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