Isometric embedding

An isometric embedding $f:X\to Y$ between metric spaces is an injective function satisfying $d_Y(f(x),f(x^{\prime }))=d_X(x,x^{\prime })$ for all $x,x^{\prime }\in X$.

An isometric extension of a connected metric manifold is a proper embedding onto an open subset of a larger connected manifold of the same dimension, preserving the metric tensor. Smooth complete connected Riemannian manifolds admit no such proper extension; Lorentzian geodesic completeness has different subtleties.