A metric space X is isometric to a metric space Y if there is a bijection f between X and Y that preserves distances. That is, d(a,b)=d(f(a),f(b)). In the context of Riemannian geometry, two manifolds M and N 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 M 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 (0,3).

See also

Isometric Latitude, Isometry, Metric Space, Riemannian Metric, Topological Space, Triangular Grid

This entry contributed by Todd Rowland

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Rowland, Todd. "Isometric." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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