  At a fixed point in spacetime (say, x_0), gravitational phase space consists
of the space of symmetric matrices F^{ab} [corresponding to the canonical
momentum pi^{ab}(x_0) and of symmetric matrices {G_{ab}}[corresponding to the
canonical metric g_{ab}(x_0), where 1 \leq a,b \leq n, and, crucially, the
matrix {G_{ab}} is necessarily positive definite, i.e. \sum u^a G_{ab}u^b > 0
whenever \sum (u^a)^2 > 0. In an alternative quantization procedure known as
Metrical Quantization, the first and most important ingredient is the
specification of a suitable metric on classical phase space. Our choice of
phase space metrics, guided by a recent study of Affine Quantum Gravity, leads
to gravitational phase space geometries which possess constant scalar curvature
and may be regarded as higher dimensional analogs of the Poincare plane, which
applies when n=1. This result is important because phase spaces endowed with
such symmetry lead naturally via the procedures of Metrical Quantization to
acceptable Hilbert spaces of high dimension.
