DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Geodesics and geodesic metric spaces

Definition

Let $(X, d)$ be a metric space and let $x, y \in X$ .

A geodesic segment from $x$ to $y$ is a map $γ : [0, ℓ] \to X$ for some real $ℓ \geq 0$ such that

$γ (0) = x, γ (ℓ) = y, d (γ (s), γ (t)) = ∣ s - t ∣ for all s, t \in [0, ℓ] .$

Necessarily $ℓ = d (x, y)$ , by substituting $s = 0$ and $t = ℓ$ .

The metric space $X$ is geodesic if every two points of $X$ are joined by a geodesic segment.

Depends on

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

Used by

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)