Global geometrycovers the geometry, in particular thetopology, 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 ageodesic manifold, free oftopological defects; relaxing either of these complicates the analysis considerably.

In general,local to global theoremsinRiemannian geometryrelate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described inThurston 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 andhyperbolic 3-spacehave the same topology but different global geometries.

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

Isinfinitein extent or, more generally, is acompact space;Has asimply or non-simply connectedtopology.

## Saturday, August 28, 2010

### Global geometry

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