Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T09:52:49.440Z Has data issue: false hasContentIssue false
  • English
  • Français

On the bundle of principal connections and the stability of b-incompleteness of manifolds

Published online by Cambridge University Press:  24 October 2008

D. Canarutto  and
C. T. J. Dodson
D. Canarutto
Affiliation:
Applied Mathematics Institute‘G. Sansone’, University of Florence
C. T. J. Dodson
Affiliation:
Department of Mathematics, University of Lancaster
Get access Rights & Permissions [Opens in a new window]

Abstract

We make use of the universal connection on the bundle of principal connections; the bundle structure is governed by the action of the group on the first jet bundle. Each section determines a connection in the principal bundle, which in the case of the frame bundle allows a metric completion projecting onto the corresponding b-completion. It is shown that b-incompleteness of the base manifold is stable under perturbations of the chosen section. Hence, for instance, for (pseudo) Riemannian manifolds including spacetimes, b-incompleteness is a stable condition under conformal deformations of the metric. This reinforces the belief that general relativistic singularities cannot be removed by quantization.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 1 , July 1985 , pp. 51 - 59
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Canarutto, D.. Metric completions and b-completions of Lorentz manifolds. Il Nuovo Cimento (1984) (in the Press).CrossRefGoogle Scholar
[2]Dodson, C. T. J.. Space-time edge geometry. Internat. J. Theoret. Phys. 17 (6) (1978), 389–504.CrossRefGoogle Scholar
[3]García, P. L.. Connections and 1-jet fiber bundles. Rend. Sem. Mat. Univ. Padova 47 (1972), 227–242.Google Scholar
[4]García, P. L.. Gauge algebras, curvature and symplectic structure. J. Differential Geom. 12 (1977), 209–227.CrossRefGoogle Scholar
[5]Gotay, M. J. and Isenberg, J. A.. Geometric quantization and gravitational collapse. Phys. Rev. D 22 (1980), 235–260.Google Scholar
[6]Mangiarotti, L. and Modugno, M.. Fibered spaces, jet spaces and connections for field theories. In Proceedings of the International Meeting ‘Geometry and Physics’, Florence 12–15October 1982, ed. Modugno, M. (Pitagora Editrice, 1983), 135–165.Google Scholar
[7]Marathe, K. B.. A condition for paracompactness of a manifold. J. Differential Geom. 7 (1972), 571–573.CrossRefGoogle Scholar
[8]Schmidt, B. G.. A new definition of singular points in General Relativity. Gen. Relativity Gravitation 1, (1971), 269–280.CrossRefGoogle Scholar
[9]Sze-Tsen, Hu. Introduction to General Topology (Holden Day, 1966).Google Scholar