Abstract
In this paper, we study biharmonic curves accordig to Sabban frame in the Heisenberg group Heis³. We characterize the biharmonic curves in terms of their geodesic curvature and we prove that all of biharmonic curves are helices in the Heisenberg group Heis³. Finally, we find out their explicit parametric equations according to Sabban Frame.
Highlights
Where Jf is the Jacobi operator of f
This study is organised as follows: Firstly, we study biharmonic curves accordig to Sabban frame in the Heisenberg group Heis[3]
We characterize the biharmonic curves in terms of their geodesic curvature and we prove that all of biharmonic curves are helices in the Heisenberg group Heis[3]
Summary
Let γ : I −→ Heis[3] be a non geodesic curve on the Heisenberg group Heis[3] parametrized by arc length. Let {T, N, B} be the Frenet frame fields tangent to the Heisenberg group Heis[3] along γ defined as follows:. T is the unit vector field γ′ tangent to γ, N is the unit vector field in the direction of ∇TT (normal to γ), and B is chosen so that {T, N, B} is a positively oriented orthonormal basis. Where κ is the curvature of γ and τ is its torsion, g (T, T) = 1, g (N, N) = 1, g (B, B) = 1, g (T, N) = g (T, B) = g (N, B) = 0
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