A metric space is isometric to a metric space if there is a bijection between and that preserves distances. That is, . In the context of Riemannian geometry, two manifolds and are isometric if there is a diffeomorphism such that the Riemannian metric from one pulls back to the metric on the other. Since the geodesics define a distance, a Riemannian metric makes the manifold a metric space. An isometry between Riemannian manifolds is also an isometry between the two manifolds, considered as metric spaces.

Isometric spaces are considered isomorphic. For instance, the circle of radius one around the origin is isometric to the circle of radius one around .