Left-invariant CR structures on 3-dimensional Lie groups

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