The Beckman-Quarles theorem for mappings from C² to C²

2004/04/30 by Apoloniusz Tyszka
Mathematics · #math.MG #msc:39B32 #msc:51B20 #msc:51M05

paper · pdf

published as Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21 (2005), no. 1, 63-69 (electronic), www.emis.de/journals · 9 pages, LaTeX2e, a new proof uses the connectivity of unit-distance graph on C^2

arxiv created 2004/11/10 · arxiv updated 2009/12/01

Abstract

Let phi: C² times C² -> C, phi((x₁,x₂),(y₁,y₂))=(x₁-y₁)²+(x₂-y₂)². We say that f:C²->C² preserves unit distance, if for each X,Y in C² phi(X,Y)=1 implies phi(f(X),f(Y))=1. We prove that each unit-distance preserving mapping f:C²->C² has a form J circ (gamma,gamma), where gamma:C->C is a field homomorphism and J:C²->C² is an affine mapping with orthogonal linear part. We also prove a more general result for any commutative field K for which x ∈ K: x²+1=0 ≠ ∅ and char(K) \not∈ 2,3,5.

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