Cartan Flat Non-degenerate CR Lie Groups

Abstract

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.