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

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.