We next turn to a more traditional approach to classical general relativity, the conventional metric formalism in the space/time splitting of Arnowitt, Deser, and Misner  [25] . As Moncrief  [206] and Hosoya and Nakao  [159] have shown, this metric formalism can also be used to give a full description of the solutions of the vacuum field equations, at least for spacetimes with the topology .
as illustrated in Figure  2 . The action then takes the usual form 2 2 In this section I use standard ADM notation: and refer to the induced metric and scalar curvature of a time slice and is the extrinsic curvature of such a slice, while the full spacetime metric and curvature are denoted by and .
with canonical momentum and the momentum and Hamiltonian constraints
To solve the constraints, we can choose the York time-slicing  [284], in which the mean (extrinsic) curvature is used as a time coordinate, . Andersson et al. have shown that this is a good global coordinate choice for classical solutions of the vacuum field equations  [20] . We next select a useful parametrization of the spatial metric and momentum. Up to a diffeomorphism, any two-metric on can be written in the form  [1121]
where are a finite-dimensional family of metrics of constant curvature ( for the two-sphere, for the torus, and for spaces of genus ). These standard metrics are labeled by a set of moduli that parametrize the Riemann moduli space of . As in Section  2.2, such constant curvature metrics can be described in terms of a geometric structure - for genus an structure - with moduli parametrizing the homomorphisms (10). We can count these just as in Section  2.2 ; now, since is three-dimensional, we find that a constant negative curvature surface of genus is described by parameters.

The corresponding decomposition of the conjugate momentum is described in  [206] : Up to a diffeomorphism, the trace-free part of can be written as a holomorphic quadratic differential , that is, a transverse traceless tensor with respect to the covariant derivative compatible with . The space of such quadratic differentials parametrizes the cotangent space of the moduli space  [1], and the reduced phase space becomes, essentially, the cotangent bundle of the moduli space.

With the decomposition of  [206], the momentum constraints become trivial, while the Hamiltonian constraint becomes an elliptic differential equation that determines the scale factor in Equation (31) as a function of and ,

where are the momenta conjugate to the moduli,
The theory of elliptic equations ensures that Equation (32) determines a unique scale factor . The action (29) then simplifies to a “reduced phase space” action, involving only the physical degrees of freedom,
with a time-dependent Hamiltonian
The classical Poisson brackets can be read off directly from Equation (34):

Three-dimensional gravity again reduces to a finite-dimensional system, albeit one with a complicated time-dependent Hamiltonian. The physical phase space is parametrized by , which may be viewed as coordinates for the cotangent bundle of the moduli space of . For a surface of genus , this gives us degrees of freedom, matching the results of Section  2.2 .

If , this correspondence can be made more explicit: For and , the space (11) of geometric structures is itself a cotangent bundle, whose base space is the space of hyperbolic structures on . This follows from the fact that the group is the cotangent bundle of . Concretely, in the first-order formalism of Section  2.3, the curvature equation (16) with implies that is a flat connection; and if is a curve in the space of such flat connections, the tangent vector satisfies the torsion equation (17). For , I know of no such direct correspondence, and the general relationship between the ADM and first-order solutions seems less transparent.