Definitions
The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should determine from context which one is intended.
Let X and Y be metric spaces with metrics dX and dY. A map ƒ : X → Y is called an isometry or distance preserving if for any a,b ∈ X one has
An isometry is automatically injective. Clearly, every isometry between metric spaces is a topological embedding.
A global isometry, isometric isomorphism or congruence mapping is a bijective isometry.
Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y. The set of bijective isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.
There is also the weaker notion of path isometry or arcwise isometry:
A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective).
This is often called just isometry and one should determine from context which one is intended.
Read more about this topic: Isometry
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