  A basic assumption in classical GR is that the metric field is nowhere
degenerate in spacetime. In particular the induced metric on Cauchy surfaces
must be nowhere degenerate. It is only under this assumption that one can
derive the hypersurace deformation algebra between the initial value
constraints which is absolutely transparent from the fact that the {\it
inverse} of the induced metric is needed to close the algebra. This statement
is independent of the density weight that one may want to equip the spatial
metric with. Accordingly, the very definition of a non-anomalous representation
of the hypersurface defomation algebra in quantum gravity has to address the
issue of non-degenracy of the induced metric that is needed in the classical
theory. In the Hilbert space representation employed in Loop Quantum Gravity
(LQG) most emphasis has been layed to define an inverse metric operator on the
dense domain of spin network states although they represent induced quantum
geometries which are degenerate almost everywhere. It is no surprise that
demonstration of closure of the constraint algebra on this domain meets
difficulties because it is a sector of the quantum theory which is classically
forbidden and which lies outside the domain of definition of the classical
hypersurface deformation algebra. Various suggestions for addressing the issue
such as non-standard operator topologies, dual spaces (habitats) and density
weights have been propposed to address this issue with respect to the quantum
dynamics of LQG. In this article we summarise these developments and argue that
insisting on a dense domain of non-degenerate states within the LQG
representation may provide a natural resolution of the issue thereby possibly
avoiding the above mentioned non-standard constructions.
