# Unfolding the singularities in superspace

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## Abstract

A method is described for unfolding the singularities in superspace,\(\mathcal{G} = \mathfrak{M}/\mathfrak{D}\), the space of Riemannian geometries of a manifold*M*. This unfolded superspace is described by the projection

where*F(M)* is the frame bundle of*M*. The unfolded space\(\mathcal{G}_{F\left( M \right)}\) is infinite-dimensional manifold without singularities. Moreover, as expected, the unfolding of\(\mathcal{G}_{F\left( M \right)}\) at each geometry [*g*_{o}] ∈\(\mathcal{G}\) is parameterized by the isometry group*Ig*_{o} (*M*) of g_{0}. 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 bundle*E* over*M*, associated with the frame bundle*F(M)*, with standard fiber\(\mathcal{G}_{F\left( M \right)}\), and with fiber at a point in*M* being the particular noncanonical unfolding of\(\mathcal{G}\) based at that point. Thus*E* is the totality of all the particular unfoldings, and so is a grand unfolding of\(\mathcal{G}\).

### Keywords

Manifold Differential Geometry Geometric Model Coordinate Transformation Riemannian Geometry## Preview

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