isometry

Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces. A function $f\colon X_{1}\to X_{2}$ is said to be an isometric mapping (or isometric embedding) if

 $d_{1}(x,y)=d_{2}(f(x),f(y))$

for all $x,y\in X_{1}$.

Every isometric mapping is injective, for if $x,y\in X_{1}$ with $x\neq y$ then $d_{1}(x,y)>0$, and so $d_{2}(f(x),f(y))>0$, and then $f(x)\neq f(y)$. One can also easily show that every isometric mapping is continuous.

An isometric mapping that is surjective (and therefore bijective) is called an isometry. (Readers are warned, however, that some authors do not require isometries to be surjective; that is, they use the term isometry for what we have called an isometric mapping.) Every isometry is a homeomorphism.

If there is an isometry between the metric spaces $(X_{1},d_{1})$ and $(X_{2},d_{2})$, then they are said to be isometric. Isometric spaces are essentially identical as metric spaces, and in particular they are homeomorphic.

Given any metric space $(X,d)$, the set of all isometries $X\to X$ forms a group under composition. This group is called the isometry group (or group of isometries) of $X$, and may be denoted by $\operatorname{Iso}(X)$ or $\operatorname{Isom}(X)$. In general, an (as opposed to the) isometry group (or group of isometries) of $X$ is any subgroup of $\operatorname{Iso}(X)$.

 Title isometry Canonical name Isometry Date of creation 2013-03-22 12:19:08 Last modified on 2013-03-22 12:19:08 Owner yark (2760) Last modified by yark (2760) Numerical id 13 Author yark (2760) Entry type Definition Classification msc 54E35 Related topic RealTree Related topic IsometricIsomorphism Defines isometric Defines isometric mapping Defines isometric embedding Defines isometry group Defines group of isometries