The Beckman-Quarles theorem for continuous mappings from Cn to Cn
Abstract
Let varphin:Cn times Cn->C, varphin((x1,...,xn),(y1,...,yn))=sumi=1n (xi-yi)2. We say that f:Cn->Cn preserves distance d>=0, if for each X,Y in Cn varphin(X,Y)=d2 implies varphin(f(X),f(Y))=d2. We prove: if n>=2 and a continuous f:Cn->Cn preserves unit distance, then f has a form I circ (rho,...,rho), where I:Cn->Cn is an affine mapping with orthogonal linear part and rho:C->C is the identity or the complex conjugation. For n >=3 and bijective f the theorem follows from Theorem 2 in [8].
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