Triharmonic Curves in 3-Dimensional Homogeneous Spaces

2020/08/24 by Stefano Montaldo, Montaldo, Stefano, Alvaro Pampano +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.2008.10571

To appear in Mediterranean Journal of Mathematics

arxiv created 2021/08/05 · arxiv updated 2021/08/06

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.

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• Polyharmonic curves in semi-Riemannian manifolds

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