Twistors but I don't think his projection created a compacted space

[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? I know Penrose had some stuff related to Twistors but I don't think his projection created a compacted space.

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!

 

[dx]

Caveat here, though, is that the real spaces should not have projected values of infinity

What does that mean?

 

[thehangedman]

Well I guess what I mean is that even though the point at infinity from the source would go to a finite point, the destination needs to be a compact space so that it is possible to parameterize it in a way where there are no infinities. Basically, the destination needs to be a finite sized hyper-sphere with no "holes". Does that make sense?

 

[dx]

Not really. What point at infinity? There's no 'point at infinity' on a general real or complex manifold. You say you want to study maps [itex]\mu : A \rightarrow B [/tex] where A is a one dimension complex manifold and B is a one dimensional real manifold. What kind of maps? Is there any reason you require holomorphic structure on A?[/itex]

 

[thehangedman]

I'm sorry for my ignorance, and as a result I don't quite know how to ask the question. My math is lacking here so it's hard for me to even know where to start. It is something I want to understand better (Twistors) but I have found all the material to be starting at a level of mathematics beyond where I currently am. I know differential geometry / calculus etc fairly well but am far less versed in the more abstract formalism of group theory etc.

So perhaps if I just explained my thoughts you could point me in the right direction... I want to explore the results of making space-time compact on it's own. I also think this is the result of some kind of projection from C4 (complex space) and I know that in Twistors there exists a projection though I'm again struggling to understand it since it's not written in lower level calculus formulas. I want to look at the results of curving this C4 space and via this new projection how space-time would curve (or not, since there are more degrees of freedom in C4 than R4).

So, I'm looking for the right projection from C4 to R4 that gives an R4 that is compact, and on small scales (or as the radius of the closed hyber-sphere goes to inifinity) gives me Minkowski's metric.