In this paper we determine all the simply connected non-degenerate CR Lie groups, which are flat with respect to the Cartan connection: in terms of associated Lie algebras, we assert that the only Cartan flat non-degenerate CR Lie algebras are $\mathfrak{ su}(2)$, $\mathfrak{ sl}(2,\mathbb{ R})$, $\mathfrak{ aff}(\mathbb{ R}) \oplus \mathbb{ R}$, and $\mathfrak{ h}_{ 2m+1}$ with its modifications, where $\mathfrak{ aff}(\mathbb{ R})$ is the affine Lie algebra of dimension 2 and $\mathfrak{ h}_{ 2m+1}$ is the Heisenberg Lie algebra of dimension $2m+1$. Furthermore, we determine all the (flat and non-flat) non-degenerate CR structures on each of these Lie groups.
Categories: math.DG