Abstract

It is shown that there is a natural metric in the fibres of the space of principal connections over the frame bundle. This metric is given by an extension of the idea of bundle metric induced by a given connection, and provides a natural way of defining the distance between two given connections. The problem of reducing this distance to a distance over the base manifold is examined and, in particular, a relation between the distance of two given connections along a curve and the bundle-incompleteness of that curve with respect to the given connections is found. This result is related to the problem of the stability of space-time singularities.

1991 MSC: 53C05, 53C07

Keywords: connections, bundle metric.