Left-invariant CR structures on 3-dimensional Lie groups

The systematic study of CR manifolds originated in two pioneering 1932 papers Élie Cartan. In the first, Cartan classifies all homogeneous 3-manifolds, most well-known case which is a one-parameter family left-invariant structures on $$\mathrm {SU}_2= S^3$$ , deforming standard ‘spherical’ structure. this paper, mostly expository, we illustrate and clarify Cartan’s results methods by providing detailed classification modern language for four 3-dimensional Lie groups. particular, find that $${\mathrm {SL}_2({\mathbb {R}})}$$ admits families structures, called elliptic hyperbolic families, characterized incidence contact distribution with null cone Killing metric. Low dimensional complex representations provide embedding or immersions these structures. same apply to other groups are illustrated descriptions {SU}_2$$ Heisenberg group, Euclidean group.