The Beckman-Quarles theorem for continuous mappings from C^n to C^n
| dc.creator | Tyszka, Apoloniusz | |
| dc.date | 2004-06-05 | |
| dc.date | 2006-09-05 | |
| dc.date.accessioned | 2026-07-07T06:36:49Z | |
| dc.date.available | 2026-07-07T06:36:49Z | |
| dc.description | 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]. | |
| dc.description | 10 pages, LaTeX2e, the version which appeared in Aequationes Mathematicae | |
| dc.identifier | https://arxiv.org/abs/math/0406093 | |
| dc.identifier | http://arxiv.org/abs/math/0406093 | |
| dc.identifier | Aequationes Mathematicae 72 (2006), no. 1-2, pp. 78-88 | |
| dc.identifier | doi:10.1007/s00010-005-2819-1 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100214 | |
| dc.subject | Metric Geometry | |
| dc.subject | 39B32, 51M05 | |
| dc.title | The Beckman-Quarles theorem for continuous mappings from C^n to C^n | |
| dc.type | text |