for all .
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 .
|Date of creation||2013-03-22 12:19:08|
|Last modified on||2013-03-22 12:19:08|
|Last modified by||yark (2760)|
|Defines||group of isometries|