Twistor CR Manifolds and Three-Dimensional Conformal Geometry

A $\text {CR}$ (i.e. partially complex) $5$-manifold is contructed as a sphere bundle over an arbitrary $3$-manifold with conformal metric. This so-called twistor $CR$ manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract $\text {CR}$ manifold to arise via the construction. The above correspondence is then used to prove that a twistor $\text {CR}$ manifold is locally imbeddable as a real hypersurface in ${{\mathbf {C}}^3}$ only if it is real-analytic with respect to a suitable atlas.