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

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