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 [31Jump To The Next Citation Point]. If we denote the real one-form by l and the complex one-form by m, then these are defined up to the transformations:

a l → a(x )l, (369 ) m → f(xa )m + g(xa)l.
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]:
l ∧ m ∧ ¯m ⁄= 0. (370 )

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 (274View Equation). 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 (365View Equation).)

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 ).

We have immediately that -- K1 = ζ = x − iy is a solution. The second solution is also easily found; we see directly from Equation (175View Equation[38],

∂ T + LT˙= 0, (373 ) (uB)
that
-- τ = T (uB,ζ,ζ),

the inverse to -- uB = X (τ,ζ,ζ), is a CR function and that we can consider ℑ+ to be embedded in the ℂ2 of -- (τ,ζ).


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