Mathematics
Scientific paper
Dec 1983
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983gregr..15.1191f&link_type=abstract
General Relativity and Gravitation, Volume 15, Issue 12, pp.1191-1198
Mathematics
12
Scientific paper
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)} = {mathfrak{M} × Fleft( M right)}/mathfrak{D} to mathfrak{M}/mathfrak{D} = mathcal{G} whereF(M) is the frame bundle ofM. The unfolded spacemathcal{G}_{Fleft( M right)} is infinite-dimensional manifold without singularities. Moreover, as expected, the unfolding ofmathcal{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 makemathcal{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 fibermathcal{G}_{Fleft( M right)}, and with fiber at a point inM being the particular noncanonical unfolding ofmathcal{G} based at that point. ThusE is the totality of all the particular unfoldings, and so is a grand unfolding ofmathcal{G}.
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