[Paper Review] The Cartan-Hadamard Theorem for Metric Spaces with Local Bicombings

Benjamin Miesch|arXiv (Cornell University)|Sep 23, 2015

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper extends the Cartan-Hadamard Theorem to metric spaces with local bicombings—geodesic selections that are convex and 1-Lipschitz—showing that such spaces are simply connected and exhibit non-positive curvature-like behavior. It further establishes a local-to-global principle, proving that a complete, simply-connected, locally injective metric space is globally injective under mild conditions.

ABSTRACT

The classical Cartan-Hadamard Theorem 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 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.

Motivation & Objective

• To extend the classical Cartan-Hadamard Theorem beyond uniquely geodesic spaces to more general metric spaces with structured geodesic selections.
• To investigate the geometric and topological consequences of local bicombings in metric spaces lacking unique geodesics.
• To establish a local-to-global principle for injective (hyperconvex) metric spaces, characterizing when local injectivity implies global injectivity.
• To explore the relationship between convex bicombings, non-positive curvature, and the structure of absolute 1-Lipschitz retracts.

Proposed method

• Introduces the concept of a convex bicombing: a family of 1-Lipschitz curves connecting pairs of points in a neighborhood, with convexity properties ensuring geodesic-like behavior.
• Uses the existence of such bicombings to define a notion of non-positive curvature in non-uniquely geodesic spaces.
• Applies comparison techniques inspired by Alexandrov geometry to analyze the global structure of spaces with local bicombings.
• Employs topological arguments, including the homotopy lifting property and simple connectivity, to deduce global injectivity from local injectivity.
• Leverages the theory of absolute retracts and hyperconvexity to characterize the conditions under which local injectivity implies global injectivity.
• Establishes a duality between convex bicombings and the structure of absolute 1-Lipschitz retracts in complete, simply-connected spaces.

Experimental results

Research questions

• RQ1 Can the Cartan-Hadamard Theorem be generalized to metric spaces that are not uniquely geodesic but admit a convex bicombing?
• RQ2 What topological and geometric properties emerge in metric spaces equipped with a convex bicombing?
• RQ3 Under what conditions does local injectivity (hyperconvexity) imply global injectivity in complete, simply-connected metric spaces?
• RQ4 How do convex bicombings relate to the structure of absolute 1-Lipschitz retracts in metric spaces?
• RQ5 To what extent can the non-positive curvature behavior of CAT(0) spaces be recovered in spaces with local bicombings?

Key findings

• The Cartan-Hadamard Theorem holds in metric spaces equipped with a convex bicombing, implying that such spaces are simply connected and exhibit non-positive curvature-like behavior.
• A complete, simply-connected metric space with a locally convex bicombing is globally injective (hyperconvex) if it is locally injective.
• The local-to-global principle for injective metric spaces is established: local injectivity implies global injectivity under the assumption of simple connectivity and completeness.
• Spaces with convex bicombings support a well-behaved notion of curvature, generalizing the CAT(0) condition to non-uniquely geodesic settings.
• The existence of a convex bicombing implies that the space is an absolute 1-Lipschitz retract if it is complete and simply connected.
• The results unify and generalize prior theorems by Ballmann and Alexander-Bishop by extending them to non-uniquely geodesic spaces via the bicombing structure.