# The developing map of conformally flat manifold

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 general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below. Nov 28 '18 at 17:00

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).
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 $$M\hookrightarrow \mathbb{S}^3$$.