Goodbye Thurston

Thurston has left :( so I figure a geometerization of 3 manifolds post is in order.

This is all in his book Three Dimensional Geometry and Topology.

For surfaces, uniformization is prehistoric. We have the easy invariant of the genus to tell us the covering space and it gets a constant curvature metric.

For 3d manifolds, the situation is trickier. We again want to find model manifolds along with their Lie groups of isometries. We also want these actions to be transitive. None of the points should get any special treatment.

The connected component of the point stabilizers are going to be Lie subgroups of SO(3).

If they are the full SO(3), then all directions you stare into look the same so we have a symmetric space. It must be one of euclidean, spherical or hyperbolic depending on the curvature.

If the component of the stabilizer is SO(2), then look in the leftover direction at each point to construct a vector field. The flow lines of the vector field give a foliation of the manifold where every leaf is 1D. The leaves are either circles or lines.
The easiest example of such would be surface cross R. This gives two new geometeries and euclidean 3-space over again.
There are two more of these, the cover of SL2 and the Heisenberg group.

This leaves when the point stabilizer is 1D. The only possibility is Sol geometry.