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

Twistor CR manifolds and three-dimensional conformal geometry