Biharmonic Curves in 3-dimensional Hyperbolic Heisenberg Group

TL;DR

This paper classifies non-geodesic non-null biharmonic curves in the 3D hyperbolic Heisenberg group, showing they are helices and providing explicit parametric equations, while also identifying the non-existence of certain types.

Contribution

It characterizes all non-geodesic non-null biharmonic curves in the 3D hyperbolic Heisenberg group and derives explicit parametric equations for specific cases.

Findings

All non-geodesic non-null biharmonic curves are helices.

Explicit parametric equations for certain biharmonic curves are obtained.

Non-existence of non-geodesic timelike horizontal biharmonic curves.

Abstract

In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmonic curves and non-geodesic spacelike horizontal biharmonic curves in 3-dimensional hyperbolic Heisenberg group, respectively. We also show that there do not exist non-geodesic timelike horizontal biharmonic curves in 3-dimensional hyperbolic Heisenberg group.