The Beckman-Quarles theorem for mappings from C^2 to C^2

Abstract

Let phi: C^2 times C^2 -> C, phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:C^2->C^2 preserves unit distance, if for each X,Y in C^2 phi(X,Y)=1 implies phi(f(X),f(Y))=1. We prove that each unit-distance preserving mapping f:C^2->C^2 has a form J circ (gamma,gamma), where gamma:C->C is a field homomorphism and J:C^2->C^2 is an affine mapping with orthogonal linear part. We also prove a more general result for any commutative field K for which {x \in K: x^2+1=0} \neq \emptyset and char(K) \not\in {2,3,5}.