Polyharmonic Helices

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 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 ≥ 4 when the ambient space is the Euclidean sphere S^m.