The Cartan-Hadamard Theorem for Metric Spaces with Local Geodesic Bicombings

TL;DR

This paper extends the Cartan-Hadamard Theorem to metric spaces with local geodesic bicombings, establishing new local-to-global principles for injective and hyperconvex spaces in a broader setting.

Contribution

It generalizes the Cartan-Hadamard Theorem to spaces with convex geodesic bicombings, not necessarily uniquely geodesic, and derives local-to-global results for injective metric spaces.

Findings

Proves Cartan-Hadamard Theorem for spaces with local convex geodesic bicombings.

Establishes local-to-global theorem for injective metric spaces.

Provides results for absolute 1-Lipschitz retracts.

Abstract

Local-to-global principles are spread all-around in mathematics. The classical Cartan-Hadamard Theorem from Riemannian geometry was generalized by W. Ballmann for metric spaces with non-positive curvature, and by S. Alexander and R. Bishop for locally convex metric spaces. In this paper, we prove the Cartan-Hadamard Theorem in a more general setting, namely for spaces which are not 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.