Ambient construction

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

${\displaystyle \delta _{\omega }g=\omega ^{2}g}$

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 g_p, then let

${\displaystyle h_{p}(X_{p},Y_{p})=g_{p}(\pi _{*}X,\pi _{*}Y).}$

This metric degenerates along the vertical directions. Furthermore, it is homogeneous of degree 2 under the R⁺ action on N:

${\displaystyle \delta _{\omega }^{*}h=\omega ^{2}h}$

Let X be the vertical vector field generating the scaling action. Then the following properties are immediate:

h(X,-) = 0

L_Xh = 2h, where L_X is the Lie derivative along the vector field X.

Let N^~ = N × (-1,1), with the natural inclusion i : N → N^~. The dilations δ_ω 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: δ_ω^* 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:

${\displaystyle h^{\sim }=t^{2}g_{ij}(x,\rho )dx^{i}dx^{j}+2\rho dt^{2}+2tdtd\rho ,\,}$

where the g_ij 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:

${\displaystyle \rho g_{ij}''-\rho g^{kl}g_{ik}'g_{jl}+{\tfrac {1}{2}}\rho g^{kl}g_{kl}'g_{ij}'+{\frac {2-n}{2}}g_{ij}'-{\tfrac {1}{2}}g^{kl}g_{kl}'g_{ij}+\mathrm {Ric} (g)_{ij}=0.}$

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

g_ij^′(x,0) = 2P_ij

where P is the Schouten tensor. Next, differentiating again and substituting the known value of g_ij^′(x,0) into the equation, the second derivative can be found to be a multiple of the Bach tensor. And so forth.