Contact Quasiconformal Immersions

Contact immersions of contact manifolds endowed with the associated Carnot–Carathéodory (CC) metric (for example, immersions of the Heisenberg group $H^3\sim \mathbb R^3_{\mathrm {CC}}$ in itself) are considered. It is assumed that the manifolds have the same dimension and the immersions are quasiconformal with respect to the CC metric. The main assertion is as follows: A quasiconformal immersion of the Heisenberg group in itself, just as a quasiconformal immersion of any contact manifold of conformally parabolic type in a simply connected contact manifold, is globally injective; i.e., such an immersion is an embedding, which, in addition, is surjective in the case of the Heisenberg group. Thus, the global homeomorphism theorem, which is well known in the space theory of quasiconformal mappings, also holds in the contact case.