In this paper we give some necessary and sufficient conditions under which an isometric immersion between two connected Riemannian manifolds of the same dimension becomes a covering projection. We also prove that, generally, even if these conditions are not satisfied, we can always associate to such an isometric immersion i:X→Y a family of covering projections. They are constructed by removing a suitable closed set from X and then restricting i to the connected components of the remaining manifolds. Some applications are given to the problem of the classification of a class of complete metrics with singularities on an analytic manifold.