The Beckman-Quarles theorem for continuous mappings from C^n to C^n

Abstract

Let varphi_n:C^n times C^n->C, varphi_n((x_1,...,x_n),(y_1,...,y_n))=sum_{i=1}^n (x_i-y_i)^2. We say that f:C^n->C^n preserves distance d>=0, if for each X,Y in C^n varphi_n(X,Y)=d^2 implies varphi_n(f(X),f(Y))=d^2. We prove: if n>=2 and a continuous f:C^n->C^n preserves unit distance, then f has a form I circ (rho,...,rho), where I:C^n->C^n 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].