Integral Equations of Biharmonic Constant Π 1 − Slope Curves according to Type-2 Bishop Frame in the Sol Space

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 (see [8]), because biharmonic maps are solutions of a fourth order strongly elliptic semilinear PDE. In this paper, we study biharmonic constant Π1− slope curves according to type-2 Bishop in the SOL. We characterize the biharmonic curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric integral equations in the SOL.


Introduction
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 (see [8]), because biharmonic maps are solutions of a fourth order strongly elliptic semilinear PDE.
In this paper, we study biharmonic constant Π 1 − slope curves according to type-2 Bishop in the SOL 3 .We characterize the biharmonic curves in terms of their Bishop curvatures.Finally, we find out their explicit parametric integral equations in the SOL 3 .

Preliminaries
Sol space, one of Thurston's eight 3-dimensional geometries with group structure (x, y, z) • (x, ỹ, z) = x + e z x, y + e −z ỹ, z + z and left invariant metric

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where (x, y, z) are the standard coordinates in R 3 .The space Sol is realized as the following solvable matrix Lie group, [7]: .
The Lie algebra sol 3 is given explicitly by Then we can take the following orthonormal basis Left-translating the basis {E 1 , E 2 , E 3 }, we obtain the following orthonormal frame field: Note that the Sol metric can also be written as: where Proposition 2.1.For the covariant derivatives of the Levi-Civita connection of the left-invariant metric g SOL 3 , defined above the following is true: where the (i, j)-element in the table above equals ∇ ei e j for our basis Integral Equations of Biharmonic Constant Π1− Slope Curves

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Lie brackets can be easily computed 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: x, e c y, z + c .Assume that {T, N, B} be the Frenet frame field along γ.Then, the Frenet frame satisfies the following Frenet-Serret equations:

Biharmonic Constant
where κ is the curvature of γ and τ 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 where Here, we shall call the set {T, M 1 , M 2 } as Bishop trihedra, k 1 and k 2 as Bishop curvatures and U (s) = arctan k2 k1 , τ (s) = U ′ (s) and κ(s

Bishop curvatures are defined by
Let γ be a unit speed regular curve in SOL 3 and (3.1) be its Frenet-Serret frame.Let us express a relatively parallel adapted frame: where We shall call this frame as Type-2 Bishop Frame.In order to investigate this new frame's relation with Frenet-Serret frame, first we write (3.5) The relation matrix between Frenet-Serret and type-2 Bishop frames can be expressed So by (3.5), we may express
Integral Equations of Biharmonic Constant Π1− Slope Curves

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With respect to the orthonormal basis {e 1 , e 2 , e 3 }, we can write Theorem 3.2.Let γ : I −→ SOL 3 be a unit speed non-geodesic biharmonic constant Π 1 − slope curves according to type-2 Bishop frame in the SOL 3 .Then, the parametric equations of γ are where R 1 , R 2 , R 3 are constants of integration.
Proof: We suppose that γ is a unit speed non-geodesic biharmonic Π 1 −slope curve.Since where E is constant angle.On the other hand, the vector Π 1 is a unit vector, we have the following equation where R 1 , R 2 ∈ R.
Integrating both sides, we have where R 3 is constant of integration.This proves our assertion.Thus, the proof of theorem is completed.✷ Π 1 −Slope Curves according to New Type-2 Bishop Frame in Sol Space SOL3

Integral
Equations of Biharmonic Constant Π1− Slope Curves 208 Theorem 3.1.γ : I −→ SOL 3 is a biharmonic curve according to Bishop frame if and only if