Is there any proper method that makes conformal one-to-one mapping (preserving the angles) from a unit disk to a unit sphere? I know with simple stereographic projection you can project a unit disk to a hemisphere.
I can imagine the solution is by bringing the points close to the center of the disk to be projected near the North pole on the sphere, while the edges of the disk go close to the South pole of the sphere.
Sorry if this question was repeated I'm no expert in differential geometry and I tried to use simple terms.
Best regards,
Mahmoud
Best Answer
A bijective conformal mapping is a homeomorphism. But the unit disk is not homeomorphic to the unit sphere. So the answer is no, no such map exists.
Of course through stereographic projection you can get a homeomorphism between the unit disk and the unit sphere minus an arbitrarily small open disk around the south pole, in the way you seem to describe.