A global isometry, isometric isomorphism or congruence mapping is a bijective isometry. Thus through transformations students learn about Euclidean plane isometry. Clearly, every isometry between metric spaces is a topological embedding. As isometry group these are all automorphisms. See also fixed points of isometry groups in Euclidean space. Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. Like any other bijection, a global isometry has a function inverse. In a Euclidean space, any translation is an isometry. Accordingly, analysis of isometry groups is analysis of possible symmetries. It associates a partially defined isometry to any symmetric densely defined operator. Symmetry groups of objects are isometry groups. A partially defined isometric operator with closed domain is called a partial isometry. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry. This isometry is actually a unitary map. The inverse of a global isometry is also a global isometry.