant helix.Then, the parametric equations of γ are x 2 (s) = a 1 cos θe sin θs k 2 + sin 2 θ (−k cos (ks + ℓ) + sin θ sin (ks + ℓ)) + a 3 , (3.12) where a 1 , a 2 , a 3 , ℓ are constants of integration and

Inextensible Flows of Tangent Developable Surfaces according to
Bishop Frame in the Special Three-Dimensional Kenmotsu Manifold K The tangent developable of γ is a ruled surface Let ̟ be the standard unit normal vector field on a surface Π defined by .
Then, the first fundamental form I and the second fundamental form II of a surface Π are defined by, respectively, On the other hand, the Gaussian curvature K and the mean curvature H are respectively.
Definition 4.1.( [9]) A surface evolution Π(s, u, t) and its flow ∂Π ∂t are said to be inextensible if its first fundamental form {E, F, G} satisfies This definition states that the surface Π(s, u, t) is, for all time t, the isometric image of the original surface Π(s, u, t 0 ) defined at some initial time t 0 .For a developable surface, Π(s, u, t) can be physically pictured as the parametrization of a waving flag.For a given surface that is rigid, there exists no nontrivial inextensible evolution.Definition 4.2.We can define the following one-parameter family of developable ruled surface Π (s, u, t) = γ (s, t) + uγ ′ (s, t) .
Hence, we have the following theorem.
Theorem 4.3.Let Π is the tangent developable surface associated with non-geodesic biharmonic B-slant helix.∂Π ∂t is inextensible if and only if where ℓ is constant of integration and θ, k 1 , k 2 are function of time t.
Proof: Assume that Π (s, u, t) be a one-parameter family of ruled surface.
On Inextensible Flows Of Tangent Developable 95 From (3.9), we have the following equation ) where ℓ is constant of integration.
On the other hand, using Bishop formulas (3.4) and (2.1), we have Using above equation and (4.5), we get .7) Furthermore, we have the natural frame {Πs, Πu} given by