Geodesics in the Heisenberg group

TL;DR

This paper offers a new elementary proof of the structure of geodesics in the Heisenberg group using an innovative isoperimetric inequality, and establishes the real analyticity of the Carnot-Carathéodory metric outside the group's center.

Contribution

It introduces a novel, elementary proof for geodesic structure in the Heisenberg group and proves the metric's real analyticity away from the center.

Findings

New proof for geodesic structure in Heisenberg group

Isoperimetric inequality for closed curves in Euclidean space

Real analyticity of the Carnot-Carathéodory metric outside the center

Abstract

We provide a new and elementary proof for the structure of geodesics in the Heisenberg group $\mathbb{H}^n$. The proof is based on a new isoperimetric inequality for closed curves in $\mathbb{R}^{2n}$. We also prove that the Carnot-Carath\'eodory metric is real analytic away from the center of the group.