General Relativity and Gravitation

, Volume 15, Issue 12, pp 1191–1198 | Cite as

Unfolding the singularities in superspace

  • Arthur E. Fischer
Research Articles

Abstract

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

$$\mathcal{G}_{F\left( M \right)} = \frac{{\mathfrak{M} \times F\left( M \right)}}{\mathfrak{D}} \to \frac{\mathfrak{M}}{\mathfrak{D}} = \mathcal{G}$$

whereF(M) is the frame bundle ofM. 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 [go] ∈\(\mathcal{G}\) is parameterized by the isometry groupIgo (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\(\mathcal{G}_{F\left( M \right)}\), 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\(\mathcal{G}\).

Keywords

Manifold Differential Geometry Geometric Model Coordinate Transformation Riemannian Geometry 

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Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • Arthur E. Fischer
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSanta Cruz

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