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

```@article{Chiu2015AnAO,
title={An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group},
author={Hung-Lin Chiu and Yen-Chang Huang and Sin Hua Lai},
journal={Symmetry Integrability and Geometry-methods and Applications},
year={2015},
volume={13},
pages={097}
}```
• Published 3 September 2015
• Mathematics
• Symmetry Integrability and Geometry-methods and Applications
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.
15 Citations

## Figures from this paper

• Mathematics
The Journal of Geometric Analysis
• 2018
In this paper, we study some global properties of curves in the Heisenberg group \$\$H_{1}\$\$H1. In particular, we obtain Fenchel-type theorem and Fáry–Milnor type theorem, together with Bray–Jauregui
• Mathieu Kohli
• Mathematics
Journal of Dynamical and Control Systems
• 2019
In this paper, we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc length in the Heisenberg group that is the simplest sub-Riemannian structure. Our goal is to
• Mathieu Kohli
• Mathematics
Journal of Dynamical and Control Systems
• 2019
In this paper, we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc length in the Heisenberg group that is the simplest sub-Riemannian structure. Our goal is to
By considering the three dimensional Heisenberg group H1 as a flat model of pseudohermitian manifolds, the authors in [8] derived the Frenet-Serret formulas for curves in H1. In this notes we show
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2021
In this paper, we introduce a curve shortening flow in a 3-dimensional pseudohermitian manifold with vanishing torsion. The flow preserves the Legendrian condition and decreases the length of curves.
By studying the group of rigid motions, \$PSH(1)\$, in the 3D-Heisenberg group \$H_1\$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain
By using the support function on the \$xy\$-plane, we show the necessary and sufficient conditions for the existence of envelopes of horizontal lines in the 3D-Heisenberg group. A method to construct
By using the support function on the xy-plane, we show the necessary and sufficient conditions for the existence of envelopes of horizontal lines in the 3D-Heisenberg group. A method to construct
• Mathematics
• 2015
We study the horizontally regular curves in the Heisenberg groups \$H_n\$. We show the fundamental theorem of curves in \$H_n\$ \$(n\geq 2)\$ and define the concept of the orders for horizontally regular

## References

SHOWING 1-10 OF 27 REFERENCES

• Mathematics
• 2013
We study the local equivalence problems of curves and surfaces in three dimensional Heisenberg group via Cartans method of moving frames and Lie groups, and find a complete set of invariants for
Abstract By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂
• Mathematics
• 2013
Abstract In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space
• Mathematics
• 2010
Abstract In this paper, we study the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group ℍ1. We discover a Codazzi-like equation for the p-area element along
C. Fefferman has shown that a real strictly pseudoconvex hypersurface in complex n-space carries a natural conformai Lorentz metric on a circle bundle over the manifold. This paper presents two
• Physics, Mathematics
• 2007
Geometric mechanics on the Heisenberg group Geometric analysis of step 4 case The geometric analysis of step \$2(k+1)\$ case Geometry on higher dimensional Heisenberg groups Complex Hamiltonian
• Mathematics
• 2003
Moving frames and exterior differential systems Euclidean geometry and Riemannian geometry Projective geometry Cartan-Kahler I: Linear algebra and constant-coefficient homogeneous systems
• Mathematics
• 2004
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously
• Mathematics
• 1996
In the Ashes of the Ether: Differential Topology.- Looking for the Forest in the Leaves: Folations.- The Fundamental Theorem of Calculus.- Shapes Fantastic: Klein Geometries.- Shapes High