Shape of The Universe - Global Geometry

Global geometry covers the geometry, in particular the topology, of the whole universe—both the observable universe and beyond. While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. For this discussion, the universe is taken to be a geodesic manifold, free of topological defects; relaxing either of these complicates the analysis considerably.

In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.

A global geometry is also called a topology, as a global geometry is a local geometry plus a topology, but this terminology is misleading because a topology does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.

Two strongly overlapping investigations within the study of global geometry are whether the universe:

• Is infinite in extent or, more generally, is a compact space;
• Has a simply or non-simply connected topology.