The Beckman-Quarles theorem for continuous mappings from Rn to Cn
Abstract
Let φ((x1,...,xn),(y1,...,yn))=(x1-y1)2+...+(xn-yn)2. We say that f:Rn -> Cn preserves distance d>=0 if for each x,y ∈ Rn φ(x,y)=d2 implies φ(f(x),f(y))=d2. We prove that if x,y ∈ Rn (n>=3) and |x-y|=(2+2/n)k · (2/n)l (k,l are non-negative integers) then there exists a finite set x,y ⊂eq S(x,y) ⊂eq Rn such that each unit-distance preserving mapping from S(x,y) to Cn preserves the distance between x and y. It implies that each continuous map from Rn to Cn (n>=3) preserving unit distance preserves all distances.
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