Can Complex Spaces Be Projected onto Closed Real Spaces without Infinity?

[thehangedman]

I am looking for good reading material and references on something. I have tried the google route and can't find anything so I thought I would ask the community of people who know...

I want to learn more about the following scenario: Suppose I start with a 1 dimensional complex space. I want to project that onto a 1 dimensional closed (compacted?) real space. Essentially, map the complex plane to the real circle. I am also interested in this mapping in higher dimensions too, so C2 mapping to the real surface of a sphere. Caveat here, though, is that the real spaces should not have projected values of infinity. I would rather the coordinates loop (spherical would work better I'd assume). I was thinking along the line of projecting the complex plane to the unit circle in the plane and using the distance around said circle to create the compacted real space, but would that work in higher dimensions and does that yield any interesting results?

Again I'm not looking for any specific "answer" but rather references of people who have studied this specific scenerio and what they came up with.

Thanks for all your help!

 

[thehangedman]

anyone?