A map between metric spaces that preserves distances exactly: for all points x and y, the distance between f(x) and f(y) equals the distance between x and y.

Principle

Isometries are rigid motions of metric spaces—structure-preserving maps that maintain the full metric structure and hence preserve lengths, diameters, and geodesic segments when defined.

Demonstration

In Euclidean space R^n, a composition of a rotation and a translation is an isometry because it preserves Euclidean distances; therefore congruent figures are related by such isometries.

Misapplication

Confusing isometry with continuous bijection or homeomorphism—continuous bijections need not preserve distances, and conformal maps preserve angles but not necessarily lengths, so they are not generally isometries.

Consequence

An isometry preserves all metric invariants: pairwise distances, completeness, boundedness, and notions of convergence defined by the metric; isometric spaces are indistinguishable from the metric viewpoint.

Reversal

The inverse of an isometry is an isometry; composing isometries yields another isometry, and the set of isometries of a space forms a group under composition.

Boundary

Isometry presupposes a metric; it is more restrictive than topological equivalence (homeomorphism) and may exclude maps that preserve only some coarse geometric features (quasi-isometries, Lipschitz embeddings).

Semantic Tension

Isometry vs quasi-isometry: isometry demands exact distance preservation, while quasi-isometry allows bounded distortion and is used in geometric group theory to compare large-scale structure.

Synthesis

An isometry is the exact-preserving morphism of metric geometry: a bijective distance-preserving map (or distance-preserving embedding) that realizes rigid equivalence of metric spaces at the level of the metric.