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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2797_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\hbox {Sol}_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>Sol</mtext> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2797_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> when the ambient space is the Euclidean sphere <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2797_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {S}}}^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation>.

Polyharmonic Helices

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2025年1月16日

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S. Montaldo,
A. Ratto

摘要

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 $n \ge 4$ $n \geq 4$ when the ambient space is the Euclidean sphere ${{\mathbb {S}}}^m$ $S^{m}$ .