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 
.
