The Cartan–Hadamard Theorem for metric spaces with local geodesic bicombings

Abstract

We prove the Cartan–Hadamard Theorem for spaces which are not necessarily uniquely geodesic but locally possess a suitable selection of geodesics, a so-called convex geodesic bicombing. Furthermore, we deduce a local-to-global theorem for injective (or hyperconvex) metric spaces, saying that under certain conditions a complete, simply-connected, locally injective metric space is injective. A related result for absolute 1-Lipschitz retracts follows.