Polyharmonic Helices

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 $$\hbox {Sol}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Sol</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> </mml:math> . 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 $$n \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> when the ambient space is the Euclidean sphere $${{\mathbb {S}}}^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:math> .