Is the Riemann sphere conformal equivalent to the 2-sphere?

Question

Today I stumbled across the calculation (mentioned in this post) of the transition maps of the stereographic projections from the 2-sphere to the plane. And I wondered about the result that the last transition map is $1/z^*$ (I guess that the $^*$ stands here for complex conjugation) and not $1/z$. That means one can only show (with these projections) that the 2-sphere $\mathbb{S}^2$ and the Riemann sphere $\mathbb{\bar C}$ are "just" diffeomorphic. But is there any biholomorphic map between $\mathbb{S}^2$ and $\mathbb{\bar C}$?

Answer 1

As mrf said, $S^2$ doesn't exactly "come with" a complex structure. The standard way to give it a complex structure is precisely to parametrize it, as the linked post tries to do (but see next paragraph), so that the transition maps between the two plane charts are holomorphic maps. When people say "the Riemann sphere", they really mean "$S^2$, given a complex structure by some parametrization in which the transition maps are holomorphic." Thus, $S^2$, given a complex structure, is tautologically biholomorphic with the Riemann sphere - it is the Riemann sphere! If $S^2$ hasn't been given a complex structure yet (say you're regarding it as the unit sphere in Euclidean 3-space, for example), it's not meaningful to ask whether it maps biholomorphically with something.

The linked question has a sign mistake somewhere. (I haven't combed through the calculation but I suspect that the equations given for $\sigma_1$ and $\sigma_1^{-1}$ are not actually inverses.) I think it is coming from indecision about whether to (A) use stereographic projections from both poles, which is a very geometrically natural parametrization but does not induce a complex structure, precisely because the transition maps are both $z\mapsto 1/z^*$, which is orientation-reversing (it is the inversion in the unit circle); or to (B) compose one of the stereographic projections with an orientation-reversal to cause the transition maps to be holomorphic, in which case both transition maps are $z\mapsto 1/z$.