Isometry

Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure".
Introduction
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;
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.
Definition
Let and be metric spaces with metrics (e.g., distances) and
A map is called an isometry or distance preserving if for any 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 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 .
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.
Isometries between normed spaces
The following theorem is due to Mazur and Ulam.
Definition: The midpoint of two elements and in a vector space is the vector .
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 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.
Manifold
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.
Definition
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.
Properties
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.
Generalizations
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 -isometry preserves distances to within and leaves no element of the codomain further than away from the image of an element of the domain. Note that -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.
See also
Beckman–Quarles theorem
The second dual of a Banach space as an isometric isomorphism
Euclidean plane isometry
Flat (geometry)
Homeomorphism group
Involution
Isometry group
Motion (geometry)
Myers–Steenrod theorem
3D isometries that leave the origin fixed
Partial isometry
Scaling (geometry)
Semidefinite embedding
Space group
Symmetry in mathematics
Footnotes
References
Bibliography
Functions and mappings
Metric geometry
Symmetry
Equivalence (mathematics)
Riemannian geometry