  The basic framework for a systematic construction of a quantum theory of
Riemannian geometry was introduced recently. The quantum versions of Riemannian
structures --such as triad and area operators-- exhibit a non-commutativity. At
first sight, this feature is surprising because it implies that the framework
does not admit a triad representation. To better understand this property and
to reconcile it with intuition, we analyze its origin in detail. In particular,
a careful study of the underlying phase space is made and the feature is traced
back to the classical theory; there is no anomaly associated with quantization.
We also indicate why the uncertainties associated with this non-commutativity
become negligible in the semi-classical regime.
