It is shown that each continuous transformation $h$ from Euclidean $m$-space ($m>1$) into Euclidean $n$-space that preserves the equality of distances (that is, fulfils the implication $|x-y|=|z-w|\Rightarrow|h(x)-h(y)|=|h(z)-h(w)|$) is a similarity map. The case of equal dimensions already follows from the Beckman--Quarles Theorem.
Leaves decompositions in Euclidean spaces
Discrete versions of the Beckman-Quarles theorem from the definability results of Raphael M. Robinson
A discrete form of the Beckman-Quarles theorem for rational eight-space
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