The Beckman-Quarles theorem for mappings from C^2 to C^2
| dc.creator | Tyszka, Apoloniusz | |
| dc.date | 2004-04-19 | |
| dc.date | 2004-11-10 | |
| dc.date.accessioned | 2026-07-07T05:07:32Z | |
| dc.date.available | 2026-07-07T05:07:32Z | |
| dc.description | 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}. | |
| dc.description | 9 pages, LaTeX2e, a new proof uses the connectivity of unit-distance graph on C^2 | |
| dc.identifier | https://arxiv.org/abs/math/0404321 | |
| dc.identifier | http://arxiv.org/abs/math/0404321 | |
| dc.identifier | Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21 (2005), no. 1, 63-69 (electronic), www.emis.de/journals | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70891 | |
| dc.subject | Metric Geometry | |
| dc.subject | 39B32, 51B20, 51M05 | |
| dc.title | The Beckman-Quarles theorem for mappings from C^2 to C^2 | |
| dc.type | text |