A CR manifold is a differentiable manifold endowed with an additional structure called its ‘CR structure’; formally this is a complex distribution (i.e., a subbundle ) which is formally integrable and almost Lagrangian [22]. More concretely, the CR structure can be described by a set of vectors or 1forms on defined up to a particular gauge freedom. In the context of this review, we are interested in the case where is a real threemanifold.
A CR structure on a real three manifold , with local coordinates , can be given intrinsically by equivalence classes of oneforms, one real, one complex and its complex conjugate [44]. 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 [44]:
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 an embeddable 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 Eq. (6.14). 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, Eq. (A.6).) The dual vectors – also describing the CR structure – areTherefore, 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 embedded into we introduce the CR equation [45]

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 Eq. (4.13) [54],
that

the inverse to , is a CR function and that we can consider to be embedded in the of .
An important structure associated to any embeddable CR manifold of codimension one is its Levi form; this determines a metric on the CR structure as a bundle on the manifold [22]. As we have just discussed, is just such a CR manifold, and one can show that its Levi form (in the CR structure induced by an asymptotically shearfree NGC) is proportional to the twist of the congruence. Hence, any CR structure on which is generated by a congruence with its source on a real world line is Leviflat [7].
In the context of this review, the important observation is that the physical content of asymptotically shearfree NGCs is encoded in the corresponding CR structure. This gives a physical interpretation for CR structures in the setting of asymptotically flat spacetimes. It would be interesting for future research to study the relationship between our findings and those of [34], which demonstrates how the Einstein equations for algebraically special spacetimes can be realized in terms of the embeddable CR structures associated with their PNDs.
http://www.livingreviews.org/lrr20121 
Living Rev. Relativity 15, (2012), 1
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