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

TL;DR

This paper extends classical Pappus-Guldin theorems to the 3D-Heisenberg group by deriving formulas for p-areas and volumes of parametric surfaces, demonstrating the theorems' validity under certain geometric conditions.

Contribution

It introduces formulas for p-areas and volumes in the Heisenberg group and generalizes Pappus-Guldin theorems to this non-Euclidean setting.

Findings

Formulas for p-areas and volumes in $ ext{H}_1$

Pappus-Guldin theorems hold under specific geometric conditions

Examples include surfaces with constant p-mean curvature

Abstract

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.