If $S$ is a hypersurface in $\mathbb{R}^n$ (like a sphere) then is $S$ a "conformally compact Einstein manifold"?
It's a compact manifold. According to the wiki, it is a conformal manifold if it is equipped with an equivalence class in which two metrics $g$ and $h$ are equivalent iff $h=\lambda^2 g$ for $\lambda:S \to \mathbb{R}$ smooth.
An Einstein manifold I find difficult to understand since I don't understand Ricci tensors.