There is not-so-rough evidence that at very large scale the universe is flat. However we see everywhere that there are local lumps of matter with positive curvature. So i have several questions regarding this:
1) Does the fact that a manifold with a) asymptotic (space) curvature zero and b) local inhomogeneities with positive (space) curvature imply that there will be regions with negative (space) curvature?
2) a Region of negative (space) curvature implies dark energy in that region?
3) assuming answer to both 1) and 2) are true: does this represent an independent confirmation of dark energy? or there is somehow an geometric relationship relating asymptotic flatness to accelerated expansion (the traditional reason to introduce dark energy in the first place)?
EDITED: to reflect distinction between space and space-time curvatures.