Riemann sphere and Maps

Question

Could somebody please clarify the following for me?

I am not too clear about the relationship between the Riemann sphere and Möbius maps. I know that we can through projection make some Möbius maps correspond to isometries of the sphere. But it is not a bijection right? Which maps have corresponding isometries and which don't, vice versa?

Thanks

Answer 1

Möbius maps $(az+b)/(cz+d)$ with $ad-bc\ne 0$ are 1-1 onto maps of the Riemann sphere. You need to add infinity to the plane to get a statement this simple.

Answer 2

It's easy to see that all Möbius maps cannot be isometries : if $M$ is an isometric Möbius map, then $\lambda M$, $\lambda \neq 1$ is also Möbius but certainly cannot be isometric.

Also, the involution $z \mapsto \overline{z}$ is an isometry of the Riemann sphere, but it is not Möbius.

Actually, the symmetry group of the sphere is $SO_3(\mathbb{R})$, and the group of Möbius transformation is $PSL_2(\mathbb{C})$.

As it has been pointed out by Chris, the real interest of the Möbius maps is that it is precisely the biholomorphisms of the Riemann sphere.

Answer 3

I think the main geometric point is that there is a difference between conformal maps (which preserve the angles between tangent vectors) and isometries (which also preserve all notions of distance -- and in particular, norms of tangent vectors).

As people have said, the Möbius group tells you what the holomorphic (complex) automorphisms are of the sphere, $S^2 = \mathbb{C} P^1$. But in one complex dimension (i.e., for Riemann surfaces) you can think of orientation-preserving, conformal (i.e., angle-preserving) transformations as the same thing as complex transformations.

So the only difference between Möbius transformations and isometries are, really, that (1) the former has to preserve orientation, and (2) the former need not preserve distances -- only angles. This explains why, for instance, scaling is a Möbius transformation, but is not an isometry.