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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
 .
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