We show how many familiar properties of complete simply connected riemannian manifolds of negative curvature generalise to incomplete manifolds satisfying certain curvature constraints. We show, for example, that such a manifold is geodesically convex, and can be canonically compactified to a topological ball on adjoining the visual sphere. Examples of such manifolds are negatively curved manifolds having bounded geometry up to local rescaling of the metric.
Pacific J. Math. 172 (1996) 1-39.