A solution of the Einstein equations is called spatially
homogeneous if there exists a group of symmetries with
three-dimensional spacelike orbits. In this case there are at
least three linearly independent spacelike Killing vector fields.
For most matter models the field equations reduce to ordinary
differential equations. (Kinetic matter leads to an
integro-differential equation.) The most important results in
this area have been reviewed in a recent book edited by
Wainwright and Ellis. See, in particular, part two of the book. There remain a host
of interesting and accessible open questions. The spatially
homogeneous solutions have the advantage that it is not necessary
to stop at just existence theorems; information on the global
qualitative behaviour of solutions can also be obtained. An
important open question concerns the mixmaster solution, as
discussed in .