Classification of homogeneous CR-manifolds in dimension 4

TL;DR

This paper classifies all locally homogeneous CR-manifolds in four dimensions, extending previous classifications in three dimensions, and explores their symmetries and realizations as affine vector fields in complex three-space.

Contribution

It provides the first complete classification of 4-dimensional homogeneous CR-manifolds and their symmetric cases, and demonstrates that any 4D real Lie algebra can be realized as affine vector fields in complex space.

Findings

Complete classification of 4D homogeneous CR-manifolds

Classification of symmetric CR-manifolds in dimension 4

Any 4D real Lie algebra can be realized as affine vector fields in 3

Abstract

Locally homogeneous CR-manifolds in dimension 3 were classified, up to local CR-equivalence, by E.Cartan. We classify, up to local CR-equivalence, all locally homogeneous CR-manifolds in dimension 4. The classification theorem enables us also to classify all symmetric CR-manifolds in dimension 4, up to local biholomorphic equivalence. We also prove that any 4-dimensional real Lie algebra can be realized as an algebra of affine vector fields in a domain in $\CC{3}$, linearly independent at each point.