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In Mathematics , an isometry, '''isometric isomorphism''' or '''congruence mapping''' is a Distance -preserving Isomorphism between Metric Spaces . Geometric figures which can be related by an isometry are called Congruent . Isometries are often used in constructions where one space is Embedded in another space. For instance, the Completion of a metric space ''M'' involves an isometry from ''M'' into M', a Quotient Set of the space of Cauchy Sequence s on ''M''. The original space ''M'' is thus isometrically isomorphic to a subspace of a Complete Metric Space , and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a Closed Subset of some Normed Vector Space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach Space . 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 guess from context which one is intended. Let and be . A global isometry is a Bijective distance preserving map. A '''path isometry''' or '''arcwise isometry''' is a map which preserves the Lengths Of Curves (not necessarily bijective). Two metric spaces ''X'' and ''Y'' are called isometric if there is an isometry from ''X'' to ''Y''. The Set of isometries from a metric space to itself forms a Group with respect to Function Composition , called the ''' Isometry Group '''. EXAMPLES
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