k-harmonic curves into a Riemannian manifold with constant sectional curvature

TL;DR

This paper derives differential equations for 3-harmonic curves in constant curvature Riemannian manifolds and demonstrates that biharmonic curves are also k-harmonic for k≥2, advancing understanding of harmonic map theory.

Contribution

It provides explicit ODEs for 3-harmonic curves and establishes the relation between biharmonic and k-harmonic curves in constant curvature spaces.

Findings

Derived differential equations for 3-harmonic curves.

Proved biharmonic curves are k-harmonic for k≥2.

Enhanced understanding of harmonic maps in constant curvature manifolds.

Abstract

J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we describe the ordinary differential equations of $3$-harmonic curves into a Riemannian manifold with constant sectional curvature, and show biharmonic curve is k-harmonic curve $(k\geq 2)$.