Abstract

It is shown that an extension of the idea of bundle-metric induced by a given connection on the frame bundle defines a metric in the fibres of the space of principal connections over the frame bundle itself. This provides a natural way of calculating 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-completeness of that curve with respect to the given connections is found. Furthermore, by using the idea of system of overconnections over the bundle of principal connections, we find some natural extensions of the above fibre metric to a metric on the whole space of principal connections. Finally, two examples are worked out.

Introduction

Connections are well known to be of fundamenal importance for physics, and particularly for gauge field theories. Furthermore, recent developments in the theory of systems of connections have provided a very rich geometric framework to work with. Roughly speaking, one has a system on a fibred manifold $E\to B$ when there is a distinguished set of connections on it, which can be realized as sections of a finite dimensional fibred manifold $C\to B$ over the same base manifold. Systems of connections arise naturally in most cases of practical interest, like principal, affine and vector bundles. This kind of structure has been studied especially by Garcia for the particular case of principal bundles. Recently, Modugno has shown that most important aspects, including geometric richness, do not actually depend on the principal bundle structure. Furthermore, by focusing our attention on the essential aspects, we can easily exploit new interesting structures. One of these is the system of overconnections, a natural system of connections over the bundle of connections of a given system. In this paper we deal essentially with systems on principal bundles, but it should be noticed that the general setting provides a very useful simplification of language. The idea is of studying if there are natural ways of defining a distance between two connections, having in mind possible applications to the stability of spacetime singularities. A first answer is provided in §4 by defining a natural metric structure on the fibres of the bundle of principal connections on the frame bundle. This definition is a generalization of the metric induced on the frame bundle by a connection. In §5 we introduce in this setting the system of overconnections which yields a system of metrics on the space of connections. This provides further extensions of the fibre metric to metrics on the whole space of connections, hence a few further natural ways of defining a distance between connections. Moreover, we introduce these techniques in the discussion about singularities and their stability and provide some examples.