Unfolding the singularities in superspace

A method is described for unfolding the singularities in superspace, (mathcal{G} = mathfrak{M}/mathfrak{D}) , the space of Riemannian geometries of a manifoldM. This unfolded superspace is described by the projection $$mathcal{G}_{Fleft( M right)} = frac{{mathfrak{M} times Fleft( M right)}}{mathfrak{D}} to frac{mathfrak{M}}{mathfrak{D}} = mathcal{G}$$ whereF(M) is the frame bundle ofM. The unfolded space (mathcal{G}_{Fleft( M right)}) is infinite-dimensional manifold without singularities. Moreover, as expected, the unfolding of (mathcal{G}_{Fleft( M right)}) at each geometry [g _o] ∈ (mathcal{G}) is parameterized by the isometry groupIg _o (M) of g₀. Our construction is natural, is generally covariant with respect to all coordinate transformations, and gives the necessary information at each geometry to make (mathcal{G}) a manifold. This construction is a canonical and geometric model of a nonrelativistic construction that unfolds superspace by restricting to those coordinate transformations that fix a frame at a point. These particular unfoldings are tied together by an infinite-dimensional fiber bundleE overM, associated with the frame bundleF(M), with standard fiber (mathcal{G}_{Fleft( M right)}) , and with fiber at a point inM being the particular noncanonical unfolding of (mathcal{G}) based at that point. ThusE is the totality of all the particular unfoldings, and so is a grand unfolding of (mathcal{G}) .