An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group - Hung-Lin Chiu; Yen-Chang Huang; Sin-Hua Lai (Symmetry, Integrability and Geometry: Methods and Applications, 2017)
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.