Generalizations of the theorems of Pappus-Guldin in the Heisenberg groups

In this paper we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group ${\mathbb{H}}_1$, which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in ${\mathbb{H}}_1$ and show that the classical result of Pappus-Guldin theorems for surface areas and volumes hold if the surfaces satisfy some geometric properties. Some examples are also provided, including the surfaces with constant p-mean curvatures.