Geodesics in the Heisenberg group

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-Carathéodory metric is real analytic away from the center of the group.

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