Triharmonic Curves in 3-Dimensional Homogeneous Spaces

TL;DR

This paper classifies triharmonic curves in various 3D Riemannian spaces, revealing their properties and specific forms, including Frenet helices, and extends understanding beyond biharmonic curves.

Contribution

It provides the first existence result for nonconstant curvature triharmonic curves and offers a complete classification in several important 3D geometries.

Findings

Existence of nonconstant curvature triharmonic curves in certain manifolds.

Classification of triharmonic curves in surfaces with constant Gaussian curvature.

Full classification of triharmonic Frenet helices in space forms and Bianchi-Cartan-Vranceanu spaces.

Abstract

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi-Cartan-Vranceanu spaces.