Isometric Embedding

An isometric embedding of metric spaces is a map preserving all pairwise distances, so that for all . Such a map is necessarily injective, and is an isometry onto its image.

For Riemannian manifolds, an isometric embedding instead means a differentiable embedding preserving the metric tensor. If , the defining condition is

where belong to the tangent space at . It preserves curve length and hence the intrinsic distances within the image, but need not preserve ambient distances. For example, the inclusion of the unit circle into the plane preserves arc length, while the Euclidean distance between opposite image points is 2 rather than the intrinsic distance . The Nash-Kuiper theorem concerns this Riemannian meaning.