Geometry & Dynamics of the Universe
Metric of Space-Time
The observed isotropy of the CMB allows one to determine that the metric of space-time is the Robertson-Walker metric to approximately the same precision as the observed isotropy of the CMB (Stoeger, Maartens, & Ellis 1995)
Isotropy of Expansion and Space-Time
One can infer the isotropy of the Hubble expansion of the Universe from the observations of the CMB. The expansion is isotropic to the order of one part in 100,000 or better.
Curvature of Space
One can make plausibility arguments about the stability of a universe with large curvature but with the small curvature fluctuations observed via the CMB. However, in the context of plausible models, (that can be tested) one can measure directly the curvature of space though the angular location of the "acoustic peaks" in the CMB anisotropy power spectrum. Preliminary results at this time show that the curvature of the Universe is small and consistent with a "flat" = Euclidean geometry. Expressed in terms of the curvature of a Friedman-Robertson-Walker model the results correspond to a density of Omega = 0.7 + 1.1 - 0.4.
Topology of the Universe
It is also possible to learn something about the topological connection of the Universe (Costa & Smoot). Simple topologies examples are: spherical universes, cylindrical universes, cubical universes with opposite edges identified, or more complicated permutations of the identifications including twists and inversions, or not opposite sides. It could be that the topology of the Universe is very complicated if quantum gravity and tunneling were important in the early epochs.
Rotation & Shear of the Universe
CMB observations allow us to put limits on the shear and vorticity of the Universe. Typically one can set a general limit on the ratio of the shear or vorticity to the Hubble expansion rate of about 10-5. Specific models can provide better tests. For example, this means that the Universe has rotated less than one second of arc in the last 10 billion years.