Part of universe mapped by coordinates

[bloby]

I read in a positively curved universe the radial r coordinate ranges from 0 to 1. Is it just a local map?

 

[bapowell]

I read in a positively curved universe the radial r coordinate ranges from 0 to 1. Is it just a local map?

Why do you suggest that it's a local map? Because the value of r ranges only from 0 to 1? Sounds to me like r is ranging over the full radius of the manifold, and is simply normalized so that r_max = 1. However, without context or more detail, I can't be of any more help.

 

[bloby]

However, without context or more detail, I can't be of any more help.

In Kolb&Turner and in my cours the coordinates of a point on a 3-sphere are the projection on an hyperplane (the equatorial plane if on a 2-sphere). They construct the Robertson-Walker metric for positive curvature this way. But then only one half of the sphere is mapped, isn't it? The huge value the line element takes when r tends to one or R(you are right it's normalized r) is simply the consequence of the choice of coordinates? (Forgive me for my bad english)((If you live on the North, to name a point, of a 2-sphere you can map each points by an angle and a length (->r*pi) or even a comoving length (r/R*pi) but these are not the usual coordinates, aren't they?))

I suggest that it's a local map because I read a constant positive curvature means the universe is "only" diffeomorphic to a 3-sphere.