0

As how I know, the Riemann sphere is a stereographic projection of the complex plane. Assuming that anyone who is familiar with projective geometry can have an intuition to do the same with complex plane, then why is the sphere named after Riemann? One possible reason is that it's a special case of Riemann surface, but it is not a satisfied explanation.

Is there any difference between a Riemann sphere and a normal stereographic projection of a complex plane?

Ooker
  • 704
  • 1
    A normal stereographic projection maps a sphere minus a point to a plane. With the Riemann sphere, we add a "point at infinity" to the plane. Think of picking up a piece of paper by its four corners so that the corners come together, and then placing a piece of tape to hold the corners together. This is the Riemann sphere, and the tape is the point at infinity (kind of) – TomGrubb Sep 25 '17 at 18:32
  • but isn't this what people have thought about at the projection of the real plane? My assumption is that in real plane the point at infinity has nothing interesting, so it isn't popularized. – Ooker Sep 25 '17 at 18:37
  • 1
    The complex numbers form an algebra (and actually a field), whereas $\mathbb{R}^2$ is simply a vector space. When you think of the plane as $\mathbb{C}$ and then form a sphere from this, you inherit additional structure on the resulting manifold. So as a set, sure, they are the "same," but the Riemann sphere is specifically a sphere with an additional complex structure on it. – TomGrubb Sep 25 '17 at 18:42
  • I see. So the Riemann sphere is special because addition and multiplication with infinity is defined, where as a normal stereographic sphere of a complex plan isn't, even when it has point of infinity, is that correct? – Ooker Sep 25 '17 at 18:55
  • 3
    I'm not sure what you mean by "the projection of the real plane", but it may be useful to note that, in projective geometry, one forms the projective plane by adjoining, to the real plane, a line at infinity, not just a single point at infinity. – Andreas Blass Oct 07 '17 at 15:08
  • @AndreasBlass what I mean is that the Riemann sphere is constructed by projecting the complex plane onto the sphere, then we can do the same with real plane (the inversion of mapping the sphere onto the plane). Is line of infinity used for the plane, and point of infinity for the sphere? – Ooker Oct 08 '17 at 13:51

0 Answers0