# Geodesic in a metric space

## Definition

### Given data

Let be a metric space with metric . Then, a curve (or or ) is termed a **geodesic** if for any , there exists an open subset of in such that the restriction of to is the shortest path between them in the metric space, for any .

If it is true that the geodesic is the shortest path between its endpoints, it is termed a minimizing geodesic.