Ces\'{a}ro condition for curves in the flat pseudo-hermitian manifolds

TL;DR

This paper explores the properties of curves in the flat pseudo-hermitian Heisenberg group, deriving applications of Frenet-Serret formulas such as Cesáro conditions, Bertrand curves, and curve classifications based on position vectors.

Contribution

It introduces new applications of Frenet-Serret formulas in pseudo-hermitian geometry, including Cesáro conditions, characterization of Bertrand curves, and classification of curves based on their position vectors.

Findings

Cesáro immobility condition for curves in the Heisenberg group

All horizontally regular curves are Bertrand curves

Classification of curves based on position vector planes

Abstract

By considering the three dimensional Heisenberg group $\mathbb{H}_1$ as a flat model of pseudo-hermitian manifolds, the authors in [8] derived the Frenet-Serret formulas for curves in $\mathbb{H}_1$. In this notes we show three applications of the Frenet-Serret formulas. The first is the Ces\'{a}ro immobility condition, which provides the criterion of curves being contained in a given rotationally symmetric surface. Secondly, we show that any horizontally regular curve is a Bertrand curve, and give all characterizations of those curves. The final application is a classification of curves depending on whether the position vector of the curve lies on the planes spanned by any pair of its unit tangent, normal, and binormal vectors.