Beckman-Quarles-Type Theorems

A fundamental theorem in Euclidean geometry is the following result of F.S. Beckman and D.A. Quarles [a1]. Let $ k $ be a fixed positive real number and let $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $ ($ n \geq 2 $) be a mapping satisfying $ d ( f ( x ) ,f ( y ) ) = k $ for all $ x,y \in \mathbf R ^ {n} $ with $ d ( x,y ) = k $. (Here, $ d ( x,y ) $ denotes the Euclidean distance

$$ d ( x,y ) = \sqrt {\sum _ {i = 1 } ^ { n } ( x _ {i} - y _ {i} ) ^ {2} } $$

of $ x = ( x _ {1}, \dots, x _ {n} ) $ and $ y = ( y _ {1}, \dots, y _ {n} ) $ in $ \mathbf R ^ {n} $.) The mapping $ x \rightarrow f ( x ) - f ( 0 ) $ is then in $ O ( n, \mathbf R ) $.

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 $ S \neq \emptyset $, a set $ W $ and a mapping $ d : {S \times S } \rightarrow W $. The element $ d ( x,y ) $ is called the distance of $ x $ and $ y $ (in this order). Beckman–Quarles-type theorems for distance spaces are, up to generalizations, statements claiming that a mapping $ f : S \rightarrow S $ preserves all occurring distances $ w $ 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 $ D ( x,y ) = ( x _ {1} - y _ {1} ) ^ {2} + \dots + ( x _ {n - 1 } - y _ {n - 1 } ) ^ {2} - ( x _ {n} - y _ {n} ) ^ {2} $, for $ x,y \in \mathbf R ^ {n} $. Let $ k \neq0 $ be a fixed real number and let $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $ ($ n \geq 2 $) be a mapping satisfying $ D ( f ( x ) ,f ( y ) ) = k $ for all $ x,y \in \mathbf R ^ {n} $ with $ D ( x,y ) = k $. Then $ x \rightarrow f ( x ) - f ( 0 ) $ is a Lorentz transformation of $ \mathbf R ^ {n} $. For $ n > 2 $ and $ k > 0 $, this was proved by J. Lester; for $ n = 2 $ and for $ n > 2 $ and $ k < 0 $ 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 $ X $ and $ Y $ be normed real vector spaces such that $ Y $ is strictly convex and the dimension of $ X $ is at least $ 2 $ (cf. also Vector space; Convex set). Let $ k > 0 $ be a fixed real number and $ N > 1 $ a fixed integer. Suppose that $ f : X \rightarrow Y $ is a mapping satisfying

$$ \left \| {a - b } \right \| = k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \leq k, $$

$$ \left \| {a - b } \right \| = N k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \geq N k , $$

for all $ a,b \in X $. Then $ f $ is an isometric operator, and hence an affine transformation (the Benz–Berens theorem, [a2]). If $ X $ and $ Y $ are, in addition, pre-Hilbert spaces (cf. Pre-Hilbert space), then $ N $ may be replaced by any real number greater than $ \sqrt 3 $ (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 $ V $ be a non-singular metric vector space of dimension $ \geq 3 $ over $ { \mathop{\rm GF} } ( p ^ {m} ) $, $ p \neq2 $, $ m \geq 3 $, and let $ k \neq0 $ be a fixed element of $ { \mathop{\rm GF} } ( p ^ {m} ) $. If $ f $ is a bijection of $ V $ preserving distance $ k $, then $ f $ is a semi-affine mapping if $ n \not\equiv0, -1, - 2 ( { \mathop{\rm mod} } p ) $.

Let $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $, $ n \geq 3 $, be a mapping satisfying

$$ \forall a,b,c \in \mathbf R ^ {n} : \Delta ( a,b,c ) = 1 \Rightarrow $$

$$ \Rightarrow \Delta ( f ( a ) ,f ( b ) ,f ( c ) ) = 1, $$

where $ \Delta ( a,b,c ) $ denotes the area of the triangle with vertices $ a,b,c $. Then $ f $ is a Euclidean motion, i.e., $ x \rightarrow f ( x ) - f ( 0 ) $ is in $ O ( n, \mathbf R ) $ (Lester's theorem), [a3]. For $ n = 2 $ the equi-affine mappings are characterized similarly.

Let $ \pi : {M ^ {n} } \rightarrow {M ^ {n} } $ be a mapping of the set $ M ^ {n} $ of lines in $ \mathbf R ^ {n} $, $ n \geq 2 $, into itself such that whenever $ a,b,c \in M ^ {n} $ are the lines making up the sides of a triangle of area $ 1 $, then $ \pi ( a ) , \pi ( b ) , \pi ( c ) $ are also the sides of a triangle of area $ 1 $. Then $ \pi $ is induced by a Euclidean motion of $ \mathbf R ^ {n} $ for $ n \geq 3 $ and by an equi-affine mapping if $ n = 2 $ (the Wen-ling Huang theorem, [a3]).

Let $ k > 0 $ be a fixed real number and $ X $ a normed real vector space of dimension $ \geq 3 $. Let $ f : X \rightarrow X $ be a function satisfying

$$ \forall x,y \in X: \left \| {x - y } \right \| = k \Rightarrow $$

$$ \Rightarrow x - y,f ( x ) - f ( y ) \textrm{ linearly independent } . $$

Then there are elements $ \lambda \in \mathbf R $ and $ t \in X $ such that $ f ( x ) = \lambda x + t $ for all $ x \in X $. This theorem was proved by Benz [a2] and, anew, by D. Laugwitz [a6].

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

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