An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group
Hung-Lin Chiu, Yen-Chang Huang, Sin-Hua Lai
TL;DR
This paper applies Cartan's moving frame method to derive fundamental theorems and invariants for curves and surfaces in the Heisenberg group, establishing links between CR and integral geometry.
Contribution
It introduces a complete set of invariants for curves and surfaces in the Heisenberg group using Cartan's method, and proves a Crofton-type formula connecting CR and integral geometry.
Findings
Complete invariants for curves and surfaces in the Heisenberg group
A Crofton-type formula related to p-area and volume variation
Connection established between CR geometry and integral geometry
Abstract
We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.
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