### 4.1 One-Killing vector reductions

Let us start by considering the simplest example of symmetry reduction corresponding to
one-dimensional spatial isometry groups. In practice one considers R or U(1) and, hence, the spacetime
metrics are required to have a single Killing vector field. These models are interesting because they retain
important features of full general relativity, in particular diffeomorphism invariance, an infinite number of
physical degrees of freedom and a non-linear character.
The local aspects of the one-Killing vector reductions were first considered by Geroch [96]. In that paper
he developed a method to dimensionally reduce gravity by defining a way to “project” 3+1 dimensional
geometric objects to the 2+1 dimensional space of orbits of the Killing field (required to have a
non-vanishing norm). Here the world “local” refers to the fact that some topological aspects are sidestepped
in a first look; however, if the quotient itself is well behaved (for example, it is Haussdorff) the projection is
globally defined and has a clear geometrical meaning. The most important result of this work was to show
that the reduced system can be interpreted as 2+1 GR coupled to certain matter fields with a concrete
geometrical meaning: the norm and twist of the four dimensional Killing vector field (a scalar
and a one-form field respectively). This link between one-Killing vector reductions and 2+1
dimensional gravity theories opened the door to quantum treatments relying on techniques
specially tailored for lower dimensional models. The Geroch method can be adapted to treat
symmetry reductions. For example it allows one to write the four-dimensional scalar curvature as a
curvature on the 2+1 dimensional orbit manifold plus some extra terms involving the norm
of the Killing. This is very useful to write the 3+1 dimensional action as a 2+1 dimensional
one.

The global aspects and the Hamiltonian formalism (for vacuum GR) have been discussed by Moncrief in
the case when the symmetry group is U(1) with compact Cauchy surfaces [175]. The spatial slices in this
case can be taken to be U(1) bundles (or rather S^{1}) over the sphere (though the analysis can be extended
to arbitrary surfaces). The discussion presented in [175] is relevant to studying some of the compact Gowdy
models, in particular those with the S^{2} × S^{1} and S^{3} spatial topologies, though it is possible to employ
other approaches that rely on the Geroch reduction as discussed in [22]. The non-compact case with
asymptotically-flat two-geometries (in the sense relevant in 2+1 gravity developed in [15]) has been studied
by Varadarajan [212].