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The Beckman-Quarles theorem for mappings from \({\mathbb C}^2\) to \({\mathbb C}^2\). (English) Zbl 1092.39023

Summary: Let \(\varphi: \mathbb C^2 \times \mathbb C^2 \to \mathbb C\), \(\varphi((x_1, x_2),(y_1,y_2)) = (x_1-y_1)^2+(x_2-y_2)^2\). We say that \(f : \mathbb C^2\to\mathbb C^2\) preserves the distance \(d \geq 0\), if for each \(X,Y \in\mathbb C^2\) \(\varphi (X,Y)= d^2\) implies \(\varphi(f(X),f(Y))=d^2\). We prove that each unit-distance preserving mapping \(f:\mathbb C^2\to \mathbb C^2\) has the form \(I \circ (\gamma,\gamma)\), where \(\gamma: \mathbb C \to \mathbb C\) is a field homomorphism and \(I:\mathbb C^2 \to \mathbb C^2\) is an affine mapping with orthogonal linear part. We prove an analogous result for mappings from \(\mathbb{K}^2\) to \(\mathbb{K}^2\), where \(\mathbb{K}\) is a field such that char\((\mathbb{K})\notin \{2, 3,5\}\) and \(-1\) is a square.

MSC:

39B32 Functional equations for complex functions
51B20 Minkowski geometries in nonlinear incidence geometry
51M05 Euclidean geometries (general) and generalizations
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