# Beckman-Quarles-type theorems

A fundamental theorem in Euclidean geometry is the following result of F.S. Beckman and D.A. Quarles [a1]. Let be a fixed positive real number and let () be a mapping satisfying for all with . (Here, denotes the Euclidean distance

of and in .) The mapping is then in .

It should be emphasized that no regularity assumption (like differentiability or continuity) is required in the theorem. For an analogue in hyperbolic geometry, see [a4] and [a5].

A distance space is a set , a set and a mapping . The element is called the distance of and (in this order). Beckman–Quarles-type theorems for distance spaces are, up to generalizations, statements claiming that a mapping preserves all occurring distances if it preserves one single distance. Beckman–Quarles-type theorems belong to a class of statements called characterizations of geometrical mappings under mild hypotheses.

Let , for . Let be a fixed real number and let () be a mapping satisfying for all with . Then is a Lorentz transformation of . For and , this was proved by J. Lester; for and for and this was proved by W. Benz (see [a2] for all these results). The proofs are different for the three cases, and no common proof is known (1996).

Let and be normed real vector spaces such that is strictly convex and the dimension of is at least (cf. also Vector space; Convex set). Let be a fixed real number and a fixed integer. Suppose that is a mapping satisfying

for all . Then is an isometric operator, and hence an affine transformation (the Benz–Berens theorem, [a2]). If and are, in addition, pre-Hilbert spaces (cf. Pre-Hilbert space), then may be replaced by any real number greater than (Radó–Andreascu–Valcán theorem).

There are generalizations for finite planes [a9], rational or constructible curves (B. Farrahi), non-Euclidean spaces [a11], and planes over fields [a10]. F. Radó [a8] has proved the following theorem. Let be a non-singular metric vector space of dimension over , , , and let be a fixed element of . If is a bijection of preserving distance , then is a semi-affine mapping if .

Let , , be a mapping satisfying

where denotes the area of the triangle with vertices . Then is a Euclidean motion, i.e., is in (Lester's theorem), [a3]. For the equi-affine mappings are characterized similarly.

Let be a mapping of the set of lines in , , into itself such that whenever are the lines making up the sides of a triangle of area , then are also the sides of a triangle of area . Then is induced by a Euclidean motion of for and by an equi-affine mapping if (the Wen-ling Huang theorem, [a3]).

Let be a fixed real number and a normed real vector space of dimension . Let be a function satisfying

Then there are elements and such that for all . This theorem was proved by Benz [a2] and, anew, by D. Laugwitz [a6].

General references for this area are [a2], [a3] and [a7].

#### References

[a1] | F.S. Beckman, D.A. Quarles, jr., "On isometries of Euclidean spaces" Proc. Amer. Math. Soc. , 4 (1953) pp. 810–815 |

[a2] | W. benz, "Geometrische Transformationen (unter besonderer Berücksichtingung der Lorentztransformationen)" , BI Wissenschaftsverlag (1992) |

[a3] | W. Benz, "Real geometries" , BI Wissenschaftsverlag (1994) |

[a4] | B. Farrahi, "A characerization of isometries of absolute planes" Resultate Math. , 4 (1981) pp. 34–38 |

[a5] | A.V. Kuz'minykh, "Mappings preserving a unit distance" Sibirsk. Mat. Zh. , 20 (1979) pp. 597–602 (In Russian) |

[a6] | D. Laugwitz, "Regular hexagons in normed spaces and a theorem of Walter Benz" Aequat. Math. , 45 (1993) pp. 163–166 |

[a7] | J. Lester, "Distance preserving transformations" F. Buekenhout (ed.) , Handbook of Incidence geometry , Elsevier (1995) |

[a8] | F. Radó, "On mappings of the Galois space" Israel J. Math. , 53 (1986) pp. 217–230 |

[a9] | H.-J. Samaga, "Zur Kennzeichnung von Lorentztransformationen in endlichen Ebenen" J. Geom. , 18 (1982) pp. 169–184 |

[a10] | H. Schaeffer, "Der Satz von Benz–Radó" Aequat. Math. , 31 (1986) pp. 300–309 |

[a11] | E.M. Schröder, "Zur Kennzeichnung distanztreuer Abbildungen in nichteuklidischen Räumen" J. Geom. , 15 (1980) pp. 108–118 |

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Beckman-Quarles-type theorems.

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