Polyharmonic helices

TL;DR

This paper studies polyharmonic Frenet helices, classifies proper r-harmonic helices in Sol_3, finds new examples in Bianchi-Cartan-Vranceanu spaces, and establishes non-existence results in Euclidean spheres.

Contribution

It provides a complete classification of proper r-harmonic helices in Sol_3 and explores their existence in various geometric spaces, introducing new examples and non-existence results.

Findings

Classified proper r-harmonic helices in Sol_3.

Discovered new examples of r-harmonic helices in Bianchi-Cartan-Vranceanu spaces.

Proved non-existence of certain Frenet helices in Euclidean spheres.

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 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 $n \geq 4$ when the ambient space is the Euclidean sphere $\s^m$.