## B CR Structures

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 sub-bundle ) which is formally integrable and almost Lagrangian [22]. More concretely, the CR structure can be described by a set of vectors or 1-forms on defined up to a particular gauge freedom. In the context of this review, we are interested in the case where is a real three-manifold.

A CR structure on a real three manifold , with local coordinates , can be given intrinsically by equivalence classes of one-forms, one real, one complex and its complex conjugate [44]. If we denote the real one-form by and the complex one-form 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 three-fold linear-independence relation between these one-forms [44]:

Any three-manifold with a CR structure is referred to as a three-dimensional CR manifold. There are special classes (referred to as embeddable) of three-dimensional CR manifolds that can be directly embedded into . We show how the choice of any specific asymptotically shear-free 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 shear-free NGC, i.e., look at the (, , , ) of Eq. (6.14). The associated dual one-forms, restricted to (after a conformal rescaling of ), become (with a slight notational dishonesty),

with , satisfying the shear-free 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 – are

Therefore, for the situation discussed here, where we have singled out a unique asymptotically shear-free 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 shear-free 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 Levi-flat [7].

In the context of this review, the important observation is that the physical content of asymptotically shear-free 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.