A Note on the Ces\`{a}ro Condition for Curves in the Three-Dimensional Heisenberg Group

Abstract

We study three consequences of the adapted Frenet--Serret formulas for horizontally regular curves in the three-dimensional Heisenberg group $\mathbb{H}_1$, viewed as the flat model in pseudo-Hermitian geometry. First, we derive the Ces\`{a}ro immobility system and solve it explicitly under the fixed standard coordinate identification $\mathbb H_1\simeq\mathbb R^3$. The solution yields identities relating the $p$-curvature and contact normality to the radial and vertical coordinates of a curve, and gives a criterion for a curve to lie on a rotationally symmetric surface. We also describe the constant $p$-curvature case and illustrate the formulas on the Euclidean sphere and the Pansu sphere. Second, we classify nontrivial adapted Bertrand mates through their horizontal projections, derive the relation between their $p$-curvatures, and describe the remaining freedom in the vertical component. Finally, we classify curves whose position vectors lie in the planes spanned by pairs of vectors in the adapted frame. The three cases yield curves contained in the $xy$-plane, a vertical plane through the $z$-axis, or a circular cylinder. These results isolate the effects of the fixed Reeb direction on several classical constructions for curves in Euclidean space.