A CR structure on a real three manifold , with local coordinates
, is given intrinsically by
equivalence classes of one-forms, one real, one complex and its complex conjugate [31
]. If we denote the real
one-form by
and the complex one-form by
, then these are defined up to the transformations:
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 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 shear-free NGC, i.e., look at
the (
,
,
,
) of Equation (274
). The associated dual one-forms, restricted
to
(after a conformal rescaling of
), become (with a slight notational dishonesty),
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 imbedded into
we introduce the CR
equation [32]
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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],
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the inverse to , is a CR function and that we can consider
to be embedded in the
of
.
http://www.livingreviews.org/lrr-2009-6 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |