bibkey: bridsonhaefliger1999midpoint
authors: Martin R. Bridson; André Haefliger
year: 1999
title: Metric Spaces of Non-Positive Curvature
doi: 10.1007/978-3-662-12494-9
url: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/bridsonhaefligerx.pdf
claim: Remark I.1.4(1) states that a complete metric space with exact midpoints for every pair is geodesic, including a local radius version.
strata_touched: []
license: citation-only
triage: anchor

The classical midpoint criterion

Remark I.1.4(1), printed page 4, states that a complete metric space is geodesic if every pair of points has an exact midpoint. It also states the radius-restricted version: midpoints for pairs at distance less than r imply r-geodesicity. The remark states these criteria without giving the midpoint-refinement proof.

The compact extended metric formulation uses only pairs at finite distance and retains infinite distances between components. Its proof can refine finite midpoint chains, round every real time to a chain index with a uniform error, intersect closed sets in the compact product of all pointwise map values, and use endpoint triangle saturation. This route does not require a convergent subsequence in an uncountable product. It is a proof of the compact case; the complete noncompact case of the classical criterion is outside that formulation. No mathematical novelty or priority is claimed.

The same book’s I.5.18–5.19, printed pages 64–65, describes the disjoint union metric and the chain quotient pseudometric. These passages do not identify a specified hyperbolic gluing with a particular formal carrier. That identification, preservation of the original quotient topology, full-orbit finiteness and the geometric hypotheses of an application need their own proofs.

Verified locator

DOI: https://doi.org/10.1007/978-3-662-12494-9. Primary text consulted: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/bridsonhaefligerx.pdf. The midpoint passage is Remark I.1.4(1), printed page 4. The disjoint-union and quotient passages are I.5.18–5.19, printed pages 64–65. The source is cited and paraphrased; no publisher text is copied.