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.

中文翻译：

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海森堡群中帕普斯-古尔丁定理的推广

在本文中，我们研究了3D-Heisenberg组\（{\ mathbb {H}} _ 1 \）中参数化曲面的面积（称为p面积）和体积，该组被视为伪埃尔米特流形的平面模型。我们推导了\（{\ mathbb {H}} _ 1 \）中参数化曲面的p面积和体积公式，并证明了如果曲面满足某些几何特性，则关于面积和体积的Pappus-Guldin定理的经典结果成立。还提供了一些示例，包括具有恒定p均值曲率的表面。