### 2.2 Geometric structures

The global geometry of vacuum spacetimes in 2+1 dimensions is described mathematically by the theory of geometric structures  [25662147] (for examples of geometric structures, see  [248] ; for a slightly more detailed version of this section, see  [72]). For simplicity, let us begin with the case of a vanishing cosmological constant. If the spacetime manifold is topologically trivial, then by Equation (1) the vacuum field equations imply that is simply a subset of ordinary Minkowski space . If is topologically nontrivial, it can still be covered by contractible coordinate patches , each isometric to , with the standard Minkowski metric on each patch. The geometry is then encoded entirely in the transition functions on the intersections , which determine how these patches are glued together. Since the metrics in and are identical, these transition functions must be isometries of , that is, elements of the Poincaré group . Similarly, if , a vacuum spacetime can be built by patching together pieces of de Sitter or anti-de Sitter space by appropriate isometries: for and or for .

Such a construction is an example of a geometric structure, in the flat case a Lorentzian or (ISO(2,1), ) structure. In general, a manifold is one locally modeled on , much as an ordinary -dimensional manifold is modeled on . More precisely, let be a Lie group that acts analytically on some -manifold , the model space, and let be another -manifold. A structure on is then a set of coordinate patches for with “coordinates” taking their values in and with transition functions in .

A fundamental ingredient in the description of a structure is its holonomy group, which can be viewed as a measure of the failure of a single coordinate patch to extend around a closed curve. Let be a manifold containing a closed path . As illustrated in Figure  1, we can cover with coordinate charts
with constant transition functions between and , i.e.,
Let us now try to analytically continue the coordinate from the patch to the whole of . We can begin with a coordinate transformation in that replaces by , thus extending to . Continuing this process along the curve, with , we will eventually reach the final patch , which again overlaps . If the new coordinate function agrees with on , we will have covered with a single patch. Otherwise, the holonomy measures the obstruction to such a covering.

It may be shown that the holonomy of a curve depends only on its homotopy class  [256] . In fact, the holonomy defines a homomorphism

is not quite uniquely determined by the geometric structure, since we are free to act on the model space by a fixed element , changing the transition functions without altering the structure of . Such a transformation has the effect of conjugating by , and it may be shown that is unique up to such conjugation  [256] . The space of holonomies is thus the quotient

Note that if we pass from to its universal covering space , we will no longer have noncontractible closed paths, and will be extendible to all of . The resulting map is called the developing map. At least in simple examples, embodies the classical geometric picture of development as “unrolling” - for instance, the unwrapping of a cylinder into an infinite strip.

The holonomies of the geometric structure in (2+1)-dimensional gravity are examples of diffeomorphism-invariant observables, which, as we shall see below, are closely related to the Wilson loop observables in the Chern-Simons formulation. It is important to understand to what extent they are complete - that is, to what extent they determine the geometry. It is easy to see one thing that can go wrong: If we start with a flat three-manifold and simply cut out a ball, we can obtain a new flat manifold without affecting the holonomy. This is a rather trivial change, though, and we would like to know whether it is the only problem.

For the case of a vanishing cosmological constant, Mess  [200] has investigated this question for spacetimes with topologies . He shows that the holonomy group determines a unique “maximal” spacetime - specifically, a domain of dependence of a spacelike surface . Mess also demonstrates that the holonomy group acts properly discontinuously on a region of Minkowski space, and that can be obtained as the quotient space . This quotient construction can be a powerful tool for obtaining a description of in reasonably standard coordinates, for instance in a time-slicing by surfaces of constant mean curvature. Similar results hold for anti-de Sitter structures. Some instructive examples of the construction of spacetimes with from holonomies are given in  [133] .

For de Sitter structures, on the other hand, the holonomies do not uniquely determine the geometry  [200] . An explicit example of the resulting ambiguity has been given by Ezawa  [117] for the case of a topology (see also Section 4.5 of  [81]). A similar ambiguity occurs for (2+ 1)-dimensional gravity with point particles, where, as Matschull has emphasized  [194], it may imply a physical difference between the metric and Chern-Simons formulations of (2+1)-dimensional gravity.

We close this section with a partial description of the space of solutions of the vacuum Einstein field equations on a manifold , where is a compact genus two-manifold, that is, a surface with “handles.” The fundamental group of such a spacetime, , is generated by pairs of closed curves , with the single relation

By Equation (11), the space of holonomies is the space of homomorphisms from to (where is for , for , or for ) modulo overall conjugation. For , this space of homomorphisms has dimension : has generators and one relation, and the identification by conjugation leaves choices of elements of a six-dimensional group 1 1 For , is trivial, and there is only one geometric structure. The case of will be discussed below in Section   2.7 . There are two subtleties that prevent the space (11) from being the exact moduli space of solutions of the vacuum field equations. First, as noted above, the holonomies do not always determine a unique geometric structure. In particular, for one may need an additional discrete variable to specify the geometry. Second, not all homomorphisms from to give geometric structures that correspond to smooth manifolds. The space of homomorphisms (11) is not connected  [148], and, in general, only one connected component gives our desired geometry. Even once these caveats are taken into account, though, we still have a -dimensional space of solutions that can, in principle, be described completely.