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 a connected manifold 
a metric
space. An isometry between Riemannian manifolds
is also an isometry between the two manifolds, considered
componentwise 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 
.
