There is one sentence I don't understand in some paper.


"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.


Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit to me?



The developing map of conformally flat manifold
Since $\mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $\mathbb{S}^3$.


If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $\mathbb{S}^3$ is a composition of inversions ([Liouville's theorem)](https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)).


So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.


The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths. 
Since $M$ is simply connected it gives a well defined conformal immersion $M\hookrightarrow \mathbb{S}^3$.


