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 ...

In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such G/K is a compact, connected, irreducible, symmetric space, and isotropy representation of G/H has exactly inequivalent, irreducible summands. We prove left metric ⟨·,·⟩t1,t2 G defined by first equation, must be an A-metric. Moreover, groups do not admit non...

The main goal is to classify 4-dimensional real Lie algebras gwhich admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the work of Barberis who classified real, 4-dimensional simply-connected Lie groups w...