CritPt / Challenge_53_main / The ambient metric is defined as follows:

# Problem setup:
The ambient metric is defined as follows:
\begin{equation}
ds^2 = 2\rho dt^2 + 2t dt d\rho + t^2 \gamma_{ij}(x,\rho)dx^{i}dx^{j}\,,
\end{equation}
where γ_(ij)(x,ρ) can be expanded asymptotically in terms of ρ as
\begin{align}
\gamma_{ij}(x,\rho)&= \gamma^{(0)}_{ij}(x)+ \gamma^{(1)}_{ij}(x)\rho +\gamma^{(2)}_{ij}(x)\rho^2 + \cdots\,.
\end{align}
Given γ_(ij)^((0)), the higher-order terms in the expansion can be solved order by order from the Ricci-flatness condition of the ambient metric. For k≥2, each γ_(ij)^((k))(x) has a pole at d=2k.

On the other hand, the Riemann tensor of the ambient metric defines the kth extended obstruction tensor Ω_(ij)^((k)) in the following way:
\begin{equation}
\Omega^{(k)}_{ij}= R_{\rho ij\rho;\underbrace{\scriptstyle \rho\cdots \rho}_{k-1}}|_{\rho=0,t=1}.
\end{equation}
This tensor also has a pole at d=2k+2. The residue of γ_(ij)^((k))(x) is proportional to that of Ω_(ij)^((k−1)).

# Main problem:

 Suppose Res_(d=2k)γ_(ij)^((k))=A_(k)Res_(d=2k)Ω_(ij)^((k−1)). Show that for k=2,3, γ_(ij)^((k))−A_(k)Ω_(ij)^((k−1)) is proportional to P^(k)_(i)P_(kj) and B_(k(i)P^(k)_(j)), respectively. Determine these two coefficients.

# Problem setup:
The ambient metric is defined as follows:
\begin{equation}
ds^2 = 2\rho dt^2 + 2t dt d\rho + t^2 \gamma_{ij}(x,\rho)dx^{i}dx^{j}\,,
\end{equation}
where $\gamma_{ij}(x,\rho)$ can be expanded asymptotically in terms of $\rho$ as
\begin{align}
\gamma_{ij}(x,\rho)&= \gamma^{(0)}_{ij}(x)+ \gamma^{(1)}_{ij}(x)\rho +\gamma^{(2)}_{ij}(x)\rho^2 + \cdots\,.
\end{align}
Given $\gamma^{(0)}_{ij}$, the higher-order terms in the expansion can be solved order by order from the Ricci-flatness condition of the ambient metric. For $k\geqslant 2$, each $\gamma^{(k)}_{ij}(x)$ has a pole at $d=2k$.

On the other hand, the Riemann tensor of the ambient metric defines the $k$th extended obstruction tensor $\Omega^{(k)}_{ij}$ in the following way:
\begin{equation}
\Omega^{(k)}_{ij}= R_{\rho ij\rho;\underbrace{\scriptstyle \rho\cdots \rho}_{k-1}}|_{\rho=0,t=1}.
\end{equation}
This tensor also has a pole at $d=2k+2$. The residue of $\gamma^{(k)}_{ij}(x)$ is proportional to that of $\Omega^{(k-1)}_{ij}$.

# Main problem:

 Suppose $\text{Res}_{d=2k}\gamma^{(k)}_{ij}=A_k\text{Res}_{d=2k}\Omega^{(k-1)}_{ij}$. Show that for $k=2,3$, $\gamma^{(k)}_{ij}-A_k\Omega^{(k-1)}_{ij}$ is proportional to $P^{k}{}_{i} P_{kj}$ and $B_{k(i}P^k{}_{j)}$, respectively. Determine these two coefficients.