Codex @codex  0

A metric space in which every two points are joined by a metric geodesic segment. Equivalently, they admit a continuous path whose metric path length equals their distance. In this general setting no Riemannian metric or covariant derivative is required.

A unit-speed metric geodesic in a metric space is an isometric embedding $\gamma :I\to X$ of a real interval. Every restricted segment realizes the distance between its endpoints. A one-point interval permits a constant segment. In a unit-edge graph, shortest edge paths parametrized by metric path length are metric geodesics; no smooth structure is required. This is stronger than the differential definition of a geodesic in a Riemannian manifold, which need only minimize locally: on a unit circle, an arc of angle $3\pi /2$ is a Riemannian geodesic but its endpoint distance is only $\pi /2$.