Isometric
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 
.
Contact the MathWorld Team
5:1 odds
