The notions of connection in a fibered bundle and of jet spaces of sections were introduced by Ehresmann. These concepts have been generalized to the case of a fibered manifold (a surjective submersion $E\to M$). The jet space associated with a fibered manifold provide an appropriate setting for defining a generalized connection. Recently, these ideas have been discussed by Libermann, Kolar, Mangiarotti and Modugno and others. An application of these generalized structures of connections to space-time edge geometry has been given in [a5, b5].

In this paper we study various kinds of connections, which are naturally associated with two-fibered manifolds. We note that a particular kind of projectable connection on two-fibered manifolds has been studied by Kolar. This general setting is suitable for studying completions of fibered manifolds. For example, if $LE\to E$ denotes the frame bundle of $E$, then $LE\to E\to M$ is a two fibered manifold. If $M$ is the space-time manifold and $E = LM$, then the fibered manifold structure of $E\to M$ is used to establish the metrizability of $M$ and its b-completion. The stability of this b-incompleteness via a structure of comlections uses essentially the two fibered manifold $K\to E\to M$ defined in sec.1. In sec.2 we briefly review the b-completions of manifolds. The stability of b-incompleteness of manifolds is discussed in sec.3. In sec. 4 we introduce connections on two-fibered manifolds and study their reductions and special bundles which are related to the study of b-completions of flbered manifolds. Sec. 5 contains a brief outline of work in progress and a discussion of some open problems.