isometry
Let and be metric spaces. A function is said to be an isometric mapping (or isometric embedding) if
for all .
Every isometric mapping is injective,
for if with then ,
and so , and then .
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 and , 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 ,
the set of all isometries forms a group under composition.
This group is called the isometry group
(or group of isometries) of ,
and may be denoted by or .
In general, an (as opposed to the) isometry group
(or group of isometries) of is any subgroup of .
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 |