Can one replace the Riemann sphere with other objects? What happens?

Question

When doing one complex variable I think we talked about the Riemann Sphere, and how it relates to Möbius transformations, the complex "point at infinity" and other things.

Is it possible (and useful) to replace this sphere with other geometric objects? And (as a follow-up-question) : Do there exist other transformations than the Möbius transformations which get intuitive explanations/interpretations on these new surfaces?

Answer 1

The projective plane has a line at infinity. Each set of parallel lines shares the same point at infinity. This has an advantage that any two lines intersect in a point, and any two points lie on a line.

Take the origin in three-space, and the plane $(x,y,1)$ sitting on top of it. Most lines through the origin intersect the plane in a point; and most planes through the origin intersect it in a line. Let a POINT be a line through the origin, and a LINE be a plane through the origin. All of two dimensional geometry works fine.

The line $(xt,yt,zt)$ intersects the plane at $(x/z,y/z,1)$. But there are lines $(xt,yt,0)$ that don't. Those are the points at infinity. Any two LINES that form parallel lines on $z=1$ are really planes that intersect in a line in $z=0$, or a POINT at infinity.

1. The point $(x,y)$ becomes the line $t(x,y,1)$
2. The line $ax+by+c=0$ becomes the plane $ax+by+cz=0$
3. Quadratics like $x^2+y^2+2x-3=0$ become $x^2+y^2+2xz-3z^2=0$. Now, instead of quadratic, linear and constant terms, all terms are quadratic.

Answer 2

Try looking up Riemann surfaces. They are surfaces equipped with a complex structure. The complex plane and the Riemann sphere are Riemann surfaces, but there are others: the cylinder $\mathbb C / \mathbb Z$, complex tori $\mathbb C / (\mathbb Z \oplus \tau \mathbb Z)$, and some other surfaces with with higher "number of handles" (genus) (and plenty of others).

To each Riemann surface you can associate a group of automorphism; the Möbius transformations form the automorphism group of the Riemann sphere. A deep theorem states that every Riemann surface except for the plane, cylinder, tori and Riemann sphere can be realized as a quotient of the unit disk under the action of some groups of Möbius transformations (called Fuchsian groups). Therefore any automorphism group of a Riemann surface can be lifted to a subgroup of Möbius transformations.