A CR structure on a real three manifold , with local coordinates , is given intrinsically by equivalence classes of oneforms, one real, one complex and its complex conjugate [31]. If we denote the real oneform by and the complex oneform by , then these are defined up to the transformations:
The are functions on : is nonvanishing and real, and are complex function with nonvanishing. We further require that there be a threefold linearindependence relation between these oneforms [31]:Any threemanifold with a CR structure is referred to as a threedimensional CR manifold. There are special classes (referred to as embeddable) of threedimensional CR manifolds that can be directly embedded into .
We show how the choice of any specific asymptotically shearfree NGC induces a CR structure on . Though there are several ways of arriving at this CR structure, the simplest way is to look at the asymptotic null tetrad system associated with the asymptotically shearfree NGC, i.e., look at the (, , , ) of Equation (274). The associated dual oneforms, restricted to (after a conformal rescaling of ), become (with a slight notational dishonesty),
with , satisfying the shearfree condition. (This same result could have been obtained by manipulating the exterior derivatives of the twistor coordinates, Equation (365).)The dual vectors – also describing the CR structure – are
Therefore, for the situation discussed here, where we have singled out a unique asymptotically shearfree NGC and associated complex world line, we have a uniquely chosen CR structure induced on .
To see how our three manifold, , can be imbedded into we introduce the CR equation [32]

and seek two independent (complex) solutions, that define the embedding of into with coordinates .
We have immediately that is a solution. The second solution is also easily found; we see directly from Equation (175) [38],
that

the inverse to , is a CR function and that we can consider to be embedded in the of .
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