### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 097, 27 pages      arXiv:1509.00950      https://doi.org/10.3842/SIGMA.2017.097

### An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

Hung-Lin Chiu a, Yen-Chang Huang b and Sin-Hua Lai a
a) Department of Mathematics, National Central University, Chung Li, Taiwan
b) School of Mathematics and Statistics, Xinyang Normal University, Henan, P.R. China

Received March 09, 2017, in final form December 09, 2017; Published online December 26, 2017

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.

Key words: CR manifolds; Heisenberg groups; moving frames.

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