Geodesics and geodesic metric spaces

Depends on

• Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric

Used by

• Comparison angles of hinges, model triangle angles, and the Alexandrov upper angle Definition
• Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles Definition
• Geodesic segments, geodesic triangles, and geodesic metric spaces Definition
• Hg toolkit slim triangles products and four point constants Definition
• Oriented geodesic rays, lines, parameters and tails Definition
• Real trees, tripod triangles, slimness and minsize Definition
• The angular path metric, the Euclidean cone and spherical joins Definition
• Uniform local radii, cyclic small-mesh polygons, mesh, length, energy, the midpoint operation and the zero-limit basin Definition
• A circle of circumference ℓ<2π fails CAT(1) Example
• A complete locally CAT(0) circle whose fundamental group prevents global CAT(0) Example
• An interval-realized tree and its discrete vertex metric Example
• Intervals and metric trees are CAT(0) Example
• The disconnected universal-Coxeter nerve and the angular truncation convention Example
• The hexagonal A₂ cell: Euclidean cell metric versus graph distance Example
• The unit circle is CAT(1) at the strict perimeter boundary Example
• FALSE: a nontrivial finitely generated group with a word metric is a geodesic metric space False statement
• Alexandrov comparison: straightening a hinge, gluing comparison triangles, and patchwork Lemma
• Bounded local displacement on a geodesic space implies coarse Lipschitz control Lemma
• Circumcenters of bounded sets and fixed sets of isometries in complete CAT(0) spaces Lemma
• Cobounded proper geodesic actions produce finite generating sets Lemma
• Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences Lemma
• Endpoint stability for local geodesics in complete locally CAT(0) spaces Lemma
• Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas Lemma
• Limits of geodesic segments, rays and lines Lemma
• Products of CAT(0) spaces, joins of CAT(1) spaces, and round spheres Lemma
• Short local geodesics in a CAT(1) space are geodesics, and closed local geodesics have length at least 2π Lemma
• The space of local geodesics, its length metric, and the covering criterion for local isometries Lemma
• The vertex set of a connected simple graph with its path metric is a (1,1)-quasi-geodesic space Proposition
• Berestovskii's cone criterion and the polyhedral link criterion Theorem
• Compact geodesic locally CAT(1) spaces are CAT(1) exactly when they contain no short circle Theorem
• Complete, simply connected, locally CAT(0) length spaces are CAT(0) Theorem
• The cone and join metrics and the local product chart of a polyhedral gluing Theorem
• The Davis complex of a finite-rank Coxeter system is CAT(0) (Moussong's theorem) Theorem
• Under the Axiom of Choice, proper polyhedral spaces admit minimizing geodesics Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.3 (standard reference, not scraped)