This section provides an overview of the construction, first of the null line bundle, and then of its ambient extension.
The null line bundle
Suppose that M is a conformal manifold, and that [g] denotes the conformal metric defined on M. Let π : N → M denote the tautological subbundle of T*M ⊗ T*M defined by all representatives of the conformal metric. In terms of a fixed background metric g0, N consists of all positive multiples ω2g0 of the metric. There is a natural action of R+ on N, given by
Moreover, the total space of N carries a tautological degenerate metric, for if p is a point of the fibre of π : N → M corresponding to the conformal representative gp, then let
This metric degenerates along the vertical directions. Furthermore, it is homogeneous of degree 2 under the R+ action on N:
Let X be the vertical vector field generating the scaling action. Then the following properties are immediate:
- h(X,-) = 0
- LXh = 2h, where LX is the Lie derivative along the vector field X.
The ambient space
Let N~ = N × (-1,1), with the natural inclusion i : N → N~. The dilations δω extend naturally to N~, and hence so does the generator X of dilation.
An ambient metric on N~ is a Lorentzian metric h~ such that
- The metric is homogeneous: δω* h~ = ω2 h~
- The metric is an ambient extension: i* h~ = h, where i* is the pullback along the natural inclusion.
- The metric is Ricci flat: Ric(h~) = 0.
Suppose that a fixed representative of the conformal metric g and a local coordinate system x = (xi) are chosen on M. These induce coordinates on N by identifying a point in the fibre of N with (x,t2g(x)) where t > 0 is the fibre coordinate. (In these coordinates, X = t ∂t.) Finally, if ρ is a defining function of N in N~ which is homogeneous of degree 0 under dilations, then (x,t,ρ) are coordinates of N~. Furthermore, any extension metric which is homogeneous of degree 2 can be written in these coordinates in the form:
where the gij are n2 functions with g(x,0) = g(x), the given conformal representative.
After some calculation one shows that the Ricci flatness is equivalent to the following differential equation, where the prime is differentiation with respect to ρ:
One may then formally solve this equation as a power series in ρ to obtain the asymptotic development of the ambient metric off the null cone. For example, substituting ρ = 0 and solving gives
- gij′(x,0) = 2Pij
where P is the Schouten tensor. Next, differentiating again and substituting the known value of gij′(x,0) into the equation, the second derivative can be found to be a multiple of the Bach tensor. And so forth.