CR Structures

B CR Structures

A CR structure on a real three manifold $𝒩$ , with local coordinates $a x$ , 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 $l$ and the complex one-form by $m$ , then these are defined up to the transformations:

The $(a,f,g)$ are functions on $𝒩$ : $a$ is nonvanishing and real, $f$ and $g$ are complex function with $f$ nonvanishing. We further require that there be a three-fold linear-independence relation between these one-forms [31 ]:

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 $2 ℂ$ .

We show how the choice of any specific asymptotically shear-free NGC induces a CR structure on $ℑ+$ . 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 ( $l∗a$ , $m ∗a$ , $-- m∗a$ , $n∗a$ ) of Equation (274 ). The associated dual one-forms, restricted to $ℑ+$ (after a conformal rescaling of $m$ ), become (with a slight notational dishonesty),

∗ ---L--- --L¯--- ¯ l = duB − 1 + ζ¯ζ dζ − 1 + ζ¯ζd ζ, (371 ) -- m ∗ = --dζ--, m-∗ = --dζ--, 1 + ζ¯ζ 1 + ζ¯ζ

with $¯ L = L(uB, ζ,ζ)$ , satisfying the shear-free 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

--- ∂ ∂ ∂ 𝔐 = P ---+ L ---- = ∂ (u ) + L ----, (372 ) ∂ζ ∂uB B ∂uB -∂- ---∂-- ---∂-- 𝔐 = P --+ L ∂u = ∂ (uB) + L ∂u , ∂ζ B B 𝔏 = --∂-. ∂uB

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 $+ ℑ$ .

To see how our three manifold, $ℑ+$ , can be imbedded into $ℂ2$ we introduce the CR equation [32 ]

𝔐K- ≡ ∂ K + L -∂--K = 0 (uB) ∂uB

and seek two independent (complex) solutions, $K = K (u ,ζ,ζ),K = K (u ,ζ,ζ) 1 1 B 2 2 B$ that define the embedding of $+ ℑ$ into $2 ℂ$ with coordinates $(K1,K2 )$ .