{
  "id": "math/0406093",
  "version": "v5",
  "published": "2004-06-05T18:59:40.000Z",
  "updated": "2006-09-05T20:02:07.000Z",
  "title": "The Beckman-Quarles theorem for continuous mappings from C^n to C^n",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "10 pages, LaTeX2e, the version which appeared in Aequationes Mathematicae",
  "journal": "Aequationes Mathematicae 72 (2006), no. 1-2, pp. 78-88",
  "doi": "10.1007/s00010-005-2819-1",
  "categories": [
    "math.MG"
  ],
  "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].",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-09-05T20:02:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B32",
      "51M05"
    ],
    "keywords": [
      "beckman-quarles theorem",
      "continuous mappings",
      "orthogonal linear part",
      "preserves unit distance",
      "preserves distance"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6093T"
    }
  }
}