In mathematics, an **isometry** (or **congruence**, or **congruent transformation**) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.^{[a]} The word isometry is derived from the Ancient Greek: ἴσος *isos* meaning "equal", and μέτρον *metron* meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry;^{[b]}
the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.

Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space involves an isometry from into a quotient set of the space of Cauchy sequences on The original space is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

An isometric surjective linear operator on a Hilbert space is called a unitary operator.

Let and be metric spaces with metrics (e.g., distances) and A map is called an **isometry** or **distance preserving map** if for any one has

^{[4]}^{[c]}

An isometry is automatically injective;^{[a]} otherwise two distinct points, *a* and *b*, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric *d*, i.e., if and only if . 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 group with respect to function 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

- Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group and Euclidean space § Isometries.
- The map in is a
*path isometry*but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective.

The following theorem is due to Mazur and Ulam.

**Definition**:^{[5]}The**midpoint**of two elements x and y in a vector space is the vector 1/2(*x*+*y*).

**Theorem ^{[5]}^{[6]}** — Let

Given two normed vector spaces and a **linear isometry** is a linear map that preserves the norms:

for all ^{[7]}
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 is affine.

A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation.

- Examples

- A linear map from to itself is an isometry (for the dot product) if and only if its matrix is unitary.
^{[8]}^{[9]}^{[10]}^{[11]}

An isometry of a manifold is any (smooth) 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 (positive-definite) 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 (pseudo-)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** (or **isometric isomorphism**), and provides a notion of isomorphism ("sameness") in the category **Rm** of Riemannian manifolds.

Let and be two (pseudo-)Riemannian manifolds, and let be a diffeomorphism. Then is called an **isometry** (or **isometric isomorphism**) if

where denotes the pullback of the rank (0, 2) metric tensor by Equivalently, in terms of the pushforward we have that for any two vector fields on (i.e. sections of the tangent bundle ),

If is a local diffeomorphism such that then is called a **local isometry**.

A collection of isometries typically form a group, the isometry group. When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fields.

The Myers–Steenrod theorem states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a Lie group.

Riemannian manifolds that have isometries defined at every point are called symmetric spaces.

- Given a positive real number ε, an
**ε-isometry**or**almost isometry**(also called a**Hausdorff approximation**) is a map between metric spaces such that- for one has and
- for any point there exists a point with

- That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.

- 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:
- is an isometry if and only if

- Note that as mentioned in the introduction this is not necessarily a unitary element because one does not in general have that left inverse is a right inverse.

- On a pseudo-Euclidean space, the term
*isometry*means a linear bijection preserving magnitude. See also Quadratic spaces.