Abstract 

The problem of the stability of b-boundary points of a manifold is examined in the context of the space of principal connections of the frame bundle. Some new results in the geometry of spaces of connections (which improve results established in a previous paper) enable us to describe a general situation in which b-incompleteness is preserved after a change of the given connection. These results seem suitable for obtaining various types of stability theorems. In fact, one such consequence is that stability with respect to the $C^0$ Whitney topology can be given a rather simple proof for points in the so-called "essential boundary".

Introduction [to be typeset by Plain TeX]

The b-completion is usually considered as the main tool for the study of the geometry of space-time singularities. This construction can be done whenever a principal connection on the frame bundle $LM$ of a manifold $M$ is given. In fact, the connection induces a riemannian metric on $LM$; the metric completion of $LM$ can be quotiented by the group action, and M is dense in this quotient, which is exactly the b-completion. We say that $M$ is b-complete, with respect to the given connection, if it coincides with its b-completion. This concept is a generalization of that of geodesic completeness, and also of that of metric completeness for riemannian manifolds. If $M$ is b-incomplete, then the b-completion is the union of $M$ with the b-boundary, which may be thougtht as the set of endpoints of b-bounded curves (i.e. curves of bounded horizontal lift in $LM$) with no endpoint in $M$.

Thus, a space-time singularity can be seen as a point in the b-boundary generated by the Levi-Civita connection associated to the Lorentz metric. In other terms, the existence of a singularity is related to the b-incompleteness of $M$. It is then clear that stability problems concerning b-incompleteness and completeness have great interest in relation to the very existence of physical singularities, both from classical and quantistic point of view.

The setting of the stability problem requires essentially two things: first, a decision about what a gravitational field is, that is, of which space it is a point (or of which bundle it is a section); second, the assignment of a precise meaning to the idea of "small change" of it. Though the most usual choice for the field is a Lorentz metric, also the connection is an important candidate, as it is suggested by the framework of gauge theories and by that of metric-affine theories of gravitation. The interest for this approach is stimulated also from the theory of the systems of connections. This framework seems promising, and in fact we are able to prove a result which can be interpretated as a kind of b-incompleteness stability, since it says that if $M$ is b-incomplete with respect to a connection, then it is also b-incomplete with respect to a new connection, which approaches the former in a certain sense. This result, which is an improvement of a previous version, may be not so expressive from an intuitive point of view, but it is rather general and seems a good startpoint for the study of the problem, since it may produce various interesting consequences.

In fact, one such consequence is obtained, in the last section, in a framework which is quite natural for stability problems, i.e. that of the Whitney topology on the space of sections of a bundle, in our case the bundle of principal connections of the frame bundle. We shall say that any property depending on the connection is stable if the set of all connections which have the property is open in this topology. Thus, we are concerned with the study of b-incompleteness stability in this context.

However, only a part of the b-boundary is usually considered as representing the physical singularities, that is the so-called "essential boundary", constituted by limit points of not (partially) trapped curves. Now, the final result of this paper is precisely that stability, in the sense of the Whitney topology, holds at least for these points.

There are still many open problems. For example, note that the choice of a different space of sections (for example Lorentz metrics) as representing the possible gravitational fields might give completely different results; in fact the map "riemannian connection" from the space of metrics to that of principal connections is not continuous. It would be important to study which is the largest possible modification of the field which preserves the existence of singularities. This could relate our approach to others in which coarser topologoes are used and some kind of instability arises. Indeed, many people think that the most physical approach would be the unexplored one based on the space of solutions of a given field equation.