Polyharmonic Helices

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Abstract The main aim of this paper is to investigate the existence of Frenet helices which are polyharmonic of order r, shortly, r-harmonic. We shall obtain existence, non-existence and classification results. More specifically, we obtain a complete classification of proper r-harmonic helices into the 3-dimensional solvable Lie group $$\hbox {Sol}_3$$ Sol 3 . Next, we investigate the existence of proper r-harmonic helices into Bianchi–Cartan–Vranceanu spaces and, in this context, we find new examples. Finally, we shall establish some non-existence results both for Frenet curves and Frenet helices of order $$n \ge 4$$ n ≥ 4 when the ambient space is the Euclidean sphere $${{\mathbb {S}}}^m$$ S m .