TITLE: The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold
QUESTION [6 upvotes]: 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!

REPLY [5 votes]: I  add a small $\varepsilon$ to Robert's answer, which is a simple explanation and a simple example to 
what he said concerning the 2 dim case. Conformal structure of signature (1,1)  on the 
surface is essentially the same as a pair of everywhere transversal
foliations. Well, up to a double cover, to be precise. 
Indeed, for a  metric, the lightline cone at every point is two straight lines in  the tangent space intersecting at the origin. The foliations are given by condition that they are tangent to these straight lines. 
It is known (see for example sect.  5.1 of http://lanl.arxiv.org/abs/1002.3934)  that for globally flat metrics these   foliations  are somehow standard: moreover, 
any  flat torus is a  quotient of $(R^2, g=dxdy)$ by a lattice. It is also easy to construct foliations that are not ``standard'', say we easily construct a foliation such that it has a Reeb component.  The conformal structure  corresponding to this foliation is conformally flat, since in dimension 2 any metric is conformally flat, but is not globally conformally flat