Let X and Y be metric spaces with metrics dX and dY. A mapf : X → Y is called an isometry or distance preserving if for any a,b ∈ X one has An isometry is automatically injective; otherwise two distinct points, a and b, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric d. This proof is similar to the proof that an order embedding between partially ordered sets is injective. Clearly, every isometry between metric spaces is a topological embedding. A global isometry, isometric isomorphism or congruence mapping is a bijective isometry. Like any other bijection, a global isometry has a function inverse. The inverse of a global isometry is also a global isometry. Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y. The set of bijective isometries from a metric space to itself forms a groupwith respect tofunction composition, called the isometry group. There is also the weaker notion of path isometry or arcwise isometry: A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. This term is often abridged to simply isometry, so one should take care to determine from context which type is intended. ;Examples
The map in is a path isometry but not an isometry. Note that unlike an isometry, it is not injective.
Isometries between normed spaces
The following theorem is due to Mazur and Ulam.
Linear isometry
Given two normed vector spaces and, a linear isometry is a linear map that preserves the norms: for all. Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective. In an inner product space, the above definition reduces to for all, which is equivalent to saying that. This also implies that isometries preserve inner products, as Linear isometries are not always unitary operators, though, as those require additionally that and. By the Mazur–Ulam theorem, any isometry of normed vector spaces over R is affine. ;Examples
An isometry of a manifold is any mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a metric on the manifold; a manifold with a metric is a Riemannian manifold, one with an indefinite metric is a pseudo-Riemannian manifold. Thus, isometries are studied in Riemannian geometry. A local isometry from one Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. When such a map is also a diffeomorphism, such a map is called an isometry, and provides a notion of isomorphism in the categoryRm of Riemannian manifolds.
Definition
Let and be two Riemannian manifolds, and let be a diffeomorphism. Then is called an isometry if where denotes the pullback of the rank metric tensor by. Equivalently, in terms of the pushforward, we have that for any two vector fields on , If is a local diffeomorphism such that, then is called a local isometry.
Generalizations
Given a positive real number ε, an ε-isometry or almost isometry is a map between metric spaces such that
# for x,x′ ∈ X one has |dY,ƒ)−dX| < ε, and
# for any point y ∈ Y there exists a point x ∈ X with dY < ε
The restricted isometry property characterizes nearly isometric matrices for sparse vectors.
Quasi-isometry is yet another useful generalization.
One may also define an element in an abstract unital C*-algebra to be an isometry:
