Global homeomorphisms and covering projections on metric spaces

For a large class of metric spaces with nice local structure, which includes Banach–Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path- liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give an application to the existence of global implicit functions. O. Gutú and J. A. Jaramillo were supported in part by D.G.E.S. (Spain) Grant BFM2003-06420.