There is a theorem :

1. 2-dim (pseudo-)Riemannian manifold must be local conformal flat;

2. 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

3. n-dim (n>3) (pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

Are there some literature or textbooks covering this question? Thanks!