Contact embeddings of 3-dimensional contact groups
Eugenio Bellini
TL;DR
This work classifies simply connected 3D contact groups with left-invariant contact structures. It shows a dichotomy: either the Lie algebra is isomorphic to $\mathfrak{su}(2)$, in which case $(G,\xi)$ is group-isomorphic to $(SU(2),\xi_{SU(2)})$, or else $\mathfrak g\not\simeq \mathfrak{su}(2)$ and $(G,\xi)$ is diffeomorphic to $({\mathbb R}^3,\xi_{\mathbb R^3})$ and embeds into the standard tight model $(\mathbb R^3,\xi_{\mathbb R^3})$. A key result is a unique factorization $p:H_1\times H_2\times H_3\to G$ by three 1D subgroups with $TH_3\subset \xi$, enabling a global diffeomorphism and an embedding into the tight model space. The paper provides a unified embedding method and proves tightness of all contact groups, with a special treatment of the universal cover of $SL(2)$. These results yield a clean, structural understanding of 3D left-invariant contact groups and their realizations inside canonical tight contact manifolds.
Abstract
A 3-dimensional contact group is a 3-dimensional Lie group endowed with a left-invariant contact structure. Making use of techniques from Riemannian geometry, we prove that any simply connected 3-dimensional contact group not isomorphic to SU(2) satisfies a unique factorization property. As an application, we develop a method to construct embeddings of 3-dimensional simply connected contact groups into one among two model tight contact manifolds.