It has been well known that every Mobius transformation can be constructed by stereographi projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back onto the plane. Related to the article: Constructing Mobius Transformations with Spheres

But I'm wondering what will specifically happened between the mapping of the two spheres on which I want to explore more; a rotation, translation etc etc. From what I understood, I think I want to explore more on 3-D version of conformal mappings.

Any suggestions to work on this task is much appreciated.

  • 2
    $\begingroup$ your question seems to be slightly on the ambiguous side. Ambiguous questions are fine sometimes but it could help if you extract at least one precise mathematical question from your conjecture. You could also clarify that you mean by "3-D". Both the sphere and the plane are 2-dimensional, if I understand correctly. On what 3-manifold are you considering a conformal structure? $\endgroup$ – user140765 Jun 1 '19 at 8:43
  • $\begingroup$ @kartop_man, apologies. Thank you for the comment. What I meant by 3-D is 3-dimensional which the sphere is. I was wondering whether to use spherical polar coordinates but couldn't fathom how. $\endgroup$ – user141340 Jun 1 '19 at 11:14
  • $\begingroup$ @kartop_man, could you explain why you said that the sphere is 2-dimensional ? I can accept the one we drew as sphere in a paper is 2-dimensional but not the "real" one. Thank you. $\endgroup$ – user141340 Jun 1 '19 at 15:18
  • $\begingroup$ no doubt terminologies can matter in mathematics, what one person means by "sphere" maybe other person's something else. However, in the context of conformal structures, I think what most of the people mean by "sphere" is the so-called Riemann sphere (there is a Wikipedia description, I think). It is a manifold of real dimension 2, complex dimension 1, no way it is 3-dimensional. $\endgroup$ – user140765 Jun 1 '19 at 15:33

The best way to realize conformal / Möbius transformation on $\mathbb{R}^n$ is to use two dimensions more and introduce indefinite quadratic form $q$ of signature $(n+1, 1)$. The sphere is then realized as projectivized nullcone (i.e. vectors $v \in \mathbb{R}^{n+2}$ such that $q(v) = 0$ considered only up to multiple by positive number). The group of conformal transformations is then the matrix group preserving $q$ $$ SO(q) = \{ A \in GL(\mathbb{R}^{n+2}) : q(Av) = q(v) \} \simeq SO(n+1, 1). $$

The Möbius transformations are then given just by matrix-vector multiplication.

See The conformal group of $S^n$. for references

| cite | improve this answer | |
  • $\begingroup$ Thank you for your suggestion. Is there anyway to look at it in terms of more simple way, using Spherical polar coordinates ? $\endgroup$ – user141340 Jun 1 '19 at 12:55
  • $\begingroup$ Simplicity is in the eye of the beholder. I would argue that having bunch of matrices multiply vectors is the simplest. But of course for specific applications you can introduce coordinates and work out explicit formulas. For that I suggest you try asking at math.stackexchange as it is basic exercise in calculus. $\endgroup$ – Vít Tuček Jun 1 '19 at 15:23