About conformal covering map from $S^3$ to $S^3$

Question

I need to understand the properties of a conformal covering map from $S^3$ to $S^3$?

Would anybody kindly recommend me some references (both books and papers are OK).

Thanks a lot.

Answer 1

The thing to know is Liouville's theorem which states that every conformal map $U\to S^n$, $n\ge 3$, defined on an open connected nonempty subset $U\subset S^n$ is the restriction of a unique Moebius transformation of $S^n$. You can find proofs in many places, in particular, in the references given in the linked Wikipedia article. Incidentally, even if you do not assume conformality, each covering map $S^n\to S^n$ ($n\ge 2$) is a homeomorphism, as follows from the fact that $S^n$ is simply-connected.