Some results about true Riemannian metrics on Lorentz manifolds are established and applied to the study of space-time edge geometry. It is shown that any such metric determines a splitting of the tangent bundle over space-time; this enables us to select a class of Riemannian metrics which satisfy certain relations with respect to the given Lorentz structure. These metrics are in one-to-one correspondence with timelike distributions. A sufficient condition for two such metrics to give homeomorphic completions is found and some examples concerning the two-dimensional Friedmann model are considered. Finally, some results about the comparison between b-completion and metric completion are established and, in particular, sufficient conditions for the equivalence of the two notions of completeness are found.

The geometrical construction of the b-boundary has solved the problem of deciding if a space-time has singularities. However, some problems still remain in the description of singularities, since, in general, the b-completion of a Lorentz manifold is neither locally compact nor a $T_1$ space. Moreover, in the case of the closed standard model of the universe, the initial and final singularities are identified.

Modifications of this construction have been proposed. One of them is Clarke's projective completion, which makes the singularities of the standard model into two distinct points. Another one is Dodson's p-completion for parallelizable manifolds, which depends on the choice of a parallelization. One may note that the choice of a parallelization $p:M\to LM$ induces a Riemannian metric on $M$ and, therefore, a metric completion; moreover, the p-completion of $M$ is homeomorphic to the metric completion of $p(M)$ as a submanifold of $LM$, with respect to the Schmidt metric. Finally, in the case of the standard model there is a canonical metric and the corresponding completion is homeomorphic to the projective completion.

These considerations lead us to discuss metric completions of Lorentz manifolds and, in particular, their relation to b-completions, for we know that each space-time admits a Riemannian structure.