Global homeomorphisms and covering projections on metric spaces

Abstract

Abstract. 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 some applications to the existence of global implicit functions.

On global inverse theorems

“…Relatively recently, Gutú and Jaramillo [22] presented an extension of some well known results in the framework of metric spaces where the continuation property plays a central role. From Example 2.2 and Theorem 2.6 of [22] we can deduce that if Y is a connected C k Banach manifold -equivalently C k path-connected [44, p. 118]-then it is enough to consider the set P as the set of the C k paths in a connected Y .…”

“…From Example 2.2 and Theorem 2.6 of [22] we can deduce that if Y is a connected C k Banach manifold -equivalently C k path-connected [44, p. 118]-then it is enough to consider the set P as the set of the C k paths in a connected Y . More precisely, let X and Y be Banach manifolds, assume Y is connected of class C k where 1 ≤ k ≤ ∞, if f : X → Y is a local homeomorphism then following statements are equivalent:…”

“…For Riemannian and Finsler manifolds the proof can be consulted in [22] and [27], respectively. So, for local diffeomorphisms condition (C8) implies that both indicators are uniformly bounded below on X.…”

“…See [64] for an alternative proof. Another generalization for metric spaces but in terms of D − x f -which includes Finsler manifolds-can be found in [22].…”

“…He also showed that the properness, closedness, and the Hadamard integral condition all imply the condition L. Since then, condition L has proved to be quite useful in global inversion theorems. In the same vein, Ioffe [26] extended the Hadamard Theorem in terms of the so-called surjection constant of the mapping f making use of the condition L. More recently, this approach was used to give necessary and sufficient conditions for a map f to be a global homeomorphism for a large class of metric spaces with nice local structure -which includes Banach-Finsler manifolds [22]. As a consequence, an extension of the Hadamard Theorem is obtained in terms of a metric version of µ(x), a kind of lower scalar Dini derivate.…”