I vaguely understand, that $f(z) = \frac{1}{z}$ is conformal on $\mathbb{C} \setminus \{0\}$, since it is a complex differentiable function.
Is it formally true? Since, basically $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a different beast from $f: \mathbb{C} \to \mathbb{C}$.
The claim seems to imply, that $f(z) = \frac{1}{z}$ preserves angles (probably reversing orientation). In the same location it quotes
For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilation and isometries (translation, reflection, rotation), which trivially preserve angles.
But this is a treatment of the $f: \mathbb{C} \to \mathbb{C}$, if I understand correctly. Why a translation on the Riemann Sphere is an isometry, I have no clue. Why an isometry on the Riemann Sphere preserves angles, also I'm missing something.
Does $f(z) = \frac{1}{z}$ as an automorphism of $\hat{\mathbb{C}}$ actually preserve angles at $z = 0$? How can I prove this? Is it that Stereographic projection preserves angles? And an inversion is an inverse of Stereographic projection to the unit sphere, then rotation of unit sphere in $\mathbb{R}^3$ and re-projection back to $\mathbb{C}$.
It follows from the Cauchy-Riemann equation that any holomorphic map $f: U \to \mathbb C$ (with $U$ an open subset of $\mathbb C$) is conformal on $U$ if and only if its derivative doesn't vanish on $U$. Here the angles are measured in the standard hermitian metric on $\mathbb C$ (which is also the euclidean metric on $\mathbb R^2$).
In particular, $f(z)=\frac{1}{z}$ is conformal on $\mathbb C^*$. Also, it preserves orientation, as do all holomorphic map.
Any isometry preserves angles; but preserving angles doesn't imply being an isometry (think of the map $f(z)=2z$ on $\mathbb C$).
A translation is certainly not an isometry for the spherical metric, but it is clearly one for the euclidean metric on $\mathbb C$.
It is possible to prove that the only holomorphic maps $f: \hat{\mathbb C} \to \hat{\mathbb C}$ whose derivatives do not vanish are Mobius transformations $M(z)=\frac{az+b}{cz+d}$.
Any pair of hermitian metrics on the same Riemann surface (in particular the Riemann sphere) define the same angles (but not necessarily the same distances). So for angle-preserving, you can actually forget about metrics (as long as you only work with hermitian metrics of course). Then it follows from the first bullet point that any holomorphic map $f: S_1 \to S_2$ between Riemann surfaces preserves angles if and only if its derivative doesn't vanish. You can work this out by using charts.