Einstein's Equations and the Embedding of 3-dim CR Manifolds

We discuss several theorems relating the connection between the local CR-embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dimensional CR manifold that is defined by the fields. Using the reduced Einstein equations, we construct two independent CR functions, which give the embedding. We also point out that the Einstein equations imply that the spacetime metric, after rescaling, become well defined on a circle bundle over the CR manifold.