Let (X1,d1) and (X2,d2) be metric spaces. A function f:X1X2 is said to be an isometric mapping (or isometric embedding) if


for all x,yX1.

Every isometric mapping is injectivePlanetmathPlanetmath, for if x,yX1 with xy then d1(x,y)>0, and so d2(f(x),f(y))>0, and then f(x)f(y). One can also easily show that every isometric mapping is continuousPlanetmathPlanetmath.

An isometric mapping that is surjectivePlanetmathPlanetmath (and therefore bijectiveMathworldPlanetmath) 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 homeomorphismPlanetmathPlanetmath.

If there is an isometry between the metric spaces (X1,d1) and (X2,d2), 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 XX forms a group under compositionMathworldPlanetmath. This group is called the isometry group (or group of isometries) of X, and may be denoted by Iso(X) or Isom(X). In general, an (as opposed to the) isometry group (or group of isometries) of X is any subgroup of 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