An isometry between metric spaces is a function between the underyling sets that respects the metrics in that . More explicitly, for any points in .
The same idea holds for extended quasi-pseudo-generalisations of metric spaces.
Global isometries are the isomorphisms of metric spaces or Riemannian manifolds. An isometry is global if it is a bijection whose inverse is also an isometry. Between metric spaces, isometries are necessarily injections and bijective isometries necessarily have isometries as inverses, so global isometries between metric spaces are also called surjective isometries; this does not work for Riemannian manifolds (where the inverse of an isometry need not be a morphism of manifolds), nor does it work for pseudometric spaces (where an isometry need not be injective).
In practice, isometries between normed vector spaces tend to be affine maps. The following theorem gives a precise meaning to this.
A norm on a vector space is strictly convex if, whenever , we have for some (hence all!) in the range . In brief, no sphere contains a line segment. Examples of strictly convex spaces include spaces of type for .
Let an isometry between normed vector spaces, and suppose is strictly convex. Then is affine.
To say that is affine means that preserves linear combinations of the form . It suffices to consider only the case where and, by continuity considerations, only the case of dyadic rationals between and . Continuing this train of thought, it suffices to prove that for all .
In the case of strict convexity, midpoints are determined in terms of the norm, as the unique point such that
The midpoint satisfies these equations for any normed vector space, but the uniqueness is a consequence of strict convexity. For if there were two such points , then for some point on the line segment between them, we would have , and by ordinary convexity. But these two inequalities taken together would violate the triangle inequality.
As a result, since is an isometry, is forced to be the midpoint between and if is strictly convex. This completes the proof.
If is not strictly convex, then isometries need not be affine. For example, consider , and equipped with the (max) norm. For any contractive map , e.g., any smooth function with , the map sending to is easily seen to be an isometry.
If however is a surjective isometry between normed vector spaces, then is affine, by the Mazur-Ulam theorem.