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On a general principle in geometry that leads to functional equations. (English) Zbl 0788.39005

One wishes many survey papers would be of this kind, shedding new light on results, putting them into a new framework. The author reverses, in a sense, Felix Klein’s “Erlanger Programm”. Given a numerical (distance, area, angle, etc.) or geometric (straight line, plane, circle, orthogonality, etc.) object, the problem is to determine those transformations (functions) under which the object is invariant on the whole domain or on part of it. This leads to functional equations.
The author presents several results on \(\mathbb{R}^ n\) (mainly for \(n=2\) or 3) and poses many intriguing problems, mainly in the case of geometries over fields.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B22 Functional equations for real functions
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
51N25 Analytic geometry with other transformation groups
51M10 Hyperbolic and elliptic geometries (general) and generalizations
51F25 Orthogonal and unitary groups in metric geometry
51M05 Euclidean geometries (general) and generalizations
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References:

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