Can Symmetries on the Riemann Sphere Deepen Our Understanding of Geometry?

[mnb96]

Hello,

in the usual 2d Euclidean plane we know we have a limited number of symmetry groups that describe certain kinds of symmetries.
Could we add richness to our "vocabulary of symmetries" by considering symmetries on the Riemann sphere, and then stereographically project onto the plane?

 

[Eynstone]

If my memory is not betraying me, mo:bius transformations are an isometry group for the Riemann sphere. We can add inversion to the planar isometry group after a fashion.

 

[mnb96]

thanks!

so if I understood correctly, the isometries in the plane are essentially given by rotations of the Riemann sphere along the north-pole/south-pole axis. Instead, the extra richness comes from using symmetries of the Riemann sphere obtained by rotations along different axes. The latter produce conformal transformation rather than isometries in the plane, and are essentially Möbius transformations.

Am I right?