A map between metric spaces that preserves distances exactly: for all x,y, d(f(x),f(y)) = d(x,y). In Euclidean spaces isometries are geometry-preserving transformations such as translations, rotations, and reflections.

Principle

Isometries commute with the metric: they are distance-preserving functions and therefore preserve metric notions such as open balls, completeness, and Cauchy sequences; in many contexts they are injective and in the bijective case are metric space isomorphisms.

Demonstration

In R^3 a rotation about an axis through the origin is an isometry: it keeps pairwise Euclidean distances the same while moving points along circular trajectories; similarly, reflections in a plane are isometries that reverse orientation.

Misapplication

Preserving some distances but not all (e.g., preserving only distances from a single point) is not an isometry; scaling by a factor ≠1 or applying shear does not qualify as an isometry.

Consequence

All metric relations (distance-based properties) are invariant under isometry; geometric statements formulated purely in terms of distances hold equally after applying an isometry; groups of isometries organize symmetries of a space.

Reversal

The opposite behavior are contractions or dilations which uniformly shrink or expand distances; functions that preserve angles but not distances (conformal maps) are distinct.

Boundary

Isometries are defined for arbitrary metric spaces, not only Euclidean ones; they need not be surjective unless specified (then they are called surjective isometries or isometric isomorphisms). Linear algebra analogues include orthogonal or unitary operators that preserve inner products and hence induced norms.

Semantic Tension

Confusion arises between isometry and congruence in geometry (congruence often implies an isometry of ambient Euclidean space) and between isometries and inner-product-preserving linear maps in vector spaces (related but context-dependent distinctions).

Synthesis

An isometry is any distance-preserving map of metric spaces: the structural notion that exactly maintains metric relations and thereby formalizes geometric sameness.