Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group

TL;DR

This paper characterizes non-geodesic biharmonic curves in the Heisenberg group, proving they are helices and providing explicit parametric formulas for these curves.

Contribution

It explicitly describes all non-geodesic biharmonic curves in the Heisenberg group, including their parametric equations, advancing understanding of biharmonic maps in this setting.

Findings

All non-geodesic biharmonic curves are helices.

Explicit parametric equations for these curves are derived.

The study enhances the classification of biharmonic curves in the Heisenberg group.

Abstract

We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.