# 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 |