isometry

Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces. A function $f:X_1\to X_2$ 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 $x\ne y$ then $d_1(x,y)>0$, and so $d_2(f(x),f(y))>0$, and then $f(x)\ne f(y)$. 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 $\mathrm{Iso}(X)$ or $\mathrm{Isom}(X)$. In general, an (as opposed to the) isometry group (or group of isometries) of $X$ is any subgroup of $\mathrm{Iso}(X)$.

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