Mappings preserving two hyperbolic distances
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Suppose that X is the set of points of a hyperbolic geometry of finite or infinite dimension \( \geq2 \), and that \( \varrho>0 \) is a fixed real number and N>1 a fixed integer. Let \( f:X\to X \) be a mapping such that for every \( x,y\in X \) if h(x, y)=\( \varrho \), then h(f,(x),f,(y)) \( \leq\varrho \), and if h(x,y) = N\( \varrho \), then h(f,(x),f,(y)) \( \geq N\varrho \), where h,(p,q) designates the hyperbolic distance of p,q \( \in X \). Then f is an isometry of X. Note that there is no regularity assumption on f, like continuity or even differentiability. Moreover, we present an example showing that the assumption that one fixed distance > 0 is preserved does not characterize hyperbolic isometries.
KeywordsReal Number Fixed Distance Regularity Assumption Hyperbolic Geometry Infinite Dimension
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